Boundary value problems for Dirac–type equations, with applications

Robert A. Bartnik^{∗} School of Mathematics and Statistics University of Canberra ACT 2601 Australia

Piotr T. Chruściel^{†} Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France

February 13, 2004

Abstract

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincaré and Schrödinger-Lichnerowicz inequalities provide asymptotic control in the non-compact case. One application yields existence of solutions for the Witten equation with a spectral boundary condition used by Herzlich in his proof of a geometric lower bound for the ADM mass of asymptotically flat 3-manifolds.

Contents

1 Introduction 2

2 The model problem 3

3 Interior Regularity 7 4 Spectral Condition 15

∗Supported in part by the Australian Research Council. Email: bartnik@ise.canberra.

edu.au

†Supported in part by grant from the Polish Committee for Scientific Research # 2 P03B 073 15 and by the French Ministry for Foreign Affairs. Email: chrusciel@univ-tours.fr, URL www.phys.univ-tours.fr\~piotr

5 Boundary Regularity 19

6 Boundary regularity for first order systems 32

7 Fredholm properties on compact manifolds 38

8 Fredholm properties on complete noncompact manifolds 42

9 Weighted Poincaré Inequalities 51

10 Examples and Applications 57

11 Positive mass theorems 60

A Fields on manifolds of W^{k+1,p} differentiability class 71

1 Introduction

Elliptic systems based on the Dirac equation arise frequently in problems in geometry and analysis. Applications to positive mass and related conjectures in general relativity motivate this paper, and involve boundary value problems on compact and non-compact domains [31, 37, 38, 64].

Previous existence and regularity results [6, 12, 16, 40, 50] are insufficient for these applications, for various reasons. The Agmon-Douglas-Nirenberg approach based on freezing coefficients and explicit kernels for the constant coefficient inverse operator, leads only to boundary conditions of Lopatinski-Shapiro type [44]. The pseudo-differential operator approach [40, 56] handles non-local boundary conditions such as the spectral projection condition of Atiyah-Patodi-Singer [6], but the assumptions of smooth coefficients and product-type boundary metric [6, 12, 16, 40] are unnatural and, as we shall show, unnecessary.

In this paper we provide an essentially elementary proof of existence and regularity for first order elliptic systems with “Dirac-type” boundary value conditions. These encompass both pointwise (Lopatinski-Shapiro) and non-local (spectral) boundary conditions, and do not require product metric structures on the boundary. We obtain explicit necessary and sufficient conditions which ensure the solvability of natural inhomogeneous boundary value problems, over both compact and non-compact manifolds with compact boundary.

The coefficient regularity conditions, for both the elliptic system and the boundary conditions, are rather general. For example, they are weaker than those in the pseudo-differential operator approach of Marschall [48]. It seems likely that the boundary conditions will admit some generalizations; the boundary data is H^{1/2} whereas there are recent results for a certain constant coefficient Dirac equation with L^{2} boundary values on a Lipschitz hypersurface [1].

Note that there is an extensive literature on applications of Dirac operators to index problems on compact and non-compact manifolds [12, 18] which we do not address, although many aspects of our results are no doubt relevant to such applications; the results here are focussed on applications to energy theorems in general relativity.

The motivating example of the Dirac (Atiyah-Singer) operator is described in some detail in §2, where the Schrödinger-Lichnerowicz identity with suitable boundary conditions combines with a Lax-Milgram argument to reduce the existence question to that of showing that a weak (L^{2}) solution of an adjoint problem is in fact a strong (H^{1}) solution. This weak-strong regularity property turns out to be the key technical step, and the focus of much of the paper. The difficult case is regularity at the boundary; interior regularity is established in §3 using standard Fourier techniques, for general first order elliptic systems.

§4 reviews conditions under which a symmetric operator has a complete set of eigenfunctions; these are used to to control the boundary operator in later sections. In §5 we prove regularity results at the boundary, for a class of operators much broader than Dirac equations, with weak assumptions on the continuity/regularity of the operator coefficients. The main technical tools are the H^{1} identity (5.15), and some basic spectral theory. The boundary conditions of §5 follow from the requirements of the arguments of the regularity theorem, and some additional work is required to apply them to first order systems. This is carried out in §6, for equations of Dirac-type near the boundary, for which the boundary operator is self-adjoint. The resulting boundary conditions are naturally presented in terms of graphs over the space of negative eigenfunctions of the boundary operator.

The boundary value problems considered have a Fredholm property, and admit an explicit solvability criteria involving solutions of the homogeneous adjoint problem. These properties are established for compact manifolds with boundary in §7, and for a large class of non-compact manifolds with boundary in §8. The analysis of the non-compact case relies on two a priori inequalities: a weighted Poincaré inequality, and a Schrödinger-Lichnerowicz inequality. These inequalities imply the manifold is non-parabolic at infinity in the sense of [18]. The weighted Poincaré inequality is established in §9 in a number of cases, including the important cases of manifolds with asymptotically flat or hyperbolic ends. The Schrödinger-Lichnerowicz inequality follows in applications from an H^{1} estimate derived from an identity of Schrödinger-Lichnerowicz type.

In section 10 we show that common pointwise and spectral boundary conditions for the Dirac equation are elliptic in the sense of our conditions. These calculations form the basis for §11, which verifies several positive mass theorems [31, 37, 64]. Appendix A collects some relevant properties of tensor and spinor fields on manifolds with W^{k+1,p} differentiable structure and W^{k,p} metric, k > n/p.

2 The model problem

In this section we use the Riemannian Dirac equation to illustrate and motivate the existence and regularity results of the following sections.

Consider an oriented manifold M with Riemannian metric g and a representation c : Cℓ(T M) → End(S) of the Clifford algebra Cℓ(T M) on some bundle S; with our conventions,

c(v)c(w) + c(w)c(v) = −2g(v, w) .

Clifford representations are discussed in detail in [5, 45]. S carries an invariant inner product, ⟨c(v)ψ, c(v)ψ⟩ = |v|^{2} ⟨ψ, ψ⟩ = |v|^{2} |ψ|^{2}, with respect to which c(v) is skew-symmetric, for all vectors v.

A Dirac connection [12, 45] is a connection on the space of sections of S which satisfies the compatibility relation

d⟨ϕ, c(v)ψ⟩ = ⟨∇ϕ, c(v)ψ⟩ + ⟨ϕ, c(v)∇ψ⟩ + ⟨ϕ, c(∇v)ψ⟩ ,  (2.1)

where ∇ also denotes the Levi-Civita connection on vector fields.

Spin manifolds provide the fundamental example, with S a bundle of spinors associated with a Spin principal bundle which double covers the Riemannian orthonormal frame bundle. In this case there is a covariant derivative ∇ defined in terms of a local orthonormal frame e_{k}, k = 1, …, n, with Riemannian connection matrix ω_{ij}(e_{k}) = g(e_{i}, ∇_{e_{k}}e_{j}), by

∇_{e_{k}}ψ = D_{e_{k}}ψ^{I} ϕ_{I} − \frac{1}{4}ψ^{I} ω_{ij}(e_{k})c(e^{i}e^{j})ϕ_{I} ,  (2.2)

where ψ = ψ^{I}ϕ_{I} and ϕ_{I}, I = 1, …, dim S, is a choice of spin frame associated with the orthonormal frame e_{k}. The expression (2.2) may be abbreviated to ∇ = d − \frac{1}{4}ω_{ij}e^{i}e^{j}. Note that there are other examples of Dirac bundles and connections, eg. [45, example II.5.8].

The Dirac operator of a Dirac connection ∇ is

Dψ = c(e^{i})∇_{e_{i}}ψ ;  (2.3)

in the spin case this is sometimes called the Atiyah-Singer operator. When the spinor representation is irreducible^{1}, a classical and very important computation [55] shows that

D^{2}ψ = ∇^{∗}∇ψ + \frac{1}{4}R(g)ψ ,  (2.4)

where R(g) is the (Ricci) scalar curvature of g. This leads to the Schrödinger-Lichnerowicz identity [46, 55]

(|∇ψ|^{2} + \frac{1}{4}R(g)|ψ|^{2} − |Dψ|^{2}) ∗1 = d (⟨ψ, (c(e_{i}e_{j}) + g_{ij})∇^{j}ψ⟩ ∗e^{i}) , (2.5)

which when integrated over the compact manifold M with boundary^{2} Y becomes

∫_{M} (|∇ψ|^{2} + \frac{1}{4}R(g)|ψ|^{2} − |Dψ|^{2}) = ∮_{Y} ⟨ψ, c(ne^{A})∇_{A}ψ⟩ .  (2.6)

Here n is the outer normal vector at Y = ∂M and {e_{A}} is a compatible orthonormal frame on Y. The boundary term may be simplified by introducing the boundary covariant derivative

¯∇ = d − \frac{1}{4}ω_{AB}c(e^{A}e^{B}) ,

1Reducible representations lead to interesting formulas with \frac{1}{4} R(g) replaced by more complicated curvature endomorphisms, c.f. §11

2Throughout this paper we use the geometer’s convention, that a manifold with boundary contains its boundary as a point set.

and the boundary Dirac operator^{3}

D_{Y} ψ = c(ne^{A}) ¯∇_{A}ψ .  (2.7)

Denoting the mean curvature by H = H_{Y} = g(n, ∇_{e_{A}}e^{A}) gives

∮_{Y} ⟨ψ, c(ne^{A})∇_{A}ψ⟩ = ∮_{Y} ⟨ψ, D_{Y} ψ + \frac{1}{2}Hψ⟩ .  (2.8)

We use conventions which give H = 2/r > 0 for M = ℝ^{3} − B(0, r), the exterior of a ball of radius r, with the outer normal n = −∂_{r}. If x is a Gaussian boundary coordinate (x ≥ 0 in M, x = 0 on Y and ∂_{x} = −n), then near the boundary we have

Dψ = −c(n)(∂_{x} + D_{Y} + \frac{1}{2}H)ψ .  (2.9)

We now seek boundary conditions for which the equation Dψ = f is solvable, following a well-known argument [32, 37, 51]. Suppose M is a compact^{4} manifold with non-negative scalar curvature, R(g) ≥ 0, and K : H^{1/2}(Y) → H^{1/2}(Y) is a bounded linear operator such that

∮_{Y} ⟨ψ, D_{Y} ψ + \frac{1}{2}Hψ⟩ ≤ 0  whenever Kψ = 0 .  (2.10)

Suppose further that M admits no parallel spinors. Define the space H_{K}^{1}(M) as the completion of the smooth spinor fields with compact support (in M ∪ Y) which satisfy the boundary condition Kψ = 0, in the norm

∥ψ∥_{H_{K}^{1}(M)}^{2} := ∫_{M}(|∇ψ|^{2} + \frac{1}{4}R(g)|ψ|^{2}) .  (2.11)

The boundary condition (2.10) combined with the Lichnerowicz identity (2.6) and the curvature condition R(g) ≥ 0 now ensures that the bilinear form

a(ψ, ϕ) = ∫_{M}⟨Dψ, Dϕ⟩, ϕ, ψ ∈ H_{K}^{1}(M) ,

is strictly coercive, a(ψ, ψ) ≥ ∥ψ∥_{H_{K}^{1}(M)}^{2}. For any spinor field f ∈ L^{2}(M), the linear functional ϕ ↦ ∫_{M}⟨f, Dϕ⟩ is bounded on H_{K}^{1}(M). Coercivity and the Lax-Milgram lemma show there is a unique ψ ∈ H_{K}^{1}(M) such that

∫_{M}⟨Dψ − f, Dϕ⟩ = 0 ∀ϕ ∈ H_{K}^{1}(M) ,

and we would like to deduce that Dψ = f. Now Ψ := Dψ − f ∈ L^{2}(M) is a weak solution of the Dirac equation; that is,

∫_{M}⟨Ψ, Dϕ⟩ = 0  ∀ ϕ ∈ H_{K}^{1}(M) .  (2.12)

3Both e^{A} → c(e^{A}) and e^{A} → c(ne^{A}) give representations of the Clifford algebra of the boundary tangent space; the choice of c(ne^{A}) is made here for convenience [37].

4The case of an (M, g) which is asymptotically flat with compact interior, which is relevant to the positive mass theorem, is discussed along similar lines in §8 and 11.

\underline{If} we could show that Ψ is in fact a strong solution, that is, Ψ ∈ H^{1}(M), then we could integrate by parts to conclude

∫_{M}⟨DΨ, ϕ⟩ + ∮_{Y} ⟨Ψ, c(n)ϕ⟩ = 0 ∀ϕ ∈ H_{K}^{1}(M) ,

and thus DΨ = 0 and ∮_{Y} ⟨Ψ, c(n)ϕ⟩ = 0 for all ϕ ∈ H^{1/2}(Y) such that Kϕ = 0. This would give the boundary condition Ψ|_{Y} ∈ c(n)(ker K)^{⊥}, which we suppose may be re-expressed as K̃Ψ = 0, for some “adjoint” boundary operator K̃. This would give Ψ ∈ H1 ; K̃, so if finally we suppose that K̃ also satisfies the boundary positivity condition (2.10), then we could conclude from a(Ψ, Ψ) = 0 and the coercivity of a(·, ·) with respect to the norm ∥ · ∥_{H1 ; K̃}, that Ψ = 0 as desired.

The key technical difficulty in this classical argument lies in establishing the “Weak-Strong” property, that weak (L^{2}) solutions lie in H^{1}. In the following sections we will prove this property for a large class of elliptic systems, under rather general boundary conditions; see §5 and §6.

Two model boundary operators illustrate the possibilities for achieving the required conditions. The APS (or spectral projection [6]) condition arose in Herzlich’s work [37]:

K = P_{+} ,  (2.13)

where P_{+} is the L^{2}(M)-orthogonal projection onto the positive spectrum eigenspace of the boundary Dirac operator D_{Y}. Using the relation c(n)D_{Y} = −D_{Y} c(n), which shows that the spectrum of D_{Y} is symmetric about 0 ∈ ℝ, we find that K̃ = K, provided there are no zero eigenvalues.

The eigenvalue estimate for Y ≃ S^{2} of Hijazi and Bär [7, 39]

|λ(D_{Y})| ≥ \sqrt{4π/area(Y)},  (2.14)

shows that in this case there are no zero eigenvalues. In addition, if we have the mean curvature condition

H_{Y} ≤ \sqrt{16π/area(Y)},  (2.15)

then K (and K̃) will satisfy the boundary positivity condition (2.10). In conclusion, if Y = ∂M ≃ S^{2} satisfies (2.15), then (assuming the Weak-Strong property can be established) the above argument shows Ψ = 0 and thus the equation Dψ = f with boundary condition P_{+}ψ = 0 is uniquely solvable, for any f ∈ L^{2}(M).

The chirality condition was used in [30, 31]. For a slightly simplified version of [31], suppose M is a totally geodesic hypersurface in a Lorentz spacetime, with future unit normal vector e_{0}, and consider the connection on spacetime spinors, restricted to M. Along Y = ∂M we define

ϵ = c(e_{0}n),  (2.16)

which satisfies the chiral conditions

\begin{aligned}ϵ^{2} & = & 1 , & ϵc(n) + c(n)ϵ & = & 0 , \\ ⟨ϕ, ϵψ⟩ & = & ⟨ϵϕ, ψ⟩ , & ϵD_{Y} + D_{Y} ϵ & = & 0 , \end{aligned}  (2.17)

and then the boundary operators

K_{±} = \frac{1}{2}(1 ± ϵ) .  (2.18)

Assuming either of the two conditions K_{±}ψ = 0 gives ϵψ = ∓ψ which implies

⟨ψ, D_{Y} ψ⟩ = ∓⟨ψ, D_{Y} ϵψ⟩ = ±⟨ψ, ϵD_{Y} ψ⟩
= ±⟨ϵψ, D_{Y} ψ⟩ = −⟨ψ, D_{Y} ψ⟩ = 0 .

If we further assume that H_{Y} ≤ 0 then (2.10) follows directly. In general relativity the condition H_{Y} ≤ 0 is the defining property for Y to be a trapped surface.

Since the K_{±}’s are complementary orthogonal projections, we have (ker K_{±})^{⊥} = ker K_{∓}, so ψ ∈ c(n)(ker K_{±})^{⊥} exactly when c(n)ψ ∈ ker K_{∓}, which gives ψ ∈ ker K_{±}, and K̃_{±} = K_{±}. In this case we conclude (still assuming the Weak-Strong property can be established) that if H_{Y} ≤ 0 then Dψ = f, with either of the boundary conditions ϵψ = ±ψ, is uniquely solvable. These examples are discussed further in §10.

3 Interior Regularity

In this section we establish regularity away from the boundary for weak (L^{2}) solutions of first order elliptic systems. We consider equations of the form

Lu := a^{j}∂_{j}u + bu = f ,  (3.1)

where u, f are sections respectively of N-dimensional real vector bundles E, F, both over an n-dimensional manifold M without boundary^{5}, and a^{j}, b, j = 1, …, n, are sections of the bundle of endomorphisms of E to F. We assume that E, F are equipped with fixed smooth inner products, denoted by ⟨·, ·⟩. The length determined by ⟨·, ·⟩ will be denoted invariably by |u|^{2} = ⟨u, u⟩. To simplify notation, the respective bundles usually will be understood, and thus L^{2}(M) will generally mean L^{2}Γ(E), the space of L^{2} sections of E, or L^{2}Γ(F), depending on context.

Remark 3.1 There is no loss of generality in considering real bundles, since complex and quaternionic bundles may be viewed simply as real bundles with additional algebraic structure. For example, a Hermitean vector space of dimension n is equivalent to a real vector space of dimension 2n with a skew endomorphism J satisfying J^{2} = −1, with the Hermitean inner product ( , ) and real inner product ⟨ , ⟩ related by (u, v) = ⟨u, v⟩ − i⟨u, Jv⟩.

Define the indices ˆ2 = ˆ2(n), n^{∗} = n^{∗}(n) by

ˆ2 = \frac{2n}{n−2}, n^{∗} = n  for n ≥ 3,
ˆ2 = 10^{6}, n^{∗} = \frac{2}{1−2/ˆ2} for n = 2,
ˆ2 = ∞ , n^{∗} = 2  for n = 1 ,

(3.2) where 10^{6} represents any large constant. Note that if M admits a Sobolev inequality with constant C_{S}

5If M has boundary ∂M ≠ ∅, then the interior M̊ = M − ∂M is a (noncompact) manifold without boundary, to which the results of this section will apply.

∥u∥_{Lˆ2} ≤ C_{S}∥u∥_{H^{1}},

then we also have

∥fu∥_{L^{2}} ≤ C_{S}∥f∥_{L^{n^{∗}}} ∥u∥_{H^{1}} .  (3.3)

Another basic fact is the inequality

∥fg∥_{W^{1,p}} ≤ C(∥f∥_{L^{∞}}∥g∥_{W^{1,p}} + ∥g∥_{L^{∞}}∥f∥_{W^{1,p}}),

which shows that W^{1,n^{∗}} ∩ C^{0} (in particular) forms a ring under addition and multiplication of functions. For n = 1, 2 the C^{0} is superfluous here, of course.

With one exception, it suffices to assume throughout that the underlying manifold has a C^{∞} differentiable structure. The exceptional point arises in §10 in the construction of approximately Gaussian coordinates in a neighbourhood of the boundary, when the metric has low regularity. The description in Appendix A of W^{k+1,p} differential structures, k > n/p, establishes the necessary consistency conditions in this case.

We will assume that a^{j}, b satisfy the regularity conditions

\begin{aligned}a^{j} & ∈ & W_{loc}^{1,n^{∗}}(M) ∩ C^{0}(M) , \\ b & ∈ & L_{loc}^{n^{∗}}(M) . \end{aligned}  (3.4)

The conditions (3.4) are preserved by bundle frame changes in W^{1,n^{∗}} ∩ C^{0}, by the above ring property. In particular, even if the bundle metrics ⟨·, ·⟩ on E, F are only in W^{1,n^{∗}} ∩ C^{0}, by the Gram-Schmidt process we may construct W^{1,n^{∗}} ∩ C^{0} frame changes which make the metric coefficients constant. Since this changes the operator coefficients a^{j}, b respectively by W^{1,n^{∗}}∩C^{0}, W^{n^{∗}} affine linear transformations, there is no loss of generality in assuming the metrics on E, F to be locally constant.

The conditions (3.4) mean that M can be covered by open neighbourhoods 𝒪_{α} with W^{1,n^{∗}} ∩ C^{0} bundle transition functions, such that the local coefficients a^{j}, b satisfy the stated regularity. Frame changes satisfying Sobolev conditions are also discussed in detail in Appendix A.

We require that a^{j} satisfy the ellipticity condition, that for each p ∈ M there is a coordinate neighbourhood p ∈ U ⊂ M and a constant η > 0 such that

η^{2}|ξ|^{2}|V |^{2} ≤ |ξ_{j}a^{j}(x)V |^{2} ≤ η^{−2}|ξ|^{2}|V |^{2} ,  (3.5)

for all x ∈ U, ξ ∈ T_{x}^{∗}M and V ∈ E_{x}, where |ξ|^{2} is measured by a fixed background metric g̊, which we may assume to be C^{∞}. Note that (3.5) implies the fibres of E, F must be of the same dimension.

A weak solution of (3.1) is u ∈ L_{loc}^{2}(E) such that

∫_{M}⟨L^{†}ϕ, u⟩ dv_{M} = ∫_{M}⟨ϕ, f⟩ dv_{M} ,  (3.6)

for all ϕ ∈ C_{c}^{∞}(M), where dv_{M} = γdx, γ > 0, is a coordinate-invariant volume measure on M with γ ∈ W^{1,n^{∗}}(U) ∩ C^{0}(U) and dx is coordinate Lebesgue measure, in any local coordinate neighbourhood U. Here the formal adjoint L^{†} is defined with respect to dv_{M} and the inner products on E, F. Thus in local coordinates,

L^{†}ϕ = −^{t}a^{j}∂_{j}ϕ + (^{t}b − γ^{−1}∂_{j}(^{t}a^{j}γ))ϕ ,  (3.7)

where the transposes ^{t}a^{j} are defined with respect to the local framing forms of the inner products of E, F.

The proof proceeds by establishing various special cases, starting with a constant coefficient operator acting on sections of a trivial bundle E over the torus 𝕋^{n}. This type of argument is very standard.

Proposition 3.2 Suppose u ∈ L^{2}(𝕋^{n}) is a weak solution of L_{0}u = f where f ∈ L^{2}(𝕋^{n}), L_{0} = a_{0}^{j}∂_{j} with a_{0}^{j} constant and satisfying the ellipticity condition (3.5). Then u ∈ H^{1}(𝕋^{n}).

Proof: We regard 𝕋^{n} = ℝ^{n}/ℤ^{n}. Fix a mollifier ϕ_{ϵ} = ϵ^{−n}ϕ((x−y)/ϵ) ∈ C^{∞}(ℝ^{n}) with ϕ(−x) = ϕ(x), and set u_{ϵ} = ϕ_{ϵ} ∗ u ∈ C^{∞}(𝕋^{n}). Then

L_{0}u_{ϵ} = ∫_{𝕋^{n}} a_{0}^{j} \frac{∂}{∂x^{j}} ϕ_{ϵ}(x − y)u(y) dy

and thus the definition of weak solution gives

∫_{𝕋^{n}}⟨ψ, L_{0}u_{ϵ}⟩dx = ∫_{𝕋^{n}} ∫_{𝕋^{n}} ϕ_{ϵ}(x − y)⟨L_{0}^{∗}ψ(x), u(y)⟩ dy dx
= ∫_{𝕋^{n}} ∫_{𝕋^{n}}⟨−^{t}a_{0}^{j} \frac{∂}{∂y^{j}} ϕ_{ϵ}(y − x)ψ(x), u(y)⟩ dy dx
= ∫_{𝕋^{n}}⟨L_{0}^{∗}(ϕ_{ϵ} ∗ ψ)(y), u(y)⟩ dy
= ∫_{𝕋^{n}}⟨ϕ_{ϵ} ∗ ψ(y), f(y)⟩ dy
= ∫_{𝕋^{n}}⟨ψ(y), ϕ_{ϵ} ∗ f(y)⟩ dy .

Thus L_{0}u_{ϵ} = f_{ϵ} = ϕ_{ϵ} ∗ f, and we note that f_{ϵ} → f strongly in L^{2}. Now the ellipticity condition (3.5) and the Plancherel theorem ensure that for all v ∈ H^{1}(𝕋^{n}),

∫_{𝕋^{n}} |∂v|^{2} dx = ∫_{𝕋^{n}} |ξ|^{2}|v̂|^{2} dξ
≤ η^{−1} ∫_{𝕋^{n}} |a_{0}^{j}ξ_{j}v̂|^{2} dξ
= η^{−1} ∫_{𝕋^{n}} |L_{0}v|^{2} dx ,

and thus

∫_{𝕋^{n}} |∂u_{ϵ}|^{2} dx ≤ η^{−1} ∫_{𝕋^{n}} |f_{ϵ}|^{2} dx .

Since u, f ∈ L^{2}, it follows that u_{ϵ} → u strongly in H^{1}.

Proposition 3.3 Under the conditions of Proposition 3.2, the map L_{0} + λ : H^{1}(𝕋^{n}) → L^{2}(𝕋^{n}) where λ = πη, is uniquely invertible, and for all u ∈ H^{1}(𝕋^{n}),

∥u∥_{H^{1}} ≤ \sqrt{5}/η∥(L_{0} + λ)u∥_{L^{2}} .  (3.8)

Proof: Write u = ∑_{k∈ℤ^{n}} u_{k}e^{2πik·x}, where the coefficients u_{k} = ∫_{𝕋^{n}} u(x)e^{−2πik·x}dx are valued in ℂ^{N}, the complexification of the real vector space modelling the fibres of E. We then have

∫_{𝕋^{n}} |(L_{0} + λ)u|^{2} dx = ∑_{k∈ℤ^{n}} |(2πik_{j}a_{0}^{j} + λ)u_{k}|^{2},

and using the vector length inequality |a + b|^{2} ≥ χ|a|^{2} − χ/(1 − χ)|b|^{2} with χ = \frac{1}{2}, we find

|(2πik_{j}a_{0}^{j} + λ)u_{k}|^{2} ≥ \frac{1}{2}|2πik_{j}a_{0}^{j}u_{k}|^{2} − λ^{2}|u_{k}|^{2}.

For k ≠ 0 this is greater than π^{2}η^{2}|u_{k}|^{2}, whilst for k = 0 we have |(2πik_{j}a_{0}^{j} + λ)u_{k}|^{2} = λ^{2}|u_{k}|^{2} = π^{2}η^{2}|u_{k}|^{2}, hence

∫_{𝕋^{n}} |(L_{0} + λ)u|^{2} dx ≥ ∑_{k∈ℤ^{n}} π^{2}η^{2}|u_{k}|^{2} = π^{2}η^{2} ∫_{𝕋^{n}} |u|^{2} dx ,

which shows L_{0} + λ has trivial kernel. Choosing χ = 1 − 1/(2|k|) shows in fact that

|(2πik_{j}a_{0}^{j} + λ)u_{k}|^{2} ≥ π^{2}η^{2}(2|k| − 1)^{2}|u_{k}|^{2},

for all k ∈ ℤ^{n} and all u_{k} ∈ ℂ^{N}. Since 2|k| − 1 ≥ 1 for all k ∈ ℤ^{n}, we obtain (3.8). Moreover, this shows also that the N × N complex matrices 2πik_{j}a_{0}^{j} + λ are invertible for any k ∈ ℤ^{n}, which gives a direct construction of the inverse of the operator L_{0} + λ.

Theorem 3.4 Suppose u ∈ L^{2}(𝕋^{n}) is a weak solution of

L_{0}u + B_{0}u + B_{1}u = f  (3.9)

where f ∈ L^{2} and L_{0} = a_{0}^{j}∂_{j} is a constant coefficient first order operator satisfying the conditions of Proposition 3.3 with ellipticity constant η, where B_{1} : L^{2} → L^{2} is bounded, and where B_{0} : H^{1} → L^{2} is a linear map satisfying

∥B_{0}∥_{H^{1}→L^{2}} ≤ η/3, ∥B_{0}^{†}∥_{H^{1}→L^{2}} ≤ η/3 ,  (3.10)

where B_{0}^{†} is the L^{2}(𝕋^{n})-adjoint of B_{0}. Then u ∈ H^{1}(𝕋^{n}) is a strong solution of (3.9), and there is a constant C, depending only on η and ∥B_{1}∥_{L^{2}→L^{2}}, such that

∥u∥_{H^{1}} ≤ C (∥f∥_{L^{2}} + ∥u∥_{L^{2}}) .  (3.11)

Proof: Construct the iteration sequence w^{(k)} ∈ H^{1}, k = 0, 1, … by defining w^{(k+1)} to be the solution of

(L_{0} + λ)w^{(k+1)} = −B_{0}w^{(k)} + f̃ ,  (3.12)

with w^{(0)} = 0, where f̃ = f + λu − B_{1}u ∈ L^{2} by the assumptions. This equation with λ = πη is uniquely solvable by Proposition 3.3. The difference v^{(k+1)} = w^{(k+1)} −w^{(k)} satisfies (L_{0} +λ)v^{(k+1)} = −B_{0}v^{(k)} and the estimate (3.8) shows that

\frac{η}{\sqrt{5}}∥v^{(k+1)}∥_{H^{1}} ≤ ∥B_{0}v^{(k)}∥_{L^{2}} ≤ \frac{η}{3}∥v^{(k)}∥_{H^{1}}.

The iteration is thus a contraction and converges in H^{1}, to w ∈ H^{1} satisfying (L_{0} + B_{0} + λ)w = f̃, and then v = u − w ∈ L^{2} is a weak solution of (L_{0} + B_{0} + λ)v = 0. Now L_{0}^{†} is also elliptic with the same ellipticity constant η, so there is z ∈ H^{1} satisfying (L_{0}^{†} + B_{0}^{†} + λ)z = v. Since v is a weak solution,

∫_{𝕋^{n}}⟨(L_{0}^{†} + B_{0}^{†} + λ)ϕ, v⟩ dx = 0 ∀ϕ ∈ H^{1}(𝕋^{n}),

we may test with ϕ = z to see that ∫ |v|^{2} = 0 and v = 0. Thus u = v + w = w ∈ H^{1} as required. By Proposition 3.3 and (3.10), we have

\frac{η}{\sqrt{5}}∥u∥_{H^{1}} ≤ ∥(L_{0} + B_{0} + B_{1})u∥_{L^{2}} + ∥B_{0}u∥_{L^{2}} + ∥B_{1}u∥_{L^{2}} + ∥λu∥_{L^{2}}
≤ ∥f∥_{L^{2}} + \frac{η}{3}∥u∥_{H^{1}} + (∥B_{1}∥_{L^{2}→L^{2}} + ηπ)∥u∥_{L^{2}} .

Since \sqrt{5} < 3, the estimate (3.11) follows.

Next we consider operators with non-constant coefficients. Let C_{S} be the 𝕋^{n} Sobolev constant

∥u∥_{Lˆ2(𝕋^{n})} ≤ C_{S}∥u∥_{H^{1}(𝕋^{n})} ,  (3.13)

where ˆ2 is defined in (3.2).

Proposition 3.5 Suppose u ∈ L^{2}(𝕋^{n}) is a weak solution of the system of equations

Lu := a^{j}∂_{j}u + bu = f  (3.14)

where f ∈ L^{2} and the coefficients a^{j} ∈ W^{1,n^{∗}} ∩ C^{0}, b ∈ L^{n^{∗}} satisfy

∥a^{j} − a_{0}^{j}∥_{L^{∞}} ≤ \frac{η}{10}  (3.15)
∥∂_{j}a^{j}∥_{L^{n^{∗}}} ≤ \frac{η}{10C_{S}}  (3.16)

where a_{0}^{j}, j = 1, …, n, are constant matrices with ellipticity constant η. Then u ∈ H^{1}(𝕋^{n}) is a strong solution of (3.14) .

Proof: It will suffice to show that L admits a decomposition satisfying the conditions of Theorem 3.4. Since L^{∞} is dense in L^{n^{∗}}, for any ϵ > 0 we may find b_{0} ∈ L^{n^{∗}}, b_{1} ∈ L^{∞}, such that b = b_{0} + b_{1} and ∥b_{0}∥_{L^{n^{∗}}} < ϵ. We choose ϵ = η/(10C_{S}). Then B_{0}u := b_{0}u + (a^{j} − a_{0}^{j})∂_{j}u satisfies

∥B_{0}u∥_{L^{2}} ≤ ∥b_{0}∥_{L^{n^{∗}}} ∥u∥_{Lˆ2} + ∥a^{j} − a_{0}^{j}∥_{L^{∞}}∥∂u∥_{L^{2}}.

Using (3.16) and the Sobolev inequality (3.13) gives

∥B_{0}u∥_{L^{2}} ≤ \frac{η}{10}(∥u∥_{H^{1}} + ∥∂u∥_{L^{2}}) ≤ \frac{η}{3}∥u∥_{H^{1}},

so ∥B_{0}∥_{H^{1}→L^{2}} ≤ η/3. Clearly B_{1}u := b_{∞}u is bounded on L^{2}, and it remains to verify the H^{1} → L^{2} bound on the adjoint operator

B_{0}^{†}w := ^{t}b_{0}w − (^{t}a^{j} −^{t}a_{0}^{j})∂_{j}w − ∂_{j}(^{t}a^{j} −^{t}a_{0}^{j})w.

Again using the Sobolev inequality and the conditions (3.15),(3.16) we find

∥B_{0}^{†}w∥_{L^{2}} ≤ C_{S}∥b_{0}∥_{L^{n^{∗}}} ∥w∥_{H^{1}} + ∥^{t}a^{j} −^{t}a_{0}^{j}∥_{L^{∞}}∥∂w∥_{L^{2}} + C_{S}∥∂_{j}a^{j}∥_{L^{n^{∗}}} ∥w∥_{H^{1}}
≤ \frac{η}{3}∥w∥_{H^{1}} ,

so the conditions of Theorem 3.4 are met and the result follows.

On a general compact manifold we define the Sobolev space H^{1}(M) by the norm

∥u∥_{H^{1}(M)}^{2} = ∫_{M}(|∇u|^{2} + |u|^{2}) dv_{M} ,  (3.17)

where the lengths |u|^{2}, |∇u|^{2} are measured using the metric ⟨ , ⟩ on sections of E and a fixed smooth background metric g̊on T M, and where ∇ is a (covariant) derivative defined in local coordinates on M and a local framing on E by

∇_{i} = ∂_{i} − Γ_{i} .  (3.18)

We assume the charts on E, M are such that

Γ_{i} ∈ L_{loc}^{n^{∗}} .  (3.19)

Note we do not require that ∇ be compatible with the metric on E. If M is compact then the space H^{1}(M) is independent of the choice of covariant derivative:

Lemma 3.6 Suppose M is compact and ∇, ˆ∇ are covariant derivatives satisfying (3.19). Then there is C > 0 such that for all u ∈ H^{1}(M),

C^{−1} ∫_{M}(|∇u|^{2} + |u|^{2})dv_{M} ≤ ∫_{M}(| ˆ∇u|^{2} + |u|^{2})dv_{M} ≤ C ∫_{M}(|∇u|^{2} + |u|^{2})dv_{M} .

(3.20)

Moreover, there is a constant C_{S}, depending on M, ∇, such that

(∫_{M} |u|ˆ2dv_{M})^{2/ˆ2} ≤ C_{S} ∫_{M}(|∇u|^{2} + |u|^{2})dv_{M} .  (3.21)

Proof: There is a finite covering of M by charts U_{α} with a corresponding partition of unity ϕ_{α}. Using the Sobolev inequality for U_{α} ⊂ ℝ^{n}, in each chart we may estimate the localisation u_{α} = ϕ_{α}u by

∫_{U_{α}} |∂u_{α}|^{2} dx ≤ C ∫_{M}(|∇u_{α}|^{2} + |u_{α}|^{2}) dv_{M} ,

where C depends also on the decomposition Γ = Γ^{∞} + Γ^{n^{∗}} ∈ L^{∞} + L^{n^{∗}}, with Γ^{n^{∗}} small. Again using the ℝ^{n} Sobolev inequality and Γ, Γ̂ ∈ L^{n^{∗}} we have

∫_{M}(| ˆ∇u|^{2} + |u|^{2}) dv_{M} ≤ C ∑_{α} ∫_{U_{α}}(| ˆ∇u_{α}|^{2} + |u_{α}|^{2}) dx
≤ C ∑_{α} ∫_{U_{α}}(|∂u_{α}|^{2} + |u_{α}|^{2}) dx
≤ C ∑_{α} ∫_{U_{α}}(|∇u_{α}|^{2} + |u_{α}|^{2}) dv_{M} ,

from which the equivalence of the norms follows easily. The Sobolev inequality follows from very similar arguments.

We may now complete the proof of interior regularity.

Theorem 3.7 Suppose M is a C^{∞} n-dimensional manifold without boundary, and E, F are real vector bundles over M, each with fibres modelled on ℝ^{N}. Suppose u ∈ L_{loc}^{2}(M) is a weak solution of Lu = f, where L is a first order operator satisfying the conditions (3.4,3.5). Then u ∈ H_{loc}^{1}(M) and u is a strong solution of Lu = f. Moreover, if M is compact there is a constant C > 0, depending on a^{j}, b and Γ, such that for all u ∈ H^{1}(M),

∥u∥_{H^{1}(M)} ≤ C(∥Lu∥_{L^{2}(M)} + ∥u∥_{L^{2}(M)}) .  (3.22)

Proof: Since L is locally of the form Lu = a^{j}∂_{j}u + bu with a^{j} ∈ W_{loc}^{1,n^{∗}} ∩ C^{0}, b ∈ L_{loc}^{n^{∗}}, for each p ∈ M there is a coordinate neighbourhood U and a constant η > 0 such that η is the ellipticity constant of a_{0}^{j} = a^{j}(p), and with respect to the local trivialisation of E|_{U} ≃ U × ℝ^{N} we have the bounds

∥a^{j} − a_{0}^{j}∥_{L^{∞}(U)} ≤ \frac{η}{10},
∥∂_{j}a^{j}∥_{L^{n^{∗}}(U)} ≤ \frac{η}{10C_{S}} ,

where we assume without loss of generality that U = Q_{R} = (0, R)^{n} is a cube of side length R ≤ 1. By paracompactness there is a locally finite countable covering {p_{α}, U_{α}}_{α∈ℤ} of M by such charts, with a subordinate C^{∞} partition of unity {ϕ_{α}}_{α∈ℤ}. Noting that supp (ϕ_{α}u) ⋐ Q_{R} and that ϕ_{α}u satisfies

L(ϕ_{α}u) = ϕ_{α}f + a^{j}∂_{j}ϕ_{α}u

weakly, we see that it suffices to consider the case where supp u ⋐ Q_{R}. Assuming this, rescaling by y = x/R, x ∈ Q_{R} and defining ũ(y) = u(x), f̃(y) = Rf(x), ã^{j}(y) = a^{j}(x) and b̃(y) = Rb(x), it follows that ũ ∈ L^{2}(𝕋^{n}) is a weak solution of

ã^{j} \frac{∂}{∂y^{j}} ũ(y) + b̃(y)ũ(y) = f̃(y).

In particular we have b̃ ∈ L^{n^{∗}}(𝕋^{n}) and

∥ã^{j} − a_{0}^{j}∥_{L^{∞}(𝕋^{n})} = ∥a^{j} − a_{0}^{j}∥_{L^{∞}(Q_{R})} ≤ η/10,
∥∂_{y^{j}}ã^{j}∥_{L^{n^{∗}}(𝕋^{n})} ≤ ∥∂_{j}a^{j}∥_{L^{n}(Q_{R})} ≤ η/10 .

The conditions of Proposition 3.5 are satisfied, so ũ ∈ H^{1}(𝕋^{n}) and thus u ∈ H_{loc}^{1}(M).

When M is compact there is a finite covering by charts {p_{α}, U_{α}}, and by Theorem 3.4, in each chart we may estimate the localisation u_{α} = ϕ_{α}u by

∥u_{α}∥_{H^{1}(U_{α})} ≤ C_{α}(∥Lu_{α}∥_{L^{2}(M)} + ∥u_{α}∥_{L^{2}(M)}),

where the H^{1}(U_{α}) norm uses the coordinate partial derivatives ∂_{i} in U_{α}. To estimate ∫_{U_{α}} |∇u_{α}|^{2}, note that the Sobolev inequality (3.3) in U_{α} gives

∫_{U_{α}} |Γu_{α}|^{2} dv_{M} ≤ C∥Γ∥_{L^{n^{∗}}(U_{α})}^{2} ∫_{U_{α}}(|∂u_{α}|^{2} + |u_{α}|^{2}) dv_{M} ,

so by the H^{1}(U_{α}) estimate we have

∥u_{α}∥_{H^{1}(M)} ≤ C_{α}(∥Lu_{α}∥_{L^{2}(M)} + ∥u_{α}∥_{L^{2}(M)}),

for some constant C_{α} depending also on ∥Γ∥_{L^{n^{∗}}(U_{α})}. Since u = ∑ u_{α} and Lu_{α} = ϕ_{α}Lu + ∂_{j}(ϕ_{α})a^{j}u, with |∂ϕ_{α}| ≤ c and |ϕ_{α}| ≤ 1, the estimate (3.22) follows easily.

The constant C of (3.22) can be controlled by ∥a^{j}∥_{W^{1,p}}, ∥b∥_{L^{p}} for any p > n^{∗}, or by otherwise controlling the decompositions ∂_{j}a^{j}, b ∈ L^{∞} + L^{n^{∗}}.

Higher regularity follows easily from Theorem 3.7 by a standard bootstrap argument:

Theorem 3.8 Suppose u ∈ L_{loc}^{2} is a weak solution of Lu = f in the situation of Theorem 3.7, where the coefficients of Lu = f satisfy the regularity conditions

a^{j} ∈ W_{loc}^{k,n^{∗}} ∩ C^{0}, b ∈ W_{loc}^{k,n^{∗}}, and f ∈ H_{loc}^{k} ,  (3.23)

for some integer k ≥ 1. Then u ∈ H_{loc}^{k+1}. If M is a compact manifold without boundary then there is a constant C = C(k, L), depending on k and ∥a^{j}∥_{W^{k,n^{∗}}}, ∥b^{j}∥_{W^{k,n^{∗}}}, such that

∥u∥_{H^{k+1}(M)} ≤ C(∥f∥_{H^{k}(M)} + ∥u∥_{H^{k}(M)}) .  (3.24)

Thus for any u ∈ L^{2}(M) such that Lu (defined weakly) satisfies Lu ∈ H^{k}(M), we have

∥u∥_{H^{k+1}(M)} ≤ C(∥Lu∥_{H^{k}(M)} + ∥u∥_{L^{2}(M)}) .  (3.25)

Proof: For simplicity we first treat the case k = 1. Theorem 3.7 shows u ∈ H_{loc}^{1}, so the vector of first derivatives ∂u ∈ L_{loc}^{2} itself is a weak solution of the system of equations

L∂u + ∂(a^{j}) ∂_{j}u = ∂f − ∂(b) u .  (3.26)

Since ∂a^{j} ∈ L^{n^{∗}} and

∥∂(b) u∥_{L^{2}} ≤ C∥b∥_{W^{1,n^{∗}}} ∥u∥_{H^{1}},

so ∥∂(b) u∥_{L^{2}} is bounded, this system satisfies the conditions of Theorem 3.7, hence u ∈ H_{loc}^{2}. The general induction step applies a similar argument: if the result is established ∀ k ≤ K − 1, and if Lu = f with coefficient conditions (3.23) with k = K, then ∂u satisfies an elliptic system (3.26) of the same form with coefficient conditions (3.23) with k = K −1, so by induction ∂u ∈ H^{K}(M) and thus u ∈ H^{K+1}(M) as required. The estimates (3.24), (3.25) follow easily by a similar argument and Theorem 3.7.

The coefficient conditions in Theorem 3.8 are not optimal in most cases. For example, if n = 3 then b ∈ W^{1,2} suffices to show u ∈ H^{2} (rather than b ∈ W^{1,3}). This follows by interpolation,

∥∂b u∥_{L^{2}} ≤ ∥∂b∥_{L^{2}}∥u∥_{L^{∞}}
≤ ϵ∥u∥_{H^{2}} + C(ϵ, ∥∂b∥_{L^{2}})∥u∥_{L^{2}} ,

which shows that ∂b u may be thought of as the sum of a small second order operator, and a large bounded operator on L^{2}. The small operator term may be absorbed as a perturbation of L, and the remainder contributes to the right hand side source term.

4 Spectral Condition

In this section we review conditions under which an operator will have a complete set of eigenfunctions. These conditions will be used in §5 to analyse boundary conditions, and thus the case of most interest concerns operators on a compact manifold without boundary, and in particular the first order elliptic systems considered in §3. However, the main result, Theorem 4.1, is stated in slightly more generality, which could be used to extend the eigenfunction representation to operators on manifolds with boundary.

Let H be a closed subspace of W^{1,2}(Y), with the induced norm, where Y is a compact manifold perhaps with boundary, and as in §3, it is understood that these spaces refer to sections of a (real) vector bundle E over Y.

The abstract spectral theorem for the map A : H → L^{2}(Y) uses the following conditions:

(C0) A : H → L^{2}(Y) is linear and bounded in the W^{1,2} topology on H.

(C1) The Gårding inequality holds: there exists a constant C such that for all ψ ∈ H we have

∥ψ∥_{H}^{2} ≤ C ∫_{Y} (⟨Aψ, Aψ⟩ + ⟨ψ, ψ⟩) dv_{Y} .  (4.1)

(C2) Weak solutions are strong solutions (“elliptic regularity”): If ϕ ∈ L^{2}(Y)

satisfies

∫_{Y} ⟨Aψ, ϕ⟩ dv_{Y} = 0 ,  ∀ ψ ∈ H,  (4.2)

then ϕ ∈ H.

(C3) A is symmetric:

∀ ϕ, ψ ∈ H  ∫_{Y} ⟨Aϕ, ψ⟩ dv_{Y} = ∫_{Y} ⟨ϕ, Aψ⟩ dv_{Y} .  (4.3)

(C4) density:

H is dense in L^{2}(Y).  (4.4)

Note that in the case ∂Y ≠ ∅, the space H must incorporate boundary conditions, and these will play an important role in verifying (C2), as will be seen in §5.

The main result of this section is the following:

Theorem 4.1 Under the conditions (C0)–(C4), there exists a countable orthonormal basis of L^{2} consisting of eigenfunctions of A, with eigenvalues all real and having no accumulation point in ℝ.

Proof: Let Ker(A) ⊂ H be the kernel of A; it is a standard fact that Ker(A) is finite dimensional when the Gårding inequality holds — we give the proof for completeness. Let {ψ_{i}}_{i=1}^{I}, I ≤ ∞, be an L^{2}–orthonormal basis of Ker(A), the equation Aψ_{i} = 0 together with (4.1) shows that {ψ_{i}}_{i=1}^{I} is bounded in W^{1,2}. The Rellich theorem [35, Theorem 7.22] implies that from the sequence ψ_{i} we can extract a subsequence ψ_{i_{j}} converging strongly in L^{2}, weakly in W^{1,2}. The Gårding inequality (4.1) with ψ replaced by ψ_{i_{j}} − ψ_{i_{k}} shows that ψ_{i_{j}} is Cauchy in W^{1,2}, hence converges in norm to some ψ ∈ W^{1,2}. By continuity of A, condition (C0), we have Aψ = 0, by continuity of L^{2} norm on W^{1,2} it holds that ∥ψ∥_{L^{2}} = 1, and it easily follows that ψ ∈ {ψ_{i}}_{i=1}^{I}. We have thus shown that {ψ_{i}}_{i=1}^{I} is compact, which yields I < ∞, as desired.

Let now

Ĥ = {ψ ∈ H : ∀ ϕ ∈ Ker(A) ∫_{Y} ⟨ϕ, ψ⟩ dv_{Y} = 0} .

For ϕ ∈ L^{2} the map H ∋ ψ → ∫_{Y} ⟨ϕ, ψ⟩ dv_{Y} ∈ ℝ is continuous in the L^{2} topology (and therefore also in the W^{1,2} topology), thus Ĥ is closed (being an intersection of closed spaces), and hence a Banach space. We note the following:

Lemma 4.2 There exists a constant C such that

∀ ψ ∈ Ĥ  ∥ψ∥_{L^{2}} ≤ C∥Aψ∥_{L^{2}} .  (4.5)

Proof: Suppose that this is not the case, then there exists a sequence ψ_{n} ∈ Ĥ such that

∥ψ_{n}∥_{L^{2}} ≥ n∥Aψ_{n}∥_{L^{2}} .  (4.6)

Rescaling ψ_{n} if necessary we can without loss of generality assume that ∥ψ_{n}∥_{L^{2}} = 1. The inequality (4.1) shows that ψ_{n} is bounded in W^{1,2} norm. By the Rellich theorem [35, Theorem 7.22] we can extract a subsequence, still denoted ψ_{n}, converging to a ψ_{∞} ∈ Ĥ, weakly in W^{1,2} and strongly in L^{2}. Equation (4.6) shows that the sequence Aψ_{n} converges to zero in L^{2}, and (4.1) with ψ replaced with ψ_{n} − ψ_{m} shows that ψ_{n} is Cauchy in the W^{1,2} norm. Continuity of A and Equation (4.6) imply that Aψ_{∞} = 0, and since A has no kernel on Ĥ we obtain ψ_{∞} = 0, which contradicts ∥ψ_{∞}∥_{L^{2}} = 1, and the lemma follows.

Returning to the proof of Theorem 4.1, define Im(A) to be the image of Ĥ under A. Then Im(A) is a closed subspace of L^{2}, which can be seen as follows: Let ψ_{i} be any sequence in Ĥ such that the sequence χ_{i} ≡ Aψ_{i} converges in L^{2} to χ_{∞} ∈ L^{2}. The inequality (4.5) shows that ψ_{i} is Cauchy in L^{2}, which together with the Gårding inequality shows that ψ_{i} is Cauchy in the W^{1,2} norm. As Ĥ is closed, it follows that there exists ψ_{∞} ∈ Ĥ such that ψ_{i} converges to ψ_{∞} in the W^{1,2} norm, and the equality χ_{∞} = Aψ_{∞} follows from continuity of A.

Let ϕ ∈ L^{2} be any element of Im(A)^{⊥}, the L^{2} orthogonal of Im(A); by definition we have

∀ ψ ∈ H ∫_{Y} ⟨ϕ, Aψ⟩ dv_{Y} = 0 .

The hypothesis (C2) of elliptic regularity implies that ϕ ∈ H, so we can use the symmetry of A to conclude

∀ ψ ∈ H ∫_{Y} ⟨Aϕ, ψ⟩ dv_{Y} = 0 .

Density of H in L^{2} implies Aϕ = 0, thus

Im(A)^{⊥} = Ker(A) .  (4.7)

Define Â : Ĥ → Im(A) by Âψ = Aψ. By the definition of all the objects involved the map Â is continuous, surjective and injective, hence bijective. Let K̂ : Im(A) → Ĥ denote its inverse, then K̂ is continuous by the open mapping theorem. Let i be the embedding of W^{1,2}(Y) into L^{2}(Y); we have i(Ĥ) ⊂ Ker(A)^{⊥} which coincides with Im(A) by (4.7). It follows that for all χ ∈ Im(A) we have i ∘ K̂(χ) ∈ Im(A), so that i ∘ K̂ defines a map of Im(A) into Im(A), which we will denote by K. Now K̂ is continuous and i compact, which implies compactness of K.

We note that Im(A) is a closed subset of the Hilbert space L^{2}, hence a Hilbert space with respect to the induced scalar product. The operator K is self–adjoint with respect to this scalar product, which can be seen as follows: let ψ_{a} = Kϕ_{a}, ϕ_{a} ∈ Im(A), a = 1, 2, thus ψ_{a} ∈ H and Aψ_{a} = ϕ_{a}. We then have

∫_{Y} ⟨ϕ_{1}, Kϕ_{2}⟩ dv_{Y} = ∫_{Y} ⟨Aψ_{1}, ψ_{2}⟩ dv_{Y} = ∫_{Y} ⟨ψ_{1}, Aψ_{2}⟩ dv_{Y} = ∫_{Y} ⟨Kϕ_{1}, ϕ_{2}⟩ dv_{Y} ,

as desired. By the spectral theorem for compact self adjoint operators [63] there exists a countable L^{2}–orthonormal basis of Im(A) consisting of eigenfunctions of K:

Kϕ_{α} = μ_{α}ϕ_{α} ,

with eigenvalues μ_{α} accumulating only at 0. Since K is invertible we have μ_{α} ≠ 0, hence

Aϕ_{α} = λ_{α}ϕ_{α} , λ_{α} = μ_{α}^{−1} .

The required basis of L^{2} is obtained by completing {ϕ_{α}} with any L^{2}–orthonormal basis of the finite dimensional kernel of A.

Definition 4.3 A is said to satisfy the spectral condition if A is an operator on C^{∞} sections of E over Y which is symmetric with respect to the L^{2} integration pairing with measure dv_{Y} and inner product ⟨·, ·⟩, and there is a countable orthonormal basis {ϕ_{α}}_{α∈Λ} of L^{2}Γ(E) consisting of eigenfunctions,

Aϕ_{α} = λ_{α}ϕ_{α}, α ∈ Λ ,  (4.8)

such that the eigenvalues λ_{α} ∈ ℝ, counted as always with multiplicity, have no accumulation point in ℝ.

Corollary 4.4 Suppose Y is a compact manifold without boundary and A : H^{1}(Y) → L^{2}(Y) is an elliptic system between sections of the bundles E, F, which satisfies the conditions (3.4,3.5) of Theorem 3.7. If A = A^{†} is formally self-adjoint (see (3.7)), then A satisfies the spectral condition, Definition 4.3.

Proof: Take H = H^{1}(Y). Condition (C0) follows from the coefficient bounds (3.4) and the inequality (3.3), and condition (C1) is conclusion (3.22) of Theorem 3.7, which also provides condition (C2). Finally, (C3) follows from the definition (3.7) of the L^{2}-adjoint A^{†}, since integration by parts is permitted in H, and (C4) is standard. The conclusions now follow from Theorem 4.1.

Corollary 4.5 Suppose Y is a compact manifold without boundary and A : H^{1}(Y) → L^{2}(Y) is an elliptic system between sections of the bundles E, F, which satisfies the conditions of Theorem 3.7. There are bases ϕ_{α} ∈ L^{2}(E), ψ_{α} ∈ L^{2}(F), α ∈ Λ, with real numbers λ_{α} having no accumulation point in ℝ, which satisfy

Aϕ_{α} = λ_{α}ψ_{α}, A^{†}ψ_{α} = λ_{α}ϕ_{α} .  (4.9)

The fields ϕ_{α}, ψ_{α} are all H^{1}(Y).

Proof: This follows directly by applying Corollary 4.4 to the formally selfadjoint operator

𝔸 = [ \begin{matrix}0 & A^{†} \\ A & 0 \end{matrix} ] ,  (4.10)

which acts between sections of the bundle E ⊕ F.

5 Boundary Regularity

In this section we introduce a broad class of boundary conditions which are elliptic in the sense that the Weak-Strong property (C2) can be established, at least for solutions supported near the boundary. When combined with the interior regularity results of §3, this will give the Weak-Strong property for compact manifolds with boundary (§6), and for a large class of noncompact manifolds with compact boundary (§8). The main result is the boundary regularity Theorem 5.11, and the primary ingredient in the arguments is the energy identity (5.15) cf. [6] and (2.6).

We consider operators which may be written abstractly in the form

L = L_{0} + B = ∂_{x} + A + B ,  (5.1)

acting on sections of a (real) vector bundle E over Y × I, where Y is a compact manifold without boundary^{6}, and for some constant δ > 0,

I = [0, δ].

Let E|_{Y} = i^{∗}E be the pullback bundle over Y, where i : Y → Y × I, y ↦ (y, 0). We assume that A is an operator on sections of E|_{Y} which is formally self-adjoint with respect to the pairing defined by integration over Y with the measure dv_{Y} and the real inner product ⟨·, ·⟩ on the fibres of E|_{Y}. The operator A and the inner product extend naturally to act on sections of E over Y × I, and we likewise extend the definition of the integration pairing by using the product measure dv_{Y} dx on Y × I. Thus, A is x–independent, but we allow B to depend upon x.

We assume also that A satisfies the spectral condition, Definition 4.3, so there is a countable index set Λ and an orthonormal basis {ϕ_{α}}_{α∈Λ} of L^{2}(E|_{Y}) consisting of eigenfunctions

Aϕ_{α} = λ_{α}ϕ_{α},  α ∈ Λ,  (5.2)

with eigenvalues λ_{α} ∈ ℝ having no accumulation points in ℝ.

A formally self-adjoint first order elliptic operator with the coefficient conditions of Theorem 3.7, will satisfy these conditions, by Corollary 4.4.

Although we have in mind primarily the case where A, B are first order differential (Dirac-type) operators, the results here will be presented in an abstract form, because they could be applied more widely. For example, A = −Δ_{Y} will also satisfy the spectral conditions, so the boundary regularity result Theorem 5.11 may also be applied to the heat equation.

We fix an eigenvalue cutoff parameter κ > 0, which is used to partition the index set Λ into

Λ^{+} = {α ∈ Λ, λ_{α} ≥ κ},  (5.3)
Λ^{−} = {α ∈ Λ, λ_{α} ≤ −κ},  (5.4)
Λ^{0} = {α ∈ Λ, |λ_{α}| < κ} ,  (5.5)

6Although many of the arguments of this section may be extended to allow ∂Y ≠ ∅, this would introduce technical complications which are not relevant to the applications we have in mind.

and we set Λ′ = Λ^{+} ∪ Λ^{−}. It will be useful also to introduce a scale parameter

θ_{0} = κ^{−1} max_{α∈Λ^{0}} |λ_{α}| < 1,  (5.6)

which measures the size of the “small” eigenvalues. For example, we could choose κ to be the smallest nonzero eigenvalue, κ = inf_{λ_{α}≠0} |λ_{α}|, in which case θ_{0} = 0. Choosing κ appropriately will lead to estimates for L which are uniform under perturbations of A which create or destroy small and zero eigenvalues.

The eigenfunction expansion u = ∑_{α∈Λ} u_{α}ϕ_{α} of u ∈ L^{2}(Y), where

u_{α} := ∮_{Y} ⟨u, ϕ_{α}⟩ dv_{Y} ,

leads to projection operators P_{+}, P_{−}, P_{0}, P′, defined by

P_{±}u = ∑_{α∈Λ^{±}} u_{α}ϕ_{α},  (5.7)
P_{0}u = ∑_{α∈Λ^{0}} u_{α}ϕ_{α} ,  (5.8)

and P′ = 1 − P_{0} = P_{+} + P_{−}.

For s ≥ 0, the Sobolev-type space H_{∗}^{s}(Y) is defined as the completion of the space of smooth sections C^{∞}(Y), with respect to the norm

∥u∥_{H_{∗}^{s}(Y)}^{2} = ∑_{α∈Λ′} |λ_{α}|^{2s}|u_{α}|^{2} + κ^{2s} ∑_{α∈Λ^{0}} |u_{α}|^{2} .  (5.9)

The space H_{∗}^{1}(Y × I) is likewise the completion of C^{∞}(Y × I) with respect to the norm

∥u∥_{H_{∗}^{1}(Y ×I)}^{2} = ∫_{0}^{δ} (∑_{α∈Λ} |u′_{α}|^{2} + ∑_{α∈Λ′} |λ_{α}|^{2}|u_{α}|^{2} + κ^{2} ∑_{α∈Λ^{0}} |u_{α}|^{2}) dx

= ∫_{0}^{δ} ∮_{Y} (|∂_{x}u|^{2} + |Au|^{2} + κ^{2}|P_{0}u|^{2}) dv_{Y} dx ,  (5.10)

where ′ = \frac{d}{dx}. Of course in the typical case where A is a first order elliptic operator, these norms will be equivalent to the usual Sobolev norms, defined using the Fourier transform. Note that the normalization (5.9) ensures that the L^{2} norm is controlled by the Sobolev norm:

∥u∥_{L^{2}(Y)} ≤ κ^{−s}∥u∥_{H_{∗}^{s}(Y)}, s ≥ 0.

In addition, this formulation leads simply to a useful trace lemma, stated in terms of the parameter ℓ,

ℓ = κδ ,  (5.11)

which measures the thickness of the boundary layer Y × [0, δ] in units of κ^{−1}.

Lemma 5.1 The restriction map r_{Y} : u ↦ r_{Y} u = u(0, ·) from C^{∞}(Y) to C^{∞}(Y × I), I = [0, δ], extends to a bounded linear map r_{Y} : H_{∗}^{1}(Y × I) → H_{∗}^{1/2}(Y) satisfying

∥r_{Y} u∥_{H_{∗}^{1/2}}^{2} ≤ c_{1}∥u∥_{H_{∗}^{1}(Y ×I)}^{2} ,  (5.12)

where c_{1} = c_{1}(ℓ) = ℓ^{−1}(1 + \sqrt{1 + ℓ^{2}}). The map x ↦ r_{Y,x}u (where r_{Y,x}u = u(x, ·) is the restriction to Y × {x}), is likewise bounded and continuous in x from I to H_{∗}^{1/2}(Y). Moreover, r_{Y} is surjective: there is an extension map e_{Y} : H_{∗}^{1/2}(Y) → H_{∗}^{1}(Y × I) such that r_{Y} e_{Y} u = u for all u ∈ H_{∗}^{1/2}(Y) and e_{Y} satisfies r_{Y,δ}e_{Y} (u) = 0 and

∥e_{Y} u∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ \frac{2}{\sqrt{3}}∥u∥_{H_{∗}^{1/2}}^{2} .  (5.13)

Proof: For x ∈ [0, ∞) set χ(x) = max(0, 1 − x). For any u ∈ C^{∞}(Y × [0, δ]) with u_{α}(x), α ∈ Λ denoting the spectral coefficients, and with χ̃(x) = χ(x/δ), we find that

|u_{α}(0)|^{2} = −2 ∫_{0}^{δ}⟨χ̃u_{α}, \frac{d}{dx}(χ̃u_{α})⟩ dx
≤ ∫_{0}^{δ} ((2χ̃|χ̃′| + ηχ̃^{2}) |u_{α}|^{2} + η^{−1}χ̃^{2}|u′_{α}|^{2}) dx

for any η > 0. For α ∈ Λ′ we take η = a|λ_{α}| and with an appropriate choice of χ we find

|λ_{α}||u_{α}(0)|^{2} ≤ ∫_{0}^{δ} (a^{−1}|u′_{α}|^{2} + |λ_{α}|^{2}(a + 2/ℓ)|u_{α}|^{2}) dx .

Likewise for α ∈ Λ^{0}, setting η = aκ gives

κ|u_{α}(0)|^{2} ≤ ∫_{0}^{δ} (a^{−1}|u′_{α}|^{2} + κ^{2}(a + 2/ℓ)|u_{α}|^{2}) dx .

Choosing a = ℓ^{−1}(\sqrt{1 + ℓ^{2}} − 1) and combining the two estimates gives (5.12) for all u ∈ C^{∞}(Y × [0, δ]). But this space is dense in H_{∗}^{1}(Y × I) by definition, and it follows easily that r_{Y} u is defined and (5.12) is valid for all u ∈ H_{∗}^{1}(Y ×I).

A very similar argument shows that for x ∈ [0, δ],

∥r_{Y,x}u∥_{H_{∗}^{1/2}}^{2} ≤ ℓ^{−1}(2 + \sqrt{4 + ℓ^{2}})∥u∥_{H_{∗}^{1}(Y ×I)}^{2}.

To establish continuity of x ↦ r_{Y,x}u as a map [0, δ] → H_{∗}^{1/2}(Y), note first that for any v ∈ H_{∗}^{1}(Y × [x_{0}, x_{1}]), x_{0} < x_{1}, the spectral coefficients v_{α} lie in H^{1}([x_{0}, x_{1}]) and we may compute:

||v_{α}(x_{1})|^{2} − |v_{α}(x_{0})|^{2}| ≤ 2 ∫_{x_{0}}^{x_{1}} |⟨v_{α}, v′_{α}⟩| dx
≤ η_{α}^{−1} ∫_{x_{0}}^{x_{1}} (|v′_{α}|^{2} + η_{α}^{2}|v_{α}|^{2}) dx .

Choosing η_{α} = |λ_{α}| for α ∈ Λ′ and η_{α} = κ for α ∈ Λ^{0} and summing gives

|∥r_{Y,x_{1}}v∥_{H_{∗}^{1/2}}^{2} − ∥r_{Y,x_{0}}v∥_{H_{∗}^{1/2}}^{2}| ≤ ∫_{x_{0}}^{x_{1}}(∑_{α∈Λ} |u′_{α}|^{2} + ∑_{α∈Λ′} |λ_{α}|^{2}|u_{α}|^{2} + ∑_{α∈Λ^{0}} κ^{2}|u_{α}|^{2}) dx
≤ ∥v∥_{H_{∗}^{1}(Y ×[x_{0},x_{1}])} .  (5.14)

Given u ∈ H_{∗}^{1}(Y × I) and x̄ ∈ [0, δ), ϵ ∈ (0, (δ − x̄)/2), we set

v(x) = u(x̄ + ϵ + x) − u(x̄ + ϵ − x),

where x ∈ [−ϵ, ϵ] and the Y -dependence of u, v is understood. Applying (5.14) with x_{0} = 0, x_{1} = ϵ, gives v(x_{0}) = 0, v(x_{1}) = u(x̄ + 2ϵ) − u(x̄) and

∥r_{Y,x̄+2ϵ}u − r_{Y,x̄}u∥_{H_{∗}^{1/2}}^{2} ≤ ∥v∥_{H_{∗}^{1}(Y ×[0,ϵ])}^{2}
≤ 2∥u∥_{H_{∗}^{1}(Y ×[x̄,x̄+2ϵ])}^{2}
= o(1) as ϵ ↘ 0 .

This establishes continuity from the right, and left continuity follows similarly.

To see that r_{Y} is surjective, we construct an extension map e_{Y} : H_{∗}^{1/2}(Y) → H_{∗}^{1}(Y × I), such that r_{Y} ∘ e_{Y} = Id. For any u ∈ H_{∗}^{1/2}(Y) let {u^{k}} be an approximating Cauchy sequence of smooth fields with spectral coefficients u_{α}^{k} and consider the sequence {ũ^{k}} defined by

ũ^{k}(x, y) = ∑_{α∈Λ} u_{α}^{k}ϕ_{α}(y)χ(x/η_{α}),

where η_{α} = \sqrt{3}/|λ_{α}| for α ∈ Λ′ and \sqrt{3}/κ for α ∈ Λ^{0}, so

∥ũ^{k}∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ ∫_{0}^{δ} ∑_{α∈Λ′} |λ_{α}|^{2}|u_{α}^{k}|^{2}(χ^{2}(x/η_{α}) + \frac{1}{3}χ^{′2}(x/η_{α})) dx

+ ∫_{0}^{δ} ∑_{α∈Λ^{0}} κ^{2}|u_{α}^{k}|^{2}(χ^{2}(x/η_{α}) + \frac{1}{3}χ^{′2}(x/η_{α})) dx .

Using the bounds ∫_{0}^{∞} χ^{2}(x)dx ≤ 1/3, ∫_{0}^{∞} χ^{′2}(x)dx ≤ 1, and noting that

∫_{0}^{δ} ψ^{2}(x/η)dx ≤ η ∫_{0}^{∞} ψ^{2}(x)dx

for any ψ, we have

∥ũ^{k}∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ \frac{2}{\sqrt{3}}\{ ∑_{α∈Λ′} |λ_{α}||u_{α}^{k}|^{2} + ∑_{α∈Λ_{0}} κ|u_{α}^{k}|^{2}\}

≤ \frac{2}{\sqrt{3}}∥ũ^{k}∥_{H_{∗}^{1/2}}^{2} .

Hence the sequence {ũ^{k}} is uniformly bounded, and a similar argument shows that it is also Cauchy, with limit ũ = e_{Y} u ∈ H_{∗}^{1}(Y ×I). It follows easily that the sequence has boundary values converging to u ∈ H_{∗}^{1/2}(Y), and u, e_{Y} u satisfy the bound (5.13).

The next result relates H^{1} estimates to boundary conditions, and is the key to understanding the nature of ellipticity for boundary data. It may be considered as a generalization either of the integration by parts formula for the Dirac operator (2.6), or of the estimate underlying the analysis in [6]. We use u(0), u(δ) to denote the restrictions r_{Y} u = u|_{Y ×{0}}, r_{Y,δ}u = u|_{Y ×{δ}} respectively.

Lemma 5.2 Suppose f ∈ L^{2}(Y × I) and u ∈ H_{∗}^{1}(Y × I) satisfies L_{0}u = f, then

∥u∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ ∥(1 − P_{0})f∥_{L^{2}(Y ×I)}^{2} + (1 + θ_{0}^{2})∥P_{0}f∥_{L^{2}(Y ×I)}^{2}

+ 3κ^{2}∥P_{0}u∥_{L^{2}(Y ×I)}^{2}

+ ∑_{α∈Λ^{+}} λ_{α} (|u_{α}(0)|^{2} − |u_{α}(δ)|^{2})

+ ∑_{α∈Λ^{−}} |λ_{α}| (|u_{α}(δ)|^{2} − |u_{α}(0)|^{2})  (5.15)

Proof: The coefficient functions u_{α}(x) are measurable and, by Fubini’s theorem, square-integrable over [0, δ]. Testing the weak formulation with ϕ(x, y) = χ(x)ϕ_{α}(y) where χ ∈ C_{c}^{∞}((0, δ)) shows that u_{α} satisfies

∫_{0}^{δ} (−u_{α}χ′ + u_{α}λ_{α}χ − f_{α}χ) dx = 0  (5.16)

for all χ ∈ C_{c}^{∞}((0, δ)). Because u ∈ H_{∗}^{1}(Y × I), the spectral coefficient u_{α}(x) is differentiable for a.e. x ∈ [0, δ], with u′_{α} square-integrable, and (5.16) shows that it satisfies the ordinary differential equation u′_{α}(x) + λ_{α}u_{α}(x) = f_{α}(x). The trace lemma also shows that the restrictions u_{α}(0), u_{α}(δ) are well defined. From the ODE we derive the fundamental identity

∫_{0}^{δ} |f_{α}|^{2} dx = ∫_{0}^{δ} |u′_{α} + λ_{α}u_{α}|^{2} dx
= ∫_{0}^{δ} (|u′_{α}|^{2} + λ_{α}^{2}|u_{α}|^{2}) dx
+ λ_{α}(|u_{α}(δ)|^{2} − |u_{α}(0)|^{2}).  (5.17)

Summing over α ∈ Λ^{+} ∪ Λ^{−} and noting that the boundary restrictions u(0), u(δ) are in H_{∗}^{1/2}(Y) by Lemma 5.1 since u ∈ H_{∗}^{1}(Y × I) by assumption, we find

∥P′u∥_{H_{∗}^{1}(Y ×I)}^{2} = ∫_{0}^{δ} ∑_{α∈Λ^{+}∪Λ^{−}}(|u′_{a}|^{2} + |λ_{α}|^{2}|u_{α}|^{2}) dx

= ∫_{0}^{δ} ∮_{Y} |P′f|^{2}dv_{Y} dx
+ ∑_{α∈Λ^{+}∪Λ^{−}} λ_{α} (|u_{α}(0)|^{2} − |u_{α}(δ)|^{2}) .  (5.18)

For α ∈ Λ^{0} we use u′_{α} = f_{α} − λ_{α}u_{α} to estimate

∥P_{0}u∥_{H_{∗}^{1}(Y ×I)}^{2} = ∫_{0}^{δ} ∑_{α∈Λ^{0}}(|u′_{a}|^{2} + κ^{2}|u_{α}|^{2}) dx

≤ ∫_{0}^{δ} ∑_{α∈Λ^{0}} ((1 + ε)|f_{α}|^{2} + (κ^{2} + (1 + ε^{−1})κ_{0}^{2})|u_{α}|^{2})
≤ 3κ^{2}∥P_{0}u∥_{L^{2}(Y ×I)}^{2} + (1 + θ_{0}^{2})∥P_{0}f∥_{L^{2}(Y ×I)}^{2} ,  (5.19)

having chosen ε = θ_{0}^{2}. Combining (5.18) and (5.19) gives (5.15).

Remark 5.3 The above proof could be generalized to allow u ∈ L^{2} and to show then that u is in H_{loc}^{1}, but this refinement is unnecessary as we soon will show a more general regularity theorem. Working with u ∈ H^{1} allows us to use the boundary terms with impunity—a freedom that is not possible with weak solutions at this stage.

The fundamental estimate (5.15) shows that in order to obtain a useful a priori elliptic estimate for a general solution of L_{0}u = f, it is necessary to impose boundary conditions which control P_{+}u(0) (and P_{−}u(δ)). Motivated by the examples of the spectral and pointwise boundary conditions for the Dirac equation (see §2), we introduce a class of boundary conditions which allow us to exploit the “good” terms in P_{−}u(0) in (5.15) to provide the required control. The effect of the parameterization below is to describe the class of admissible boundary data as graphs over the complementary subspace of “good” data (1 − P_{+})H_{∗}^{1/2}(Y). The first justification of this approach is the following existence result and its corresponding elliptic estimate (5.24).

Lemma 5.4 Let P = P_{Λ^{+}∪Λ̂} be the spectral projection determined by Λ^{+} and some subset Λ̂ ⊂ Λ^{0} of the set of small eigenvalues. Let σ ∈ P H_{∗}^{1/2}(Y) and f ∈ L^{2}(Y × I) be given, and suppose K : (1 − P)H_{∗}^{1/2}(Y) → P H_{∗}^{1/2}(Y) is a continuous linear operator, so there is a constant k ≥ 0 such that for all w ∈ H_{∗}^{1/2}(Y),

∥K(1 − P)w∥_{H_{∗}^{1/2}} ≤ k∥(1 − P)w∥_{H_{∗}^{1/2}}.  (5.20)

Then there exists a solution u ∈ H_{∗}^{1}(Y × I) to the boundary value problem

L_{0}u = f  (5.21)
P u(0) = σ + K(1 − P)u(0)  (5.22)
(1 − P)u(δ) = 0.  (5.23)

Moreover, the solution u satisfies the estimate

∥u∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ c_{4}(∥f∥_{L^{2}(Y ×I)}^{2} + ∥σ∥_{H_{∗}^{1/2}}^{2})  (5.24)

where the constant c_{4} depends on ℓ, θ_{0} and k.

Proof: The solution to an ordinary differential equation u′(x) + λu = f may be written in either of the two forms

u(x) = \{ \begin{matrix}e^{−λx}u(0) + ∫_{0}^{x} e^{λ(s−x)}f(s)ds \\ e^{λ(δ−x)}u(δ) − ∫_{x}^{δ} e^{λ(s−x)}f(s)ds. \end{matrix}  (5.25)

Consider first the spectral coefficients u_{α}(x) for α ∈ Λ^{−} ∪ Λ̂′, where Λ̂′ = Λ^{0}\Λ̂. The boundary condition (5.23) is achieved for u_{α} by letting u_{α}(δ) = 0, so we define

u_{α}(x) = − ∫_{x}^{δ} e^{λ_{α}(s−x)}f_{α}(s) ds,  α ∈ Λ^{−} ∪ Λ̂′,  (5.26)

where f_{α}(x) is the spectral coefficient of f. Note that f_{α} ∈ L^{2}(I), so the integral in (5.26) is well-defined. The identity (5.17) and u_{α}(δ) = 0 shows that

∫_{0}^{δ}(|u′_{α}|^{2} + λ_{α}^{2}|u_{α}|^{2}) dx = λ_{α}|u_{α}(0)|^{2} + ∫_{0}^{δ} |f_{α}|^{2} dx , ∀α ∈ Λ^{−} ∪ Λ̂′ . (5.27)

It follows that for all α ∈ Λ^{−},

|λ_{α}||u_{α}(0)|^{2} ≤ ∫_{0}^{δ} |f_{α}|^{2}dx − ∫_{0}^{δ}(|u′_{α}|^{2} + λ_{α}^{2}|u_{α}|^{2}) dx.  (5.28)

To control the small eigenvalues α ∈ Λ̂′ we use an elementary lemma, the proof of which is an exercise:

Lemma 5.5 For any f ∈ L^{2}([0, δ]) and λ ∈ ℝ,

∫_{0}^{δ} (∫_{x}^{δ} e^{λ(s−x)}f(s)ds)^{2} dx ≤ \{ \begin{matrix}\frac{1}{2}δ^{2}e^{2λδ} ∫_{0}^{δ} f^{2}(x)dx & λ > 0 \\ \frac{1}{2}δ^{2} ∫_{0}^{δ} f^{2}(x)dx & λ ≤ 0. \end{matrix}  (5.29)

From (5.26) and Lemma 5.5 it follows that for α ∈ Λ̂′,

∫_{0}^{δ} κ^{2}|u_{α}|^{2}dx ≤ \frac{1}{2}ℓ^{2}e^{2ℓθ_{0}} ∫_{0}^{δ} |f_{α}|^{2} dx .

Using u′_{α} = f_{α} − λ_{α}u_{α} as in (5.19) we obtain

∫_{0}^{δ}(|u′_{α}|^{2} + κ^{2}|u_{α}|^{2})dx ≤ (1 + c_{2}) ∫_{0}^{δ} |f_{α}|^{2}dx ,  (5.30)

where

c_{2} = c_{2}(ℓ, θ_{0}) = θ_{0}^{2} + \frac{3}{2}ℓ^{2}e^{2ℓθ_{0}}.  (5.31)

Combining (5.27) and (5.30) shows that

u^{(−)}(x, y) := ∑_{α∈Λ^{−}∪Λ̂′} u_{α}(x)ϕ_{α}(y)

is a sum converging in H_{∗}^{1}(Y × I), and u^{(−)} satisfies

∥u^{(−)}∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ ∥(1 − P)f∥_{L^{2}(Y ×I)}^{2} + c_{2}∥PΛ̂′f∥_{L^{2}(Y ×I)}^{2}
− ∑_{α∈Λ^{−}} |λ_{α}| |u_{α}(0)|^{2} ,  (5.32)

where 1 − P = P_{−} + PΛ̂′. For α ∈ Λ̂′ such that λ_{α} ≠ 0, using (5.26) we find that

|u_{α}(0)|^{2} ≤ (∫_{0}^{δ} e^{λ_{α}s}f_{α}(s)ds)^{2}
≤ \frac{1}{2λ_{α}} (e^{2λ_{α}δ} − 1) ∫_{0}^{δ} |f_{α}|^{2}dx ,

while if λ_{α} = 0 then |u_{α}(0)|^{2} ≤ δ ∫_{0}^{δ} |f_{α}|^{2}dx. Defining

c_{3} = ℓe^{2ℓθ_{0}},  (5.33)

it follows that

κ|u_{α}(0)|^{2} ≤ c_{3} ∫_{0}^{δ} |f_{α}|^{2}dx , ∀ α ∈ Λ̂′.

Thus, combining with (5.32) we have

∥u^{(−)}∥_{H_{∗}^{1}(Y ×I)}^{2} + ∥u^{(−)}(0)∥_{H_{∗}^{1/2}}^{2}
≤ ∥(1 − P)f∥_{L^{2}(Y ×I)}^{2} + (c_{2} + c_{3})∥PΛ̂′f∥_{L^{2}(Y ×I)}^{2}.  (5.34)

For α ∈ Λ^{+} ∪ Λ̂ we use (5.25) to define u_{α} by

u_{α}(x) = e^{−λ_{α}x}(σ_{α} + K_{α}u^{(−)}(0)) + ∫_{0}^{x} e^{λ_{α}(s−x)}f_{α}(s) ds,  (5.35)

where σ_{α}, K_{α}u^{(−)}(0) denote the ϕ_{α} coefficients of σ and Ku^{(−)}(0) respectively. Note in particular that (5.34) shows that u^{(−)}(0) ∈ H_{∗}^{1/2}(Y), so Ku^{(−)}(0) ∈ H_{∗}^{1/2}(Y) by the hypothesis (5.20), hence the coefficients K_{α}u^{(−)}(0) are well defined.

For α ∈ Λ^{+} we estimate using (5.17) and (5.22):

∫_{0}^{δ}(|u′_{α}|^{2} + λ_{α}^{2})dx ≤ λ_{α}|σ_{α} + K_{α}u^{(−)}(0)|^{2} + ∫_{0}^{δ} |f_{α}|^{2}dx
≤ 2λ_{α}(|σ_{α}|^{2} + |K_{α}u^{(−)}(0)|^{2}) + ∫_{0}^{δ} |f_{α}|^{2}dx . (5.36)

For α ∈ Λ̂ we estimate directly from (5.35):

κ^{2} ∫_{0}^{δ} |u_{α}|^{2}dx ≤ 3κ^{2}(|σ_{α}|^{2} + |K_{α}u^{(−)}(0)|^{2}) ∫_{0}^{δ} e^{−2λ_{α}x}dx
+ 3κ^{2} ∫_{0}^{δ} (∫_{0}^{x} e^{λ_{α}(s−x)}f_{α}(s)ds)^{2} dx
≤ 3c_{3}κ(|σ_{α}|^{2} + |K_{α}u^{(−)}(0)|^{2}) + \frac{3}{2}ℓc_{3} ∫_{0}^{δ} |f_{α}|^{2}dx , (5.37)

where Lemma 5.5 has been used to control the final term. Using u′_{α} = f_{α}−λ_{α}u_{α} to estimate |u′_{α}|^{2} ≤ (1 + ε)|f_{α}|^{2} + (1 + ε^{−1})λ_{α}^{2}|u_{α}|^{2} with ε = θ_{0}^{2}, (5.37) gives for α ∈ Λ̂,

∫_{0}^{δ} (|u′_{α}|^{2} + κ^{2}|u_{α}|^{2}) dx ≤ 9c_{3}κ(|σ_{α}|^{2} + |K_{α}u^{(−)}(0)|^{2})
+ (1 + 3c_{2}) ∫_{0}^{δ} |f_{α}|^{2}dx .  (5.38)

Combining (5.36) and (5.38) we have (setting u^{(+)} = ∑_{Λ^{+}∪Λ̂} u_{α}ϕ_{α})

∥u^{(+)}∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ ∥P f∥_{L^{2}(Y ×I)}^{2} + 3c_{2}∥PΛ̂f∥_{L^{2}(Y ×I)}^{2}
+ 2(∥P_{+}σ∥_{H_{∗}^{1/2}}^{2} + ∥P_{+}Ku^{(−)}(0)∥_{H_{∗}^{1/2}}^{2})
+ 9c_{3}(∥PΛ̂σ∥_{H_{∗}^{1/2}}^{2} + ∥PΛ̂Ku^{(−)}(0)∥_{H_{∗}^{1/2}}^{2}) , (5.39)

where P_{+} = P_{Λ^{+}}, P = P_{+} +PΛ̂. Since we have already shown that K_{α}u^{(−)}(0) ∈ H_{∗}^{1/2}(Y), all terms on the right hand side of (5.39) are bounded, which shows that u^{(+)} is well-defined in H_{∗}^{1}(Y × I).

With u = u^{(+)} + u^{(−)}, we add an appropriate multiple of (5.34) to (5.39) to control the bad terms in Ku^{(−)}(0) with the good term u^{(−)}(0) of (5.34). This gives the elliptic estimate (5.24):

∥u∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ ∥u^{(+)}∥_{H_{∗}^{1}(Y ×I)}^{2} + max(1, 2k^{2} , 9c_{3}k^{2})∥u^{(−)}∥_{H_{∗}^{1}(Y ×I)}^{2}
≤ c_{4}^{2}(∥f∥_{L^{2}(Y ×I)}^{2} + ∥σ∥_{H_{∗}^{1/2}}^{2}) ,

where c_{4} = c_{4}(ℓ, θ_{0}, k) as required. The definitions (5.26), (5.35) ensure u is a solution satisfying the boundary conditions (5.22), (5.23).

Explicitly, we may take c_{4}^{2} = 3c_{2} + (k^{2} + 1)(2 + 9c_{3}) in general, and c_{4}^{2} = 2 max(1, k^{2}) if θ_{0} = 0.

The next result is the key to handling operators with coefficients depending on x. Recall that the operator norm ∥B∥_{op} of a linear map B : X_{1} → X_{2} between Banach spaces is the smallest constant such that

∥Bu∥_{X_{2}} ≤ ∥B∥_{op}∥u∥_{X_{1}}, ∀u ∈ X_{1} .  (5.40)

Lemma 5.6 Suppose L_{0}, A, f, σ, K are as in Lemma 5.4, and suppose B : H_{∗}^{1}(Y × I) → L^{2}(Y × I) is a linear map satisfying

c_{4} ∥B∥_{op} < 1.  (5.41)

Then there exists u ∈ H_{∗}^{1}(Y × I) satisfying

(L_{0} + B)u = f

and the boundary conditions (5.22),(5.23), such that

∥u∥_{H_{∗}^{1}(Y ×I)}^{2} ≤ \frac{c_{4}}{1 − c_{4}∥B∥_{op}} (∥f∥_{L^{2}(Y ×I)} + ∥σ∥_{H_{∗}^{1/2}}).  (5.42)

Proof: Let u^{(0)} ∈ H_{∗}^{1}(Y × I) be any function satisfying the boundary conditions (5.22),(5.23); the trace lemma 5.1 ensures the existence of a suitable u^{(0)}. Construct a sequence {u^{(k)}} ⊂ H_{∗}^{1}(Y × I) by solving the problems

L_{0}u^{(k)} = f − Bu^{(k−1)}  (5.43)
P u^{(k)}(0) = σ + K(1 − P)u^{(k)}(0)  (5.44)
(1 − P)u^{(k)}(δ) = 0, n = 1, 2, ….  (5.45)

Lemma 5.4 ensures this problem has a solution for every n ≥ 1, and the difference w^{(k)} = u^{(k)} − u^{(k−1)} satisfies

L_{0}w^{(k)} = −Bw^{(k−1)}
P w^{(k)}(0) = K(1 − P)w^{(k)}(0)
(1 − P)w^{(k)}(δ) = 0 .

The elliptic estimate (5.24) gives

∥w^{(k)}∥_{H_{∗}^{1}(Y ×I)} ≤ c_{4}∥Bw^{(k−1)}∥_{L^{2}(Y ×I)}.

If ∥B∥_{op} < 1/c_{4} then the iteration is a contraction and thus the sequence u^{(k)} is Cauchy, converging to u = lim_{n→∞} u^{(k)} strongly in H_{∗}^{1}(Y × I). Taking the limit of (5.43) shows that (L_{0}+B)u = f, and boundedness of the trace operator r_{Y} shows that u satisfies the boundary conditions (5.22)–(5.23). The elliptic estimates (5.24) satisfied by u^{(k)} are preserved in the limit, so u satisfies

∥u∥_{H_{∗}^{1}(Y ×I)} ≤ c_{4}(∥f − Bu∥_{L^{2}(Y ×I)} + ∥σ∥_{H_{∗}^{1/2}}),

from which (5.42) follows easily.

Observe that the proof of Lemma 5.6 relies on just two properties of the operator L_{0}; namely, the solvability of the problem (5.43) with boundary conditions (5.44), (5.45), and the elliptic estimate (5.24), which provides the size bound (5.41) for the perturbation B. This suggest that it should be possible to extend this existence result to more general operators L = L_{0} + B, for which a strictly coercive estimate such as (5.24) can be established.

Consider, for example, the case where E has a complex structure J : E → E, J^{2} = −1, and A is a normal operator ([A, A^{∗}] = 0), so A = A_{0} + JA_{1} where A_{0}, A_{1} are self-adjoint and commuting, A_{0} satisfies the spectral condition, and both commute with J. Then A admits an eigenfunction basis

Aϕ_{α} = (λ_{α} + μ_{α}J)ϕ_{α}, λ_{α}, μ_{α} ∈ ℝ, ∀α ∈ Λ,

and the results of this section extend with only minor modifications, provided the eigenvalues λ_{α}, μ_{α} satisfy the sectorial condition

sup_{α∈Λ} |μ_{α}|/(1 + |λ_{α}|) < ∞.

Before stating the main uniqueness theorem, a definition of weak solution with boundary conditions is required. Note that although the definition is consistent with just L^{2} boundary data, the regularity theorem 5.11 will require data in H_{∗}^{1/2}.

Definition 5.7 Suppose L = ∂_{x} + A + B where A satisfies the spectral conditions (Definition 4.3) and B : H_{∗}^{1}(Y × I) → L^{2}(Y × I) is a bounded linear operator for which there exists an L^{2}-adjoint B^{†} : H_{∗}^{1}(Y × I) → L^{2}(Y × I) such that:

∫_{Y ×I}⟨Bu, v⟩ dv_{Y} dx = ∫_{Y ×I}⟨u, B^{†}v⟩ dv_{Y} dx , ∀ u, v ∈ H_{∗}^{1}(Y × I). (5.46)

Suppose further that P = P_{+} +PΛ̂ (as in Lemma 5.4), that K : (1−P)L^{2}(Y) → P L^{2}(Y) is a bounded linear map with L^{2}-adjoint K^{†} : P L^{2}(Y) → (1−P)L^{2}(Y), and let σ ∈ P L^{2}(Y), f ∈ L^{2}(Y × I) be given. A weak solution of the boundary value problem

Lu = f  (5.47)
P u(0) = σ + K(1 − P)u(0)  (5.48)
(1 − P)u(δ) = 0  (5.49)

is a field u ∈ L^{2}(Y × I) satisfying (with L^{†} = −∂_{x} + A + B^{†})

∫_{Y ×I}⟨u, L^{†}ϕ⟩ dv_{Y} dx = ∫_{Y ×I}⟨f, ϕ⟩ dv_{Y} dx + ∮_{Y} ⟨σ, ϕ(0)⟩ dv_{Y} ,  (5.50)

for all ϕ ∈ H_{∗}^{1}(Y × I) satisfying the adjoint boundary conditions

(1 − P + K^{†}P)ϕ(0) = 0  (5.51)
P ϕ(δ) = 0 .  (5.52)

The boundary values ϕ(0), ϕ(1) both lie in H_{∗}^{1/2}(Y) by the trace lemma, so the adjoint boundary conditions are well-defined on the space of test fields. Since C^{∞} fields are dense in H_{∗}^{1}(Y × I) and in H_{∗}^{1/2}(Y), to verify the weak equation (5.50) it suffices to test just with C^{∞} fields ϕ; however the uniqueness argument of Lemma 5.10 requires the use of an H_{∗}^{1} test field.

The structure of the adjoint boundary condition (5.51) is explained by the next lemma, which is applied with v = u(0) − σ and H = L^{2}(Y).

Lemma 5.8 If H is a Hilbert space, P : H → H is an orthogonal projection and K : ker P → range P is bounded, and if v ∈ H satisfies

⟨v, ϕ⟩_{H} = 0 ∀ ϕ ∈ ker(1 − P + K^{†}P)  (5.53)

(where K^{†} is the adjoint of K in H), then v ∈ ker(P − K(1 − P)).

Proof: Since ker P = range (1 − P) ⊥ range P, it follows that P is self-adjoint and there is an orthogonal decomposition H = (1 − P)H ⊕ P H. Setting ϕ_{1} = (1 − P)ϕ, ϕ_{2} = P ϕ, the condition ϕ = ϕ_{1} + ϕ_{2} ∈ ker(1 − P + K^{†}P) is equivalent to ϕ_{1} = −K^{†}ϕ_{2}, which exhibits ker(1 − P + K^{†}P) as a graph over P H. Similarly decomposing v = v_{1} + v_{2}, the condition ⟨v, ϕ⟩ = 0 is equivalent to ⟨v_{2}−Kv_{1}, ϕ_{2}⟩ = 0. Since this holds for all ϕ_{2} ∈ P H, it follows that v_{2} = Kv_{1}, or equivalently, v ∈ ker(P − K(1 − P)).

In other words, H = ker(P −K(1−P))⊕ker(1−P +K^{†}P) is an orthogonal splitting of H, where P − K(1 − P), 1 − P + K^{†}P are projections, which are not orthogonal in general.

Using Lemma 5.8 we next show that an H^{1} weak solution of the boundary value problem (5.47)-(5.49), in fact satisfies the equation (5.47) and the boundary conditions (5.48,5.49) in the strong sense:

Lemma 5.9 If u ∈ H_{∗}^{1}(Y × I) is a weak solution of (5.47) with the boundary conditions (5.48), (5.49), then u satisfies the equation Lu = f in the sense of strong (H^{1}) derivatives, and the restrictions u(0) = r_{Y} (u), u(δ) = r_{Y,δ}u satisfy the boundary conditions (5.48),(5.49) in L^{2}(Y). Conversely, if u ∈ H_{∗}^{1}(Y × I) is a strong solution of (5.47,5.48,5.49), then u is also a weak solution.

Proof: Integration by parts gives

∫_{Y ×I}⟨u, L^{†}ϕ⟩ = ∫_{Y ×I}⟨Lu, ϕ⟩ + ∮_{Y} ⟨u(0), ϕ(0)⟩ − ∮_{Y} ⟨u(δ), ϕ(δ)⟩

for any u, ϕ ∈ H_{∗}^{1}(Y ×I). Testing u with arbitrary ϕ ∈ C_{c}^{∞}(Y ×I) shows that a H_{∗}^{1} weak solution satisfies Lu = f in the sense of strong derivatives. Comparing this formula with (5.50) shows also that

∮_{Y} ⟨u(0) − σ, ϕ(0)⟩ = 0, ∮_{Y} ⟨u(δ), ϕ(δ)⟩ = 0,

for all ϕ(0), ϕ(δ) satisfying the adjoint boundary conditions (5.51), (5.52). Since H_{∗}^{1/2}(Y) is dense in ker(1 − P + K^{†}P) ⊂ L^{2}(Y), Lemma 5.8 may be applied with v = u(0) − σ to show the boundary condition (5.48) holds in L^{2}(Y), and (5.49) follows similarly.

To show the converse, integration by parts again gives

∫_{Y ×I}⟨u, L^{†}ϕ⟩ − ⟨Lu, ϕ⟩
= ∮_{Y} ⟨u(0), ϕ(0)⟩ − ⟨u(δ), ϕ(δ)⟩
= ∮_{Y} ⟨σ + (1 + K)(1 − P)u(0), ϕ(0)⟩ − ⟨P u(δ), ϕ(δ)⟩
= ∮_{Y} ⟨σ, ϕ(0)⟩ + ⟨u(0), (1 − P + K^{†}P)ϕ(0)⟩ − ⟨u(δ), P ϕ(δ)⟩ ,

and the final two terms vanish by the adjoint boundary conditions (5.51,5.52).

By solving an adjoint problem, we now show that weak solutions of (5.47)- (5.49) are unique.

Lemma 5.10 Let u ∈ L^{2}(Y ×I) be a weak solution of the boundary value problem (5.47)-(5.49), with σ ∈ L^{2}(Y) and f ∈ L^{2}(Y × I). Suppose that the operator L = ∂_{x} + A + B satisfies the conditions of Lemma 5.6 and Definition 5.7, and the L^{2}(Y × I)-adjoint B^{†} : H_{∗}^{1}(Y × I) → L^{2}(Y × I) satisfies

c_{4}∥B^{†}∥_{op} < 1.  (5.54)

Suppose also that the boundary operators K, K^{†} of Definition 5.7 satisfy

∥K(1 − P)w∥_{H_{∗}^{1/2}} ≤ k ∥(1 − P)w∥_{H_{∗}^{1/2}} ,  (5.55)
∥K^{†}P w∥_{H_{∗}^{1/2}} ≤ k ∥P w∥_{H_{∗}^{1/2}}.  (5.56)

for some constant k ≥ 0 and all w ∈ H_{∗}^{1/2}(Y). Then u is unique.

Proof: It will suffice to show that any weak solution ũ of (5.47)–(5.49) with σ = 0, f = 0, must vanish. Consider the adjoint problem L^{†}ϕ = ũ with boundary conditions (5.51),(5.52); writing L^{†}ϕ = ũ as (∂_{x} − A − B^{†})ϕ = −ũ, we see that L^{†}, K^{†} satisfy the conditions required by Lemma 5.6, since interchanging A ↔ −A means replacing P by 1 − P, and perhaps changing a finite number of eigenfunctions in Λ̂ (without modifying Λ_{0}). The elliptic estimate (5.42) does not depend on Λ̂. Thus, by Lemma 5.6 there exists a solution ϕ ∈ H_{∗}^{1}(Y × I) of this boundary value problem. By construction, ϕ satisfies the boundary conditions required of test functions in (5.50), so testing ũ in (5.50) with ϕ gives

∫_{Y ×I} |ũ|^{2} = ∫_{Y ×I}⟨ũ, L^{†}ϕ⟩ = 0

and thus ũ = 0.

It is easy to check that (5.56) is equivalent to requiring that K : (1 − P)H_{∗}^{−1/2}(Y) → P H_{∗}^{−1/2}(Y) is bounded, with constant k.

We now obtain the main result on boundary regularity of weak solutions. Note that although the definition of weak solution assumes boundary data σ ∈ L^{2}(Y) only, and uniqueness of weak solutions holds also in this generality, this condition is incompatible with regularity u ∈ H_{∗}^{1}(Y ×I), which would imply (by simple restriction) that σ ∈ H_{∗}^{1/2}(Y). However, some results for L^{2} boundary conditions on domains with uniformly Lipschitz boundary are known [1, 2], so it is plausible that the results here could be extended.

Theorem 5.11 Suppose u ∈ L^{2}(Y × I) is a weak solution of the boundary value problem (5.47)-(5.49) with operator L = ∂_{x} + A + B_{0} + B_{1}, where A satisfies the spectral conditions (Definition 4.3), B_{0} satisfies the size condition (5.41) with L^{2}-adjoint B_{0}^{†} satisfying (5.54), and B_{1} : L^{2}(Y × I) → L^{2}(Y × I) is bounded. Further suppose the boundary operators K, K^{†} satisfy (5.55,5.56), and σ ∈ P H_{∗}^{1/2}(Y). Then u ∈ H_{∗}^{1}(Y × I) (so u is a strong solution) and u satisfies the a priori estimate

∥u∥_{H_{∗}^{1}(Y ×I)} ≤ \frac{c_{4}}{1 − c_{4}∥B_{0}∥_{op}} (∥f∥_{L^{2}(Y ×I)} + ∥σ∥_{H_{∗}^{1/2}} + ∥B_{1}∥_{L^{2}→L^{2}}∥u∥_{L^{2}(Y ×I)}).

(5.57)

Proof: Since ∥B_{1}u∥_{L^{2}(Y ×I)} ≤ ∥B_{1}∥_{op}∥u∥_{L^{2}(Y ×I)}, u satisfies L̃u := (∂_{x} + A + B_{0})u = f̃ where f̃ = f − B_{1}u ∈ L^{2}(Y × I). Lemma 5.6 constructs a solution ū ∈ H_{∗}^{1}(Y × I) of L̃ū = f̃ satisfying the same boundary conditions, and it follows that ū is also a weak solution. By the Uniqueness Lemma 5.10 we have u = ū and thus u ∈ H_{∗}^{1}(Y × I), as required. The estimate (5.42) of Lemma 5.6 (with f replaced by f̃) leads directly to (5.57).

6 Boundary regularity for first order systems

In this section we determine conditions under which the boundary regularity results of §5 apply to a first order equation of Dirac type (see (6.7), (6.9)) at the boundary, with suitable boundary operator, to show H^{1} regularity of an L^{2} weak solution.

We assume M is a smooth manifold with compact boundary Y, E → M and F → M are real vector bundles over M with scalar products, and L is a first order elliptic operator on sections of E to sections of F, which in local coordinates x^{j}, j = 1, …, n takes the form

Lu = a^{j}∂_{j}u + bu ,  (6.1)

where a^{j}, b are homomorphisms of E to F as before. Note that we do not assume Y to be connected.

To apply the preceding results, the coefficients must satisfy the interior regularity conditions (3.4), and the boundary restrictions must be defined and satisfy the corresponding conditions in dimension n − 1,

\begin{aligned}a^{j}|_{Y} & ∈ & W^{1,(n−1)^{∗}}(Y) ∩ C^{0}, j = 1, …, n, \\ b|_{Y} & ∈ & L^{(n−1)^{∗}}(Y) . \end{aligned}  (6.2)

Conditions (3.4,3.5) and (6.2) will be assumed throughout this section.

Remark 6.1 The H^{s} conditions

\begin{aligned}a^{j} & ∈ & W_{loc}^{s,2}(M) ∩ C^{0}(M) , \\ b & ∈ & W_{loc}^{s−1,2}(M) , \end{aligned}  (6.3)

where

s = n/2 for n ≥ 4 ,
s > 3/2 for n = 3 ,
s = 3/2 for n = 2 ,

(6.4)

imply the interior (3.4) and boundary (6.2) coefficient regularity conditions, through the Sobolev embedding and trace theorems [60]. The C^{0} condition in (6.3) is superfluous for n = 2, 3.

Let x = x^{n} be a boundary coordinate, defining a tubular neighbourhood Y × [0, 1] ⊂ M of Y with local coordinates (y^{i}, x) ∈ Y × [0, 1] (where we identify Y with Y × {0}). Let dv_{M} be a volume measure on M, and define dv_{Y} = (−1)^{n−1}∂_{x} dv_{M}|_{Y} on Y × {0}. The local coordinate integration factor γ is defined in Y × [0, 1] by dv_{M} = γ dy dx, where dy dx is coordinate Lebesgue measure. We assume the local coordinate condition^{7}

γ ∈ (W^{1,n^{∗}} ∩ C^{0})(Y × [0, 1]) .  (6.5)

In order to directly apply the results of the previous section, we assume that γ = γ(0) is independent of x in Y × [0, 1], so dv_{M} = dv_{Y} dx. This involves no loss of generality, as the x-dependence of γ in the integral form (6.15) of the equation can be absorbed through a rescaling of the coefficients a^{i}, b. Since the restrictions to Y × {0} are unchanged, this does not affect the boundary operator A.

7As in §3, this means that we can cover Y by a finite number of coordinate charts 𝒪_{α} so that γ has the stated regularity in the local coordinates on 𝒪_{α} × [0, 1].

To minimize confusion with the outer unit normal n = −∂_{x} we set

ν := ^{t}a^{n} ,  (6.6)

and for simplicity we assume

tν = ν^{−1} .  (6.7)

This is satisfied by the Dirac operator. Equation (6.7) can also be achieved in several other situations of interest by pre-multiplying L by a suitable homomorphism, or by making a frame change in F (this can be done e.g. when F is trivial). Note that such a pre-multiplication does not affect the values of ã^{i}, b̃, where we define

ã^{i} := νa^{i}|_{Y} , i = 1, …, n − 1, b̃:= νb|_{Y} .  (6.8)

By extending independent of x we regard ã^{i}, b̃ as defined on Y × [0, 1]. We assume the important boundary symmetry condition

ã^{i} = −^{t}ã^{i}, i = 1, …, n − 1 ,  (6.9)

where the transpose ^{t}ã^{i} is taken with respect to the inner product on E. We wrap up these conditions into a definition.

Definition 6.2 A first order system (6.1) is of boundary Dirac type if the coefficients a^{j} satisfy the conditions (6.7,6.9) in some neighbourhood of the boundary.

As discussed in §10, this class includes the examples of §2 and §11. These conditions will be assumed henceforth.

Using (6.9) we define the boundary operator

Au := ∑_{i=1}^{n−1} ã^{i}∂_{i}u + ã_{0}u + b̃_{0}u ,  (6.10)

where

ã_{0} = \frac{1}{2} ∑_{i=1}^{n−1} (∂_{i}ã^{i} + ã^{i}∂_{i} log γ) ,  (6.11)

and b̃_{0} = b_{0}|_{Y} for some symmetric endomorphism b_{0} of E. We require that b_{0} ∈ L^{n^{∗}}(Y ×I)) and b̃_{0} ∈ L^{(n−1)^{∗}}(Y), compare (3.4), (6.2). Then A is formally self-adjoint (A^{†} = A) on the bundle E|_{Y} over Y, with respect to the measure dv_{Y}.

Note that the choice of zero-order term b_{0} gives some freedom in the definition of the boundary operator A, which is thus not uniquely determined by L. Near the boundary, Lu = ^{t}ν(∂_{x} + A + B), which may be expressed as

L̃u := (∂_{x} + A + B)u = f̃ ,  (6.12)

where L̃u = νLu, f̃ = νf, and B denotes the difference

Bu = ∑_{i=1}^{n−1}(νa^{i} − ã^{i})∂_{i}u + (νb − ã_{0} − b̃_{0})u .  (6.13)

By Corollary 4.4, under the above conditions, A will satisfy the spectral condition 4.3. Denote the eigenvalue index set by Λ and fix a cutoff κ, as in §5. Similarly let Λ^{+} = {α ∈ Λ : λ_{α} ≥ κ}, fix some subset Λ̂ ⊂ {α ∈ Λ : |λ_{α}| < κ} and let P = P_{Λ^{+}} + PΛ̂ be the associated spectral projection operator. Note that Theorem 3.7 (see (3.22)) shows that the H_{∗}^{s} norms defined using A will be equivalent to the corresponding H^{s} norms defined on Y and Y × I, at least for s = 0, 1; it seems likely that this will hold, by interpolation, for all s ∈ [0, 1]. If the coefficient regularity allows an H^{k+1} elliptic estimate (cf. Theorem 3.8) then this should extend to s ∈ [0, k + 1].

The definition of weak solution on a manifold with boundary which we are about to give is slightly simpler than the tubular neighbourhood Definition 5.7 of §5, since the conditions (5.49), (5.52) may be imposed by localising with a boundary cutoff function; see the proof of Theorem 6.4. Let H_{c}^{1}(M) denote the dense subspace of H^{1}(M) consisting of functions of compact support, where we recall that because M is a manifold with boundary Y = ∂M, H_{c}^{1}(M) includes functions which are non-zero on Y. As in §5, the boundary condition is expressed using a positive spectrum projection P : L^{2}(Y) → L^{2}(Y) and a bounded linear map K : (1 − P)L^{2}(Y) → P L^{2}(Y) and its L^{2}(Y) adjoint K^{†}.

Definition 6.3 Let f ∈ L_{loc}^{2}(M) and σ ∈ P L^{2}(Y) be given. A section u ∈ L_{loc}^{2}(M) is a weak solution of Lu = f with boundary condition

P u_{0} = σ + K(1 − P)u_{0} ,  (6.14)

if

∫_{M}⟨u, L^{†}ϕ⟩ dv_{M} = ∫_{M}⟨f, ϕ⟩ dv_{M} + ∮_{Y} ⟨σ, νϕ_{0}⟩ dv_{Y} ,  (6.15)

for all ϕ ∈ H_{c}^{1}(M) satisfying the boundary condition

(1 − P + K^{†}P)(νϕ_{0}) = 0 ,  (6.16)

where L^{†} is the L^{2} adjoint given by (3.7).

Note we are using the notation u_{0}, ϕ_{0}, etc., to denote the restriction (trace) on the boundary Y. The additional term ν in (6.15,6.16) (cf. (5.50,5.51)) arises from the relation L = ^{t}ν(∂_{x} + A + B) between L and the boundary form ∂_{x} + A + B used in §5.

The boundary condition (6.14) restricts u_{0} = u|_{Y} to lie in the affine subspace of L^{2}(Y) given by the graph of x ↦ σ+Kx over the negative spectrum subspace x ∈ (1 − P)L^{2}(Y). It will be useful to re-express (6.14) as

Ku_{0} = σ ,  (6.17)

where we have introduced the operator K on L^{2}(Y),

K := P − K(1 − P) ,  (6.18)

and likewise to re-express the “adjoint” boundary condition (6.16) as

K^{†}ϕ_{0} = 0 ,  (6.19)

where we define

K^{†} := ν^{−1}(1 − P + K^{†}P)ν .  (6.20)

The next result generalizes the interior weak-strong Theorem 3.7 to boundary value problems.

Theorem 6.4 Suppose L and A satisfy the conditions (3.4,3.5,6.2, 6.5,6.7,6.9), and suppose σ ∈ P H_{∗}^{1/2}(Y), f ∈ L_{loc}^{2}(M). Further suppose K : (1−P)L^{2}(Y) → P L^{2}(Y) is bounded linear and satisfies (5.55), with L^{2} adjoint K^{†} satisfying (5.56). Assume u ∈ L_{loc}^{2}(M) is a weak solution of Lu = f with the boundary condition Ku_{0} = σ (6.17). Then u ∈ H_{loc}^{1}(M) and u is a strong solution. Moreover, there are constants δ ∈ (0, 1], c_{5}, depending only on κ, k and a^{i}, b, and intervals I′ = [0, δ/2], I = [0, δ] such that

∥u∥_{H^{1}(Y ×I′)} ≤ c_{5}(∥f∥_{L^{2}(Y ×I)} + ∥σ∥_{H_{∗}^{1/2}(Y)} + ∥u∥_{L^{2}(Y ×I)}) .  (6.21)

Proof: Theorem 3.7 ensures u ∈ H_{loc}^{1}(M̊), where M̊ is the interior of M, so it suffices to consider u compactly supported in Y × [0, δ), for any choice of δ ∈ (0, 1). In particular, because u then vanishes near Y × {δ}, it follows from (6.14) that u is a weak solution with boundary conditions, in the sense of Definition 5.7.

It will suffice to show, for a sufficiently small choice of δ > 0, that we may decompose B = B_{0} + B_{1} into pieces satisfying the size conditions of Theorem 5.11. Write B = β^{i}∂_{i} + β where

β^{i}(y, x) = (a^{n}(y, x))^{−1}a^{i}(y, x) − (a^{n}(y, 0))^{−1}a^{i}(y, 0), i = 1, …, n − 1,

so β^{i} ∈ W^{1,n^{∗}} ∩ C^{0} and β^{i}(y, 0) = 0. Since the constant c_{4} of Lemma 5.4 depends only on θ_{0} < 1, ℓ = κδ and the constant k of (5.55,5.56), it is bounded uniformly in δ ≤ 1. Consequently for any ϵ > 0 there is δ_{0} > 0 such that c_{4}∥β^{i}∥_{L^{∞}(Y ×[0,δ])} < ϵ for all δ ≤ δ_{0}.

Likewise, since γ ∈ W^{1,n^{∗}} (6.5), we have β ∈ L^{n^{∗}} and there is a decomposition β = β_{0} + β_{1} with c_{4} C_{S} ∥β_{0}∥_{L^{n^{∗}}(Y ×[0,1])} ≤ ϵ, where C_{S} is the Sobolev constant on Y × [0, 1], and β_{1} ∈ L^{∞}. Then B_{0} = β^{i}∂_{i} + β_{0} satisfies (5.41), as does B_{0}^{†} (possibly after decreasing δ), and B_{1} = β_{1} is bounded on L^{2}. Theorem 5.11 now applies and shows u ∈ H^{1}(Y × [0, δ]), since H^{1}(Y × I) = H_{∗}^{1}(Y × I) as remarked above. The elliptic estimate (6.21) follows by applying (5.57) to ũ = χu, where χ = χ(x) is a cutoff function, χ(x) = 1 for 0 ≤ x ≤ δ/2, χ(x) = 0 for x ≥ \frac{3}{4}δ.

Corollary 6.5 Suppose L, A, K, K satisfy the conditions of Theorem 6.4. Then for all u ∈ H_{loc}^{1}(M) we have the boundary estimate

∥u∥_{H^{1}(Y ×I′)} ≤ c_{5}(∥Lu∥_{L^{2}(Y ×I)} + ∥Ku_{0}∥_{H_{∗}^{1/2}(Y)} + ∥u∥_{L^{2}(Y ×I)}) .  (6.22)

Proof: If u ∈ H_{loc}^{1}(M) then f := Lu ∈ L_{loc}^{2}(M), σ := Ku_{0} ∈ H_{∗}^{1/2}(Y), and u is a strong solution. Since integration by parts may be applied to show u satisfies the weak equation (6.15), Theorem 6.4 applies and gives (6.22).

A bootstrap argument, slightly more complicated than that used for the interior bounds Theorem 3.8, leads to higher, H^{1+k}, regularity. Rather than stating complicated conditions for general k, we describe the details only for the case k = 1 (u ∈ H^{2}). The coefficient regularity conditions are most likely not optimal.

Theorem 6.6 In the setting of Theorem 6.4 let u ∈ H^{1}(Y × I) be the solution and suppose the following additional regularity conditions are satisfied,

a^{j} , b ∈ W^{1,∞}(Y × I) ,  (6.23)
γ ∈ W^{2,n^{∗}}(Y × [0, 1]) ,  (6.24)
f ∈ H^{1}(Y × I), σ ∈ H_{∗}^{3/2}(Y) ,  (6.25)
[A, K](1 − P) : (1 − P)H_{∗}^{1/2}(Y) → P H_{∗}^{1/2}(Y) is bounded.  (6.26)

Then there exists δ′′ ≤ δ/2 such that u ∈ H^{2}(Y × I′′), where I′′ = [0, δ′′], and there is a constant c_{6} depending on the coefficient bounds (6.23)-(6.26), such that

∥u∥_{H^{2}(Y ×I′′)} ≤ c_{6}(∥f∥_{H^{1}(Y ×I)} + ∥σ∥_{H_{∗}^{3/2}(Y)} + ∥u∥_{L^{2}(Y ×I)}) .  (6.27)

Remark 6.7 For the APS and chiral boundary conditions (2.13), (2.18), A commutes with K and thus (6.26) is trivially satisfied.

Proof: The idea is to show that Au satisfies a similar boundary value problem. For convenience, let H̃_{0}^{1}(Y × I), I = [0, δ], denote the H^{1} completion of the C^{∞} functions of compact support in Y × [0, δ), where δ is the constant of Theorem 6.4. In particular, functions in H̃_{0}^{1}(Y × I) have vanishing trace on Y × {δ}. For any v, ψ ∈ C_{c}^{∞}(Y × [0, δ)), we have the identity

∫_{Y ×I}⟨Av, L̃^{†}ψ⟩ dx dv_{Y} = ∫_{Y ×I} (⟨[L̃, A]v, ψ⟩ + ⟨L̃v, Aψ⟩) dx dv_{Y}
+ ∮_{Y} ⟨v_{0}, Aψ_{0}⟩ dv_{Y} .  (6.28)

Since A is formally self-adjoint with respect to dv_{Y}, this formula follows by direct calculation. The terms with L̃^{†}ψ, L̃v are well-defined for v, ψ ∈ H̃_{0}^{1}(Y ×I). Since v_{0}, ψ_{0} ∈ H_{∗}^{1/2}(Y) by Lemma 5.1, the boundary integral extends also by writing it as

∮_{Y} ⟨v_{0}, Aψ_{0}⟩ dv_{Y} = ∮_{Y} ⟨Jv_{0}, J^{−1}Aψ_{0}⟩ dv_{Y} .  (6.29)

Here J = (1 + |A|)^{1/2} is defined as Ju = ∑_{α∈Λ} u_{α}(1 + |λ_{α}|)^{1/2}ψ_{α}, with u = ∑_{α∈Λ} u_{α}ψ_{α}; J^{−1} is defined similarly. This shows that

|∮_{Y} ⟨v_{0}, Aψ_{0}⟩ dv_{Y} | ≤ c∥v_{0}∥_{H_{∗}^{1/2}(Y)}∥ψ_{0}∥_{H_{∗}^{1/2}(Y)},

for all v_{0}, ψ_{0} ∈ H_{∗}^{1/2}(Y), where the constant c is determined by A.

More care is required to control the commutator [L̃, A], which takes the form

[L̃, A]v = α^{ij}∂_{ij}^{2}v + α_{1}^{i}∂_{i}v + α_{2}v

where

α^{ij} = [νa^{i}, ã^{j}],
α_{1}^{i} = νa^{j}∂_{j}ã^{i} − ã^{j}∂_{j}(νa^{i}) + [νa^{i}, ã_{0}],
α_{2} = νa^{i}∂_{i}ã_{0} − ã^{i}∂_{i}(νb) .

Observe that α^{ij}|_{Y} = 0. The coefficient conditions (6.23) and v ∈ H^{1} ensure that α_{1}^{i}∂_{i}v is in L^{2} and hence may be combined with the source term Af̃. Likewise (6.23) ensures α_{2}v is bounded in L^{2}.

Let λ_{0} ∈ ℝ\Λ, so A − λ_{0} has trivial kernel and satisfies elliptic estimates ∥v∥_{H^{s+1}} ≤ c∥(A − λ_{0})v∥_{H^{s}} for s = 0, 1 at least. Since A is self-adjoint and elliptic, the cokernel of A − λ_{0} is also trivial, so A − λ_{0} : H^{s+1} → H^{s} is invertible. Now decompose α^{ij}∂_{ij}^{2}v = B_{2}Av + B_{3}v where

B_{2} = α^{ij}∂_{ij}^{2}(A − λ_{0})^{−1}

is bounded from H^{1} → L^{2}, and

B_{3} = −λ_{0}α^{ij}∂_{ij}^{2}(A − λ_{0})^{−1}

is also bounded from H^{1} → L^{2}. Note that by perhaps decreasing δ we may ensure that B_{2} and B_{2}^{†} satisfy a smallness condition similar to (5.41).

Direct calculation (noting that [A, P] = 0) establishes the boundary formula

∮_{Y} ⟨(1 + K)(1 − P)v_{0}, Aψ_{0}⟩ dv_{Y} = ∮_{Y} ⟨Av_{0}, (1 − P + K^{†}P)ψ_{0}⟩ dv_{Y}

+ ∮_{Y} ⟨[A, K](1 − P)v_{0}, ψ_{0}⟩ dv_{Y} , (6.30)

for all v_{0}, ψ_{0} ∈ H_{∗}^{1/2}(Y), since K satisfies (6.26) and (5.55) by assumption.

Now u ∈ H̃_{0}^{1}(Y × I) satisfies L̃u = f̃ and u_{0} = σ + (1 + K)(1 − P)u_{0}. Substituting u for v in (6.28) and using these relations, shows that u satisfies

∫_{Y ×I}⟨Au, (L̃ − B_{2})^{†}ψ⟩ dx dv_{Y} = ∫_{Y ×I}⟨Af̃+ α_{1}^{i}∂_{i}u + α_{2}u, ψ⟩ dx dv_{Y}

+ ∮_{Y} ⟨Aσ + [A, K](1 − P)u_{0}, ψ_{0}⟩ dv_{Y}

+ ∮_{Y} ⟨Au_{0}, (1 − P + K^{†}P)ψ_{0}⟩ dv_{Y} . (6.31)

In particular, if ψ_{0} ∈ ker(1 − P + K^{†}P) ∩ H_{∗}^{1/2}(Y) then w = Au ∈ L^{2}(Y × I) satisfies

∫_{Y ×I}⟨w, (L̃ − B_{2})^{†}ψ⟩ dx dv_{Y} = ∫_{Y ×I}⟨f̃_{1}, ψ⟩ dx dv_{Y}
+ ∮_{Y} ⟨σ_{1}, ψ_{0}⟩ dv_{Y} ,  (6.32)

for all ψ ∈ H^{1}(Y × I) such that ψ ∈ ker(1 − P + K^{†}P), where

f̃_{1} = Af̃+ α_{1}^{i}∂_{i}u + α_{2}u ∈ L^{2},
σ_{1} = Aσ + [A, K](1 − P)u_{0} ∈ H_{∗}^{1/2}(Y) .

In other words, w ∈ L^{2}(Y × I) is a weak solution of the problem

(L̃ − B_{2})w = f̃_{1},
P w_{0} = σ_{1} + K(1 − P)w_{0} .

By shrinking the boundary layer we may assume ∥B_{2}∥_{H^{1}→L^{2}} and ∥B_{2}^{†}∥_{H^{1}→L^{2}} are sufficiently small that the conditions of Theorem 6.4 are met, so w = Au ∈ H^{1}(Y × [0, δ]). The equation now gives ∂_{x}u = f̃ − Au − Bu ∈ H^{1} and thus u ∈ H^{2}(Y × [0, δ]).

7 Fredholm properties on compact manifolds

The interior and boundary estimates of § 6 lead to solvability (Fredholm) results, by standard arguments. The main interest lies in identifying the cokernel, and we give a simple necessary and sufficient condition for solvability, in Theorem 7.3. This section treats only compact manifolds, leaving the more difficult case of non-compact manifolds to the following section. Because more detailed descriptions are given in § 8, some of the arguments are only briefly summarised here.

Throughout this section we assume the coefficients a^{j} , j = 1, …, n and b of L satisfy the conditions of § 6, namely (3.4),(3.5), (6.2), (6.5), (6.7), (6.9); L^{†} is given by (3.7), and the boundary operators K, K^{†} satisfy (5.55,5.56), where P = P_{Λ} + PΛ̂ is a positive spectrum projection of A (6.10), and K , K^{†} are defined by (6.18,6.20).

Recall the Sobolev space H^{1}(M) of sections of E over M is defined by the norm (3.17)

∥u∥_{H^{1}(M)}^{2} = ∫_{M}(|∇u|^{2} + |u|^{2}) dv_{M} ,  (7.1)

where lengths are measured using the metric ⟨ , ⟩ on E and a fixed smooth background metric g̊ on T M, and the connection ∇ satisfies (3.18,3.19). Note again that ∇ need not be compatible with the metric on E, and the space H^{1}(M) is independent of the choice of ∇.

The following basic elliptic estimate extends (3.22) of Theorem 3.7 to manifolds with boundary, using the boundary neighbourhood estimate (6.21) of Theorem 6.4.

Proposition 7.1 There is a constant C > 0 depending on a^{j} , b, Γ and K such that for all u ∈ H^{1}(M),

∥u∥_{H^{1}(M)} ≤ C(∥Lu∥_{L^{2}(M)} + ∥Ku_{0}∥_{H_{∗}^{1/2}(Y)} + ∥u∥_{L^{2}(M)}) .  (7.2)

Proof: The argument used in Theorem 3.7 to prove the interior estimate (3.22) may be applied using Theorem 6.4, estimate (6.21), to estimate ∥u_{α}∥_{H^{1}(U_{α})} over boundary neighbourhoods U_{α}. The remaining details are unchanged.

Theorem 7.2 The linear operator

(L, K) : H^{1}(M) → L^{2}(M) × P H_{∗}^{1/2}(Y)  (7.3)

is semi-Fredholm (i.e. has finite dimensional kernel and closed range).

Proof: Suppose {u_{k}}_{1}^{∞} is a sequence in ker(L, K), normalised by ∥u_{k}∥_{H^{1}(M)} = 1. To show the kernel is finite dimensional, it suffices to show there is a subsequence converging in H^{1}(M). By Rellich’s lemma there is a subsequence (which we also denote u_{k}) which converges strongly in L^{2}(M), to ū ∈ L^{2}(M) say. The elliptic estimate applied to the differences u_{j} − u_{k} shows that the sequence is Cauchy in H^{1}(M) and thus converges strongly to ū ∈ H^{1}(M). Since (7.3) is bounded, it follows that ū ∈ ker(L, K), so the unit ball in the kernel is compact and hence the kernel is finite dimensional.

To show the range is closed, let H̊^{1}(M) be the finite codimension subspace of H^{1}(M) defined by the condition

∫_{M}(⟨∇u, ∇ϕ⟩ + ⟨u, ϕ⟩) dv_{M} = 0 ∀ ϕ ∈ ker(L, K).

A Morrey-type argument by contradiction using (7.2) shows there is a constant C > 0 such that for all u ∈ H̊^{1}(M),

C^{−1} ∫_{M} |u|^{2} dv_{M} ≤ ∫_{M} |Lu|^{2} dv_{M} + ∮_{Y} |JKu_{0}|^{2} dv_{Y} ,  (7.4)

where J = (1+|A|)^{1/2}. Now suppose {u_{k}}_{1}^{∞} ⊂ H^{1}(M) is such that Lu_{k} = f_{k} → f ∈ L^{2}(M) and K(u_{k})_{0} = s_{k} → σ ∈ P H_{∗}^{1/2}(Y). (Note that by the definition (6.18) of K, the range of K is a subspace of P H_{∗}^{1/2}(Y)). Since the kernel is finite dimensional we may normalise u_{k} ∈ H̊^{1}(M), and then (7.4) and (7.2) show that {u_{k}}_{1}^{∞} is bounded in H^{1}(M). It then follows as above that there is a subsequence converging strongly in H^{1}(M) to ū, and that Lū = lim_{k→∞} f_{k} = f and Kū_{0} = lim_{k→∞} s_{k} = σ, so the range of (L, K) is closed.

The general boundary value problem

\{ \begin{matrix}Lu & = & f & in M \\ Ku_{0} & = & σ & on Y \end{matrix}  (7.5)

is solvable for u ∈ H^{1}(M) provided (f, σ) satisfies the condition (7.6) of the following main result.

Theorem 7.3 (f, σ) ∈ L^{2}(Y) × P H_{∗}^{1/2}(Y) lies in the range of (L, K) (that is, (7.5) admits a solution u ∈ H^{1}(M)), if and only if

∫_{M}⟨f, ϕ⟩ dv_{M} + ∮_{Y} ⟨σ, νϕ_{0}⟩ dv_{Y} = 0 ∀ ϕ ∈ ker(L^{†}, K^{†}) .  (7.6)

Proof: If u ∈ H^{1}(M) satisfies Lu = f and Ku_{0} = σ then u is also a weak solution. Condition (7.6) then follows directly from the definition 6.3 of weak solution, hence (7.6) is a necessary condition for solvability.

To establish the converse, consider first the case σ = 0. Thus we suppose f ∈ L^{2}(M) satisfies ∫_{M}⟨f, ϕ⟩ dv_{M} = 0 for all ϕ ∈ ker(L^{†}, K^{†}), and we must find u ∈ H^{1}(M) satisfying Lu = f, Ku_{0} = 0.

By Lemma 5.1 the trace map r_{Y} : u ↦ u_{0} is bounded, hence

H̊_{K}^{1} := {u ∈ H^{1}(M) : Ku_{0} = 0, and
∫_{M}(⟨∇u, ∇ϕ⟩ + ⟨u, ϕ⟩) dv_{M} = 0 ∀ ϕ ∈ ker(L, K)} (7.7)

is a closed subspace of H^{1}(M). The argument of Theorem 7.2 (ii) shows there is a constant C such that

∫_{M}(|∇u|^{2} + |u|^{2}) dv_{M} ≤ C ∫_{M} |Lu|^{2} dv_{M}  (7.8)

for all u ∈ H̊_{K}^{1}(M). In particular, ∫_{M} |Lu|^{2} dv_{M} is strictly coercive on H̊_{K}^{1}, so the Lax-Milgram lemma gives u ∈ H̊_{K}^{1} satisfying

∫_{M}⟨f, Lϕ⟩ dv_{M} = ∫_{M}⟨Lu, Lϕ⟩ dv_{M}

for all ϕ ∈ H̊_{K}^{1}. This equality also holds if ϕ ∈ ker(L, K), so Ψ = Lu−f satisfies

∫_{M}⟨Ψ, Lϕ⟩ dv_{M} = 0 ∀ ϕ ∈ H^{1}(M) , Kϕ_{0} = 0 .  (7.9)

Lemma 5.8 and the identity

∫_{M}⟨Ψ, Lϕ⟩ dv_{M} = ∫_{M}⟨L^{†}Ψ, ϕ⟩ dv_{M} − ∮_{Y} ⟨νΨ_{0}, ϕ_{0}⟩ dv_{M}  (7.10)

show that (7.9) is the weak form of the adjoint problem

L^{†}Ψ = 0, K^{†}Ψ_{0} = 0 .  (7.11)

By (3.7), L^{†} is elliptic with boundary representation

L^{†} = −ν(∂_{x} + Â + B̂),

where Â = −ν^{−1}Aν since A^{†} = A, and B̂ = −ν^{−1}B^{†}ν. By (6.8,6.9) the leading terms in Â are ã^{i}∂_{i} so Â is elliptic on Y, and self-adjoint by (6.7). Since Â(ν^{−1}ϕ_{α}) = −λ_{α}ν^{−1}ϕ_{α} if Aϕ_{α} = λ_{α}ϕ_{α}, we see that Â satisfies the spectral conditions, and spec Â = −spec A. (Note that in the usual case of Dirac operators, Â = A and the spectrum is symmetric). Now P̂ := 1−ν^{−1}P ν is a positive eigenspace projector for Â, with eigenvalues −λ_{α} for α ∈ Λ^{−} ∪ (Λ^{0}\Λ̂), and the boundary operator satisfies

K^{†}Ψ_{0} = (P̂ + ν^{−1}K^{†}ν(1 − P̂))Ψ_{0} .  (7.12)

Since K̂ = −ν^{−1}K^{†}ν maps negative eigenvectors (of Â) to positive eigenvectors, it follows that K^{†}Ψ_{0} = 0 is an elliptic boundary condition for L^{†}. The boundedness conditions (5.55,5.56) for K̂ follow from the corresponding conditions for K.

Since (L^{†}, K^{†}) is elliptic and satisfies the conditions for Theorem 6.4, we conclude that Ψ ∈ H^{1}(M) and Ψ satisfies the strong form (7.11).

Since Ψ ∈ ker(L^{†}, K^{†}), assumption (7.6) with σ = 0 gives

∫_{M}⟨f, Ψ⟩ dv_{M} = 0.

By construction Ku_{0} = 0, so we may use u as a test function in the weak form (7.9) of the equation satisfied by Ψ, giving

∫_{M}⟨Lu, Ψ⟩ dv_{M} = 0.

It follows from Ψ = Lu − f that Ψ = 0 and thus u is the required solution.

Now consider the case σ ≠ 0. By Lemma 5.1 there is an extension v = e_{Y} (σ) ∈ H^{1}(M) supported in a neighbourhood of Y such that v_{0} = σ, ∥v∥_{H^{1}(M)} ≤ 2∥σ∥_{H_{∗}^{1/2}}. Let f̃ = f − Lv and consider the equation

Lũ = f̃, Kũ_{0} = 0 .  (7.13)

The previous case shows there is a solution provided f̃ satisfies

∫_{M}⟨f̃, ψ⟩ dv_{M} = 0 ∀ ψ ∈ ker(L^{†}, K^{†}) ⊂ H^{1}(M).

Now (7.10) shows that for all ψ ∈ ker(L^{†}, K^{†}),

∫_{M}⟨f̃, ψ⟩ dv_{M} = ∫_{M}⟨f, ψ⟩ dv_{M} − ∫_{M}⟨v, L^{†}ψ⟩ dv_{M} + ∮_{Y} ⟨v_{0}, νψ_{0}⟩ dv_{Y}
= ∫_{M}⟨f, ψ⟩ dv_{M} + ∮_{Y} ⟨σ, νψ_{0}⟩ dv_{Y} .

Thus if (7.6) is satisfied then there exists a solution ũ of (7.13), and then u = ũ + v is the required full solution. This establishes sufficiency for the condition (7.6).

We note two important consequences of Theorems 7.2, 7.3.

Corollary 7.4 (7.5) admits a solution for all (f, σ) ∈ L^{2}(M) × P H_{∗}^{1/2}(Y) if and only if ker(L^{†}, K^{†}) = {0}.

Corollary 7.5 (L, K) : H^{1}(M) → L^{2}(M) × P H_{∗}^{1/2}(Y) is Fredholm.

Proof: The argument of Theorem 7.3 shows that (L^{†}, K^{†}) is elliptic and thus has finite dimensional kernel by Theorem 7.2. Now (7.6) shows that the range of (L, K) has finite codimension.

8 Fredholm properties on complete noncompact manifolds

In this section we establish conditions under which the Fredholm and existence results of the previous section for the operator (L, K) : H^{1}(M) → L^{2}(M) × H_{∗}^{1/2}(Y), may be extended to non-compact manifolds. This includes in particular, a generalisation of the solvability criterion (7.6) of Theorem 7.3. Results of this type may be applied to establish positive mass results in general relativity, for example.

The non-compactness of M causes some difficulties not found in the compact case. A classical result [45, 65] shows that a Dirac operator D on a non-compact manifold is essentially self-adjoint on L^{2}(M). However, this elegant result is useless for our purposes, since it implies only that {(ϕ, Dϕ) : ϕ ∈ dom D ⊂ L^{2}(M)} is closed in the graph topology on L^{2}(M)×L^{2}(M). This is weaker than the closed range property, which is necessary for useful solvability criteria. In fact, because L^{2}(M) often does not encompass natural decay rates of solutions, the self-adjoint closure may not have closed range. In such cases the Dirac operator defined on L^{2}(M) will not be semi-Fredholm. This is shown explicitly in the following example.

Consider the self-adjoint closure \overline{D} : dom \overline{D} ⊂ L^{2}(ℝ^{3}) → L^{2}(ℝ^{3}) of the constant coefficient Dirac operator D = γ^{i}∂_{i} and let f = Du, u = (1−χ)|x|^{−1}ψ, where χ(r) is a smooth compactly supported function identically one around 0 and ψ is a constant spinor on ℝ^{3}. Clearly f ∈ L^{2}(ℝ^{3}) but u ∉ L^{2}(ℝ^{3}), so in particular, u ∉ dom \overline{D}. However, f still lies in the closure of the range of \overline{D}, since D(χ_{R}u) = χ_{R}f + Dχ_{R}u → f in L^{2}(ℝ^{3}), where χ_{R}(x) = χ(x/R), but χ_{R}u can not converge in L^{2}(ℝ^{3}). Clearly (u, f) ∉ graph \overline{D} since u ∉ L^{2}(ℝ^{3}), and it can be shown (using the corresponding Schrödinger-Lichnerowicz identity) that there is no ū ∈ L^{2}(ℝ^{3}) satisfying Dū = f. Thus the self-adjoint closure \overline{D} does not have closed range.

In order to obtain an operator with closed range, it is thus necessary to enlarge the domain, which raises the question of determining the appropriate decay rate. We sidestep this problem by using the L^{2} size of the covariant derivative as a norm. To obtain sufficient control on the L_{loc}^{2} behaviour, we then must postulate a weighted Poincaré inequality (8.3). The existence of such inequalities can be established for the applications of most interest in general relativity; see Proposition 8.3 and §9.

The elliptic estimate (7.2) plays a central role in the analysis over a compact manifold, but its noncompact analogue cannot be obtained directly by similar localisation arguments. However, in cases of geometric interest an identity of Schrödinger-Lichnerowicz form (generalising (2.5)) is available, and can be used to construct suitable global estimates.

The weighted Poincaré and Schrödinger-Lichnerowicz estimates are the two additional ingredients needed for establishing solvability and Fredholm properties on a non-compact manifold.

For ease of further reference, let us summarize the hypotheses which will be made throughout this section:

Hypotheses 8.1 M is a non-compact manifold with compact boundary Y, which is complete with respect to a C^{∞} background metric g̊. The case Y = ∅ is admitted. The operator L = a^{j}∂_{j} + b satisfies the global uniform ellipticity and boundedness condition

η^{2}|V |^{2} ≤ g̊_{jk}⟨a^{j}(x)V, a^{k}(x)V ⟩ ≤ η^{−2}|V |^{2} ,  (8.1)

for some η > 0, for all V ∈ E_{x} and all x ∈ M. The coefficients of L satisfy the interior regularity conditions (3.4), and the boundary regularity and structure conditions of §6, namely (6.2), (6.5), (6.7), (6.9). Let A be the boundary operator and P its associated positive spectrum projection, as in §6. The boundary operator K : (1 − P)L^{2}(Y) → P L^{2}(Y) satisfies (5.55,5.56), and K , K^{†} are defined in (6.18,6.20). The connection

∇ = ∂ − Γ  (8.2)

satisfies (3.18,3.19) and we note again that ∇ need not be compatible with the metric on E — this is important in some applications.

We may express L in terms of ∇ by

L = a^{j}∇_{j} + (b + a^{j}Γ_{j}) = a^{j}∇_{j} + β,

where β ∈ L_{loc}^{n^{∗}}(M). Additional, rather weak, decay conditions will be imposed on β (8.8), on the negative part of the curvature endomorphism ρ (8.13), and on Γ^{S} = \frac{1}{2}(Γ +^{t}Γ) in §9.

Definition 8.2 The covariant derivative ∇ on E over M admits a weighted Poincaré inequality if there is a weight function w ∈ L_{loc}^{1}(M) with ess inf_{Ω} w > 0 for all relatively compact Ω ⋐ M, such that for all u ∈ C_{c}^{1}(M) we have

∫_{M} |u|^{2} w dv_{M} ≤ ∫_{M} |∇u|^{2} dv_{M} .  (8.3)

Here the length |∇u|^{2} is measured by the metric on E and the background Riemannian metric g̊ on M, and dv_{M} is the volume measure of g̊. It is clear that the weight function w can be chosen to be smooth.

The semi-norm

∥u∥_{ℍ}^{2} = ∫_{M} |∇u|^{2} dv_{M}  (8.4)

on C_{c}^{∞}(M) may be completed to form the space

ℍ := ∥ · ∥_{ℍ}-completion of C_{c}^{∞}Γ(E) ,  (8.5)

which consists of equivalence classes of ℍ-convergent sequences in C_{c}^{∞}(M). The weighted Poincaré inequality (8.3) ensures that an ℍ-convergent sequence converges locally in L^{2}, so the equivalence classes may be identified with crosssections in the usual Lebesgue sense: with cross-sections having coefficient functions agreeing dv_{M}–a.e.

If there is no weighted Poincaré inequality, then it may be that ℍ can not be identified with a space of Lebesgue-measurable cross-sections in this sense. For example, the trivial spinor bundle over M = 𝕋^{2} × ℝ with the flat connection ∇_{i} = ∂_{i} admits a global parallel spinor ∇_{i}ψ = 0 which is approximated in the ℍ seminorm by ψ_{k} = χ(x/k)ψ for χ ∈ C_{c}^{∞}(ℝ), χ = 1 on [−1, 1]. Now ∫_{M} |∇ψ_{k}|^{2} dv_{M} → 0, but lim_{k→∞} ψ_{k} = ψ ≠ 0, so the ℍ-equivalence class [0] contains ψ ≠ 0 everywhere. In other words, (8.4) does not define a norm on spinors in this example. This shows, inter alia, that (8.3) will not hold in all cases.

More generally, a weighted Poincaré inequality fails for manifolds of the form N × ℝ, where N is compact and itself admits a parallel spinor. It follows from the proof of Theorem 9.3 below that in such cases the orthogonal complement in ℍ of the subspace of all parallel spinors will admit a weighted Poincaré inequality. Note also that the presence of a weighted Poincaré inequality (8.3) does not imply there are no global parallel spinors — ℝ^{3} provides a simple counterexample.

However, weighted Poincaré inequalities can be demonstrated in many cases of interest. In the next section we will prove:

Proposition 8.3 A covariant derivative ∇ on E admits a weighted Poincaré inequality if any one of the following conditions holds:

1. there is a relatively compact domain Ω ⊂ M and a constant c > 0 such

that

∫_{Ω} |u|^{2} dv_{M} ≤ c ∫_{M} |∇u|^{2} dv_{M}  (8.6)

for all u ∈ C_{c}^{∞}(M);

2. there are no nontrivial globally parallel sections (∇u = 0 ⇒ u = 0);

3. M has a weakly asymptotically flat end M̃ (see Definition 9.4), with dim M ≥ 3;

4. M has a weakly asymptotically hyperboloidal end (see Definition 9.9), with dim M ≥ 2.

When M is non-compact, the global Gårding inequality (generalizing (7.2)) cannot be constructed from local estimates. Motivated by some classical and fundamental identities, we instead introduce the following definition.

Definition 8.4 The operator pair (L, K) admits a Schrödinger-Lichnerowicz estimate if there is C > 0 and a non-negative function ρ such that^{8} for all u ∈ C_{c}^{1}(M),

8The function ρ here should not be confused with the energy density arising in general relativity: in Section 11 ρ will be zero.

C^{−1} ∫_{M} |∇u|^{2} dv_{M} ≤ ∫_{M}(|Lu|^{2} + ρ|u|^{2}) dv_{M} + ∮_{Y} |JKu_{0}|^{2} dv_{Y} ,  (8.7)

where J = (1 + |A|)^{1/2}.

Lemma 8.5 Suppose that the Schrödinger-Lichnerowicz estimate (8.7) holds for all u ∈ C_{c}^{1}(M) with ρ and β = L − a^{j}∇_{j} satisfying ρ ∈ L_{loc}^{n^{∗}/2}, β ∈ L_{loc}^{n^{∗}}, and

lim_{R→∞} sup_{M\M_{R}} \frac{ρ + |β|^{2}}{w} < ∞ ,  (8.8)

where {M_{R}}_{R→∞} is an exhaustion of M. Then L : ℍ → L^{2}(M) is bounded and (8.7) holds for all u ∈ ℍ.

Proof: It will suffice to show that the individual terms of the right-hand-side of (8.7) are bounded by ∥u∥_{ℍ}^{2}. Now

∫_{M} |Lu|^{2} ≤ C ∫_{M} |∇u|^{2} + 2 ∫_{M} |β|^{2}|u|^{2} ,  (8.9)

and we use (8.8) and (8.3) to estimate

∫_{M\M_{R}}(ρ + |β|^{2})|u|^{2} ≤ sup_{M\M_{R}} \frac{ρ + |β|^{2}}{w} ∫_{M\M_{R}} |u|^{2}w
≤ sup_{M\M_{R}} \frac{ρ + |β|^{2}}{w} ∫_{M\M_{R}} |∇u|^{2}
≤ C ∫_{M} |∇u|^{2} ,

for some R < ∞. Let χ_{R} ∈ C_{c}^{∞}(M) be a cut-off function with support contained in M_{2R}, χ_{R} = 1 on M_{R}. Then

∫_{M_{R}}(ρ + |β|^{2})|u|^{2} ≤ ∫_{M_{2R}}(ρ + |β|^{2})|χ_{R}u|^{2}
≤ (∥ρ∥_{L^{n^{∗}/2}(M_{2R})} + ∥β∥_{L^{n^{∗}}(M_{2R})}^{2}) ∥χ_{R}u∥_{L^{n^{∗}}(M_{2R})}^{2} .

Applying the Sobolev inequality for ∇ on the compact set M_{2R} and the weighted Poincaré inequality show that the last term is controlled by ∫_{M} |∇u|^{2}. Finally, the K-bound (5.55) and the restriction Lemma 5.1 show that the boundary term is also controlled by ∫_{M} |∇u|^{2}.

Schrödinger-Lichnerowicz identities hold for many common examples, and can easily be adapted to produce estimates of the form (8.7). We will not attempt to give general conditions which imply such inequalities — it is simpler to ask only that (8.7) be established separately in any particular case of interest.

For example, consider the classical Dirac operator D of the metric g as in §2, on a non-compact spin manifold M. Combining (2.6) and (2.8) gives

∫_{M} |∇ψ|^{2} dv_{M} = ∫_{M}(|Dψ|^{2} − \frac{1}{4}R(g)|ψ|^{2}) dv_{M} + ∮_{Y} ⟨ψ_{0}, (D_{Y} + \frac{1}{2}H_{Y})ψ_{0}⟩ dv_{Y} ,

(8.10)

for any C_{c}^{1} spinor field on M. Suppose the boundary operator is K = P_{+}, the orthogonal projection onto the positive spectrum eigenspinors of D_{Y}. If the boundary mean curvature H_{Y} satisfies H_{Y} ≤ \sqrt{16π/Area(Y)}, then the argument in §2 shows that the boundary term in (8.10) is not greater than

∮_{Y} ⟨P_{+}ψ_{0}, D_{Y} P_{+}ψ_{0}⟩ dv_{Y} ≤ ∥Kψ_{0}∥_{H_{∗}^{1/2}(Y)}^{2} ,

and (8.7) follows immediately, with

ρ = max(0, −\frac{1}{4}R(g)) .  (8.11)

Since β = 0 in this example, the inequality holds for all u ∈ ℍ provided ρ satisfies (8.8). For general mean curvatures H_{Y} ∈ L^{∞}(Y), note again that

∮_{Y} ⟨ψ_{0}, D_{Y} ψ_{0}⟩ dv_{Y} ≤ ∥P_{+}ψ_{0}∥_{H_{∗}^{1/2}(Y)}^{2} .

If H_{Y} ∈ L^{∞}(Y) then ∮_{Y} H_{Y} |ψ_{0}|^{2} ≤ ∥H_{Y} ∥_{L^{∞}(Y)}∥Jψ_{0}∥_{H_{∗}^{1/2}(Y)}^{2}. Using a frac-

tional Sobolev inequality, the control on H_{Y} may be weakened to H_{Y} ∈ L^{p}(Y), p = n−1 for n ≥ 3 and p > 1 for n = 2. Lemma 5.1 shows that ∥Jψ_{0}∥_{H_{∗}^{1/2}(Y)} ≤

c∥ψ̃_{0}∥_{H^{1}(Y ×I′)}, where ψ̃ = χψ and χ = χ(x) is a cutoff function supported in I′ = [0, δ/2], as in the proof of Theorem 6.4. Now Corollary 6.5 shows that

C^{−1}∥ψ̃∥_{H_{∗}^{1}(Y ×I′)}^{2} ≤ ∫_{Y ×I}(|Dψ|^{2} + |ψ|^{2}) dv_{M} + ∮_{Y} |JP_{+}ψ_{0}|^{2} dv_{Y} ,

which provides the required Schrödinger-Lichnerowicz estimate (8.7).

In applications, a Schrödinger-Lichnerowicz estimate is usually obtained in the special case of homogeneous boundary data (Ku_{0} = 0). The above trick shows that the estimate in the homogeneous case implies the general estimate (8.7):

Lemma 8.6 Under the hypotheses of Lemma 8.5, suppose there is C̄ > 0 such that for all u ∈ ℍ with Ku_{0} = 0 we have

C̄^{−1} ∫_{M} |∇u|^{2} dv_{M} ≤ ∫_{M}(|Lu|^{2} + ρ̄|u|^{2}) dv_{M} ,  (8.12)

for some ρ̄. Then there is C > 0 such that (8.7) holds for all u ∈ ℍ.

Proof: Suppose u ∈ ℍ and let ũ = u − χu, where χ = χ(x) ∈ C^{∞}(M) is a cutoff function supported in Y × I′ as in the proof of Theorem 6.4. Then Kũ_{0} = 0 so (8.12) applies to ũ, giving

∫_{M} |∇u|^{2} dv_{M} ≤ 2 ∫_{M}(|∇ũ|^{2} + |∇(χu)|^{2}) dv_{M}
≤ C ∫_{M}(|Lũ|^{2} + ρ̄|ũ|^{2}) dv_{M} + 2 ∫_{Y ×I′} |∇(χu)|^{2} dv_{M}
≤ C ∫_{M}(|Lu|^{2} + (ρ̄ + |dχ|^{2})|u|^{2}) dv_{M} + 2 ∫_{Y ×I′} |∇(χu)|^{2} dv_{M} .

Now it follows easily from Corollary 6.5 that

C^{−1} ∫_{Y ×I′} |∇(χu)|^{2} dv_{M} ≤ ∫_{Y ×I}(|Lu|^{2} + |u|^{2}) dv_{M} + ∮_{Y} |JKu_{0}|^{2} dv_{Y} ,

which gives the required inequality.

Theorem 8.7 Under the hypotheses 8.1, suppose (M, ∇, L, K) admits a weighted Poincaré inequality (8.3) and a Schrödinger-Lichnerowicz inequality (8.7) with ρ and β satisfying the conditions of Lemma 8.5. If ρ ∈ L_{loc}^{p}(M) for some p > n^{∗}/2, and if

lim_{R→∞} sup_{M\M_{R}} \frac{ρ}{w} = 0 ,  (8.13)

where {M_{R}}_{R→∞}, is any exhaustion of M, then

(L, K) : ℍ → L^{2}(M) × H_{∗}^{1/2}(Y)  (8.14)

is semi-Fredholm.

Proof: Lemma 8.5 gives Lu ∈ L^{2}(M) for u ∈ ℍ. We first show the unit ball in the kernel is compact. Let {u_{k}}_{k=1}^{∞} be a sequence in the kernel of (L, K), normalised by ∥u_{k}∥_{ℍ} = 1. Weak compactness of bounded sets in ℍ shows there is ū ∈ ℍ and a subsequence, which we also denote by u_{k}, such that u_{k} ⇀ ū ∈ ℍ and ∥ū∥_{ℍ} ≤ lim inf ∥u_{k}∥_{ℍ} = 1.

Since (8.13) is independent of the choice of exhaustion, we may suppose for definiteness that M_{R} = {x ∈ M : d(x) < R} where d(x) is the smoothed distance function from some fixed base point. Let χ ∈ C_{c}^{∞}(ℝ) satisfy χ(x) = 1 for x ≤ 1, χ(x) = 0 for x ≥ 2 and 0 ≤ χ(x) ≤ 1, |χ′(x)| ≤ 2 for all x. Then the functions χ_{R}(x) = χ(d(x)/R) form support functions for the exhaustion M_{R} which satisfy supp χ_{R} ⊂ M_{2R}, χ_{R} = 1 on M_{R} and |dχ_{R}| ≤ 2. Using the weighted Poincaré inequality we have

∫_{M} |∇(χ_{R}u_{k})|^{2} dv_{M} ≤ 2 ∫_{M_{2R}\M_{R}} |dχ_{R}|^{2}|u_{k}|^{2} dv_{M} + 2 ∫_{M_{2R}} |∇u_{k}|^{2} dv_{M}
≤ 2(1 + 2 sup_{M_{2R}\M_{R}} w^{−1}) ∫_{M} |∇u_{k}|^{2} dv_{M} ,

which shows that for any R > 1 the sequence χ_{R}u_{k} is bounded in H^{1}(M_{2R}). Since χ_{R}u_{k} ⇀ χ_{R}ū in H^{1}(M_{2R}), the Rellich lemma implies χ_{R}u_{k} → χ_{R}ū strongly in L^{q}(M_{2R}) for any q < ˆ2 = 2n/(n − 2) and any R > 1.

Applying (8.7) to any difference u_{j} − u_{k} gives

∫_{M} |∇(u_{j} − u_{k})|^{2} dv_{M} ≤ ∫_{M} ρ |u_{j} − u_{k}|^{2} dv_{M}
≤ ∥ρ∥_{L^{p}(M_{R})}∥u_{j} − u_{k}∥_{L^{q}(M_{R})}^{2}
+ sup_{M\M_{R}} \frac{ρ}{w} ∫_{M} |u_{j} − u_{k}|^{2} w dv_{M} (8.15)

where, since p > n^{∗}/2, we have q = 2p/(p − 1) < ˆ2. Now (8.3) and ∥u_{k}∥_{ℍ} = 1 combine to show that

∫_{M} |u_{j} − u_{k}|^{2} w dv_{M} ≤ 4,

so by (8.13), for any ϵ > 0 there is R = R(ϵ) such that the second term of (8.15) is less than ϵ/2 for all j, k. Since u_{k} converges in L^{q}(M_{R}) there is N = N(ϵ, R) such that the first term is less than ϵ/2 for all j, k ≥ N. This shows u_{k} is a Cauchy sequence, hence strongly convergent to ū, in ℍ.

As noted above, ∥Lu∥_{L^{2}(M)} ≤ C∥u∥_{ℍ} and thus

∫_{M} |Lū|^{2} dv_{M} = ∫_{M} |L(ū − u_{k})|^{2} dv_{M}
≤ C ∫_{M} |∇(ū − u_{k})|^{2} dv_{M}
→ 0 as k → ∞ ,

which shows that Lū = 0. Similarly, since K : H_{∗}^{1/2}(Y) → H_{∗}^{1/2}(Y) is bounded, for any u ∈ ℍ we have

∮_{Y} |JKu_{0}|^{2} dv_{Y} ≤ c∥Ku_{0}∥_{H_{∗}^{1/2}(Y)} ≤ ck∥u_{0}∥_{H_{∗}^{1/2}(Y)}
≤ C ∫_{M} |∇u|^{2} dv_{M} ,

by (5.55) and the trace lemma 5.1. Choosing u = ū − u_{k} gives

∮_{Y} |JKū_{0}|^{2} dv_{Y} ≤ c ∫_{M} |∇(ū − u_{k})|^{2} dv_{M} = o(1),

which shows also that Kū_{0} = 0. Thus ū ∈ ker(L, K) and the kernel is finite dimensional.

To show the closed range property, observe that by (8.13) and (8.3), the elliptic estimate (8.7) may be strengthened to

C^{−1} ∫_{M} (|∇u|^{2} + |u|^{2}w) dv_{M} ≤ ∫_{M} |Lu|^{2}dv_{M} +∫_{Ω} ρ|u|^{2} dv_{M} +∮_{Y} |JKu_{0}|^{2} dv_{Y} ,

(8.16)

for some relatively compact domain Ω ⋐ M. Now we claim there is a constant C > 0 such that

∫_{Ω} ρ|u|^{2} dv_{M} ≤ C (∫_{M} |Lu|^{2}dv_{M} + ∮_{Y} |JKu_{0}|^{2} dv_{Y} ) ,  (8.17)

for all u ∈ ℍ such that

∫_{M}⟨∇u, ∇ϕ⟩ dv_{M} = 0 ∀ ϕ ∈ ker(L, K) .  (8.18)

Suppose (8.17) fails, so there is a sequence u_{k} ∈ ℍ, k = 1, 2, …, such that (8.18) holds for each u_{k}, and

∫_{Ω} ρ|u_{k}|^{2} dv_{M} = 1, ∫_{M} |Lu_{k}|^{2}dv_{M} + ∮_{Y} |JK(u_{k})_{0}|^{2} dv_{Y} ≤ 1/k.

The sequence is bounded in ℍ by (8.16), so by passing to a subsequence we may assume u_{k} converges weakly to ū ∈ ℍ and strongly in L^{q}(Ω), q = 2p/(p−1) < ˆ2 as before. Applying (8.16) to u_{j} − u_{k} shows the sequence is Cauchy and thus converges strongly in ℍ. It follows that

∫_{M} |Lū|^{2}dv_{M} + ∮_{Y} |JKū_{0}|^{2} dv_{Y} = 0,

so ū ∈ ker(L, K). Strong convergence shows that (8.18) is also satisfied by ū, so testing (8.18) for ū with ϕ = ū shows that ū = 0. However, strong convergence in L^{q}(Ω) shows that ∫_{Ω} ρ|ū|^{2}dv_{M} = 1, which is a contradiction and establishes the claim (8.17).

Combining (8.17) with (8.16) gives

∫_{M} (|∇u|^{2} + |u|^{2}w) dv_{M} ≤ C (∫_{M} |Lu|^{2}dv_{M} + ∮_{Y} |JKu_{0}|^{2} dv_{Y} ) (8.19)

for all u ∈ ℍ satisfying (8.18). Now suppose u_{k} ∈ ℍ is a sequence such that Lu_{k} = f_{k} → f ∈ L^{2}(M) and K(u_{k})_{0} = s_{k} → s ∈ H_{∗}^{1/2}(Y). These convergence properties are retained if we replace u_{k} by u_{k} + y_{k} for any convergent sequence y_{k} ∈ ker(L, K), so we may assume the u_{k} all satisfy (8.18). In particular, applying (8.19) to u_{j} − u_{k} shows that u_{k} is Cauchy in ℍ and converges to ū satisfying Lū = f, Kū_{0} = s. This shows (L, K) has closed range.

By Definition 6.3, u is a weak solution of

Lu = f, Ku_{0} = σ ,  (8.20)

for f ∈ L^{2}(M), σ ∈ P H_{∗}^{1/2}(Y), if u ∈ L_{loc}^{2}(M) and

∫_{M}⟨u, L^{†}ϕ⟩ dv_{M} = ∫_{M}⟨f, ϕ⟩ dv_{M} + ∮_{Y} ⟨σ, νϕ_{0}⟩ dv_{Y} ,  (8.21)

for all ϕ ∈ H_{c}^{1}(M) such that K^{†}ϕ_{0} = 0. Similarly, the argument of Theorem 7.3 shows that the weak form of the adjoint problem

L^{†}u = g, K^{†}u_{0} = τ ,  (8.22)

for g ∈ L^{2}(M), τ ∈ P̂H_{∗}^{1/2}(Y), P̂ = 1 − ν^{−1}P ν, is that u ∈ L_{loc}^{2}(M) and

∫_{M}⟨u, Lϕ⟩ dv_{M} = ∫_{M}⟨g, ϕ⟩ dv_{M} − ∮_{Y} ⟨τ, ν^{−1}ϕ_{0}⟩ dv_{Y}  (8.23)

for all ϕ ∈ H_{c}^{1}(M) such that Kϕ_{0} = 0.

We now extend the solvability criterion (Fredholm alternative) of Theorem 7.3 to the non-compact case.

Theorem 8.8 Under the conditions of Theorem 8.7, suppose the formal adjoint (L^{†}, K^{†}) also satisfies a Schrödinger-Lichnerowicz estimate (8.7) with the same covariant derivative ∇ and with a curvature term ρ̂ satisfying (8.13). Then the system (8.20) with (f, σ) ∈ L^{2}(M) × P H^{1/2}(Y) has a solution u ∈ ℍ if and only if (f, σ) satisfies

∫_{M}⟨f, ϕ⟩ dv_{M} + ∮_{Y} ⟨σ, νϕ_{0}⟩ dv_{Y} = 0 ,  (8.24)

for all ϕ ∈ ℍ ∩ L^{2}(M) satisfying L^{†}ϕ = 0, K^{†}ϕ_{0} = 0. In particular, the system (8.20) is solvable for all (f, σ) ∈ L^{2}(M) × P H^{1/2}(Y) if and only if there are no 0 ≠ Ψ ∈ ℍ ∩ L^{2}(M) satisfying L^{†}Ψ = 0, K^{†}Ψ_{0} = 0.

Remark 8.9 We emphasise that in Theorem 8.8 it is not necessary to impose conditions on ρ̂ other than (8.13), and no conditions on the map β̂ := L^{†}−^{t}a^{i}∇_{i} are needed.

Remark 8.10 See Theorem 11.9 for an example where L^{†} ≠ L, with L^{†} satisfying two Lichnerowicz-Schrödinger identities with respect to two different connections.

Proof: The necessity of (8.24) follows immediately from the weak form (8.21). To show sufficiency, the argument of Theorem 7.3 applies to reduce to the case σ = 0, which we now consider.

Let ℍ_{K} = {u ∈ ℍ : Ku_{0} = 0}. The elliptic estimate (8.7) gives

∫_{M} |∇u|^{2} dv_{M} ≤ C ∫_{M}(|Lu|^{2} + ρ|u|^{2}) dv_{M} , ∀ u ∈ ℍ_{K} .

The arguments used to show (8.16) and (8.17) apply and give

∫_{M} |∇u|^{2} dv_{M} ≤ C ∫_{M} |Lu|^{2} dv_{M} ∀ u ∈ ℍ̊_{K} ,  (8.25)

where we define

ℍ̊_{K} := {u ∈ ℍ_{K} : ∫_{M}⟨∇u, ∇ϕ⟩ dv_{M} = 0 ∀ ϕ ∈ ker(L, K)} .  (8.26)

Thus the bilinear form u ↦ ∫_{M} |Lu|^{2} dv_{M} is strictly coercive on the Hilbert space ℍ̊_{K}, and for each f ∈ L^{2}(M) the map ϕ → ∫_{M}⟨f, Lϕ⟩ dv_{M} is bounded on ℍ̊_{K}. The Lax-Milgram lemma shows there is u ∈ ℍ̊_{K} satisfying

∫_{M}⟨Lu, Lϕ⟩ dv_{M} = ∫_{M}⟨f, Lϕ⟩ dv_{M} ∀ ϕ ∈ ℍ̊_{K} .

Thus setting Ψ = Lu − f we have

∫_{M}⟨Ψ, Lϕ⟩ dv_{M} = 0 ∀ ϕ ∈ ℍ_{K} ,  (8.27)

since ϕ ∈ ker(L, K) will also satisfy the relation (8.27). Lemma 8.5 shows that Ψ ∈ L^{2}(M) and from Definition 6.3 and (8.27) we see that Ψ is a weak solution of

L^{†}Ψ = 0, K^{†}Ψ = 0 .

If there are no such non-trivial Ψ then Lu = f, and u is the required solution. The arguments of Theorem 7.3 show that (L^{†}, K^{†}) is elliptic and Theorem 6.4 applies to show Ψ ∈ H_{loc}^{1}(M). Let M_{R} be the exhaustion of M constructed in Theorem 8.7, with associated cutoff functions χ_{R} ∈ C_{c}^{∞}(M), and let Ψ_{k} = χ_{k}Ψ ∈ H_{c}^{1}(M) ⊂ ℍ. The assumed Schrödinger-Lichnerowicz estimate (8.7) for (L^{†}, K^{†}) gives (with L^{†} curvature term ρ̂)

∫_{M} |∇(Ψ_{k} − Ψ_{l})|^{2} dv_{M} ≤ C ∫_{M} (|L^{†}(Ψ_{k} − Ψ_{l})|^{2} + ρ̂|Ψ_{k} − Ψ_{l}|^{2}) dv_{M} . (8.28)

Since L^{†}Ψ = 0 we have

∫_{M} |L^{†}(Ψ_{k} − Ψ_{l})|^{2} dv_{M} ≤ c ∫_{M} (|dχ_{k}|^{2}|Ψ|^{2} + |dχ_{l}|^{2}|Ψ|^{2}) dv_{M} → 0 ,

because Ψ ∈ L^{2}(M), |dχ_{k}| ≤ 2 and supp dχ_{k} ⊂ M_{2k}\M_{k}. Now

∫_{M} ρ̂|Ψ_{k} − Ψ_{l}|^{2} dv_{M} ≤ ϵ ∫_{M} |Ψ_{k} − Ψ_{l}|^{2} w dv_{M}

by the condition (8.13) on ρ̂, for sufficiently large k, l. By the weighted Poincaré inequality (8.3), this is in turn bounded by ϵ times the left side of (8.28) and may therefore be discarded in (8.28) by choosing ϵ sufficiently small. It follows that Ψ_{k} is a Cauchy sequence in ℍ, so Ψ ∈ ℍ∩L^{2} and thus L^{†}Ψ = 0, K^{†}Ψ_{0} = 0. If there is no such Ψ ≠ 0 then Lu = f, and u is the required solution. More generally we have Lu = f +Ψ, u ∈ ℍ̊_{K}, and since ∫_{M}⟨Lu, Ψ⟩ dv_{M} = 0 by (8.27), the condition (8.24) (with σ = 0 and ϕ = Ψ) shows that Ψ = 0 and we have solved Lu = f, as required.

9 Weighted Poincaré Inequalities

Before proceeding with the analysis, define the symmetric part Γ^{S} of the connection ∇ by the formula

⟨ϕ, Γ^{S}(X)ψ⟩ := \frac{1}{2}(X⟨ϕ, ψ⟩ − ⟨ϕ, ∇_{X}ψ⟩ − ⟨∇_{X}ϕ, ψ⟩) ,  (9.1)

for all smooth sections ϕ, ψ of E and all smooth vector fields X: One easily checks that (9.1) defines a linear map Γ^{S}(X) from fibers of E to fibers of E, symmetric with respect to the scalar product ⟨·, ·⟩, with the map X → Γ^{S}(X) being linear as well. Clearly, ∇ is compatible with ⟨·, ·⟩ if and only if Γ^{S} vanishes. If Γ is defined by (8.2), then

Γ^{S} = \frac{1}{2}(Γ +^{t}Γ) .

We establish Proposition 8.3 via a special case, based on an argument of Geroch–Perng [32]:

Lemma 9.1 Let Ω, Ω̃ be any two relatively compact domains in M, and assume that

Γ^{S} ∈ L_{loc}^{n^{∗}}(M) .  (9.2)

There is a constant ϵ > 0 such that for all sections u ∈ H_{loc}^{1}(M) of E we have

ϵ ∫Ω̃ |u|^{2} dv_{M} ≤ ∫_{Ω} |u|^{2} dv_{M} + ∫_{M} |∇u|^{2} dv_{M} .  (9.3)

Proof: Let q be any point of Ω̃, fix p ∈ Ω and let r_{p} be small enough that the g̊-geodesic ball B(p, r_{p}) of radius r_{p} and centred at p, lies within Ω. Let X be a C^{∞} compactly supported vector field, such that the associated flow ϕ_{t} satisfies ϕ_{1}(B(p, r_{p})) ⊃ B(q, r_{q}) for some r_{q} > 0. (Since M is C^{∞} and connected, it is always possible to construct such an X.) Let Ω_{t} = ϕ_{t}(B(p, r_{p})).

By direct calculation and Hölder’s inequality we have, for any u ∈ H_{loc}^{1}(M),

\frac{d}{dt} ∫_{Ω_{t}} |u|^{2}dv_{M} = ∫_{Ω_{t}} (2⟨u, (∇_{X} + Γ_{X}^{S})u⟩ + |u|^{2}divg̊X) dv_{M}
≤ C (∫_{Ω_{t}} (|u|^{2} + |∇u|^{2}) dvg̊ + ∥Γ^{S}∥_{L^{n^{∗}/2}(Ω_{t})}∥u∥_{Lˆ2(Ω_{t})}^{2}) ,

where C depends on ∥X∥_{L^{∞}}, ∥divg̊X∥_{L^{∞}}. By the Sobolev inequality in the coordinate ball Ω_{t} for functions, ∥f∥_{Lˆ2(Ω_{t})} ≤ C(∥∂f∥_{L^{2}(Ω_{t})} + ∥f∥_{L^{2}(Ω_{t})}). Applying this to f = |u| gives ∥u∥_{Lˆ2(Ω_{t})} ≤ C(∥Du∥_{L^{2}(Ω_{t})} + ∥u∥_{L^{2}(Ω_{t})}), where D is any metric-compatible connection. Since Γ^{S} ∈ L^{n^{∗}} may be written as Γ_{1} + Γ_{2}, Γ_{1} ∈ L^{∞}, ∥Γ_{2}∥_{L^{n^{∗}}} ≤ ϵ, the Sobolev inequality gives

∥u∥_{Lˆ2(Ω_{t})} ≤ C (∥∇u∥_{L^{2}(Ω_{t})} + ∥u∥_{L^{2}(Ω_{t})}) ,

for some constant C depending on Γ. Defining F (t) = ∫_{Ω_{t}} |u|^{2}dvg̊, we have

\frac{d}{dt}F (t) ≤ CF (t) + C ∫_{M} |∇u|^{2}dv_{M} ,

and Gronwall’s lemma gives F (1) ≤ e^{C}(F (0) + ∫_{M} |∇u|^{2}dv_{M}). Thus there is ϵ > 0 such that

ϵ ∫_{B(q,r_{q})} |u|^{2} dv_{M} ≤ ∫_{Ω} |u|^{2} dv_{M} + ∫_{M} |∇u|^{2} dv_{M} .

Since Ω̃ has compact closure, it is covered by finitely many such balls B(q, r_{q}) and (9.3) follows.

Corollary 9.2 Under condition (9.2), if there is a domain Ω ⊂ M and a constant ϵ > 0 such that

ϵ ∫_{Ω} |u|^{2} dv_{M} ≤ ∫_{M} |∇u|^{2} dv_{M}  (9.4)

for all u ∈ C_{c}^{1}(M), then M admits a weighted Poincaré inequality (8.3).

Proof: By paracompactness and Lemma 9.1, there is a countable locally finite covering of M by domains Ω_{k} and constants 1 ≥ ϵ_{k} > 0, k ∈ ℤ^{+}, such that for each k,

ϵ_{k} ∫_{Ω_{k}} |u|^{2} dv_{M} ≤ ∫_{Ω} |u|^{2} dv_{M} + ∫_{M} |∇u|^{2} dv_{M} .

This is in turn bounded uniformly by (9.4), so the function

w(x) = ∑_{k:x∈Ω_{k}} \frac{2^{−k}ϵϵ_{k}}{1 + ϵ}  (9.5)

is bounded, strictly positive, and satisfies

∫_{M} |u|^{2} w dv_{M} ≤ ∫_{M} |∇u|^{2} dv_{M} ,

which is the required weighted Poincaré inequality.

This establishes part (i) of Proposition 8.3, and we next turn to the proof of part (ii).

Theorem 9.3 Suppose that M has a locally finite cover such that

∇_{i} = ∂_{i} − Γ_{i} ,  with Γ_{i} ∈ L_{loc}^{n^{∗}} .  (9.6)

If there are no global ∇-parallel sections of the bundle E, then M admits a weighted Poincaré inequality. Equivalently, if M does not admit a weighted Poincaré inequality then M admits a global ∇-parallel section.

Proof: Assume M does not admit a weighted Poincaré inequality, so by Corollary 9.2, for each domain Ω ⋐ M and each constant ϵ > 0, there is u ∈ H_{loc}^{1}(M) such that (9.4) fails. In particular, fixing Ω, for each k > 0 there is u_{k} ∈ H_{loc}^{1}(M) such that

∫_{Ω} |u_{k}|^{2} dv_{M} = 1, ∫_{M} |∇u_{k}|^{2} dv_{M} ≤ k^{−1} .  (9.7)

It follows that ∇u_{k} → 0 strongly in L^{2}(M). Under (9.6) Rellich’s lemma holds, so there is a subsequence converging strongly to u ∈ L^{2}(Ω). Then ∇u = 0 and u ≠ 0 in Ω.

Now let M_{j}, j = 1, 2, … be the exhaustion of M from Theorem 8.7, and let u_{j} ∈ H^{1}(M_{j}) be the corresponding parallel spinors, constructed in the preceding paragraph. Since u_{j} ≠ 0 there is M′_{j} ⋐ M_{j} such that ∫_{M′_{j}} |u_{j}|^{2} ≠ 0. Lemma 9.1 applied with M_{j} replacing M shows there is η_{j} > 0 such that for all v ∈ H_{loc}^{1}(M_{j}),

η_{j} ∫_{M′_{j}} |v|^{2} dv_{M} ≤ ∫_{M_{0}} |v|^{2} dv_{M} + ∫_{M_{j}} |∇v|^{2} dv_{M} .

In particular this implies ∫_{M_{1}} |u_{j}|^{2} dv_{M} ≠ 0 and we may impose the normalisation ∫_{M_{1}} |u_{j}|^{2} dv_{M} = 1. By Rellich’s lemma there is ū_{1} ∈ H^{1}(M_{1}) and a subsequence, also denoted by u_{j}, such that u_{j} → ū_{1} in H^{1}(M_{1}) and ∫_{M_{1}} |ū_{1}|^{2} = 1, ∇ū_{1} = 0.

Again by Lemma 9.1, for each k ≥ 1 there is ϵ_{k} > 0 such that

ϵ_{k} ∫_{M_{k}} |v|^{2} dv_{M} ≤ ∫_{M_{1}} |v|^{2} dv_{M} + ∫_{M_{k+1}} |∇v|^{2} dv_{M} , ∀ v ∈ H_{loc}^{1}(M_{k+1}) .

Setting v = u_{i} − u_{j}, i, j > k, shows that the sequence u_{j} is Cauchy in L^{2}(M_{k}) and therefore converges strongly in L^{2}(M_{k}) for all k ≥ 1 to some nontrivial ū ∈ L_{loc}^{2}(M), and ∇ū = 0.

Another application of Corollary 9.2 leads to Proposition 8.3 part 3, for asymptotically flat manifolds. In fact the proof works for a much broader class of manifolds:

Definition 9.4 A weakly asymptotically flat end M̃ ⊂ M of a Riemannian manifold M with metric g is a connected component of M\K for some compact set K, such that M̃ ≃ ℝ^{n}\B(0, 1) and there is a constant η > 0 such that

η δ_{ij}ξ^{i}ξ^{j} ≤ g_{ij}(x)ξ^{i}ξ^{j} ≤ η^{−1}δ_{ij}ξ^{i}ξ^{j} ;

for all x ∈ ℝ^{n}\B(0, 1) and all vectors ξ ∈ ℝ^{n}.

Theorem 9.5 Suppose (M, g) is a (connected) Riemannian manifold of dimension n ≥ 3, g ∈ C^{0}(M), and M has a weakly asymptotically flat end M̃. Suppose also the connection ∇_{i} = ∂_{i} − Γ_{i} on E satisfies Γ ∈ L_{loc}^{n^{∗}}(M) and the decay conditions

∥r^{−1}Γ^{S}∥_{L^{n/2}(M̃)} + ∥Γ^{S}∥_{L^{n}(M̃)} < ∞ ,  (9.8)

where Γ^{S} is the symmetric, scalar product incompatible, component of ∇ defined by Equation (9.1). Then M admits a weighted Poincaré inequality.

Remark 9.6 The restriction dim M ≥ 3 is rather harmless as far as the applications to the positive mass theorems are concerned, since the notion of asymptotic flatness for two dimensional manifolds, relevant to general relativistic applications, has to be defined in a completely different way. An adequate analogue of mass here when dim M = 2 is provided by the Shiohama theorem [57].

Remark 9.7 The decay condition (9.8) is independent of the choice of flat background metric g̊_{ij} = δ_{ij}: Equation (9.1) shows that Γ^{S} is a tensor. By comparison with the g-distance function from any chosen point p, the function r is equivalent to this distance function, which implies the result.

Remark 9.8 The proof below establishes the inequality (9.4) for spinors supported in Ω := ℝ^{3} \ B(0, R) for some R without assuming that Γ ∈ L_{loc}^{n^{∗}}(M).

Proof: Let r = (∑(x^{i})^{2})^{1/2} ∈ C^{∞}(M̃) and χ = χ(r) ∈ C_{c}^{1}(M̃) satisfy, for some R_{0} > 1 and k ≥ 10,

χ(r) = \frac{log(r/R_{0})}{log k}, 2R_{0} ≤ r ≤ (k − 1)R_{0}

and χ(r) = 1 for r > kR_{0}, χ(r) = 0 for r ≤ R_{0}. Then |χ′(r)| ≤ 2/(r log k), so for any section u ∈ C_{c}^{1}(M)

∫_{M} |∇(χu)|^{2} dv_{M} ≤ 2 ∫_{M} |∇u|^{2} dv_{M} + \frac{4}{(log k)^{2}} ∫_{R_{0}≤r≤kR_{0}} \frac{1}{r^{2}} |u|^{2} dv_{M} . (9.9)

Now Δ_{0}(r^{2−n}) = 0 for r ≥ 1 in ℝ^{n}, n ≥ 3, so for any v ∈ C_{c}^{1}(ℝ^{n}\B(0, R_{0})) we have

0 = − ∫_{ℝ^{n}} ∂_{i}(∂_{i}(r^{2−n}) |v|^{2}r^{n−2}) dx
= (n − 2)^{2} ∫_{ℝ^{n}} r^{−2}|v|^{2} dx + (n − 2) ∫_{ℝ^{n}} r^{−1}2⟨v, (∇_{r} + Γ_{r}^{S})v⟩ dx ,

where Γ_{r}^{S} = r^{−1}x^{i}Γ_{i}^{S} and lengths are measured by g̊and the metric on E. Using Hölder’s inequality we obtain

\frac{(n − 2)^{2}}{4} ∫_{ℝ^{n}} r^{−2}|v|^{2} dx ≤ ∫_{ℝ^{n}} |∇v|^{2} dx + (n − 2) ∫_{ℝ^{n}} r^{−1}|v|^{2}|Γ_{r}^{S}| dx .

The Sobolev inequality in ℝ^{n}, n ≥ 3,

(∫_{ℝ^{n}} |v|ˆ2 dx)^{1−2/n} ≤ C_{S} ∫_{ℝ^{n}} |Dv|^{2} dx ,

where D = ∇ + Γ^{S} is the metric-compatible connection, gives the estimate

∫_{ℝ^{n}} |Dv|^{2} dx ≤ 2 ∫_{ℝ^{n}}(|∇v|^{2} + |v|^{2}|Γ^{S}|^{2}) dx
≤ 2 ∫_{ℝ^{n}} |∇v|^{2} dx + 2 C_{S} ∥Γ^{S}∥_{L^{n}(ℝ^{n}\B(0,R_{0}))} ∫_{ℝ^{n}} |Dv|^{2} dx
≤ 4 ∫_{ℝ^{n}} |∇v|^{2} ,

provided 2C_{S}∥Γ^{S}∥_{L^{n}(ℝ^{n}\B(0,R_{0}))} ≤ \frac{1}{2}. Now (9.8) implies there is R_{0} < ∞ such that this condition will be satisfied, so for any v ∈ C_{c}^{1}(ℝ^{n}\B(0, R_{0})) we have

∫_{ℝ^{n}} r^{−1}|Γ_{r}^{S}| |v|^{2} dx ≤ ∥r^{−1}Γ^{S}∥_{L^{n/2}(ℝ^{n})} C_{S} ∫_{ℝ^{n}} |Dv|^{2} dx
≤ 4C_{S}∥r^{−1}Γ^{S}∥_{L^{n/2}(ℝ^{n})} ∫_{ℝ^{n}} |∇v|^{2} dx .

Hence there is ϵ > 0 such that for all v ∈ C_{c}^{1}(M̃ ∩ {r > R_{0}}),

ϵ ∫M̃ r^{−2}|v|^{2} dv_{M} ≤ ∫M̃ |∇v|^{2} dv_{M} .  (9.10)

Combining (9.10) with v = χu and (9.9) gives

∫_{{r>kR_{0}}} r^{−2}|u|^{2} dv_{M} ≤ ∫M̃ r^{−2}|χu|^{2} dv_{M}
≤ C ∫M̃ |∇(χu)|^{2} dv_{M}
≤ C ∫M̃ |∇u|^{2} dv_{M} + \frac{C}{(log k)^{2}} ∫M̃ r^{−2}|u|^{2} dv_{M} ,

where now |∇u|^{2} = g^{ij}⟨∇_{i}u, ∇_{j}u⟩. If k is chosen so that C/(log k)^{2} ≤ \frac{1}{2} then the last term may be absorbed into the left hand side, giving

∫_{r≥kR_{0}} r^{−2}|u|^{2} dv_{M} ≤ C ∫_{M} |∇u|^{2} dv_{M} .  (9.11)

Lemma 9.1 now applies and gives the required weighted Poincaré inequality.

In order to prove part 4. of Proposition 8.3 the following Definition is needed:

Definition 9.9 A weakly hyperboloidal end M̃ ⊂ M is a connected component of M\K for some compact set K, such that M̃ ≃ (0, x_{0}) × N, where (N , h) is a (boundaryless) compact Riemannian manifold with continuous metric h, with g|M̃ being uniformly equivalent to

g̊ ≡ x^{−2}(dx^{2} + h) .

Here x is the coordinate running along the (0, x_{0}) factor of (0, x_{0}) × N.

We have the following hyperboloidal counterpart of Theorem 9.5:

Theorem 9.10 Suppose (M, g) is a (connected) Riemannian manifold of dimension n ≥ 2, g ∈ C^{0}(M), and M has a weakly hyperboloidal end M̃. Suppose also the connection ∇_{i} = ∂_{i} − Γ_{i} on E satisfies Γ ∈ L_{loc}^{n^{∗}}(M) and the decay condition

lim sup_{x→0} |xΓ_{x}^{S}| < \frac{n − 1}{2}  (9.12)

in M̃, where Γ_{x}^{S} is the symmetric part of ∇_{∂_{x}}, with norm understood as that of an endomorphism of fibres of E. Then M admits a weighted Poincaré inequality.

Proof: This is essentially McKean’s inequality [49]; we follow the proof in [22]. Let, first, f be a function in C^{1}([0, x_{0}] × N) with f = 0 at {x = 0}; we have

f^{2}(x, v) = 2 ∫_{0}^{x} f(s, v)\frac{∂f(x, v)}{∂x}ds
≤ \frac{n − 1}{2} ∫_{0}^{x} \frac{f^{2}(s, v)}{s}ds + \frac{2}{n − 1} ∫_{0}^{x} s (\frac{∂f}{∂x}(s, v))^{2} ds . (9.13)

Here we use the symbol v to label points in N. Integrating on [0, x_{0}] × N, a change of the order of integration in x and s together with some obvious manipulations gives

∫_{[0,x_{0}]×N} f^{2} x^{−n}dx dμ_{h} ≤ \frac{4}{(n − 1)^{2}} ∫_{[0,x_{0}]×N} (x\frac{∂f}{∂x})^{2} x^{−n}dx dμ_{h}
≤ \frac{4}{(n − 1)^{2}} ∫_{[0,x_{0}]×N} g̊(df, df) x^{−n}dx dμ_{h} .(9.14)

This is the desired inequality on M̃ with metric g̊ for functions, with weight function w = (n − 1)^{2}/4. The result for general weakly asymptotically hyperboloidal metrics and for functions follows immediately from the above, using uniform equivalence of g with g̊on the asymptotic region, and using Lemma 9.1.

Let, finally, v be a smooth compactly supported section of a Riemannian bundle with not-necessarily-compatible connection ∇. Let ϕ be any smooth compactly supported function equal to 1 on the support of v, set

f_{ϵ} = ϕ\sqrt{ϵ + ⟨v, v⟩} .

We have

|\frac{∂f_{ϵ}}{∂x} |^{2} = |d_{x}ϕ|^{2}(ϵ + ⟨v, v⟩) + ϕ^{2} \frac{⟨v, (∇_{x} + Γ_{x}^{S})v⟩⟨v, (∇_{x} + Γ_{x}^{S})v⟩}{ϵ + ⟨v, v⟩}
≤ ϵ|dϕ|^{2} + ϕ^{2}|(∇_{x} + Γ_{x}^{S})v|^{2} .

The first line of (9.14) yields

∫_{M} f_{ϵ}^{2} x^{−n}dx dμ_{h} = ∫_{M} ϕ^{2}(ϵ + ⟨v, v⟩) x^{−n}dx dμ_{h}
≤ \frac{4}{(n − 1)^{2}} ∫_{M} x^{2} (ϵ|dϕ|^{2} + ϕ^{2}|(∇_{x} + Γ_{x}^{S})v|^{2}) x^{−n}dx dμ_{h} .

Passing with ϵ to zero gives

∫_{M}⟨v, v⟩ x^{−n}dx dμ_{h} ≤ \frac{4}{(n − 1)^{2}} ∫_{M} x^{2}|(∇_{x} + Γ_{x}^{S})v|^{2} x^{−n}dx dμ_{h}
≤ \frac{4}{(n − 1)^{2}} ∫_{M} ((1 + \frac{1}{δ})|∇v|2 ; g̊ + x^{2}(1 + δ)|Γ_{x}^{S}v|^{2}dx) x^{−n}dx dμ_{h} ,

for any δ > 0, and if condition (9.12) holds the last term can be carried over to the left hand side, leading to

C^{−1} ∫_{M}⟨v, v⟩dv_{M} ≤ ∫_{M} |∇v|_{g}^{2}dv_{M} .

Lemma 9.1 gives then the desired inequality, with a weight function w equal to 1/C in the asymptotic region.

10 Examples and Applications

The structure and regularity conditions may be readily verified in situations of interest, which we illustrate by considering the Dirac operator examples of §2.

Suppose M is a Riemannian spin manifold. Fix local coordinates (x^{μ}) and a local orthonormal framing e_{i} = e_{i}^{μ}∂_{μ} of the tangent bundle T M, and let ϕ_{I}, I = 1, …, dim S denote an associated spinor frame, determined by some choice of representation c : Cℓ_{n} → End(S). The Dirac operator D defined by (2.2)-(2.3) is

D = e_{i}^{μ}γ^{i}∂_{μ} − \frac{1}{4}ω_{ij}(∂_{μ})e_{k}^{μ}γ^{k}γ^{i}γ^{j} ,  (10.1)

where the skew-symmetric matrices γ^{i} = c(e_{i}) ∈ End(S) are constant in the local spinor frame and satisfy the Clifford relation

γ^{i}γ^{j} + γ^{j}γ^{i} = −2δ^{ij} .  (10.2)

Clearly D has the form a^{μ}∂_{μ} + b where

\begin{aligned}a^{μ} & = & ∑_{i=1}^{n}e_{i}^{μ}γ^{i} , \\ b & = & − \frac{1}{4} ∑_{i,j,k,μ=1}^{n}ω_{ij}(∂_{μ})e_{k}^{μ}γ^{k}γ^{i}γ^{j} . \end{aligned}  (10.3)

If g_{μν} ∈ W_{loc}^{1,n^{∗}} ∩ C_{loc}^{0} then by the Gram-Schmidt construction, the local orthonormal frame may be chosen so that the coefficients also satisfy e_{i}^{μ} ∈ W_{loc}^{1,n^{∗}} ∩ C_{loc}^{0}, and then a^{μ}, b also satisfy (3.4), cf. Proposition A.9, Appendix A. The Clifford identity shows |ξ_{μ}a^{μ}V |^{2} = g^{μν}ξ_{μ}ξ_{ν}|V |^{2} pointwise, which implies (3.5) so Theorem 3.7 may be applied to establish the H_{loc}^{1} interior regularity of L^{2} weak solutions.

For boundary regularity we assume there is a diffeomorphism of Y × I with a neighbourhood of Y × {0} = ∂M in M, such that for any chart of Y, the associated chart of adapted coordinates (y^{A}, x), x ≥ 0 satisfies

\begin{aligned}g_{xA}(y, 0) & = & 0, A = 1, …, n − 1 , \\ g_{xx}(y, 0) & = & 1 , \end{aligned}  (10.4)

and

g_{μν}|_{Y} ∈ W^{1,(n−1)^{∗}}(Y) ∩ C^{0}(Y) .  (10.5)

(Recall that the symbol “|_{Y} ” stands for “|_{Y ×{0}}”.)

For smooth metrics this follows easily using Gaussian coordinates about Y. Geodesic uniqueness may not be available in the more general case of a W^{k+1,p} manifold with metric g_{μν} ∈ W^{k,p} for k > n/p; in this case the existence of boundary coordinates satisfying (10.4) is guaranteed by Proposition A.10.

Then (6.5) follows from the regularity conditions on g_{μν}, (6.2) follows from (10.5), and (6.9) follows from (10.4), the Clifford relations and the skew-symmetry of the γ^{i}. In terms of an adapted frame e_{i}, where e_{n}|_{Y} = −∂_{x^{n}} is the outer normal at Y = ∂M, the boundary operator (6.10) may be taken as

A = D_{Y} = − ∑_{i=1}^{n−1} e_{i}^{μ}γ^{n}γ^{i}∂_{μ} + \frac{1}{4} ∑_{i,j,k=1}^{n−1} ω_{ij}(∂_{μ})e_{k}^{μ}γ^{n}γ^{k}γ^{i}γ^{j} ,  (10.6)

by an appropriate choice of b̃_{0} in (6.10). Again note that other choices of A are possible, such as D_{Y} + F for any function or symmetric endomorphism F.

The boundary condition (2.13)

P_{+}ψ = σ,  σ ∈ P_{+}H_{∗}^{1/2}(Y) ,  (10.7)

where P_{+} is the orthogonal projection to the positive eigenspace of A, corresponds via (6.14) to K = 0, which clearly satisfies the conditions of Theorem 6.4 for K, K^{†}. From Theorem 6.4 it follows that the weak-strong property and the elliptic estimate (6.21) hold at the boundary for the boundary condition (10.7).

Boundary operators of the type (2.18) were used in [30, 31, 47], for example. More generally suppose there is an endomorphism ϵ : S → S acting on sections of E, which satisfies

ϵ^{2} = 1, ^{t}ϵ = ϵ, ϵA + Aϵ = 0 .  (10.8)

Note that if there is a splitting E = E_{0} ⊕ F_{0} such that A has the form (4.10)

then ϵ = [ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ] will satisfy (10.8). Conversely, if (10.8) holds then setting

E_{0}, F_{0} equal to the ±1 eigenspaces of ϵ shows that A may be written in the supersymmetric form (4.10).

If Aψ = λψ then A(ϵψ) = −ϵAψ = −λϵψ, so the spectrum of A is symmetric and ϵ interchanges the positive and negative eigenspaces. Let P_{+} (P_{−}) be orthogonal projection to the positive (negative) eigenspaces, and consider the eigenspace splitting L^{2}(Y) = H_{+} ⊕ H_{−} ⊕ H_{0}^{+} ⊕ H_{0}^{−}, where H_{0} = ker A and H_{0} = H_{0}^{+} ⊕ H_{0}^{−} is the decomposition into ±1 eigenspaces of ϵ. There is an isometric isomorphism e : H_{+} → H_{−} such that ϵ has the block decomposition

ϵ = [ \begin{matrix}& e^{−1} \\ e \\ & & 1 \\ & & & −1 \end{matrix} ], acting on = H_{+} ⊕ H_{−} ⊕ H_{0}^{+} ⊕ H_{0}^{−}.

It follows that the action of K_{+} = \frac{1}{2}(1 + ϵ) is given by

K_{+}ψ = \frac{1}{2}(1 + ϵ) [ \begin{matrix}p \\ q \\ r^{+} \\ r^{−} \end{matrix} ] = [ \begin{matrix}\frac{1}{2}(p + e^{−1}q) \\ \frac{1}{2}(ep + q) \\ r^{+} \\ 0 \end{matrix} ] .  (10.9)

Hence the boundary condition K_{+}ψ = σ where σ = ϵσ = ^{t}[σ_{+}, eσ_{+}, σ_{0}^{+}, 0], is equivalent to the component conditions \frac{1}{2}(p + e^{−1}q) = σ_{+} and r^{+} = σ_{0}^{+}. This may be expressed in the form (6.14) if we define P, K by

P = P_{+} + P_{0}^{+} = [ \begin{matrix}1 \\ & 0 \\ & & 1 \\ & & & 0 \end{matrix} ] , K = [ \begin{matrix}& e^{−1} \\ 0 \\ & & 0 \\ & & & 0 \end{matrix} ] ,  (10.10)

since then

P ψ − K(1 − P)ψ = [ \begin{matrix}p \\ 0 \\ r^{+} \\ 0 \end{matrix} ] + [ \begin{matrix}e^{−1}q \\ 0 \\ 0 \\ 0 \end{matrix} ] = [ \begin{matrix}2σ_{+} \\ 0 \\ σ_{0}^{+} \\ 0 \end{matrix} ]  (10.11)

is equivalent to K_{+}ψ = σ as above. Clearly K is bounded on both L^{2}(Y) and H_{∗}^{1/2}(Y) as required by the regularity theorems of §5. Similarly we find that the boundary condition K_{−}ψ = σ is equivalent to \frac{1}{2}(p − e^{−1}q) = σ_{−} and r^{−} = σ_{0}^{−}, so an equivalent spectral projection condition may be constructed using P̂ = P_{+} + P_{0}^{+} and K̂ = −K, so

P̂ψ − K̂(1 − P̂)ψ = [ \begin{matrix}p − e^{−1}q \\ 0 \\ 0 \\ r^{−} \end{matrix} ] = [ \begin{matrix}2σ_{+} \\ 0 \\ 0 \\ σ_{0}^{−} \end{matrix} ] .  (10.12)

11 Positive mass theorems

Our motivation for the present work arose from positive energy theorems, and we shall present here some such theorems which follow from our work elsewhere in the paper. The main point is to give a complete proof of Herzlich’s inequality, cf. Theorem 11.7 below. In addition, our remaining results in this section improve the previous ones [8, 11, 19, 20, 25, 37, 38, 51, 54, 62, 64, 67] in two respects:

1) the low differentiability of the metric; 2) we do not require M to have a compact interior. This second aspect of our results is critical for some applications of the positive mass theorem to black holes uniqueness theory [17, 23, 24, 53]. Some of the arguments already presented in other sections will be repeated in the proofs below, whenever useful for the clarity of presentation.

A set (M, g, K) will be called initial data for Einstein equation if (M, g) is a three^{9} dimensional Riemannian manifold, and K is a symmetric tensor on M; this is a slight abuse of terminology as we are not requiring any constraints to be satisfied. Only complete (M, g)’s will be considered, with boundary either compact or empty. Given such a triple we set

μ := R − |K|^{2} + (trK)^{2} ,  (11.1a)
ν^{j} := 2D_{i}(K^{ij} − trKg^{ij}) .  (11.1b)

In a field theoretic framework one should provide some further initial data for the complete system of equations in the model under consideration; μ and ν correspond then to the energy and momentum densities of the matter fields while Equations (11.1) become constraint equations. Now, we are not assuming anything about matter fields, so μ and ν should be thought of as derived from the data (M, g, K), as in (11.1).

Throughout this section we shall be working in a space of spinors which carries a representation c of the Clifford algebra associated with the (n + 1)- dimensional Lorentzian metric

γ := − θ^{0} ⊗ θ^{0} + ∑_{i=1}^{n} θ^{i} ⊗ θ^{i} = − θ^{0} ⊗ θ^{0} + g ,  (11.2)

with θ^{i} denoting a co-frame dual to a frame e_{i}. We assume that the c(e_{i})’s are antisymmetric, and that c(e_{0}) is symmetric. The symbol D will be used to denote the canonical spinor connection on (M, g) defined by Equation (2.2) (and denoted by ∇ there). The connection

∇_{j} := D_{j} + \frac{1}{2}K^{i}_{j}c(e_{i})c(e_{0})  (11.3)

9The results here generalize without any difficulties to spin manifolds of higher dimensions, so that the restriction n = 3 is only made for simplicity of presentation.

will be called the space-time spin connection on M; ∇ is sometimes referred to as the Sen connection. We note that ∇ is not compatible with the positive definite metric ⟨·, ·⟩:

∂_{j}⟨ϕ, ψ⟩ = ⟨∇_{j}ϕ, ψ⟩ + ⟨ϕ, ∇_{j}ψ⟩ − ⟨ϕ, K^{i}_{j}c(e_{i})c(e_{0})ψ⟩ .  (11.4)

However, ∇ is compatible with the Lorentz-invariant (and hence indefinite) inner product (ϕ, ψ) := ⟨ϕ, c(e_{0})ψ⟩. The identity associated with ∇, which replaces the identity (2.6) of Schrödinger-Lichnerowicz, takes the form

∫_{Ω} (|∇ψ|^{2} + \frac{1}{4} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i}e_{0})ψ⟩) − |𝒟ψ|^{2}) = ∮_{∂Ω}⟨ψ, c(ne^{A})∇_{A}ψ⟩ ,

(11.5)

where

𝒟 := g^{ij}c(e_{i})∇_{j} .  (11.6)

For sufficiently differentiable (g, K)’s, as will be made precise below, the identity (11.5) holds in the following circumstances:

1. If ψ is a H_{loc}^{1} spinor field, we may take as Ω a domain in M with compact closure and differentiable boundary ∂Ω;

2. If ψ is a compactly supported C^{1} spinor field, then Equation (11.5) holds with Ω = M, and ∂Ω = ∂M; in particular no boundary term is present if M has no boundary;

3. Suppose (M, ∇) admits a weighted Poincaré inequality, assume ∂M = ∅, and let ℍ be the space defined in (8.5). We then have the following:

Lemma 11.1 Suppose that ∂M = ∅. The function

C_{c}^{1}(M) ∋ ψ → G(ψ) := ∫_{M} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i})c(e_{0})ψ⟩)

extends by continuity to a continuous function on ℍ, still denoted by the same symbol.

Proof: For ψ, χ ∈ C_{c}^{1}(M) the identity (11.9) gives

G(ψ) − G(χ) = 4 ∫_{M} (−⟨∇ψ, ∇ψ⟩ + ⟨∇χ, ∇χ⟩ + ⟨𝒟ψ, 𝒟ψ⟩ − ⟨𝒟χ, 𝒟χ⟩)
= 4 ∫_{M} (⟨∇χ − ∇ψ, ∇ψ + ∇χ⟩ + ⟨𝒟χ + 𝒟ψ, 𝒟ψ − 𝒟χ⟩
≤ C∥ψ + χ∥_{ℍ}∥ψ − χ∥_{ℍ} ;

in the last step we have used the fact that the |c(e_{i})| are uniformly bounded.

Lemma 11.1 implies that the left-hand-side of (11.5) is continuous on ℍ for any measurable Ω ⊂ M. Since compactly supported C^{1} fields are by definition dense in ℍ, one easily checks, using continuity, that (11.5) holds with Ω = M and with vanishing right-hand-side for all ψ ∈ ℍ. (Nonempty compact boundaries ∂M will be considered shortly.)

We shall say that M_{ext} ⊂ M is an asymptotically flat end if M_{ext} is diffeomorphic to ℝ^{3} \ B(0, R) for some R, with

r^{−1}(g_{ij} − δ_{ij}) , ∂_{k}g_{ij} , K_{ij} ∈ L^{2}(M_{ext}) ,  (11.7a)
∂_{k}g_{ij} = o(r^{−3/2}) , K_{ij} = o(r^{−3/2}) ,  (11.7b)
μ , ν ∈ L^{1}(M_{ext}) ,  (11.7c)

compare Definition 9.4. Those conditions guarantee that the ADM four-momentum of the data set is finite and well defined, as follows from what is said in [21] (compare [8]): (11.7a) and (11.7c) guarantee convergence of the mass and momentum integrals, while (11.7b) guarantees geometric invariance.

One of the ingredients of Witten-type proofs of positive energy theorems is the introduction of appropriate boundary conditions on the spinor field in the asymptotic regions. In the asymptotically flat case this is straightforward: one chooses a g-orthonormal triad such that, in the coordinate system of (11.7),

dx^{k}(e_{i} − ∂_{i}) →_{r→∞} 0 , ∂_{i} (dx^{k}(e_{j})) ∈ L^{2}(M_{ext}) ;

this is easily achieved by a Gram-Schmidt orthonormalisation of the frame {∂_{i}}. Then a spin frame on M_{ext} is introduced, such that the c(e_{i})’s are represented by constant matrices, as in Section 10. The boundary condition then is that the spinor field ψ, which will be required to solve the generalized Dirac equation,

𝒟ψ = g^{ij}c(e_{i})∇_{j}ψ = 0 ,  (11.8)

asymptotes, as r tends to infinity, to a spinor ψ_{∞} which, for r ≥ R for some R, has constant entries in the spin frame above. It is convenient to choose ψ_{∞} so that ψ_{∞} is smooth, and supported in M_{ext}. The procedure is somewhat more delicate in the asymptotically hyperboloidal setting; an elegant geometric framework for such constructions has been provided in [4, 14].

Consider the identity (11.5) with a spinor field ψ = ψ_{∞} +χ, with χ differentiable and compactly supported, while ∂Ω = S_{R}, a coordinate sphere of radius R in the exterior region, with R large enough that χ vanishes there. A classical calculation along the lines of [8] shows that the boundary term in (11.5) is then proportional to

4πp_{α}⟨ψ_{∞} , c(e^{α}e^{0})ψ_{∞}⟩ + o(1) ,

where p^{α} is the ADM four-momentum of (M_{ext}, g), with o(1) → 0 as R tends to infinity. Passing to this limit we thus have

∫_{M} (|∇ψ|^{2} + \frac{1}{4} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i})c(e_{0})ψ⟩) − |𝒟ψ|^{2}) = 4πp_{α}⟨ψ_{∞} , c(e^{α}e^{0})ψ_{∞}⟩ ,

(11.9)

still for C^{1} compactly supported χ’s. But the left-hand-side of (11.9) is continuous on ℍ, which is shown by a calculation similar to that in Lemma 11.1: Let F (χ) denote the left-hand-side of Equation (11.9) with ψ = ψ_{∞} + χ there, let χ_{i} ∈ ℍ converge in ℍ to χ ∈ ℍ, so we have

F (χ) − F (χ_{i}) = ∥χ∥_{ℍ}^{2} − ∥χ_{i}∥_{ℍ}^{2}

+2 ∫_{M}⟨∇^{k}ψ_{∞} , ∇_{k}(χ − χ_{i})⟩

−2 ∫_{M}⟨𝒟ψ_{∞} , 𝒟(χ − χ_{i})⟩

+\frac{1}{2} ∫_{M}⟨ψ_{∞} , (μ + ν^{k}c(e_{k})c(e_{0})) (χ − χ_{i})⟩ .

Because ∇ψ_{∞} ∈ L^{2}(M), the first three terms above converge to zero as i → ∞. The convergence of the final term can be justified by applying the Cauchy-Schwarz inequality |⟨u, Qv⟩| ≤ \sqrt{⟨u, Qu⟩}\sqrt{⟨v, Qv⟩} whenever Q is positive:

|∫_{M}⟨ψ_{∞} , (μ + ν^{j}c(e_{j})c(e_{0})) (χ − χ_{i})⟩|
≤ (∫_{M}⟨ψ_{∞} , (μ + ν^{j}c(e_{j})c(e_{0})) ψ_{∞}⟩)^{1/2}
× (∫_{M}⟨(χ − χ_{i}), (μ + ν^{j}c(e_{j})c(e_{0})) (χ − χ_{i})⟩)^{1/2}
≤ C(∥μ∥_{L^{1}} + ∥ν∥_{L^{1}})(∥χ + χ_{i}∥_{ℍ}∥χ − χ_{i}∥_{ℍ})^{1/2} ;

in the last step of the calculation of Lemma 11.1 has been used. Now, F (χ_{i}) = F (0), and density implies that (11.9) remains true for any ψ of the form ψ_{∞}+χ, with χ ∈ ℍ.

We are ready now to prove the following version of the positive energy theorem, the regularity conditions of which have been chosen as a compromise between those needed for solvability of the Dirac equation (11.8) (cf. Remark 6.1, p. 32), those needed for a well defined notion of ADM mass, and those needed for a Banach manifold structure for the set of solutions of the general relativistic vacuum constraint equations.

Theorem 11.2 Let (M, g, K) be initial data for the Einstein equations with g ∈ W_{loc}^{2,2}, K ∈ W_{loc}^{1,2}, with (M, g) complete (without boundary). Suppose that M contains an asymptotically flat end and let p_{α} = (m, p⃗) be the associated ADM four-momentum.^{10} If

μ ≥ |ν|_{g} ,  (11.10)

then

m ≥ |p⃗|_{δ} ,  (11.11)

10There is a signature-dependent ambiguity in the relationship between p^{0}, p_{0} and the mass m: in the space-time signature (−, +, +, +) used in this paper this sign is determined by the fact that p_{0}, obtained by Hamiltonian methods, is usually positive in Lagrangean theories on Minkowski space-time such as the Maxwell theory, while the mass m is a quantity which is expected to be positive.

with equality if and only if m vanishes. Further, in that last case there exists a non-trivial covariantly constant (with respect to the space-time spin connection) spinor field on M.

Remark 11.3 Under the supplementary assumption of smoothness of g and K, it has been shown in [10] that the existence of a covariantly constant spinor implies that the initial data can be isometrically embedded into Minkowski space-time, cf. also [66]. We expect this result to remain true under the current hypotheses, but we have not attempted to prove this.

Proof: Suppose that for all ψ_{∞} we can establish existence of χ = χ[ψ_{∞}] ∈ ℍ such that ψ_{∞} + χ satisfies the Dirac equation (11.8). Equation (11.9) would then show that the quadratic form

ψ_{∞} → 4πp_{α}⟨ψ_{∞} , c(e^{α}e^{0})ψ_{∞}⟩

is non-negative, and the Theorem follows by a standard calculation. The existence of χ will be a consequence of Theorem 8.8, provided that the relevant hypotheses are met. We have

𝒟 = c(e^{i})D_{i} − \frac{1}{2}tr_{g}Kc(e_{0}) ⟹ 𝒟^{†} = 𝒟 .

Equation (11.9) with ψ_{∞} = 0 shows that the Schrödinger-Lichnerowicz estimate of Definition 8.4, with Y = ∅ and ρ = 0, holds both for L := 𝒟 and its formal adjoint 𝒟^{†} = 𝒟. Next, we note that the symmetric part Γ^{S} of the connection (11.3) is

Γ^{S} = \frac{1}{2}K^{i}_{j}c(e_{i})c(e_{0}) ⊗ dx^{j}

which does not vanish for non-zero K’s, but satisfies nevertheless the fall-off condition (9.8) by (11.7b). It follows from Theorem 9.5 that the weighted Poincaré inequality holds. The regularity conditions on the metric imply that the requirements of Hypothesis 8.1 with g̊ = g are met: for trivial bundles, or for smooth initial data, this is a straightforward calculation, compare Remark 6.1; for non-trivial bundles Proposition A.9, Appendix A, has to be invoked. The map β of Equation (8.9) is zero, as is the curvature term ρ in the Schrödinger-Lichnerowicz inequality (8.7), by the energy condition μ ≥ |ν|_{g}. From what has been said it follows that spinor fields in ℍ which are also in the kernel of 𝒟 are covariantly constant; they then have constant length, and are not in L^{2} if they are non-zero. Theorem 8.8 now shows that for any ψ_{∞} there exists a solution χ ∈ ℍ of the equation

𝒟χ = −𝒟ψ_{∞} ,

and the existence of the desired ψ follows.

Let us now turn our attention to manifolds with boundary. We shall say that a boundary ∂M is future-trapped if

θ_{+} := H + ∑_{A=2,3} K(e_{A}, e_{A}) ≤ 0.  (11.12)

Here H is the mean curvature of ∂M with respect to an inner-pointing normal, while the e_{A}’s form an ON basis for T ∂M. A future-trapped boundary in the sense above is future-trapped in the usual sense [36] for a surface in space-time. The following result generalises one by Herzlich^{11} [38]:

Theorem 11.4 Under the remaining hypotheses of Theorem 11.2, suppose instead that M has a differentiable, compact, future-trapped boundary ∂M. Then the conclusions of Theorem 11.2 hold.

Remark 11.5 One expects that the equality case cannot occur in (11.11), and a possible argument could proceed as follows: First, the existence of a covariantly constant spinor implies existence of a non-spacelike, covariantly constant, Killing vector field in the associated space-time. Further, if the metric is C^{2} and K is C^{1}, then the space-time metric fulfills the Einstein equations with a null fluid as a source [10, Appendix B]; this conclusion is expected to hold under the weaker differentiability conditions considered here. By reduction of the field equations, this should imply smoothness of the metric. (Alternatively, one could assume at the outset that g is C^{3} and K is C^{2}, in which case the argument presented in the current remark settles the issue). Topological censorship results [28] applied to the Killing development [10] of the initial data show that the boundary is then the union of a finite number of spheres. Arguing as in the proof of Theorem 4.6 of [25], the restriction to the boundary of the covariantly constant spinor would be harmonic, which is impossible by the Hijazi-Bär inequality (2.14) [7, 39].

Remark 11.6 Past-trapped boundaries are defined by changing the sign of K in (11.12); as the remaining hypotheses of Theorem 11.4 are invariant under this change of sign, an identical result holds for compact past-trapped ∂M’s.

Proof: The proof follows closely that of Theorem 11.2, the main difference being the need to impose suitable boundary conditions. Indeed, when ∂M is non-empty Equation (11.9) becomes

∫_{M} (|∇ψ|^{2} + \frac{1}{4} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i})c(e_{0})ψ⟩) − |𝒟ψ|^{2})
= 4πp_{α}⟨ψ_{∞} , c(e^{α}e^{0})ψ_{∞}⟩ + ∮_{∂M}⟨ψ, c(n)c(e^{A})∇_{A}ψ⟩
= 4πp_{α}⟨ψ_{∞} , c(e^{α}e^{0})ψ_{∞}⟩ +
∮_{∂M}⟨ψ, 𝒟_{∂M}ψ + \frac{1}{2}(H − ∑_{A}(K_{AA}c(n) − K_{A1}c(e_{A}))c(e_{0})ψ⟩ , (11.13)

for, say, continuously differentiable ψ’s of the form ψ = ψ_{∞} + χ, with χ compactly supported, and ψ_{∞} as in the proof of Theorem 11.2. Further, 𝒟_{∂M} is the Dirac boundary operator defined by Equation (2.7), and Equation (2.8) has been used. Finally, e_{i} is an ON frame on ∂M with n ≡ e_{1} normal to ∂M. Following [33] we impose the boundary condition

11The proof in [38] is the rigorous version of an argument proposed in [31]; it also extends that argument, as in [31] only marginally trapped boundaries are considered.

K_{−} := \frac{1}{2}(1 − ϵ)ψ = 0 on ∂M ,  (11.14)

where ϵ := −c(n)c(e_{0}). We then have

⟨ψ, c(e_{A})c(e_{0})ψ⟩ = −⟨ψ, c(e_{A})c(e_{0})c(n)c(e_{0})ψ⟩
= −⟨ψ, c(e_{0})c(n)c(e_{A})c(e_{0})ψ⟩
= −⟨c(n)^{t}c(e_{0})^{t}ψ, c(e_{A})c(e_{0})ψ⟩
= ⟨c(n)c(e_{0})ψ, c(e_{A})c(e_{0})ψ⟩ = −⟨ψ, c(e_{A})c(e_{0})ψ⟩ ,

which shows that the last term in the last line of Equation (11.13) vanishes. Then

𝒟_{∂M}ϵ = −ϵ𝒟_{∂M}

and ϵ^{t} = ϵ, so

⟨ψ, 𝒟_{∂M}ψ⟩ = ⟨ψ, 𝒟_{∂M}ϵψ⟩
= −⟨ψ, ϵ𝒟_{∂M}ψ⟩
= −⟨ϵψ, 𝒟_{∂M}ψ⟩ = −⟨ψ, 𝒟_{∂M}ψ⟩ ,

which shows that the first term in the last line of Equation (11.13) vanishes. Next,

⟨ψ, (H − ∑_{A} K_{AA}c(n)c(e_{0}))ψ⟩ = ⟨ψ, (H + ∑_{A} K_{AA})ψ⟩ = θ_{+}⟨ψ, ψ⟩ ,

which shows that the sum of the second and third term in the last line of Equation (11.13) gives a non-positive contribution when ∂M is trapped. When μ ≥ |ν|_{g}, ψ_{∞} = 0, and (11.14) holds, from Equation (11.13) we obtain

∫_{M} (|∇ψ|^{2} + \frac{1}{4} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i})c(e_{0})ψ⟩)) − \frac{1}{2} ∮_{∂M} θ_{+}⟨ψ, ψ⟩ = ∫_{M} |𝒟ψ|^{2} ,

so conditions (11.10), (11.12) give

∫_{M} |∇ψ|^{2} ≤ ∫_{M} |𝒟ψ|^{2} ,  (11.15)

for all ψ ∈ C_{c}^{1}(M) which satisfy (11.14). Define

ℍ_{K_{−}} := {ψ ∈ ℍ, K_{−}ψ = 0 on ∂M} ,  (11.16)

where ℍ is defined in (8.5). If we let

G′(ψ) := ∫_{M} (\frac{1}{4} (μ|ψ|^{2} + ν^{i}⟨ψ, c(e_{i})c(e_{0})ψ⟩)) − \frac{1}{2} ∮_{∂M} θ_{+}⟨ψ, ψ⟩ , (11.17)

then the calculation of the proof of Lemma 11.1 shows that G′(ψ) can be extended by continuity to a continuous function on ℍ_{K_{−}}. The boundary integral in (11.17) is continuous, so the volume integral is also continuous on ℍ_{K_{−}}. which implies that Equations (11.15) holds for all ψ ∈ ℍ_{K_{−}}. Lemma 8.6 establishes the Schrödinger-Lichnerowicz estimate (8.7) with ρ = 0 for (𝒟, K_{−}).

As explained in Section 10, the boundary value problem determined by Equation (11.14) belongs to the family of problems considered in Theorem 8.8. If both M and ∂M are simultaneously parallelizable, and if the metric g is a product near ∂M, then the regularity conditions of Theorem 8.8 are met by hypothesis; the general case is handled by Propositions A.9 and A.10, Appendix A. Repeating now the arguments of the proof of Theorem 11.2 gives the non-negativity of p_{0}.

It is expected that the positivity statement of Theorem 11.4 can be strengthened to the so-called Penrose inequality when trapped boundaries occur. This question remains wide open, except in the special^{12} case K_{ij} = 0 [15, 42]. An interesting related inequality has been, essentially, proved by Herzlich [37]; however, the arguments of that last reference do not include a sufficient justification of existence of the required spinor field, except in the rather special case of a smooth metric which is a product near the boundary, as analyzed by Bunke [16]. Here we fill this gap and establish the following:

Theorem 11.7 Let (M, g) be a complete Riemannian manifold with g ∈ W_{loc}^{2,2}, and suppose that M has a boundary ∂M diffeomorphic to S^{2}, with non-positive inwards pointing mean curvature. Suppose that the curvature scalar R(g) of the metric g is non-negative, and that M contains an asymptotically flat end with mass m. If σ is the dimensionless quantity defined as

σ := \sqrt{\frac{Area(∂M)}{π}} inf_{f∈C_{c}^{∞}(\overline{M}),f≢0} \frac{∥df∥_{L^{2}(M)}^{2}}{∥f∥_{L^{2}(∂M)}^{2}} ,

then

m ≥ \frac{σ}{1 + σ} \sqrt{\frac{Area(∂M)}{4π}} .

Moreover, if the metric is smooth, then equality is achieved if and only if (M, g) can be isometrically embedded in the Schwarzschild space-time with mass \sqrt{Area(∂M)/16π}.

Remark 11.8 If M is the union of a compact set with a finite number of asymptotically flat ends, then σ > 0.

Proof: The details of the argument follow closely those of the proof of Theorem 11.4, the pointwise boundary conditions (11.14) being replaced by the spectral boundary conditions (2.10) with K given by (2.13); compare the discussion of Section 2, as well as that in the paragraph following Equation (10.7). The main elements missing in the arguments of [37] are provided by the boundary regularity results of Section 5; those are the key to the proof of Theorem 8.8.

12Similarly to Theorem 11.7, for the results in [15, 42] it actually suffices that R ≥ 0 ⇔ μ ≥ −|K|_{g}^{2} + (tr_{g}K)^{2}, and that tr_{h}K vanishes on ∂M, where tr_{h} is the trace of the restriction of K to ∂M.

The reader is referred to [38] and [23, p. 679] for the analysis of the equality case, cf. also [9].

Following [31, 34], let us pass now to inequalities with an electric charge contribution. A set (M, g, K, E, B) will be called initial data for the Einstein-Maxwell equations if (M, g) is a three dimensional Riemannian manifold, K is a symmetric tensor on M, while E and B are vector fields on M; as before, this is a serious abuse of terminology, as we are not requiring any constraint equations to be satisfied. Given such a triple we set

div E := D_{i}E^{i} ,
div B := D_{i}B^{i} ,
μ := R − |K|_{g}^{2} + (trK)^{2} − 2|E|_{g}^{2} − 2|B|_{g}^{2} ,
ν_{i} := 2D_{j}(K^{j}_{i} − trKδ_{i}^{j}) + 4ϵ_{ijk}E^{j}B^{k} .  (11.18)

Here D is the Levi-Civita connection associated with the metric g. In a general relativistic context, div E is the electric charge density, div B is the magnetic charge density (usually zero, whether electro-vacuum or not), μ is the energy density remaining after subtracting the electro-magnetic contribution, and ν_{i} is the left-over matter current; μ, div E, div B and J_{i} vanish when the Einstein-Maxwell constraint equations hold.

Let a new connection ∇ be defined as

∇_{i} := D_{i} + \frac{1}{2}K_{ij}c(e^{j})c(e_{0})− \frac{1}{2}c(E)c(e_{i})c(e_{0})− \frac{1}{4}ϵ_{jkℓ}B^{j}c(e^{k})c(e^{ℓ})c(e_{i}) ; (11.19)

The connection ∇ will be called the space-time Einstein-Maxwell spin connection on M. ∇ is again not metric compatible, with symmetric part Γ^{S} given by

Γ^{S} = (\frac{1}{2}K_{ij}c(e^{j})c(e_{0}) − E_{i}c(e_{0}) − \frac{1}{4}ϵ_{jkℓ}B^{j}c(e^{k})c(e^{ℓ})c(e_{i})) ⊗ θ^{i} , (11.20)

where, as before, θ^{i} is the co-frame dual to e_{i}. In this context the asymptotic flatness conditions have to be complemented by conditions on E and B: we shall require

E, B ∈ L^{2}(M_{ext}) , div E, div B ∈ L^{1}(M_{ext}) , E = o(r^{−1}) , B = o(r^{−1}) .

(11.21)

Following an argument proposed by Gibbons and Hull [34] we have:

Theorem 11.9 Let (M, g, K, E, B) be initial data for Einstein-Maxwell equations with g ∈ W_{loc}^{2,2}, K, E, B ∈ W_{loc}^{1,2}, with (M, g) complete (without boundary). Suppose that M contains an asymptotically flat end M_{ext} with E and B satisfying the fall-off conditions (11.21) there, and let p_{α} = (m, p⃗) be the associated ADM four-momentum. Let Q and P be the total electric and magnetic charge of M_{ext},

Q = lim_{R→∞} \frac{1}{4π} ∮_{r=R} E^{i}dS_{i} , P = lim_{R→∞} \frac{1}{4π} ∮_{r=R} B^{i}dS_{i} .

If

μ ≥ \sqrt{|ν|_{g}^{2} + |div E|^{2} + |div B|^{2}} ,  (11.22)

then

m ≥ \sqrt{|p⃗|_{δ}^{2} + Q^{2} + P^{2}} ,  (11.23)

where |p⃗|_{δ} ≡ \sqrt{∑(p^{i})^{2}}, with equality if and only if there exists a spinor field on M which is covariantly constant with respect to the Einstein-Maxwell space-time spin connection (11.19).

Remark 11.10 Under the hypothesis of smoothness of the metric, Tod [61] has found the local form of the metrics which admit covariantly constant spinors as above; however, no classification of globally regular such space-times is known. It is expected that the only singularity-free solutions here have vanishing Maxwell field, or belong to the standard Majumdar-Papapetrou family (cf., e.g., [27]). It would be of interest to fill this gap.

Remark 11.11 Charged matter might violate (11.22); however, there might exist a constant α ∈ (0, 1) such that

μ ≥ \sqrt{|ν|_{g}^{2} + α^{2}(|div E|^{2} + |div B|^{2})} .  (11.24)

Replacing in (11.19) the fields E and B by αE and αB, an essentially identical argument leads to

m ≥ \sqrt{|p⃗|_{δ}^{2} + α^{2}(Q^{2} + P^{2})} .  (11.25)

Proof: For the connection (11.19) the identity (11.9) becomes [34]

∫_{M} (|∇ψ|^{2} − |𝒟ψ|^{2}
+\frac{1}{4}⟨ψ, (μ + ν^{i}c(e_{i})c(e_{0}) − div Ec(e_{0}) − div Bc(n)c(e_{2})c(e_{3})) ψ⟩)
= 4π⟨ψ_{∞} , [p_{α}c(e^{α})c(e^{0}) + Qc(e_{0}) − P c(n)c(e_{2})c(e_{3})] ψ_{∞}⟩ , (11.26)

again for ψ of the form ψ_{∞} + χ, with C^{1} compactly supported χ’s. The Dirac operator

𝒟 := c(e^{i})∇_{i} = c(e^{i})D_{i} − \frac{1}{2}(tr_{g}K − c(E))c(e_{0}) − \frac{1}{4}ϵ_{jkℓ}B^{j}c(e^{k})c(e^{ℓ})

is not formally self-adjoint, we have instead

𝒟^{†} = c(e^{i})D_{i} − \frac{1}{2}(tr_{g}K − c(E))c(e_{0}) + \frac{1}{4}ϵ_{jkℓ}B^{j}c(e^{k})c(e^{ℓ}) .

This shows that the adjoint of 𝒟 coincides with 𝒟 modulo the replacement

B → −B .

The arguments follow now the previous ones, basing on the identity (11.26). We simply note that (11.22) implies non-negativity of the quadratic form appearing in the second line of (11.26). Similarly, positivity of the quadratic form defined by the third line of Equation (11.26) implies (11.23). Some comments are in order here, related to the fact that L is not formally self-adjoint when the magnetic field does not vanish. Since we are not assuming interior compactness of M, M could have other asymptotic regions in which B could grow in an uncontrollable way, so that L^{†} will not map ℍ into L^{2}. Now, L^{†} differs from L by a change of the sign of B, which implies that L^{†} also satisfies a Schrödinger-Lichnerowicz identity with a connection in which B is replaced by −B. The arguments already given show that the weak equation L^{†}ϕ = 0 has no L^{2} solutions, and Corollary 8.8 provides the desired isomorphism property of L.

In the presence of boundaries we have:

Theorem 11.12 Under the remaining hypotheses of Theorem 11.9, suppose instead that M has a compact future-trapped boundary ∂M. Then the conclusions of Theorem 11.9 hold.

Proof: This is a repetition of the argument of the proof of Theorem 11.4; one imposes again the boundary condition (11.14), and we only need to check that (11.15) still holds. This is indeed the case, which is established as follows: the electromagnetic field leads to a supplementary contribution

∮_{∂M}⟨ψ, [E^{i}c(e_{0}) − B^{i}c(e^{1})c(e^{2})c(e^{3})] ψ⟩n_{i}

to the boundary integral (11.13). When (11.14) holds we have

⟨ψ, c(e_{0})ψ⟩ = −⟨ψ, c(e_{0})c(n)c(e_{0})ψ⟩
= ⟨ψ, c(n)ψ⟩
= −⟨c(n)ψ, ψ⟩
= −⟨c(n)c(e_{0})c(e_{0})ψ, ψ⟩
= ⟨c(e_{0})c(n)c(e_{0})ψ, ψ⟩
= −⟨c(e_{0})ψ, ψ⟩ = −⟨ψ, c(e_{0})ψ⟩ ,

hence

⟨ψ, c(e_{0})ψ⟩ = 0.

Similar manipulations show that Equation (11.14) implies

⟨ψ, c(e^{1})c(e^{2})c(e^{3})ψ⟩ = 0 ,

and the result follows.

We finish this section by noting that positive energy results follow by identical arguments for asymptotically hyperboloidal manifolds [4, 25, 26, 29, 33, 52, 62, 67]; here Theorem 9.10 should be used instead of Theorem 9.5. The definition of mass in that case is considerably more delicate, we refer the reader to [25, 26] for details.

A Fields on manifolds of W^{k+1,p} differentiability class

Consider a smooth manifold M; on such a manifold one can define in a geometrically invariant way tensor fields which are of C^{∞} differentiability class, or of C^{k} class, or of W_{loc}^{k,p} class. For example, one says that a tensor field is of W_{loc}^{k,p} class if there exists a covering of M by coordinate patches such that the coordinate components of the tensor in question are in W_{loc}^{k,p} in each of the coordinate patches. Since the transition functions when going from one coordinate system to another are smooth, this property will be true in any coordinate system.

Let, now, (M, g) be a smooth manifold with a pseudo–Riemannian metric g which is of W_{loc}^{k,p} differentiability class. For various arguments it is convenient to use local coordinate systems which are adapted to the metric, such as geodesic coordinates, or harmonic coordinates. In this case the transition functions to the adapted coordinate system will not belong to the original smooth atlas on M in general. At this point there are two strategies possible: either to enlarge the atlas on M to contain those new coordinate systems, or to ignore this issue and try to analyze the problems that arise on an ad hoc basis. For nearly all of this paper the ad hoc approach, working entirely within a C^{∞} structure on M, is quite adequate. However, the proofs of Theorems 11.7 and 11.12 require the existence of approximately Gaussian coordinates near a boundary or near a hypersurface of M (cf. Lemma A.10 below). Direct construction of such coordinates with respect to the W_{loc}^{k,p} metric g produces a coordinate change which is not C^{∞}, which forces us to analyse the problems involved when constructing systematically manifolds of W_{loc}^{k+1,p} differentiability class. For this reason we will present such a construction here. For technical reasons we shall always assume that

p ∈ [1, ∞] ,  k ∈ ℕ ,  kp > n ;  (A.1)

these restrictions are more than sufficient for our purposes. Generalising the condition k ∈ ℕ to k ∈ ℝ^{+} would require an analogue of Lemma A.2 for non-integral k, ℓ, which seems not to be available. Condition (A.1) and the Sobolev embedding W_{loc}^{k+1,p}(Ω) ⊂ C^{1}(Ω) (for appropriately regular open domains Ω ⊂ ℝ^{n}) mean that we will consider only manifolds which are at least of C^{1} differentiability class.

Consider, thus, a connected paracompact Hausdorff manifold M of C^{1} differentiability class. We shall say that M is of W_{loc}^{k+1,p} differentiability class if M has an atlas for which all the transition functions are of W_{loc}^{k+1,p} differentiability class. Unless indicated otherwise, the Lebesgue measure in local coordinates is used.

A tensor field with components which are C^{∞} with respect to some coordinate chart (belonging to a C^{∞} sub-atlas of the W_{loc}^{k+1,p} atlas), will not generally have smooth components in all W_{loc}^{k+1,p} charts. In this situation a C^{∞} tensor field “comes equipped” with a preferred atlas of coordinate charts in which it has smooth coordinate components. This is a priori the case for tensor fields of any differentiability class on W_{loc}^{k+1,p} manifolds, and it is of interest to single out those classes of tensor fields, the coordinate components of which will be of a prescribed differentiability class in every coordinate system of the W_{loc}^{k+1,p} atlas on M. Differentiability classes of this type will be referred to as invariantly defined. Our next result describes some such classes of tensor fields. It is convenient to introduce the following notation: let x, y ∈ ℝ, we shall write x >_{∗} y if the following holds:

x >_{∗} y ⟺ \{ \begin{matrix}x ≥ y , & if y > 0 , \\ x > y , & if y ≤ 0 . \end{matrix}  (A.2)

(We note that for x ≥ 0 the only value of x at which “>_{∗}” does not coincide with “≥” is x = 0.) In this notation the the Sobolev embedding theorem can be stated as:

W_{loc}^{s,t} ⊂ W_{loc}^{u,v} ⟺ u ≤ s and \frac{1}{v} >_{∗} \frac{1}{t} − \frac{s − u}{n} .  (A.3)

Proposition A.1 Let (M, g) be a W_{loc}^{k+1,p} manifold, kp > n, p ∈ [1, ∞].

1. Let (ℓ, q) be such that the Sobolev embedding

W_{loc}^{k+1,p} ⊂ W_{loc}^{ℓ,q}

holds (which is equivalent to the condition

\frac{1}{q} >_{∗} \frac{1}{p} + \frac{ℓ − k − 1}{n} ,  (A.4)

with >_{∗} defined in (A.2)). Then the space of W_{loc}^{ℓ,q} scalar fields on M is invariantly defined.

2. Let (ℓ, q) be such that the Sobolev embedding

W_{loc}^{k,p} ⊂ W_{loc}^{ℓ,q}

holds (which is equivalent to the condition

\frac{1}{q} >_{∗} \frac{1}{p} + \frac{ℓ − k}{n} .)  (A.5)

Then the space of W_{loc}^{ℓ,q} tensor fields on M is invariantly defined.

Proof: Point 1 is a straightforward consequence of the following Lemma:

Lemma A.2 Let Ω, U ⊂ ℝ^{n} and let ψ : Ω → U be a C^{1} diffeomorphism such that ψ ∈ W_{loc}^{k+1,p}(Ω; ℝ^{n}), kp > n. If (ℓ, q) is such that the Sobolev embedding W_{loc}^{k+1,p} ⊂ W_{loc}^{ℓ,q} holds, cf. Equations (A.4) and (A.2), then for all F ∈ W_{loc}^{ℓ,q}(U) we have

F ∘ ψ ∈ W_{loc}^{ℓ,q}(Ω) .

Remark A.3 In [13, 58] some partial results can be found concerning sharpness of this result.

Proof: We have, for 0 ≤ |α| ≤ ℓ(≤ k + 1),

∂^{α}(F ∘ ψ) = ∑ C(α_{1}, …, α_{m})∂^{α_{1}}ψ ⋯ ∂^{α_{m}}ψF^{(m)} ∘ ψ ,

where the sum is taken over sets (α_{1}, …, α_{m}) satisfying α_{1} + ⋯ + α_{m} = α, with |α_{i}| ≥ 1. For any compact K ⊂ Ω it follows that

∥∂^{α}(F ∘ ψ)∥_{L^{q}(K)} ≤ C ∑ ∥∂^{α_{1}}ψ∥_{L^{s(α_{1})}(K)} ⋯ ∥∂^{α_{m}}ψ∥_{L^{s(αm)}(K)}∥F^{(m)}∥_{L^{t}(ψ(K))} , (A.6)

\frac{1}{s(α_{1})} + ⋯ + \frac{1}{s(α_{m})} + \frac{1}{t} ≤ \frac{1}{q} .  (A.7)

Here we have used the generalized Hölder inequality, and the change of variables theorem to pass from ∥F^{(m)} ∘ ψ∥_{L^{t}(K)} to ∥F^{(m)}∥_{L^{t}(ψ(K))}. By Sobolev’s embedding we have F^{(m)} ∈ L^{r}(ψ(K)) for all r satisfying

\frac{1}{r} >_{∗} \frac{1}{q} + \frac{m − ℓ}{n} .  (A.8)

Consider, first, those terms in (A.6) for which the right hand side of (A.8) is positive (if any). Let r be defined by Equation (A.8) with >_{∗} replaced by =. Set

\frac{1}{s(α_{i})} = \frac{|α_{i}| − 1}{n} ;  (A.9)

since kp > n we have

\frac{1}{s(α_{i})} > \frac{1}{p} − \frac{k}{n} + \frac{|α_{i}| − 1}{n} ,  (A.10)

and Sobolev’s embedding theorem implies that ∥∂^{|α_{i}|}ψ∥_{L^{s(α_{i})}(K)} is finite. With this choice of r and of the s(α_{i})’s we have

∑ \frac{1}{s(α_{i})} + \frac{1}{r} = \frac{|α| − m}{n} + \frac{1}{q} + \frac{m − ℓ}{n}
= \frac{|α| − ℓ}{n} + \frac{1}{q} ≤ \frac{1}{q} ,

so that those terms will give a finite contribution to the right hand side of (A.6) by setting t = r.

Consider, next, those terms in (A.6) for which the right hand side of (A.8) vanishes. If all the α_{i}’s have length one the term in question will give a finite contribution to the right hand side of (A.6) by setting t = q. If one of the α_{i}’s, say α_{1}, has length large than 1, for i ≥ 2 we choose the α_{i}’s as in (A.9), while we set 1/s(α_{1}) = (|α_{1}| − 1)/n − ϵ > 0, with 0 < ϵ so chosen that (A.10) still holds, ϵ < (kp − n)/2pn. Choosing 1/t = ϵ will lead to a finite contribution in (A.6).

It remains to consider those terms in (A.6) for which the right hand side of (A.8) is negative. In this case we set t = ∞, and

\frac{1}{s(α_{i})} = \frac{|α_{i}|}{ℓq} ⟹ ∑ \frac{1}{s(α_{i})} = \frac{|α|}{ℓq} ≤ \frac{1}{q} .  (A.11)

By Sobolev’s embedding ∥∂^{α_{i}}ψ∥_{L^{s(α_{i})}(K)} will be finite when (A.10) holds. Now Equation (A.10) with s(α_{i}) defined by (A.11) is equivalent to

\frac{|α_{i}|}{ℓ}(\frac{1}{q} − \frac{ℓ}{n}) > \frac{1}{p} − \frac{k − 1}{n} .  (A.12)

The right hand side of (A.12) is negative. If the left hand side is positive or vanishes there is nothing to check. If both sides are negative the worst case is obtained with |α_{i}| = ℓ, and the inequality holds when (A.4) is an inequality. The simple analysis of the case of equality in (A.4) is left to the reader.

Before returning to the proof of Proposition A.1 we need one more Lemma:

Lemma A.4 Let 0 ≤ m ≤ ℓ ≤ k, q, p ∈ [1, ∞], kp > n. Suppose that (ℓ, q) is such that the Sobolev embedding W_{loc}^{k,p} ⊂ W_{loc}^{ℓ,q} holds, cf. Equations (A.5) and (A.2). Then the product map

W_{loc}^{k−m,p} × W_{loc}^{ℓ,q} ∋ (f, g) ⟶ fg ∈ W_{loc}^{ℓ−m,q}

is continuous.

Proof: For any 0 ≤ |α| ≤ ℓ − m the Leibniz rule gives

∂^{α}(fg) = ∑_{α_{1}+α_{2}=α} C_{α_{1},α_{2}} ∂^{α_{1}}f ∂^{α_{2}}g ,

so that on any compact set K ⊂ M the Hölder inequality gives

∥∂^{α}(fg)∥_{L^{q}(K)} ≤ C ∑_{α_{1}+α_{2}=α}∥∂^{α_{1}}f∥_{L^{s(α_{1})}(K)}∥∂^{α_{2}}g∥_{L^{s(α_{2})}(K)} , (A.13)
\frac{1}{s(α_{1})} + \frac{1}{s(α_{2})} ≤ \frac{1}{q} .  (A.14)

Let

a(α_{1}) = \frac{1}{p} + \frac{|α_{1}| − k + m}{n} ,
a(α_{2}) = \frac{1}{q} + \frac{|α_{2}| − ℓ}{n} .

By Sobolev’s embedding we will have ∂^{α_{1}}f ∈ L^{s(α_{1})}(K), ∂^{α_{2}}g ∈ L^{s(α_{2})}(K) when

\frac{1}{s(α_{i})} >_{∗} a(α_{i}) .  (A.15)

We have the following cases:

• If a(α_{1}) ≤ 0 and a(α_{2}) ≤ 0 we set s(α_{i}) = q/2, and we obtain

∥∂^{α_{1}}f∥_{L^{s(α_{1})}(K)}∥∂^{α_{2}}g∥_{L^{s(α_{2})}(K)} ≤ C∥f∥_{W^{k−m,p}(K)}∥g∥_{W^{ℓ,q}(K)} (A.16)

• If a(α_{1}) > 0 and a(α_{2}) > 0 we set 1/s(α_{i}) = a(α_{i}) so that (A.15) holds, and we obtain

\frac{1}{s(α_{1})} + \frac{1}{s(α_{2})} = \frac{1}{q} + \frac{1}{p} + \frac{|α| − k − ℓ + m}{n} ≤ \frac{1}{q} + \frac{1}{p} − \frac{k}{n} < \frac{1}{q} ,

since kp > n, so that (A.14) holds. We note that (A.16) is again satisfied.

• If a(α_{1}) = 0 and a(α_{2}) > 0 we have

\frac{|α_{2}|}{n} = \frac{|α| − |α_{1}|}{n} = \frac{|α| − k + m}{n} + \frac{1}{p} ,

so that

a(α_{2}) = \frac{1}{q} + \frac{1}{p} + \frac{|α| − k − ℓ + m}{n} ≤ \frac{1}{q} + \frac{1}{p} − \frac{k}{n} < \frac{1}{q} .  (A.17)

Let ϵ be any number satisfying 0 < ϵ < (kp − n)/np, set

\frac{1}{s(α_{1})} = \frac{ϵ}{2} , \frac{1}{s(α_{2})} = a(α_{2}) + \frac{ϵ}{2} .

Decreasing ϵ if necessary we will have s(α_{2}) > 0. Then (A.14), (A.15) and (A.16) hold by the calculation in Equation (A.17). A similar analysis takes care of the case a(α_{1}) > 0 and a(α_{2}) = 0.

• If a(α_{1}) < 0 and a(α_{2}) > 0 we set s(α_{1}) = ∞ and s(α_{2}) = q.

• If a(α_{1}) > 0 and a(α_{2}) < 0 we set s(α_{1}) = q and s(α_{2}) = ∞; (A.14) obviously holds, while

\frac{1}{s(α_{1})} − a(α_{1}) = \frac{1}{q} − \frac{1}{p} + \frac{k − m − |α_{1}|}{n} ≥ \frac{1}{q} − \frac{1}{p} + \frac{k − ℓ}{n}

which is non-negative by (A.5), hence (A.15) and (A.16) hold again.

This establishes that fg ∈ W_{loc}^{ℓ−m,q}. The continuity follows immediately from the inequality

∥fg∥_{W^{ℓ−m,q}(K)} ≤ C∥f∥_{W^{k−m,p}(K)}∥g∥_{W^{ℓ,q}(K)}

which has been established during the proof.

We can pass now to the proof of point 2 of Proposition A.1. Let two coordinate systems on M be given related to each other by a map ψ ∈ W_{loc}^{k+1,p}, set χ ≡ ψ^{−1}. Let t^{α_{1}...α_{k}}_{β_{1}...β_{s}} and t̂^{μ_{1}...μ_{k}}_{ν_{1}...ν_{s}} be the coordinate components of a tensor field t, with t^{α_{1}...α_{k}}_{β_{1}...β_{s}} ∈ W^{ℓ,q}. We have the transformation rule

t̂^{μ_{1}...μ_{k}}_{ν_{1}...ν_{s}}(x) = t^{α_{1}...α_{k}}_{β_{1}...β_{s}}(ψ(x))\frac{∂ψ^{β_{1}}}{∂x^{ν_{1}}} (x) ⋯ \frac{∂ψ^{β_{s}}}{∂x^{ν_{s}}} (x)\frac{∂χ^{μ_{1}}}{∂x^{α_{1}}} (ψ(x)) ⋯ \frac{∂χ^{μ_{k}}}{∂x^{α_{k}}} (ψ(x)) .

(A.18)

Now the matrix \frac{∂χ^{μ}}{∂x^{α}} ∘ ψ is the inverse matrix to\frac{∂ψ^{β}}{∂x^{ν}}, so the components of the former are rational function of those of the latter. We recall the Gagliardo– Moser–Nirenberg inequalities (cf., e.g. [41, Corollaries 6.4.4 and 6.4.5])

∀ f, g ∈ W^{ℓ,q} ∩ L^{∞} ∥fg∥_{W^{ℓ,q}} ≤ C_{1}(∥f∥_{L^{∞}}∥g∥_{W^{ℓ,q}} + ∥f∥_{W^{ℓ,q}}∥g∥_{L^{∞}}) ,(A.19)

∀ f ∈ W^{ℓ,q} ∩ L^{∞}  ∥F (f)∥_{W^{ℓ,q}} ≤ C_{2}(∥f∥_{L^{∞}})(1 + ∥f∥_{W^{ℓ,q}})  , (A.20)

for some f and g independent constant C_{1}, and for some constant C_{2}(∥f∥_{L^{∞}}) depending upon f only through its L^{∞} norm. Here one assumes that F is a smooth function of its argument which is allowed to take values in ℝ^{N}, and the integrals are taken over compact sets. Equations (A.19)–(A.20) show that the \frac{∂χ^{μ}}{∂x^{α}} ∘ ψ’s are W_{loc}^{k,p} functions of their arguments. Lemma A.4 implies that

\frac{∂ψ^{β_{1}}}{∂x^{ν_{1}}} (x) ⋯ \frac{∂ψ^{β_{s}}}{∂x^{ν_{s}}} (x)\frac{∂χ^{μ_{1}}}{∂x^{α_{1}}} (ψ(x)) ⋯ \frac{∂χ^{μ_{k}}}{∂x^{α_{k}}} (ψ(x)) ∈ W_{loc}^{k,p} .

It follows that the right hand side of Equation (A.18) is of the form

t ∘ ψ A , t ∈ W_{loc}^{ℓ,q} , ψ ∈ W_{loc}^{k+1,p} , A ∈ W_{loc}^{k,p} .

Lemma (A.2) implies that t∘ψ ∈ W_{loc}^{ℓ,q}, and Lemma A.4 with m = 0 shows that the right hand side of Equation (A.18) is in W_{loc}^{ℓ,q}, as desired.

We wish to extend the above discussion to spinor fields; this requires the introduction of orthonormal frames, and hence of the metric. Consider, then, a W_{loc}^{k+1,p} manifold M with a strictly positive definite symmetric two-covariant tensor field g. We shall say that (M, g) is a pseudo–Riemannian W_{loc}^{k+1,p} manifold if M is a W_{loc}^{k+1,p} manifold and if g is a pseudo–Riemannian metric of W_{loc}^{k,p} differentiability class. This is an invariantly defined notion by Proposition A.1. Before proceeding further we note the following:

Proposition A.5 Let (M, g) be a W_{loc}^{k+1,p} manifold with a pseudo–Riemannian metric of W_{loc}^{k,p} differentiability class, kp > n, p ∈ [1, ∞]. Then the following hold

1. In any coordinate system in the W_{loc}^{k+1,p} atlas the Christoffel coefficients

Γ_{jk}^{i} satisfy

Γ_{jk}^{i} ∈ W_{loc}^{k−1,p} .

2. The Riemann tensor is of W_{loc}^{k−2,p} differentiability class.

3. The curvature scalar R ≡ g^{ij}R^{k}_{ikj} is of W_{loc}^{k−2,p} differentiability class.

4. Assume that ℓ ≥ 1 and suppose that (ℓ, q) is such that the Sobolev embedding W_{loc}^{k,p} ⊂ W_{loc}^{ℓ,q} holds, cf. Equations (A.5) and (A.2). Let t be a tensor field of W_{loc}^{ℓ,q} differentiability class, then for any vector field X ∈ W_{loc}^{k,p} we have

X^{i}∇_{i}t ∈ W_{loc}^{ℓ−1,q} .

Proof: 1. By definition of the Γ_{jk}^{i}’s we have

Γ_{jk}^{i} = \frac{1}{2}g^{iℓ}(\frac{∂g_{ℓj}}{∂x^{k}} + \frac{∂g_{ℓk}}{∂x^{j}} − \frac{∂g_{jk}}{∂x^{ℓ}} ) .

By (A.19) we have g^{iℓ} ∈ W_{loc}^{k,p}, by definition the derivatives of the metric are in W_{loc}^{k−1,p}, and lemma A.4 with m = 0 and (ℓ, q) = (k − 1, p) gives the result.

2. By definition of the curvature tensor we have

R^{i}_{jkℓ} = \frac{∂Γ_{jℓ}^{i}}{∂x^{k}} − \frac{∂Γ_{jk}^{i}}{∂x^{ℓ}} + Γ_{mk}^{i}Γ_{jℓ}^{m} − Γ_{mℓ}^{i}Γ_{jk}^{m} ,

Lemma A.4 with m = 1 and (ℓ, q) = (k − 1, p) shows that the product terms above are in W_{loc}^{k−2,p}, and the result follows from point 1.

3. This claim follows immediately from point 2, from g^{iℓ} ∈ W_{loc}^{k,p} and from Lemma A.4 with m = 0 and (ℓ, q) = (k − 2, p).

4. Let t be a tensor field of W_{loc}^{ℓ,q} differentiability class, in local coordinates we have

∇_{i}t^{α_{1}...α_{k}}_{β_{1}...β_{s}} = ∂_{i}t^{α_{1}...α_{k}}_{β_{1}...β_{s}} + Γ_{iσ}^{α_{1}}t^{σ...α_{k}}_{β_{1}...β_{s}} + … − Γ_{iβ_{1}}^{σ}t^{α_{1}...α_{k}}_{σ...β_{s}} − ….

The Γ’s are in W_{loc}^{k−1,p} by point 1, thus the product terms are in W_{loc}^{ℓ−1,q} by Lemma A.4 with m = 1. The claim about X^{i}∇_{i}t follows again from Lemma A.4.

Let (M, g) be a W_{loc}^{k+1,p} pseudo–Riemannian manifold and let ON_{M,g} be the bundle of g–orthonormal frames on M. We can equip ON_{M,g} with a W_{loc}^{k,p} structure by considering only those g–orthonormal sets of vector fields which are all of W_{loc}^{k,p} differentiability class. Let us start by showing that the set of such (locally defined) frames is not empty. On O, the domain of a coordinate system (x^{i}), we can construct a g–orthonormal frame e_{j} = e_{j}^{i}∂/∂x^{i} by performing a Gram–Schmidt orthonormalisation of the basis {∂/∂x^{i}}. By construction the coordinate coefficients e_{j}^{i} of the vector fields e_{j} are smooth functions of g_{ij} (at least on a neighborhood of the range of values taken by g_{ij}), where the g_{ij}’s are the coordinate coefficients of the metric g, g = g_{ij}dx^{i}dx^{j}. Since kp > n, (A.19)–(A.20) applied to the e_{j}^{i} considered as functions of the g_{ij} shows that the vector fields e_{j} are indeed of W_{loc}^{k,p} differentiability class, as desired.

The following shows that the W_{loc}^{k,p} structure of ON_{M,g} is an invariantly defined property of a W_{loc}^{k+1,p} pseudo–Riemannian manifold:

Proposition A.6 Any two (globally or locally defined) g–orthonormal frames of W_{loc}^{k,p} differentiability class are related to each other by a O(n)–rotation of W_{loc}^{k,p} differentiability class.

Proof: Consider two locally defined g–orthonormal frames e_{i} and f_{i}, i = 1, …, n = dim M, of W_{loc}^{k,p} differentiability class. In particular each of the e_{i} and f_{i}’ is a vector field of W_{loc}^{k,p} differentiability class, which is invariantly defined by Proposition A.1, so that it is sufficient to prove the result in any coordinate system in the W_{loc}^{k+1,p} atlas on M. In such a coordinate system {x^{i}} we can write e_{j} = e_{j}^{i} \frac{∂}{∂x^{i}}, f_{j} = f_{j}^{i} \frac{∂}{∂x^{i}}, for some functions e_{j}^{i}, f_{j}^{i} ∈ W_{loc}^{k,p}. Since both frames are orthonormal there exists an O(n)–valued function w_{i}^{j} such that

e_{i} = w_{i}^{j}f_{j} .  (A.21)

It follows that

w_{i}^{j} = e_{i}^{k}f^{j}_{k} ,

where f^{j}_{k} is the matrix inverse to f_{j}^{k}. We have f^{j}_{k} ∈ W_{loc}^{k,p} by (A.20), thus w_{i}^{j} ∈ W_{loc}^{k,p} by (A.19), hence the result.

Now suppose that M has a spin structure, namely a Spin–principal bundle ℱ̃ which double-covers the principal bundle ℱ of g–orthonormal frames of (M, g):

0 → ℤ_{2} → ℱ̃ →^{π} ℱ → M .  (A.22)

We note that the obstruction to the existence of such structures is purely topological, (cf., e.g., [45, Chapter II]) and therefore independent of the choice of the metric and differentiability class. A bundle of spinors 𝒱 = ℱ̃ ×_{T} V is a vector bundle associated to ℱ̃ and a representation

T : Spin → End(V) ,

for some finite-dimensional vector space V. A choice of g–orthonormal frame e = (e_{i}) of W_{loc}^{k,p} differentiability class defined on an open set O ⊂ M determines a local section of ℱ. This lifts to a section of ℱ̃, which in turn is associated with a local orthonormal frame ϕ = (ϕ_{I}) in 𝒱 . Let U be another open set with a g–orthonormal frame e′ = (e′_{i}), so by Proposition A.6 there exists an O(n) valued map w = (w_{i}^{j}) of W_{loc}^{k,p} differentiability class such that the frames e, e′ are related by e_{i} = w_{i}^{j}e′_{j} on O ∩ U. The map w : O ∩ U → SO lifts to w̃ : O ∩ U → Spin. This lift is not unique, but the possible lifts differ only by a fixed nontrivial element z of the centre Z(Spin) ≃ ℤ_{2}. The corresponding spin frames ϕ, ϕ′ are related by ϕ = T (w̃)ϕ′ or T (zw̃)ϕ′. Analyticity of the local inverse π^{−1}(·) and the inequality (A.20) show that T (w̃), T (zw̃) ∈ W_{loc}^{k,p}, and it follows from (A.19) that the spin frames ϕ on U and ϕ′ on O are W_{loc}^{k,p} compatible on U ∩ O. This establishes the following result:

Proposition A.7 Let (M, g) be a W_{loc}^{k+1,p} spin manifold with a pseudo–Riemannian metric of W_{loc}^{k,p} differentiability class. Then every spinor bundle carries a natural W_{loc}^{k,p} differentiable structure.

An argument similar to that of Proposition A.1 shows:

Proposition A.8 Let (M, g) be a W_{loc}^{k+1,p} spin manifold with a pseudo–Riemannian metric of W_{loc}^{k,p} differentiability class, kp > n, p ∈ [1, ∞]. Let (ℓ, q) be such that the Sobolev embedding

W_{loc}^{k,p} ⊂ W_{loc}^{ℓ,q}

holds, cf. Equations (A.5) and (A.2). Then the space of W_{loc}^{ℓ,q} spinor fields is invariantly defined.

To proceed further, we recall the definition of the covariant derivative of a spinor field. Let V, e_{i}, O ⊂ M and γ_{i} be as before, and let ϕ = (ϕ_{I}) be the spinor frame corresponding to the orthonormal frame e = (e_{i}). This defines a preferred local spin frame, with respect to which the Clifford action is represented by locally constant matrices γ_{i}. Let ψ be a spinor field over O; the spinor covariant derivative of ϕ is given in terms of the orthonormal frame connection matrix ω_{ij}(e_{k}) = g(e_{i}, ∇_{e_{k}}e_{j}) and the spinor frame components ψ = ψ^{I}ϕ_{I} by (2.2), and the Dirac operator of ∇ on S is defined by

𝒟ψ = γ^{i}∇_{e_{i}}ψ .  (A.23)

Proposition A.9 Let (M, g) be a W_{loc}^{k+1,p} manifold with a pseudo–Riemannian metric of W_{loc}^{k,p} differentiability class, kp > n, p ∈ [1, ∞]. Then the following hold

1. Let e^{i} be any g–orthonormal frame of W_{loc}^{k,p} differentiability class, then the spin connection coefficients ω_{k} defined as ∇_{e_{k}}ψ = e_{k}(ψ) + ω_{k}ψ satisfy

ω_{k} ∈ W_{loc}^{k−1,p} .

2. If (ℓ, q) is as in Proposition A.8 with ℓ ≥ 1, and if X is a vector field of W^{k,p} differentiability class, then ∇_{X} maps continuously W_{loc}^{ℓ,q} spinor fields to W_{loc}^{ℓ−1,q} spinor fields:

W_{loc}^{ℓ,q} ∋ ϕ ⟶ ∇_{X}ϕ ∈ W_{loc}^{ℓ−1,q} .

In particular the Dirac operator maps continuously W_{loc}^{ℓ,q} to W_{loc}^{ℓ−1,q}.

Proof: To prove point 1 choose a spin frame in which the c(e^{i})’s are point independent matrices. Then

ω_{k} ≡ −\frac{1}{4}c(e^{i})c(e^{j})ω_{ij}(e_{k}) ,

with ω_{ij}(e_{k}) ≡ g(e_{i}, ∇_{e_{k}}e_{j}). The claim that ω_{ij}(e_{k}) ∈ W_{loc}^{k−1,p} follows immediately from point 4 of Proposition A.5 and from Lemma A.4. The result in any spin frame follows from the transformation rule of the connection coefficients under changes of frames and from Lemma A.4. The proof of point 2 follows that of point 4 of Proposition A.5 and will be omitted.

Given a smooth metric in a neighbourhood of a compact boundary ∂M, geodesics normal to the boundary determine a diffeomorphism of Y × I with a neighbourhood of ∂M ≃ Y, such that in adapted coordinates v = (y^{A}, x), x ∈ [0, x_{0}), y^{A} ∈ U_{i} we have

g^{xx} ≡ 1, g^{xA} ≡ 0.  (A.24)

The diffeomorphism determines a tubular neighbourhood of ∂M and the resulting coordinates are called Gaussian coordinates. If the metric has only low differentiability then uniqueness of the geodesic equation may fail, and the existence of Gaussian coordinates becomes problematic. However, for our applications it is sufficient for (A.24) to hold only approximately near ∂M, in which case we may rely on the following result.

Proposition A.10 (Almost Gaussian tubular neighbourhood coordinates for ∂M). Let k ∈ ℕ, ℓ ∈ ℕ ∪ {0}, and suppose (M, g) be a W_{loc}^{k+1,p} Riemannian manifold with metric g ∈ W_{loc}^{k,p}(M), (k − ℓ)p > n. Let Y ⊂ M be a compact connected component of the boundary of M with Y of W_{loc}^{k+1,p} differentiability class. There is a neighbourhood O of Y ⊂ M and x ∈ W^{k+1,p}(O), and a diffeomorphism O ≃ Y × I, I = [0, x_{0}), which determines coordinates (v^{i}) = (y^{A}, x) ∈ O such that Y ∩ O = {x = 0} and

g(dv^{i}, dv^{j}) = g^{ij} ∈ W^{k,p}(O) ,  (A.25)
g(dx , dx) − 1 = O(x^{ℓ+σ}),  (A.26)
g(dx , dy^{A}) = O(x^{ℓ+σ}) .,  (A.27)

for some σ > 0.

Remarks: 1. A similar result for C^{k,1} metrics, k ≥ 1, follows from [3, Appendix B].

2. Similar results hold for pseudo–Riemannian manifolds provided Y is non–characteristic, and when Y is a hypersurface in M. Proof: If k = ∞ we can use Gauss coordinates near Y, and the result follows. Suppose thus that k < ∞, let x be any defining function for Y and let O_{α} be any conditionally compact coordinate neighborhood of Y, with g^{ij} ≡ g(dv^{i}, dv^{j}) ∈ W^{k,p}(O_{α}). Passing to a subset of O_{α} if necessary without loss of generality we may assume O_{α} ≈ [0, x_{0}) × U_{α}, U_{α} ⊂ Y. Coordinate systems of this form will be called cylindrical.

We construct a suitable W^{k+1,p} coordinate change (ȳ^{A}, x̄) in O_{α} × I by noting first that dx̄ = \frac{∂x̄}{∂x}dx + \frac{∂x̄}{∂y^{A}} dy^{A}, dȳ^{A} = \frac{∂ȳ^{A}}{∂x} dx + \frac{∂ȳ^{A}}{∂y^{B}} dy^{B}. Thus if x̄ is also a boundary coordinate, so x̄(y, 0) = 0 and \frac{∂x̄}{∂y^{A}} = 0 on Y, then the metric coefficients satisfy

g^{x̄x̄} = g(dx̄, dx̄)
= (\frac{∂x̄}{∂x})^{2} g^{xx} + 2\frac{∂x̄}{∂x} \frac{∂x̄}{∂y^{A}} g^{xA} + \frac{∂x̄}{∂y^{A}} \frac{∂x̄}{∂y^{B}} g^{AB}
= (\frac{∂x̄}{∂x})^{2} g^{xx} on Y = \{x = 0\},
g^{x̄Ā} = g(dx̄, dȳ^{A})
= \frac{∂x̄}{∂x} (\frac{∂ȳ^{A}}{∂x} g^{xx} + \frac{∂ȳ^{A}}{∂y^{B}} g^{xB}) on Y .

Since g^{xx} ∈ W^{k,p}, the restriction g^{xx}|_{Y} lies in the Besov space Λ_{k−1/p}^{p,p}(Y) (see [59, §VI.4.4], or [43, Theorem VII.1]) and there is an extension x̄ = x̄(y, x) ∈ W^{k+1,p}(U_{α} × I) satisfying the conditions

x̄(y, 0) = 0, \frac{∂x̄}{∂x}(y, 0) = (g^{xx})^{−1/2}(y, 0) ,

for y = (y^{A}) ∈ O_{α} ([59, §VI.6], [43, Theorem VII.3]). This implies

g^{x̄x̄} = g(dx̄, dx̄) ∈ W^{k,p}(U_{α} × I)

and g^{x̄x̄} = 1 when x = 0. Similarly, there is f^{A}(y, x) ∈ W^{k+1,p}(U_{α} × I) such that

f^{A}(y, 0) = 0, \frac{∂f^{A}}{∂x} (y, 0) = − g^{xA}/g^{xx}(y, 0) ,

so the coordinates (ȳ, x̄), ȳ^{A}(y, x) = y^{A} + f^{A}(y, x) also satisfy

g^{Āx̄}(y, 0) = g(dx̄, dȳ^{A}) = 0

on Y, since dȳ^{A} = dy^{A} − g^{xA}/g^{xx} dx.

As shown in Proposition A.1, in the new coordinate system (v̄^{i}) ≡ (ȳ^{A}, x̄) we still have g(dv̄^{i}, dv̄^{j}) ∈ W^{k,p}(O_{α}), so that by embedding theorems the metric coefficients are σ–Hölder continuous on O_{α}, for some σ > 0, and

g^{x̄x̄} − 1 = O(x^{σ}), g^{x̄Ā} = O(x^{σ}) .  (A.28)

Let ϕ, f^{A} ∈ W^{k+1,p}(O_{α}), and consider the effect of the change

x̄ = x + ϕ(y, x),  (A.29)
ȳ^{A} = y^{A} + f^{A}(y, x),  (A.30)

where ϕ(y, 0) = 0, f^{A}(y, 0) = 0. Then

g^{x̄x̄} = g^{xx} + 2g^{xi} \frac{∂ϕ}{∂v^{i}} + g^{ij} \frac{∂ϕ}{∂v^{i}} \frac{∂ϕ}{∂v^{j}} ,  (A.31)
g^{x̄Ā} ≡ g(dx̄, dȳ^{A}) = g^{xA} + g^{xi} \frac{∂f^{A}}{∂v^{i}} + g^{Ai} \frac{∂ϕ}{∂v^{i}} + g^{ij} \frac{∂ϕ}{∂v^{i}} \frac{∂f^{A}}{∂v^{j}} .  (A.32)

Suppose that for some ℓ ≥ 0 we have

g^{xx} − 1 = O(x^{ℓ+σ}), g^{xA} = O(x^{ℓ+σ}) .  (A.33)

This holds for ℓ = 0 by (A.28) and we establish the general case by induction. Again by restriction and extension results [43, 59] there exist ϕ, f^{A} ∈ W^{k+1,p}(O_{α}) satisfying

\frac{∂^{ℓ+1}ϕ}{∂x}|_{(y,0)} = − \frac{1}{2} \frac{∂^{ℓ}g^{xx}}{∂x^{ℓ}} |_{(y,0)} ,
\frac{∂^{ℓ+1}f^{A}}{∂x^{ℓ+1}} |_{(y,0)} = − \frac{∂^{ℓ}g^{xA}}{∂x^{ℓ}} |_{(y,0)} ,

while all the lower order x–derivatives of ϕ, f^{A} vanish at x = 0. Passing to coordinates (ȳ, x̄) on a (possibly smaller) cylindrical neighborhood O_{α}, one finds from (A.31)–(A.32) that (A.33) still holds and moreover,

g^{x̄x̄} − 1 = O(x^{ℓ+1+σ}),
g^{x̄Ā} = O(x^{ℓ+1+σ}),

Dropping bars one finds that (A.33) holds with ℓ replaced by ℓ + 1, and the induction step is complete.

Finally we show that the local charts can be combined to form a tubular neighbourhood diffeomorphism. It follows from (A.31) that if x, x̄ both satisfy (A.28) and vanish on Y, then g(dx, dx̄) = 1. In particular, by combining the functions x̄_{α} from each of the local coordinate charts O_{α} using a subordinate partition of unity ϕ_{α}, the function x = Σ_{α}ϕ_{α}x̄_{α} satisfies x ∈ W^{k+1,p}(Y ×I) and x = 0, g^{xx} = 1 on Y.

In order to construct a diffeomorphism with Y × I we need to construct a similar averaging of the ȳ^{A} coordinate functions. Fix a smooth embedding Φ : Y → ℝ^{K} and let Π_{Φ(Y)} : N ⊂ ℝ^{K} → Φ(Y) be the orthogonal projection in ℝ^{K} from a tubular neighbourhood N back to Φ(Y). Let y_{α} = (y_{α}^{A}) : U_{α} ⊂ Y → ℝ^{n−1} denote both the coordinates of a C^{∞} chart on Y, and their natural extension to y_{α} = (y_{α}^{A}) : O_{α} = U_{α} × I ⊂ Y × I → ℝ^{n−1}. Let ȳ_{α} = (ȳ_{α}^{A}) : O_{α} → ℝ^{n−1} be the functions constructed above, so there is a neighbourhood Õ_{α} ⊂ O_{α} containing Y = Y × {0} such that y_{α}^{−1} ∘ ȳ_{α} : Õ_{α} → Y. Choose a finite covering Õ_{α} of Y × I with subordinate partition of unity ϕ̃_{α} and define Ψ : Y × I → Y,

Ψ(p) = Φ^{−1} ∘ Π_{Φ(Y)} (_{∑α}ϕ̃_{α}(p)Φ(y_{α}^{−1} ∘ ȳ_{α}(p))) .

Since ȳ_{α}^{A}(y, 0) = y^{A}, Ψ|_{Y ×{0}} = Id and (Ψ, x) defines a diffeomorphism of Y × I. Now for any C^{∞} chart y = (y^{A}) on Y, ỹ^{A} := y^{A} ∘ Ψ defines a chart on Y × I by p ↦ (ỹ^{A}(p), x(p)), which satisfies ỹ^{A}(y, 0) = y^{A}. Moreover, dỹ^{A}(y, 0) = dȳ^{A}(y, 0), so g^{xÃ}(y, 0) = 0. The condition g^{xx}(y, 0) = 1 is not affected by changes in the y-coordinate, so (Ψ, x) defines the required tubular neighbourhood.

References

[1] J Hogan, A. M^{c}Intosh, A. Axelsson, R. Grognard, Harmonic analysis of Dirac operators in Lipschitz domains, Clifford analysis and its applications (Prague 2000) (V. Soucek F. Brackx, J. S. R. Chisholm, ed.), NATO Sci. Ser. II, Kluwer, 2001, pp. 231–246.

[2] D. Mitrea, A. M^{c}Intosh and M. Mitrea, Rellich type estimates for one-sided mongenic functions in lipschitz domains and applications, Analytical and Numerical Methods in Quaternionic and Clifford algebras (K. Gürlbeck and W. Sprössig, eds.), 1996, pp. 135–143.

[3] L. Andersson and P.T. Chruściel, On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”, Dissert. Math. 355 (1996), 1–100.

[4] L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Annals of Global Anal. and Geom. 16 (1998), 1–27, dg-ga/9707017.

[5] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), 3–38.

[6] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.

[7] C. Bär, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39–46.

[8] R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661–693.

[9] R. Beig, Conformal properties of static spacetimes, Class. Quantum Grav. 8 (1991), 263–271.

[10] R. Beig and P.T. Chruściel, Killing vectors in asymptotically flat space– times: I. Asymptotically translational Killing vectors and the rigid positive energy theorem, Jour. Math. Phys. 37 (1996), 1939–1961, gr-qc/9510015.

[11] P. Bizoń. and E. Malec, On Witten’s positive-energy proof for weakly asymptotically flat spacetimes, Class. Quantum Grav. 3 (1986), L123–L128.

[12] B. Booß and K. P. Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhauser Boston, 1993.

[13] G. Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math. 104 (1991), 435–446.

[14] J-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Commun. Math. Phys. 144 (1992), 581–599.

[15] H. Bray, Proof of the Riemannian Penrose conjecture using the positive mass theorem, Jour. Diff. Geom. 59 (2001), 177–267, math.DG/9911173.

[16] U. Bunke, Comparison of Dirac operators on manifolds with boundary, Rend. Circ. Mat. Palermo (2) Suppl. (1993), 133–141, Proceedings of the Winter School “Geometry and Physics” (Srní, 1991).

[17] G. Bunting and A.K.M. Masood–ul–Alam, Nonexistence of multiple black holes in asymptotically euclidean static vacuum space-time, Gen. Rel. Grav. 19 (1987), 147–154.

[18] G. Carron, Un théorème de l’indice relatif, Séminaire de Théorie Spectrale et Géométrie, No. 15, Année 1996–1997, Sémin. Théor. Spectr. Géom., vol. 15, Univ. Grenoble I, Saint, 1996–97, pp. 193–202.

[19] Y. Choquet-Bruhat, Positive-energy theorems, Relativity, groups and topology, II (Les Houches, 1983) (B.S. deWitt and R. Stora, eds.), North-Holland, Amsterdam, 1984, pp. 739–785.

[20] P.T. Chruściel, A remark on the positive energy theorem, Class. Quantum Grav. 33 (1986), L115–L121.

[21] , Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological Properties and Global Structure of Space–Time (P. Bergmann and V. de Sabbata, eds.), Plenum Press, New York, 1986, pp.

49–59, URL http://www.phys.univ-tours.fr/~piotr/scans.

[22] , Quelques inégalités dans les espaces de Sobolev à poids,

Tours preprint, unpublished, http://www.phys.univ-tours.fr/~piotr/
papers/wpi, 1987.

[23] , The classification of static vacuum space–times containing an asymptotically flat spacelike hypersurface with compact interior, Class. Quantum Grav. 16 (1999), 661–687, gr-qc/9809088.

[24] , Towards the classification of static electro–vacuum space–times containing an asymptotically flat spacelike hypersurface with compact interior, Class. Quantum Grav. 16 (1999), 689–704, gr-qc/9810022.

[25] P.T. Chruściel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, (2001), dg-ga/0110035.

[26] P.T. Chruściel, J. Jezierski, and S. Łȩcki, The Trautman-Bondi mass of hyperboloidal initial data sets, in preparation (2002).

[27] P.T. Chruściel and N.S. Nadirashvili, All electrovacuum Majumdar– Papapetrou spacetimes with non–singular black holes, Class. Quantum Grav. 12 (1995), L17–L23, gr-qc/9412044.

[28] P.T. Chruściel and R.M. Wald, On the topology of stationary black holes, Class. Quantum Grav. 11 (1994), L147–152.

[29] M. Dahl, The positive mass theorem for ALE manifolds, Chruściel, Piotr T. (ed.), Mathematics of gravitation. Part I: Lorentzian geometry and Einstein equations. Proceedings of the workshop on mathematical aspects of theories of gravitation, Warsaw, Poland, February 29–March 30, 1996. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 41(1), 133-142 .

[30] A. J. Dougan and L. J. Mason, Quasi-local mass constructions with positive gravitational energy, Phys. Rev. Lett. 67 (1991), 2119–2123.

[31] G. T. Horowitz G. W. Gibbons, S. W. Hawking and M. J. Perry, Positive mass theorems for black holes, Commun. Math. Phys. 88 (1983), 295–308.

[32] R. Geroch and S.-M. Perng, Total mass-momentum of arbitrary initial data sets in general relativity, J. Math. Physics 35 (1994), 4157–4177, grqc/9403057.

[33] G.W. Gibbons, S.W. Hawking, G.T. Horowitz, and M.J. Perry, Positive mass theorem for black holes, Commun. Math. Phys. 88 (1983), 295–308.

[34] G.W. Gibbons and C.M. Hull, A Bogomolny bound for general relativity and solitons in N = 2 supergravity, Phys. Lett. 109B (1982), 190–194.

[35] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer Verlag, 1977.

[36] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973.

[37] M. Herzlich, A Penrose-like inequality for the mass on Riemannian asymptotically flat manifolds, Commun. Math. Phys. 188 (1997), 121–133.

[38] , The positive mass theorem for black holes revisited, Jour. Geom. Phys. 26 (1998), 97–111.

[39] O. Hijazi, Première valeur propre de l’opérateur de Dirac et nombre de Yamabe, C.R. Acad. Sci. Paris 313 (1991), 865–868.

[40] L. Hörmander, The analysis of partial differential operators, iii, Grundlehren vol. 224, Springer, 1985.

[41] , Lectures on nonlinear hyperbolic differential equations, Math. & Appl. 26, Springer, 1997.

[42] G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Int. Math. Res. Not. 20 (1997), 1045–1058.

[43] A. Jonsson and H. Wallin, Function spaces on subsets of ℝ^{n}, Mathematical Reports, vol. 2, Harwood, 1984.

[44] C. B. Morrey Jr., Multiple integrals in the calculus of variations, Springer Verlag, 1966.

[45] H. B. Lawson and M. L. Michelsohn, Spin geometry, Princeton Math. Series vol. 38, Princeton UP, 1989.

[46] A. Lichnerowicz, Spineurs harmonique, C.R. Acad. Sci. Paris Sér. A-B 257 (1963), 7–9.

[47] M. Ludvigsen and J. A. G. Vickers, A simple proof of the positivity of Bondi mass, J. Phs. A: Math. Gen. 15 (1982), L67–L70. 1983.

[48] J. Marschall, Pseudo-differential operators with coefficients in Sobolev spaces, Trans. AMS 307 (1988), 335–361.

[49] H.P. McKean, An upper bound to the spectrum of Δ on a manifold of negative curvature, Jour. Diff. Geom. 4 (1970), 359–366.

[50] C.B. Morrey, Multiple integrals in the calculus of variations, Springer Verlag, Berlin, Heidelberg, New York, 1966.

[51] T. Parker and C. Taubes, On Witten’s proof of the positive energy theorem, Commun. Math. Phys. 84 (1982), 223–238.

[52] O. Reula and K.P. Tod, Positivity of the Bondi energy, Jour. Math. Phys. 25 (1984), 1004–1008.

[53] P. Ruback, A new uniqueness theorem for charged black holes, Class. Quantum Grav. 5 (1988), L155–L159.

[54] R. Schoen and S.-T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1981), 231–260.

[55] E. Schrödinger, Diracsches elektron im Schwerfeld, Preuss. Akad. Wiss. Phys.-Math. 11 (1932), 436–460.

[56] R. I. Seeley, Singular integrals and boundary problems, Am. J. math. 88 (1966), 781–809.

[57] K. Shiohama, Total curvature and minimal area of complete open surfaces, Proc. Am. Math. Soc. 94 (1985), 310–316.

[58] W. Sickel, Composition operators acting on Sobolev spaces of fractional order. — A survey on sufficient and necessary conditions., Function spaces, differential operators and nonlinear analysis (J. Rakosnik, ed.), Prometheus Publishing House, Prague, 1996, Proceedings of the conference held in Paseky nad Jizerou, September 1995, pp. 159–182, URL: ftp://

cirm.univ--mrs.fr/pub/EMIS/proceedings/Paseky95/sickel.ps.gz.

[59] E. Stein, Singular integrals and differentiability properties of functions, Princeton UP, 1970.

[60] M. E. Taylor, Partial differential equations III, Applied Mathematical Sciences, vol. 117, Springer, 1996.

[61] K.P. Tod, All metrics admitting super–covariantly constant spinors, Phys. Lett. 121B (1983), 241–244.

[62] X. Wang, Mass for asymptotically hyperbolic manifolds, Jour. Diff. Geom. 57 (2001), 273–299.

[63] J. Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer Verlag, New York, Heidelberg, Berlin, 1980.

[64] E. Witten, A simple proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381–402.

[65] J. Wolf, Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611–640.

[66] P.F. Yip, A strictly-positive mass theorem, Commun. Math. Phys. 108 (1987), 653–665.

[67] X. Zhang, A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds I, (2001), preprint.