Generalized Dirac Operators on Nonsmooth

Manifolds and Maxwell’s Equations

Marius Mitrea ∗

Abstract

We develop a function theory associated with Dirac type operators on Lipschitz subdomainsof Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems,and our aim is to identify the geometric and analytic assumptions guaranteeing the validity ofbasic results from complex function theory in this general setting. For example, we study Plemelj-Calderon-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’monogenics and show that this problem has finite index. We also consider Szego projections and thecorresponding Lp-decompositions. Our approach relies on an extension of the classical Calderon-Zygmund theory of singular integral operators which allows one to consider Cauchy type operatorswith variable kernels on Lipschitz graphs. In the second part, where we explore connections withMaxwell’s equations, the main novelty is the treatment of the corresponding electro-magnetic bound-ary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

1 Introduction

Since the introduction in 1928 by the physicist P. M. Dirac of a first-order linear differential operatorwhose square is the wave operator, Dirac type operators have become of central importance in manybranches of mathematics such as PDE’s, differential geometry and topology. See, e.g., the monographs[6], [4], [17], [28], [44] and the references therein. At the heart of the matter lies the fact that ellipticsystems of the first order generalizing the classical Cauchy-Riemann system give rise to a natural, richfunction theory.

The general aim of this paper is to develop such a function theory for a general Dirac operator Don a manifold M under minimal smoothness assumptions. A special emphasis is placed on studyingHardy type spaces associated with D,

Hp(Ω, D) := u; Du = 0 in Ω, N (u) ∈ Lp(∂Ω), (1.1)

in an arbitrary Lipschitz subdomain Ω of M ; here N (u) stands for the nontangential maximal functionof u (more precise definitions will be given shortly).

When the underlying domain is the unit disk or even a more general but smooth domain in thecomplex plane, this topic is classical and a great deal of information is known; cf. the excellent accountsin [12], [18]. The study of Hardy spaces in nonsmooth subdomains of the complex plane originates in[25], based on conformal mapping techniques. Subsequent developments, emphasizing real methods,are in [13], [26], [11]. One suitable replacement for the Cauchy-Riemann operator ∂x + i∂y in higherdimensions is the Dirac operator

∑ej∂j within the context of a Clifford algebra generated by the

∗Supported in part by NSF grant DMS #98700181991 Mathematics Subject Classification. Primary 31C12, 42B20, 35F15, 42B30; Secondary 58G20, 42B25, 78A25.Key words. Dirac operators, Hardy spaces, Maxwell’s equations, Lipschitz domains.

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(anticommuting) imaginary units ejj . Hardy spaces in Lipschitz domains of Rn associated with suchoperators have been studied in [29], [17], [35].

In the present paper we continue this line of research and take the next natural step by consideringgeneralized Dirac operators with variable coefficients in the context of Lipschitz domains on manifolds.These are first order, elliptic differential operators so that

D and D∗ have the unique continuation property. (1.2)

In fact, our entire theory of Hardy spaces in § 4 is developed based solely on ellipticity and this uniquecontinuation property assumption. As is well known, Dirac operators naturally associated with Cliffordalgebra structures automatically satisfy (1.2). Thus, from this perspective, the primary role of Cliffordalgebras is to provide natural examples of operators D for which (1.2) holds. On the other hand, eachoperator D satisfying (1.2) as well as certain extra algebraic hypotheses arises precisely in this fashion;cf. the discussion in § 5 for a more precise statement.

The highlights of the Hardy space theory we develop at this level of generality include two decom-position theorems which we now proceed to describe. Let Ω be a Lipschitz domain in the manifold Mand set Ω+ := Ω, Ω− := M \ Ω. Also, consider the boundary Hardy spaces Hp±(∂Ω, D) := u|∂Ω; u ∈Hp(Ω±, D). As is well known, when D := ∂, the Cauchy-Riemann operator and M := C, the Plemelj-Calderon decomposition

Lp(∂Ω) = Hp−(∂Ω, D)⊕Hp+(∂Ω, D), 1 < p <∞, (1.3)

plays a basic role in complex and harmonic analysis. In particular, (1.3) is equivalent to the Lp-boundedness of the classical Cauchy singular integral operator on the Lipschitz curve ∂Ω; cf. thediscussion in [32]. A natural question is the extent to which (1.3) remains valid when D is a generalizedDirac operator and the interface ∂Ω a Lipschitz submanifold (of codimension one) of M . We shall provethat (1.3) continues to hold in this general context but modulo finite dimensional spaces. Specifically,(Hp−(∂Ω, D),Hp+(∂Ω, D)

)is a Fredholm pair (cf. [23]) and

Index(Hp−(∂Ω, D),Hp+(∂Ω, D)

)= IndexD. (1.4)

In particular, the index of D on M can be read off data living on ∂Ω. See Theorem 4.4 for a completestatement; this result has been announced in [38]. Let us point out that a version of this theorem inthe smooth context has first been proved (cf. [6]) via techniques which do not work in the contextof nonsmooth domains. At the heart of the matter is the fact that, in the presence of boundaryirregularities, the relevant operators are longer pseudodifferential and only belong to the class of singularintegrals.

Another important consequence of (1.4) is that the transmission boundary problem

(TBV P )

u+ ∈ Hp(Ω+, D), u− ∈ Hp(Ω−, D),

u+|∂Ω − u−|∂Ω = f ∈ Lp(∂Ω)(1.5)

is Fredholm solvable and its index is precisely IndexD. In particular, the index of (TBV P ) is in-dependent of the particular Lipschitz domain Ω (for a more complete statement see Corollary 4.5).Interesting examples are offered by Hodge Dirac operators, signature operators, etc.

To state the second decomposition result alluded to earlier, let D be as before and denote by σ(D; ξ)the principal symbol of D at ξ ∈ T ∗M . Fix Ω arbitrary Lipschitz domain in M and denote by ν theoutward unit normal to ∂Ω. Then there exists ε = ε(Ω, D) > 0 so that

Lp(∂Ω) = Hp(∂Ω, D)⊕ iσ(D∗; ν)Hp(∂Ω, D∗) (1.6)

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for any 2− ε < p < 2 + ε, where the sum is direct and topological. Moreover, when σ(DD∗; ξ) is scalarand ν ∈ vmo (∂Ω), the class of functions of vanishing mean oscillations on ∂Ω, then (1.6) is valid forany 1 < p <∞.

As explained in §4, the decomposition (1.6) is essentially equivalent to the statement that theorthogonal projection of L2(∂Ω) onto H2(∂Ω, D), the so-called Szego projection, extends to a boundedoperator on Lp(∂Ω). In particular, it is illuminating to point out that (1.6) contains Riesz’s classicalestimate ∥∥∥∥∥

+∞∑n=0

cneinθ

∥∥∥∥∥Lp(−π,π)

≤ Cp

∥∥∥∥∥+∞∑

n=−∞cne

inθ

∥∥∥∥∥Lp(−π,π)

, 1 < p <∞, (1.7)

as a very special case. We prove (1.6) from the analysis of a Kerzman-Stein type formula which wededuce in the present context (cf., e.g., [2] for more on this in smoother settings).

We also consider Lp-based Hodge type decompositions for Dirac type operators in Lipschitz do-mains. Specifically, we show that there exists ε = ε(Ω, D) > 0 so that, for any 2− ε < p < 2 + ε,

Lp(Ω) = Ker (D∗; Lp(Ω))⊕DH1,p0 (Ω), (1.8)

where the direct sum is topological. Furthermore, if D is actually Dirac (i.e. D∗D has scalar principalsymbol), then (1.8) is shown to hold for the larger range 3/2− ε < p < 3 + ε. A key ingredient in thelatter result is the recent solution of the Poisson problem for the Laplace-Beltrami operator in Lipschitzdomains from [41].

In the second part of the paper we consider boundary problems for Dirac type operators in Lipschitzdomains. Here we limit the discussion to specific classes of Dirac operators and/or boundary conditions.For example, for an arbitrary symmetric Dirac operator D, an arbitrary Lipschitz domain Ω ⊂M andwith P± standing for 1

2(I ± σ(D; ν)), I being the identity operator, we discuss in §5 the followingboundary problem:

(BV P±)

u ∈ Hp(Ω, D),

P±(u|∂Ω) prescribed in Lp(∂Ω).(1.9)

The trace u|∂Ω is taken in the pointwise (nontangential) sense; this is meaningful a.e. on ∂Ω. At thepresent time there is no analogue of the concept of regular elliptic problem in the nonsmooth settingbut, broadly speaking, each such problem is teated via ad-hoc methods. Here we identify the rightspaces and operators for the method of layer potentials to apply.

Next, we consider what we call the Maxwell-Dirac operator

IDk := d+ δ + k dt· (1.10)

where d and δ are the exterior derivative operator and its adjoint, respectively, k is a complex parameter,t is the “time” variable (on the Fourier side) and dt· acts as a Clifford algebra multiplier. The aim isto initiate a detailed investigation of boundary problems of the type

(BV P )

u ∈ Hp(Ω, IDk),

utan or unor prescribed in Lp(∂Ω).(1.11)

Here, again, Ω is an arbitrary Lipschitz subdomain of M and utan, unor are, respectively, the tangentialand the normal component of u on ∂Ω. It is therefore natural to think of (1.11) as half-Dirichletproblems for IDk.

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When all structures involved are smooth and solutions are sought in a sufficiently regular class offunctions, such problems are regular elliptic and, hence, Fredholm solvable (cf. [44]). In this scenario,pseudodifferential operator techniques play a crucial role. The nature of the problem at hand changeswhen the smoothness assumptions are significantly relaxed. In particular, the method of layer potentials(which we employ in this paper) leads to considering singular integrals in place of pseudodifferentialoperators. One notable difficulty in the case we are interested, i.e. Lipschitz boundaries, metric tensorswith a very limited amount of smoothness, is the absence of an algebra structure and the lack of asymbolic calculus within the class of general singular integral operators.

Our approach utilizes an array of tools from harmonic analysis which have been successful in thetreatment of second-order, constant coefficient elliptic boundary problems in Lipschitz domains of theEuclidean space. See [25] and the references cited there for a survey of the state of the art in this fieldup to early 1990’s. A more recent line of research, initiated in [39], [34] (and further developed in [40],[41], [42]), is the use of layer potentials in order to solve boundary problems for general second-order,variable coefficient, elliptic systems in non-smooth manifolds. The present paper, dealing with variablefirst-order elliptic systems, is a natural continuation of this work. A basic goal of this program is toachieve an “elliptization” of (the non-coercive) Maxwell’s equations

(Maxwell)

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω,(1.12)

with N (E), N (H) ∈ Lp(∂Ω) and Etan or Enor prescribed in Lp(∂Ω), by embedding (1.12) into alarger, “half-Dirichlet” problem for a suitable Dirac type operator. Indeed, as explained in last partof our paper, (1.12) can be understood as a particular manifestation of the half-Dirichlet problem forMaxwell-Dirac operator (1.10). In this case, under the identification u = H − i dt · E, (1.11) becomes

(Generalized

Maxwell

) E,H ∈ C0(Ω,GC

M ),

δE + dE − ikH = 0 in Ω,

δH + dH + ikE = 0 in Ω,(1.13)

with N (E), N (H) ∈ Lp(∂Ω) and Etan, Htan or Enor, Hnor prescribed in Lp(∂Ω), and we give necessaryand sufficient conditions on the boundary data guaranteeing that (1.11) and (1.13) are equivalent. Ofcourse, for this result to have practical value, we first need to give a thorough solution to (1.11) tobegin with. See Theorem 7.1 for this.

Following the work in the three-dimensional case in [37], the boundary problem (1.12) has beenfirst solved on Lipschitz domains in [36], [20], [34] via integral equation methods. The philosophy of theapproach in these papers is to reduce (1.12) to boundary problems for the (perturbed) Hodge-Laplacianwith absolute and relative boundary conditions. In particular, one transforms it into a second orderPDE. Here we provide an alternative approach based on working directly with the more general, firstorder system (1.13).

Ultimately, implementing this idea requires understanding connections between Dirac operators andMaxwell’s equations in non-smooth domains. In the flat, Euclidean setting and for constant coefficientoperators, this direction of research has been initiated in [31] and [30]. A basic ingredient in [31] is aRellich type estimate to the effect that

‖utan‖L2(∂Ω) ≈ ‖unor‖L2(∂Ω), (1.14)

for two-sided monogenic functions in Ω, i.e. elements in H2(Ω, IDk) which are also annihilated by theaction of IDk to the right. A step forward in the direction of dealing with one-sided Clifford modules

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(as in the case of manifolds) was taken in [30] where the monogenicity assumptions were relaxed. Inthe present paper we continue this program by showing that (1.14) holds for any u ∈ H2(Ω, IDk) andΩ arbitrary Lipschitz subdomain of a Riemannian manifold.

In somewhat greater detail, the organization of the paper is as follows. Section 2 contains adiscussion of the global invertibility properties of D and Laplacians associated with D. A functiontheory associated with a general, first-order, variable coefficient elliptic system, with a special emphasison Hardy type spaces and Cauchy like operators in Lipschitz domains is developed in §3-4. How Diractype operators fit in this general framework makes the subject of Section 5. Boundary value problems forHodge-Dirac operators of the form d+αδ, α ∈ R, in Lipschitz subdomains of Riemannian manifolds arestudied in Section 6. The first part of Section 7 is reserved for a similar discussion, this time pertainingto the Maxwell-Dirac operator (1.10). Finally, in the second part of this section we elaborate onthe connections between the half-Dirichlet problem for the perturbed Dirac operator (1.10) and theMaxwell system (1.12).

Acknowledgments. I am grateful to Alan McIntosh for sharing his ideas with me and for the manyspirited conversations we have had during his visit at UMC in the Fall of 1997. His constructivesuggestions led to several improvements in § 7. I also thank Bernhelm Booß-Bavnbek for his insightfulcomments, David Calderbank for giving me a copy of his thesis [8] and Michael Taylor for an inspiringdiscussion during his visit at UMC in the Fall of 1998.

2 Inverting generalized Dirac operators and Laplacians

Let M be a compact, boundaryless, smooth, orientable manifold, of real dimension m. ConsiderE ,F →M smooth vector bundles and

D : C1(M, E) −→ Meas (M,F) (2.1)

a first order, elliptic (i.e. with an invertible symbol) differential operator mapping C1 sections of E intomeasurable sections of F . Assume that, in local coordinates,

Du =∑

aαβj ∂juβfα +

∑bαβuβfα, (2.2)

where (eβ)β, (fα)α are local frames for E and F , respectively, and u =∑uβeβ, with

aαβj ∈ Cγ , γ > 0, and bαβ ∈ Lr, r > m. (2.3)

The first order of business is to study the global action of the operator D on the manifold M . Inthis context, we shall prove that D is invertible modulo finite dimensional spaces, i.e. it is a Fredholmoperator. To state the main result in this direction, denote by Hs,p the usual scale of Sobolev spaces.Also, recall the index r from (2.3).

Theorem 2.1 Granted (2.3), the operator

D : H1,p(M, E) −→ Lp(M,F) (2.4)

is Fredholm for any 1 < p < r. In particular, it has closed range and a finite dimensional kernel.Furthermore, for τ + γ > 1, τ ∈ [0, 1), 1 < p ≤ q < r, the following regularity result holds:

u ∈ Hτ,p, Du ∈ Lq ⇒ u ∈ H1,q. (2.5)

In particular, if γ > 1, then

5

u ∈ Lp and Du = 0 =⇒ u ∈ Cα for some α > 0. (2.6)

In the smooth context, this is well known. The primary interest in this result stems from the lowregularity assumptions which we make on the coefficients. For similar results on Zygmund spaces see[45].

Proof. Work in local coordinates and use a symbol decomposition as in [Ta2]. Make all subsequentpseudodifferential operators properly supported. Then, via a partition of unity, we can write

D = D# +Db +B (2.7)

with

D# ∈ OPC∞S11,δ, Db ∈ OPCγS1−γδ

1,δ , B ∈ L∞(M,Hom(E ,F)) (2.8)

for some 0 < δ < 1. Let E ∈ OPC∞S−11,δ be a two-sided parametric for the elliptic operator D#. Then

ED = I + EDb + EB modulo a smoothing operator. (2.9)

Now, if 1− γ < s < 1 + γ(1− δ), it follows that

Db : Hs,p −→ Hs−1+γδ,p. (2.10)

See [7], [27] and [43] for a discussion of mapping properties of pseudodifferential operators whosesymbols have a limited amount of smoothness. In particular, for s as before,

EDb : Hs,p −→ Hs+γδ,p. (2.11)

Hence, EDb is a compact operator from H1,p into itself.Going further, assume p ≤ m and observe that EB : H1,p → H1,p factors as

H1,p → Lp′ −−−B→ Lt −−−E→ H1,t → H1,p (2.12)

where 1p′ := 1

p −1m , 1

t := 1p′ + 1

r , and the first inclusion is the usual Sobolev embedding. Note thatr > m entails t > p so that the last inclusion is well-defined and compact. The case p > m is similarand requires r > p. Hence, in any event, D in (2.4) has a quasi-inverse to the left.

Similarly, DE = I +DbE +BE modulo smoothing operators, and the composition

Lp −−−E→ H1,p −−−Db

→ Hγδ,p → Lp (2.13)

is compact. Also, if p ≤ m, then

Lp −−−E→ H1,p → H1,p′ → Lp′′ −−−B→ Lt

′→ Lp (2.14)

where p′ < p, 1p′′ = 1

p′ −1m and 1

t′ = 1p′′ + 1

r . Since the first inclusion is compact, so is the entirecomposition. When p > m it follows that q′ = ∞, t′ = r and the last inclusion holds if r > p. Thisproves the first part of the theorem. With regard to (2.5), we have from (2.9)

u = E(Du)− EDbu− EBu, modC∞. (2.15)

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Now, u ∈ Hτ,p with τ + γ > 1 implies EDbu ∈ Hτ+γδ,p. If p ≤ mτ we see, as in (2.12) but starting

with Hτ,p in place of H1,p, that EBu ∈ H1,t with 1t = 1

p −τm + 1

r . If p > mτ , then EBu ∈ H1,r. Also,

Du ∈ Lq ⇒ E(Du) ∈ H1,q. Hence,

u ∈ H1,q +⋂δ<1

Hτ+δγ,p +H1,t0 ,

where t0 := t if p ≤ mτ , and t0 := r otherwise. Since r > maxm, q (by our hypotheses), this is an

improvement over the original regularity assumption on u and the procedure can be iterated sufficientlymany times to yield (2.5). 2

Next, we take up the task of studying the kernel of D in (proper) Lipschitz subdomains of M whenno boundary conditions are imposed. As we shall see momentarily, this leads to a natural concept ofHardy spaces associated with D. Before we do so, however, we make the following definition. Theoperator D is said to have the unique continuation property (abbreviated UCP henceforth) if

u ∈ H1,2(M, E), Du = 0 on M ⇒ u ≡ 0 or suppu = M. (2.16)

If this holds, we simply write D ∈ UCP.Assume that the Riemannian metric on M has H2,r coefficients for some r > m = dimM , i.e.

g ∈ H2,r(M,Hom(TM ⊗ TM,R)), (2.17)

and denote by dVol the corresponding volume element on M . Also, equip E and F with H2,r Hermitianstructures. Let D be as in (2.1)-(2.2). From now on, strengthen (2.3) to

aαβj ∈ H2,r, bαβ ∈ H1,r for some r > m. (2.18)

An important observation is that, under the current assumptions, the coefficients of D∗, the formaladjoint of D, also satisfy (2.18).

Fix V ∈ C∞(M), a scalar, positive, non-identically zero function with M\ suppV 6= ∅ and considerthe second-order differential operator

L := DD∗ + V. (2.19)

Locally, if u =∑uβeβ,

Lu =∑∑

∂jaαβjk ∂ku

βfα +∑

bαβj ∂juβfα +

∑∂j(c

αβj uβ)fα +

∑dαβuβfα. (2.20)

Then L is a formally self-adjoint, strongly elliptic operator whose coefficients satisfy

aαβjk ∈ C1+γ , bαβj , cαβj ∈ H

1,r, dαβ ∈ Lr, (2.21)

for some γ > 0, r > m.

Proposition 2.2 With the above notation and hypotheses, we have

D∗ ∈ UCP⇒ L : H1,2(M,F)→ H−1,2(M,F) is invertible. (2.22)

In a similar fashion, if

L := D∗D + V (2.23)

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then

D ∈ UCP⇒ L : H1,2(M, E)→ H−1,2(M, E) is invertible. (2.24)

Proof. Indeed, based on a deformation argument, it suffices to show that L in (2.22) has trivial kernel.To this end, if u ∈ H1,2 is so that Lu = 0, it follows that

0 = 〈Lu, u〉 =∫ ∫

M(|D∗u|2 + V |u|2) dVol . (2.25)

Hence, D∗u = 0 in M and u = 0 in suppV . Since D∗ ∈ UCP, these imply u ≡ 0 in M , as desired.Thus, (2.22) is proved. The treatment of (2.24) is similar and this finishes the proof. 2

Next we tackle the issue of invertibility for D itself under the additional hypothesis that D isformally selfadjoint.

Proposition 2.3 Let D : E → E be a first order elliptic differential operator whose coefficients satisfy(2.3) locally. Assume that D ∈ UCP and that D is formally selfadjoint. Also, fix some open set O inM . There exists a smoothing operator P on M whose integral kernel is supported in O×O and so that

D − P : H1,2(M, E) −→ L2(M, E) (2.26)

is an invertible operator.

Proof. Consider the mapping

Φ : C∞comp(O) −→[Ker (D : H1,2(M, E)→ L2(M, E))

]∗(2.27)

given by Φ(ϕ) :=∫∫O〈·, ϕc〉 dVol. Since D ∈ UCP, it follows that Φ is onto. Based on this and the fact

that, by Theorem 2.1, Ker (D : H1,2(M, E) → L2(M, E)) is finite dimensional, one can select a finitedimensional subspace V ⊆ C∞comp(O) so that

Φ : V −→[Ker (D : H1,2(M, E)→ L2(M, E))

]∗is an isomorphism. (2.28)

Pick now P : L2(M, E) → L2(M, E) to be the orthogonal projection onto V. There remains to showthat the operator (2.26) is invertible. To this end, observe first that, since P is of finite rank, (2.26) isFredholm, thanks to Theorem 2.1. Next we prove that it is also one-to-one. Indeed, this follows moreor less directly from the fact that

V ∩ ImD = 0 (2.29)

since, as (2.28) implies, KerP ∩ KerD = 0. In turn, (2.29) is a simple consequence of (2.28) andGreen’s formula (note that the selfadjointness of D is used here). This concludes the proof of theinjectivity of the operator (2.26).

Passing to adjoint, we get that

D − P : L2(M, E) −→ H−1,2(M, E) is onto. (2.30)

Now, if f ∈ L2(M, E), it follows from (2.30) that there exists u ∈ L2(M, E) so that (D − P )u = f .Thus, since P is smoothing, Du = f + Pu ∈ L2(M, E) and, further, u ∈ H1,2(M, E), by the regularityresult in Theorem 2.1. This proves that the operator (2.26) is surjective. 2

8

3 Cauchy type operators on Lipschitz domains

Let D : E → F be a first order elliptic operator as in (2.1)–(2.2) and whose coefficients satisfy (2.18).For the duration of this section, we make the standing assumption that

D ∈ UCP, D∗ ∈ UCP. (3.1)

As is well known, (3.1) always holds for Dirac type operators with reasonably smooth coefficients;we shall discuss this in greater detail in § 5. Thus, it seems reasonable to call a first order ellipticdifferential operator D satisfying (3.1) a generalized Dirac operator. It is interesting to point out inthis connection that, according to an old conjecture of L. Schwartz, the two conditions in (3.1) are infact equivalent (in the smooth context).

In order to continue, we need some notation. Let Ω ⊂M be a fix Lipschitz domain. This means that,in appropriate local coordinates, ∂Ω is locally described by graphs of (Euclidean) Lipschitz functions.Denote by dσ the surface measure on ∂Ω and by ν ∈ T ∗M the unit (outward) conormal to ∂Ω. We setΩ+ := Ω, Ω− := M \Ω, and let ·|∂Ω± be the nontangential boundary trace operators on ∂Ω±. That is,

u|∂Ω±(x) := lim

y∈γ±(x)u(y), x ∈ ∂Ω, (3.2)

where γ±(x) ⊆ Ω± are appropriate nontangential approach regions. Finally, we let N stand for thenontangential maximal operator defined for sections u defined in Ω+ or Ω− by setting

Nu(x) := sup |u(y)|; y ∈ γ±(x), x ∈ ∂Ω, (3.3)

(the choice of the sign ± depends on where u is defined). See, e.g., [39] for more details.In the sequel, we shall assume that V (introduced in connection with (2.19)) also satisfies suppV ∩

Ω = ∅. For a fixed such V , we consider:

E the Schwartz kernel of L−1, E ∈ D′(M ×M,F ⊗ F), (3.4)

E the Schwartz kernel of L−1, E ∈ D′(M ×M, E ⊗ E), (3.5)

Γ(x, y) := (D∗x ⊗ Idy)E(x, y), Γ ∈ D′(M ×M, E ⊗ F), (3.6)

Γ(x, y) := (Idx ⊗ Dy)E(x, y), Γ ∈ D′(M ×M, E ⊗ F). (3.7)

In (3.7), Du := [Duc]c, where [...]c denotes complex conjugation.Next, we define Cauchy type operators. One intriguing aspect is that there are actually two Cauchy

operators naturally associated with D: one which has a “holomorphic” kernel and one which reproduces“holomorphic” functions. As we shall see momentarily, they satisfy similar properties (such as Lp

boundedness and jump relations) to the ordinary Cauchy operator on Lipschitz curves of the complexplane as discussed in [9]. For now, if f : ∂Ω→ E is an arbitrary section set:

Cf(x) :=∫∂Ω〈Γ(x, y), iσ(D; ν(y))f(y)〉y dσ(y) for x /∈ ∂Ω,

Cf(x) :=∫∂Ω〈Γ(x, y), iσ(D; ν(y))f(y)〉y dσ(y) for x /∈ ∂Ω,

Cf(x) := p.v.∫∂Ω〈Γ(x, y), iσ(D, ν(y))f(y)〉y dσ(y) for x ∈ ∂Ω,

9

Cf(x) := p.v.∫∂Ω〈Γ(x, y), iσ(D; ν(y))f(y)〉y dσ(y) for x ∈ ∂Ω.

Here “p.v.” indicates that the integral is taken in the Cauchy principal value sense, i.e. by removinggeodesic balls. See [39] and [34] for a discussion. Also, σ(D; ξ) stands for the principal symbol of D atξ ∈ T ∗xM \ 0, x ∈M , and i =

√−1.

While the operators C, C (and C, C) are, in many respects, similar, there are some importantdifferences. Most notably, generic elements in the range of C satisfy a first order PDE in Ω, whereasthose in the range of C satisfy a second order PDE in Ω. Another way of understanding C is asfollows. Note that the integral kernel of C is iσ(Dt; ν(y))DyE(x, y). The first order differential operatorσ(Dt; ν)D can be thought of as a natural conormal derivative for L = D∗D + V and, hence, C can bethought of as a natural double layer potential operator associated with L.

The next theorem collects several basic properties of these Cauchy type operators. As such, thisextends results from the Euclidean context and for standard constant coefficient Dirac operators in [9],[24], [17] and [35].

Theorem 3.1 Let Ω be a Lipschitz subdomain of M . With the above notation and hypotheses, thefollowing are true:

(1) C and C are bounded operators on Lp(∂Ω, E) for 1 < p <∞;

(2) ‖N (Cf)‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω,E) uniformly for f ∈ Lp(∂Ω, E), and we have DCf = 0 in Ω;

(3) ‖N (Cf)‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω,E) uniformly for f ∈ Lp(∂Ω, E) and LcCf = 0 in Ω;

(4) Cf |∂Ω± = (±12I + C)f and Cf |∂Ω± = (±1

2I + C)f for any f ∈ Lp(∂Ω, E), 1 < p < ∞ (Plemeljjump formulas).

Proof. The point (1) follows directly from Theorem 2.9 in [34] (where some of the main results of [9]are extended to manifolds). To see (2), we first note the readily verified identity

(Dx ⊗ Idy)Γ(x, y) = δx(y) for x ∈ Ω. (3.8)

Hence, D(Cf) = 0 in Ω. The estimate ‖N (Cf)‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω,E) is also a direct consequence ofTheorem 2.9 in [34]. The point (3) follows similarly. As for (4), the general jump formulas from [34]give

Cf |∂Ω±(x) = ∓12 i σ(D∗; ν(x))σ(L; ν(x))−1iσ(D; ν(x))f(x) + Cf(x),

for any f ∈ Lp(∂Ω, E), 1 < p <∞ and a.e. x ∈ ∂Ω. Upon noticing that

σ(D∗; ν)σ(L; ν)−1σ(D; ν) = σ(D∗; ν)σ(D∗; ν)−1σ(D; ν)−1σ(D; ν) = Id,

the first part of (4) follows. The case of C is handled analogously. 2

In a series of theorems below we address the issue of the global interior regularity for the operatorC. This is going to be measured either on the Sobolev spaces Hs,p or the Besov spaces Bp,q

s . A goodreference for the latter class (considered on Ω or on ∂Ω) is [21]. To state our first result recall thata ∨ b := max a, b.

10

Theorem 3.2 Let Ω be a Lipschitz domain in M and let D be an operator as at the beginning of § 3.Then, for any 1 < p <∞, the operator

C : Lp(∂Ω, E) −→ Bp,p∨21/p (Ω, E) (3.9)

is well defined and bounded.

Proof. Assume that the metric tensor g is given locally by∑gjkdxj ⊗ dxk and set (gjk)j,k :=

[(gjk)j,k]−1, g := det [(gjk)j,k]. As in §2 of [34], locally we decompose

E(x, y) =1√g(y)

e0(y, x− y) + e1(x, y)

, (3.10)

where e0(y, x − y) is the Schwartz kernel of the classical pseudodifferential operator E0(D,x) ∈ØPC1+µS−2

cl , some µ > 0, whose symbol is

E0(ξ, y) := −[(∑j,k,γ

aαγj (y)aβγk (y)c ξjξk)αβ]−1

, (3.11)

and e1(x, y) is a residual term. Thanks to (2.3), in local coordinates the latter satisfies

|e1(x, y)||x− y|2 + |∇xe1(x, y)||x− y|+ |∇ye1(x, y)||x− y|

+|∇x∇ye1(x, y)| ≤ Cε|x− y|−(m−1+ε), ∀ ε > 0. (3.12)

The decomposition (3.10), at the level of kernels, naturally induces a splitting C = C0 + C1. ThatC1 maps Lp(∂Ω, E) into Bp,p∨2

1/p (Ω, E) for each 1 < p <∞ is elementary and follows from (3.12). As for

C0, the following result from [41] applies. Let q(D,x) ∈ ØPC0S−1cl have an odd principal symbol. Then

the integral operator whose kernel is the Schwartz kernel of q(D,x) maps Lp(∂Ω, E) into Bp,p∨21/p (Ω, E)

for each 1 < p <∞. 2

We continue by discussing the action of C on the Sobolev scale H1,p(∂Ω, E), 1 < p <∞.

Theorem 3.3 Let Ω be an arbitrary Lipschitz subdomain of M . Also, assume that the coefficients ofthe operator D and the metric tensors are C1+γ for some γ > 0. Then, for each 1 < p < ∞, thereexists C = C(p,Ω) > 0 so that

‖N (∇Cf)‖Lp(∂Ω) ≤ C‖f‖H1,p(∂Ω,E) (3.13)

for any f ∈ H1,p(∂Ω, E).In particular, the operator C : H1,p(∂Ω, E)→ H1,p(∂Ω, E) is well-defined and bounded.

Proof. We first present a proof which works in the case when the coefficients of D and the metrictensors are C∞. Then we indicate how this can be modified in the case of structures exhibiting alimited amount of smoothness.

To this end, with an eye on (3.13), fix f ∈ H1,p(∂Ω, E) for some 1 < p < ∞. Also, recall the localdescription of D in (2.2). Working in local coordinates and with orthonormal frames we see that theα-component of Cf is, modulo lower order terms, given by∫

Σaµβj (y)caµγk (y)nk(y)

(∂

∂yjeαβ(x, y)

)fγ(y)

√g(y) dσΣ(y) (3.14)

11

where the superscript c denotes complex conjugation. Hereafter, the summation convention is tacitlyused. Also,

∑⊆ Rm is the image of ∂Ω in local coordinates, dσΣ the area element on

∑inherited

from the Riemannian metric, n = (nk)k is the unit normal to∑

with respect to the Euclidean metric,(eαβ)α,β are the entries in E(x, y), and dVol =

√g dx, where dx is the ordinary Lebesgue measure in

Rm.Before we proceed with the main arguments, we make an important observation to the effect that,

for any 1 < p <∞ and j ∈ 1, ...,m,

∇x

(∂E

∂yj(x, y) +

∂E

∂xj(x, y)

)and ∇y

(∂E

∂yj(x, y) +

∂E

∂xj(x, y)

)

are kernels which yield bounded operators on Lp(Σ). (3.15)

Indeed, take for instance the first expression in (3.15). With [·, ·] standing for the usual commutatorbracket, this is the Schwartz kernel of ∇x[L−1, ∂/∂xi]. Now, if p(x, ξ) ∈ S−2

cl is the principal symbol ofL−1 and if p1, p2 := ∂ξjp1∂xjp2 − ∂xjp1∂ξjp2 denotes the Poisson bracket, then the principal symbolof [L−1, ∂/∂xi] ∈ OPS−2

cl is (cf., e.g., [44], Vol. 2, pp. 13)

i ξj , p(x, ξ) = i∂p

∂xj(x, ξ). (3.16)

Since ∂p∂xj

(x, ξ) ∈ S−2cl is even, Proposition 1.4 in [39] can be invoked to finish the proof of (3.15).

Turning now to the analysis of∇x(Cf), we need to consider the effect of applying ∂/∂xs, 1 ≤ s ≤ m,to (3.14). In this regard, there are two cases to study. First, when ∂/∂xs hits the lower order terms, thehighest singularity comes from terms of the form ∂xs eαβ(x, y). The contribution from these kernels canbe handled directly by the theory developed in [34]; the conclusion is that the corresponding integraloperators are bounded on Lp(Σ) for any 1 < p <∞.

Second, there is the case when ∂/∂xs hits the first part of the expression (3.14). This time, the mainsingularities are contained in terms of the form ∂xs∂yj eαβ(x, y). In the sequel, we find it convenient toreplace these by ∂ys∂yj eαβ(x, y). By (3.15), this can be arranged modulo operators bounded on Lp(Σ)which, of course, suits our purposes. Next, consider the substitute terms in the larger content of (3.14).Specifically, for each fixed s, we write

aµβj (y)caµγk (y)nk(y)∂

∂ys

(∂eαβ∂yj

(x, y)

)fγ(y)

= aµβj (y)caµγk (y)(nk(y)

∂

∂ys− ns(y)

∂

∂yk

)(∂eαβ∂yj

(x, y)

)fγ(y)

+aµβj (y)caµγk (y)ns(y)

(∂

∂yk

∂

∂yjeαβ(x, y)

)fγ(y) =: Is + IIs. (3.17)

Observe that Is contains a tangential derivative. Hence, when Is is integrated against∫

Σ dσΣ, thistangential derivative can be passed on to the other factors. The resulting terms obviously obey thetype of estimate we are after since we assume that f belongs to H1,p(Σ). There remains IIs. In orderto treat this term, we shall used the PDE satisfied by E. The point is that

IIs = (LtyE(x, y))αγns(y)fγ(y) + residual terms. (3.18)

12

The source of main singularities in the residual terms is ∇yE(x, y) and, once again by [34], the integraloperators with kernels of this type can be controlled in the desired fashion. Finally, the fact thatLtyE(x, y) = 0 for x 6= y takes care of IIs. This concludes the proof of (3.13).

The claim about the boundedness of C on H1,p(∂Ω, E) then follows from (3.13) and the point (4)in Theorem 3.1.

Turning now attention to the case when the coefficients of D and the metric tensors are only C1+γ ,we first remark that (3.15) is satisfied by the most singular part of E in the decomposition (3.10).As for the contribution from e1(x, y), matters are readily reduced (cf. also (3.34), (3.36) below) toanalyzing integrals of the type Tf(x) :=

∫∂Ω∇∇e1(x, y)[f(y)− f(x)]dσ(y), where f is a scalar-valued

function in H1,p(∂Ω) and ∇∇e1 stands for two arbitrary derivatives on e1. In this context, the keyestimate, provided by the analysis in [34], is that |∇∇e1(x, y)| ≤ Cε|x−y|−(n−1+ε) for any ε > 0. Basedon this, an elementary interpolation argument gives that T : Bp,p

θ (∂Ω) → Lp(∂Ω) for any 1 ≤ p ≤ ∞and θ > 0. The desired conclusion now follows easily since H1,p(∂Ω) → Bp,p

θ (∂Ω) for 1 < p < ∞ andθ ∈ (0, 1). 2

The next result is a natural extension of Theorem 3.2 to a larger scale of spaces.

Theorem 3.4 Retain the same hypotheses as in Theorem 3.3. Then, for any 1 < p <∞ and 0 ≤ s ≤ 1,the operator

C : Hs,p(∂Ω, E) −→ Bp,p∨2s+1/p(Ω, E) (3.19)

is well defined and bounded. Also, for 1 < p, q <∞ and 0 < s < 1

C : Bp,qs (∂Ω, E) −→ Bp,q

s+1/p(Ω, E), C : Bp,qs (∂Ω, E) −→ Bp,q

s (∂Ω, E) (3.20)

are well defined and bounded.

Proof. It only remains to treat the case s = 1; the full result then follows from this, Theorem 3.2 andcomplex interpolation (cf. [3]). The problem localizes and, working in local coordinates, it suffices toshow that ∂x` C sends H1,p(∂Ω) boundedly into Bp,p∨2

1/p (Ω) for each 1 < p < ∞, ` = 1, ...,m. Takinginto account (3.14), (3.15) (with appropriate modifications when dealing with structures which are nolyC1+γ) and the identity (3.17) this can be proved much as we did for Theorem 3.3. Now, (3.19) andreal interpolation gives (3.20) as long as 1 < p, q <∞, 0 < s < 1. 2

Our final results in this section improve on (3.20) when p = q.

Theorem 3.5 Again, retain the same hypotheses as in Theorem 3.3. Then, for each 1 ≤ p ≤ ∞ and0 < s < 1, the operator

C : Bp,ps (∂Ω, E) −→ Bp,p

s+1/p(Ω, E) (3.21)

is well defined and bounded. In particular,

C : Bp,ps (∂Ω, E)→ Bp,p

s (Ω, E) (3.22)

is well defined and bounded for each 0 < s < 1 and 1 ≤ p ≤ ∞.

Proof. Our strategy is to treat the end-point cases p = 1 and p = ∞ separately and then useinterpolation in order to cover the whole range 1 ≤ p ≤ ∞. To this effect, assume first that p = 1 andobserve that the problem at hand localizes. Also, recall from [16] that each f ∈ B1

s (∂Ω) has an atomicdecomposition of the form

13

f =∑j

λjaj , (λj)j ∈ `1, aj B1s (∂Ω)-atom, ‖f‖B1

s (∂Ω) ≈∑j

|λj |. (3.23)

In (3.23), each B1s (∂Ω)-atom a satisfies

supp a ⊆ Sr, ‖∇tana‖L∞(∂Ω) ≤ rs−m, (3.24)

where Sr is a surface ball on ∂Ω of radius r > 0. We shall now proceed to analyze the action of C onan individual B1

s (∂Ω)-atom f . Our aim is to show that there exists C(s,Ω) > 0, independent of f sothat

‖dist (·, ∂Ω)1−s|∇2Cf | ‖L1(Ω) + ‖∇Cf‖L1(Ω) + ‖Cf‖L1(Ω) ≤ C(s,Ω). (3.25)

As in [21], it follows then from (3.25) and (3.23) that (3.21) holds with p = 1, s ∈ (0, 1).Consider now the membership of dist (·, ∂Ω)1−s|∇2Cf | to L1(Ω). First, based on the identity (3.17),

locally we can write

∇Cf(x) =∫∂Ω〈k0(x− y, y),∇tanf(y)〉 dσ(y)

+∫∂Ω〈k1(x− y, y), f(y)〉 dσ(y) =: A0 +A1, (3.26)

where k0(x− y, y), k1(x− y, y) are kernels which behave similarly to Schwartz kernels of pseudodiffer-ential operators in ØPC∞S−1

cl possessing odd principal symbols.In order to continue, let us invoke a result to the effect that if K is an integral operator (mapping

sections from ∂Ω into sections over Ω) whose kernel k(x, y) satisfies

|∇ix∇jyk(x, y)| ≤ Cdist (x, y)−(m−2+τ+i+j), 0 ≤ i ≤ N, 0 ≤ j ≤ 1, (3.27)

for some positive integers N , τ , then

‖dist (·, ∂Ω)s−1+µ+τ |∇1+µKf | ‖L1(Ω) + ‖∇µKf‖L1(Ω)

+‖Kf‖L1(Ω) ≤ C(s,Ω)‖f‖(B∞s (∂Ω))∗ (3.28)

for µ = 0, 1, ..., N − 1. This is a slight generalization of a lemma in [41] which, in turn, extends someEuclidean estimates from [14]. Two key observations which allow us to use this result in the presentcontext are as follows. First, (3.27) with N = 2, τ = 1, holds for k0(x− y, y) and, second,

‖∇tanf‖(B∞s (∂Ω))∗ ≤ C(s,Ω) <∞ (3.29)

is valid uniformly for f B1s (∂Ω)-atom. These take care of the contribution coming from A0 (cf. (3.26))

in ∇2Cf in the context of (3.25). As for the contribution of A1, note that

dist (x, ∂Ω)∇x∫∂Ω〈k1(x− y, y), f(y)〉 dσ(y) (3.30)

has a kernel which exhibits a Poisson-like decay. In particular, in absolute value, (3.30) does not exceeda (fixed) multiple of Mf(x), where M is the Hardy-Littlewood maximal operator on ∂Ω and x is theprojection of x onto ∂Ω (along a suitable smooth transversal direction). Consequently, the contributionof A1 (cf. (3.26)) to

∫Ω dist (x, ∂Ω)1−s|∇2Cf(x)| dVol(x) can be controlled by

14

(∫ diam (Ω)

0t−s dt

)(∫∂Ω

(Mf)(x) dσ(x)), (3.31)

where t plays the role of dist (x, ∂Ω). Now, since B1s (∂Ω) → Lτ(s)(∂Ω) with τ(s) > 1 (in fact, 1/τ(s) =

1− s/(m−1)), the boundedness ofM on Lτ(s)(∂Ω) can be used to obtain the desired conclusion. Thisconcludes the proof of the fact that

‖dist (·, ∂Ω)1−s|∇2Cf | ‖L1(Ω) ≤ C(s,Ω). (3.32)

The remaining terms in (3.25) are easier to handle and we omit the details; this finishes the proof of(3.21) when p = 1.

We now concentrate on the case p =∞, s ∈ (0, 1) in (3.21). This time, the goal is to prove that

‖dist (·, ∂Ω)1−s|∇Cf | ‖L∞(Ω) + ‖Cf‖L∞(Ω) ≤ C(s,Ω)‖f‖B∞s (∂Ω), (3.33)

uniformly for f ∈ B∞s (∂Ω). To this end, assume that f has support in U , a small open subset of M ,and that ψαα is a frame of E over U . Fix an arbitrary x0 ∈ U and denote by x0 its projection on∂Ω along some smooth, a priori fixed transversal field. In particular, d := dist (x0, ∂Ω) ≈ dist (x0, ∂Ω),uniformly in x0. Going further, let θ ∈ C∞c (U) be a cut off function which is identically one near x0

and, if f =∑α fαψα in U , introduce f(x) := θ(x)(

∑α fα(x0))ψα(x) for x ∈ U . Clearly, f agrees with

f at x0 and ‖f‖Lip ≤ C‖f‖L∞ .With an eye toward (3.33) we write

∇Cf(x0) = ∇C(f − f)(x0) +∇Cf(x0) =: I + II. (3.34)

To treat I, for a large constant C, split the domain of integration into y ∈ ∂Ω : dist (y, x0) ≤ Cdand y ∈ ∂Ω : dist (y, x0) > Cd. In the first resulting integral majorize the kernel of ∇C by Cd−m,while in the second one by C dist (y, x0)−m. That this works, is guaranteed by the expansion (3.10)and the estimates that e0, e1 satisfy.

As for II in (3.34), the idea is to use the identity (3.17) in order to absorb the gradient inside.Once this is done, there remains to estimate the action of an integral operator T whose kernel k(x, y)satisfies |k(x, y)| ≤ Cdist (x, y)−(m−1), x ∈ Ω, y ∈ ∂Ω, on L∞(∂Ω). A crude estimate gives

|Tf(x)| ≤ C|log (dist (x, ∂Ω))| ‖f‖L∞(∂Ω), ∀x ∈ Ω, (3.35)

and this suffices, in the context of (3.33).There remains to treat the second term in (3.33). Making use again of a decomposition similar to

(3.34), it is enough to consider ‖Cφ‖L∞(Ω) and seek a bound of the order of ‖φ‖Lip(Ω). The idea is tointegrate by parts in Cφ and write this as φ plus a Newtonian potential. That is,

Cφ(x) =∫∂Ω〈Γ(x, y), iσ(D; ν(y))f(y)〉y dσ(y)

= φ(x)−∫ ∫

Ω〈Γ(x, y), Dφ(y)〉 dVol(y). (3.36)

In this latter form, the desired estimate is obviously satisfied thanks to (3.10)–(3.12). This finishes theproof of (3.21) corresponding to p =∞, s ∈ (0, 1). 2

15

Theorem 3.6 Retain the same hypotheses as in Theorem 3.3. Then, for each 1 < p < ∞ and0 < s < 1, the operator

C : Bp,ps (∂Ω, E) −→ Lps+1/p(Ω, E) (3.37)

is well defined and bounded.

Proof. If 1 < p ≤ 2, this is a direct consequence of (3.21) and classical embedding results (cf. [3]).The full range (s, p) ∈ (0, 1)× (1,∞) then follows by interpolating (via the complex method) betweenthis region and the one described by 1 < p <∞ and 0 < s < 1− 1/p. The crux of the matter is that,in this latter case, Ker L∩Bp,p

s+1/p(Ω) ⊆ Lps+1/p(Ω); see for this Lemma 4.5 and Proposition 4.4 in [41].The proof is finished. 2

In closing, we would like to point out that all our results about C in this section continue to holdunder the weaker assumptions that D ∈ UCP has an injective symbol.

4 Hardy spaces and generalized Dirac operators

In this section we study Hardy spaces associated with generalized Dirac operators. Topics include:(pointwise) boundary behavior theory, an Lp maximum principle, (a sharp form of) Cauchy’s vanishingformula, decomposition theorems and (global) regularity issues.

Through the section, Ω will denote an arbitrary Lipschitz domain in M . In relation to D, an ellipticfirst order differential operator as in (2.1)–(2.2) and so that (2.21), (3.1) are satisfied, we introducesome Hardy type spaces. Specifically, for 0 < p ≤ ∞, we set

Hp(Ω, D) := u ∈ Lp(Ω, E); Du = 0 in Ω and Nu ∈ Lp(∂Ω) (4.1)

and equip it with the “norm” ‖u‖Hp(Ω,D) := ‖Nu‖Lp(∂Ω). As such, Hp(Ω, D) becomes a Banach spaceif 1 < p ≤ ∞.

Theorem 4.1 Let D and Ω be as above. Then, for each 1 < p <∞, there holds

Hp(Ω, D) → Bp,p∨21/p (Ω). (4.2)

Also, for any u ∈ Hp(Ω, E) there exists u|∂Ω in the nontangential pointwise sense and

‖u|∂Ω‖Lp(∂Ω,E) ≈ ‖Nu‖Lp(∂Ω), (4.3)

uniformly for u ∈ Hp(Ω, D) (the Lp-version of the maximum principle). In particular, the boundaryversion Hardy spaces

H(∂Ω, D) := u|∂Ω; u ∈ Hp(Ω, D) (4.4)

are well defined, closed subspaces of Lp(∂Ω, E) for each 1 < p <∞.

Proof. The crux of the matter is Cauchy’s reproducing formula

u = C(u|∂Ω) in Ω, (4.5)

valid for any u ∈ Hp(Ω, D) which is also continuous up to and including the boundary of Ω. Oncethis is available, the extra hypothesis of continuity can be eliminated via a routine limiting argument.

16

Then, the existence of u|∂Ω follows from (4.5) and (4) in Theorem 3.1. Also, (4.2) is a consequence of(4.5) and Theorem 3.2.

Thus, there remains (4.5) which is going to be a direct consequence of a Pompeiu type representationformula to the effect that

u(x) = C(u|∂Ω) +∫ ∫

Ω〈Γ(x, y), (Du)(y)〉y dVol(y), x ∈ Ω, (4.6)

for any u ∈ C1(Ω, E). Indeed, (4.6) follows based on Idx ⊗ DyΓ(x, y) = δx(y) and∫ ∫Ω〈Du, v〉 dVol =

∫ ∫Ω〈u,Dtv〉 dVol−

∫∂Ω〈iσ(D; ν)u, v〉 dσ. (4.7)

In turn, (4.7) follows from the usual integration by parts formula applied to a sequence of smoothapproximating domains and the boundary behavior theory which we established for functions belongingto Hardy spaces. Going further, (4.6), the fact that ∃ Cf |∂Ω plus a limiting argument show, much asin the Euclidean setting (cf., e.g., [35]) that u|∂Ω exists a.e. on ∂Ω and u = C(u|∂Ω) in Ω for anyu ∈ Hp(Ω, D). This finishes the proof. 2

We now introduce Hardy spaces whose elements are more regular functions. Specifically, set

Hp1(Ω, D) := u ∈ Hp(Ω, D); N (∇u) ∈ Lp(∂Ω). (4.8)

Theorem 4.2 Let D be as in Theorem 3.3. Then, for any 1 < p <∞,

u ∈ Hp(Ω, D) has u|∂Ω ∈ H1,p(∂Ω, E)⇐⇒ u ∈ Hp1(Ω, D) (4.9)

and

‖N (∇u)‖Lp(∂Ω) ≈ ‖u|∂Ω‖H1,p(∂Ω,E). (4.10)

Moreover, for any 1 < p <∞,

Hp1(Ω, D) → Bp,p∨21+1/p(Ω). (4.11)

Proof. The left-to-right implication in (4.9) together with the inequality “” in (4.10) follow directlyfrom (4.5) and Theorem 3.3. The opposite implication in (4.9) as well as the opposite inequality in(4.10) are general phenomena. Also, (4.11) is a consequence of (4.5) and Theorem 3.4. 2

We next present a sharp form of Cauchy’s vanishing formula. To state this result, we let [...] standfor the annihilator of [...] under the pairing (u, v) =

∫∂Ω〈u, vc〉 dσ.

Theorem 4.3 Let D, Ω be as at the beginning of §4 and fix 1 < p, q < ∞, two conjugate exponents.Then the mapping

Φ : Hp(∂Ω, D∗)→ [Hq(∂Ω, D)] , Φ(u) := iσ(D∗; ν)u, (4.12)

is an isomorphism.

Proof. That Φ is well-defined is immediate from Theorem 4.1 and an integration by parts. Also, notethat Φ is one-to-one because of the ellipticity of D. There remains the fact that Φ is also onto.

We approach this problem as follows. The first observation is that there is no loss of generality inassuming that E = F and D = D∗, the adjoint of D. This is seen by “symmetrizing” the operator D,i.e. by working with

17

ID :=

(0 DD∗ 0

), on F ⊕ E , (4.13)

since the original claim can be ultimately recovered from the corresponding one for ID. Next, weproduce an invertible perturbation D of D. More specifically, there exists a symmetric, smoothingoperator P ∈ L−∞ supported away from Ω so that D := D−P is invertible, say, from H1,2(M, E) ontoL2(M, E). That this is possible follows from Proposition 2.3.

Denote by Θ(x, y) the Schwartz kernel of D−1 and introduce the corresponding Cauchy integraloperators

CDf(x) :=∫∂Ω〈Θ(x, y), iσ(D; ν)(y)f(y)〉y dσ(y), x ∈ Ω, (4.14)

CDf(x) := p.v.∫∂Ω〈Θ(x, y), iσ(D; ν)(y)f(y)〉y dσ(y), x ∈ ∂Ω. (4.15)

The point is that CD satisfies properties similar to those enjoyed by the Cauchy operators in Theorem3.1. As a consequence, we have

Im(

12I + CD; Lp(∂Ω, E)

)= Hp(∂Ω, D) = Ker

(−1

2I + CD; Lp(∂Ω, E)), (4.16)

for any 1 < p < ∞. One final property we want to single out is that the adjoint of CD acting onLp(∂Ω, E) is

(CD)∗ = −iσ(D∗; ν)CD∗ [iσ(D∗; ν)]−1 on Lq(∂Ω, E), 1/p+ 1/q = 1. (4.17)

Here CD∗ is the singular integral operator constructed analogously to CD but with D∗ replacing D. Inparticular, (4.17) remains valid with D replaced by D∗. Thus, from (4.17) and the second equality in(4.16), we have

Ker(

12I + (CD)∗;Lp(∂Ω, E)

)= Ker

(12I − iσ(D∗; ν)CD∗ [iσ(D∗; ν)]−1;Lp(∂Ω, E)

)= iσ(D∗; ν)Ker

(12I − CD∗ ;L

p(∂Ω, E))

= iσ(D∗; ν)Hp(∂Ω, D∗). (4.18)

Returning to the study of Φ in (4.12), we may write

Im Φ = iσ(D∗; ν)Hp(∂Ω, D∗) = Ker((1

2I + CD)∗;Lp(∂Ω, E))

=[Im

(12I + CD;Lq(∂Ω, E)

)]= [Hq(∂Ω, E)] . (4.19)

Hence Φ in (4.12) is onto also. 2

The Cauchy reproducing formula (4.5) involves the operator C which does not map all Lp(∂Ω, E)into Hp(Ω, D), as C does. An important case when matters can be arranged so that C = C occurs for

E = F , D2 = (Dt)2, D2 : H1,2(M, E)→ H−1,2(M, E) invertible. (4.20)

Note that (4.20) is automatically satisfied when D = Dt and the kernel Ker (D : H1,2(M, E) →L2(M, E)) is trivial.

18

The idea is that, granted (4.20), L(= D2) is invertible from H1,2(M, E) into H−1,2(M, E) and wemay base the construction of the Cauchy operators C, C on the kernels

Γ := (Dx ⊗ Idy)E(x, y) and Γ := (Idx ⊗Dty)E(x, y), (4.21)

where E is the Schwartz kernel of L−1. That these kernels (and, hence, the associated) operatorscoincide, is a consequence of the commutation identity DL−1 = L−1D read at the level of Schwartzkernels. In this case, C2 = 1

4I and, if we set

Hp±(∂Ω, D) := u|∂Ω; u ∈ Hp(Ω±, D) (4.22)

then the Plemelj-Calderon type decomposition

Lp(∂Ω, E) = Hp+(∂Ω, E)⊕Hp−(∂Ω, D) (4.23)

is valid for any Lipschitz domain Ω and any 1 < p <∞. This follows more or less directly from (3)-(4)in Theorem 3.1.

To describe what happens with the decomposition (4.23) in the general case, we first need a def-inition. Recall from [23] that (A,B) is called a Fredholm pair for the Banach space X if A,B areclosed subspaces of X so that dim (A ∩ B) <∞ and dim (X/(A+B)) <∞. In this case, one definesIndex (A,B) := dim (A ∩B)− dim (X/(A+B)).

Making use of Theorem 2.1 and Theorem 3.1, the following result has been announced in [38]. Tostate it, recall the smoothness index r from (2.18).

Theorem 4.4 Let E ,F → M , D : E → F be as in the previous theorem and consider an arbitraryLipschitz subdomain Ω of M .

Then the Hardy spaces(Hp−(∂Ω, D),Hp+(∂Ω, D)

)are a Fredholm pair for Lp(∂Ω, E) and

Index(Hp−(∂Ω, D),Hp+(∂Ω, D)

)= Index

(D : H1,p(M, E)→ Lp(M,F)

)(4.24)

for each 1 < p <∞.

The point is that in the general case the analogue of (4.23) is valid only modulo finite dimensionallinear spaces. A version of Theorem 4.4 corresponding to p = 2 in the smooth case has been firstconjectured by B. Bojarski in mid 1970’s (cf. [5]) and has been subsequently proved by B. Booß-Bavnbek and K. Wojciechowski in mid 1980’s (see [6]).

Proof. Consider two positive, scalar-valued functions V± ∈ C∞(M), non-identically zero and so thatΩ± ∩ suppV± = ∅. Also, set L± := DD∗ + V±. As before, it follows that L± : H1,2(M,F) →H−1,2(M,F) are invertible and we denote by E± ∈ D′(M ×M,F ⊗ F) the Schwartz kernels of L−1

± .Also, introduce Γ±(x, y) := (D∗x⊗Idy)E±(x, y), Γ± ∈ D′(M×M, E⊗F) and the Cauchy type operatorsacting on arbitrary sections f : ∂Ω→ E by

C±f(x) :=∫∂Ω〈Γ±(x, y), iσ(D; ν(y))f(y)〉y dσ(y), for x /∈ ∂Ω, (4.25)

and

C±f(x) := p.v.∫∂Ω〈Γ±(x, y), iσ(D)(y, ν(y))f(y)〉y dσ(y), for x ∈ ∂Ω. (4.26)

There are several properties of these operators which are going to be of importance for us in thesequel. First, C± : Lp(∂Ω, E)→ Hp(Ω±, D) are well defined and bounded for each 1 < p <∞. Second,for all f ∈ Lp(∂Ω, E),

19

C+f |∂Ω± = (±12I + C+)f, C−f |∂Ω± = (±1

2I + C−)f a.e. on ∂Ω, (4.27)

and, third,

C± are bounded on Lp(∂Ω, E), 1 < p <∞. (4.28)

One immediate conclusion is that

Im(±1

2I + C±;Lp(∂Ω, E))⊆ Hp±(∂Ω, D), 1 < p <∞. (4.29)

Going further, a key observation is that the main singularity in Γ±(x, y), as described in (3.10), isindependent of V±. In particular, if m := dimM , then Γ+(x, y) − Γ−(x, y) = O(|x − y|−(m−2+ε))for any ε > 0. Hence, the integral operator K := C+ − C− is compact from Lp(∂Ω, E) into itself,1 < p <∞. Now, since I +K = (1

2I + C+)− (−12I + C−), it follows from (4.29) that

Im (I +K;Lp(∂Ω, E)) ⊆ Hp+(∂Ω, D) +Hp−(∂Ω, D). (4.30)

Consequently, since I+K is Fredholm, it follows that Hp+(∂Ω, D) +Hp−(∂Ω, D) is closed and has finitecodimension in Lp(∂Ω, E).

Now, if f ∈ Hp−(∂Ω, D) ∩ Hp+(∂Ω, D), then there exist (unique, by Theorem 4.1) functions u± ∈Hp(Ω±, D) so that u+|∂Ω = f = u−|∂Ω. Define uf ∈ Lp(M, E) by setting uf := u± in Ω± so that uf ∈Ker

(D : H1,p(M, E)→ Lp(M,F)

). Consequently, the assignment f 7→ uf is linear, well-defined and, by

Theorem 4.1, one-to-one fromHp−(∂Ω, D)∩Hp+(∂Ω, D) into the space Ker(D : H1,p(M, E)→ Lp(M,F)

).

Invoking Theorem 2.1, it is clear that this is also onto (note that (2.21) implies (2.3) with γ > 1 andr =∞). Hence,

dim(Hp−(∂Ω, D) ∩Hp+(∂Ω, D)

)= dim Ker

(D : H1,p(M, E)→ Lp(M,F)

)<∞. (4.31)

At this point, the first part in Theorem 4.4 follows. There remains (4.24) to which we now turn. Acombination of Theorem 2.1, (4.31) and the fact that the mapping (4.12) is an isomorphism, allows usto write

dim Ker (D∗;Lq(M,F)) = dim Ker (D∗;H1,q(M,F))= dim

(Hq−(∂Ω, D∗) ∩Hq+(∂Ω, D∗)

)= dim

(Hp−(∂Ω, D) ∩Hp+(∂Ω, D)

)= dim

(Hp−(∂Ω, D) +Hp+(∂Ω, D)

) (4.32)

where 1/p + 1/q = 1. Now, the index formula (4.24) follows from this, the fact that Hp−(∂Ω, D) +Hp+(∂Ω, D) is closed in Lp(∂Ω, E) and (4.31). This completes the proof of the theorem. 2

A natural, direct consequence of Theorem 4.4 is the following.

Corollary 4.5 With the previous notation and hypotheses, the transmission problem

(TBV P )

u ∈ Hp(Ω+, D), v ∈ Hp(Ω−, D),

u|∂Ω − v|∂Ω = f ∈ Lp(∂Ω, E),(4.33)

is Fredholm solvable for any 1 < p < ∞ and its index is the same as the index of the Fredholm pair(Hp−(∂Ω, D),Hp+(∂Ω, D)

)in Lp(∂Ω, E).

20

Furthermore, (TBV P ) is uniquely solvable for any boundary data in Lp(∂Ω, E), 1 < p <∞, if andonly if

D : H1,p(M, E)→ Lp(M,F) is invertible. (4.34)

In particular, if (TBV P ) is uniquely solvable for some Lipschitz domain Ω then, in fact, (TBV P ) isuniquely solvable for any Lipschitz domain.

Next, we study the Poisson problem for the operator D (with full-Dirichlet boundary conditions)in arbitrary Lipschitz domains.

Theorem 4.6 Assume that E ,F → M , D : E → F are as before and let Ω be an arbitrary Lipschitzsubdomain of M . Then, for any 1 < p < r, the Poisson problem for D

(Poisson)

u ∈ H1,p(Ω, E) +Hp(Ω, D),

Du = w ∈ Lp(Ω,F),

u|∂Ω = f ∈ Lp(∂Ω, E),

(4.35)

is solvable if and only if the compatibility condition∫ ∫Ω〈w, v〉 dVol +

∫∂Ω〈iσ(D; ν)f, v〉 dσ = 0, (4.36)

is satisfied for each v ∈ Hq(Ω, Dt), 1/p + 1/q = 1. In this case, the solution is unique and naturalestimates hold.

Proof. The necessity of (4.36) in order for (4.35) to be solvable follows easily from (4.7). Conversely,assuming that this compatibility condition holds, we now proceed to show that (4.35) admits a solution.First, so we claim, there exists ω0 ∈ H1,p(Ω, E) so that Dω0 = w. Indeed, this can be arranged bytaking

ω0(x) :=∫ ∫

Ω〈Γ(x, y), w(y)〉 dVol(y), x ∈ Ω, (4.37)

where Γ(x, y) is as in (3.6). See [34] for a more detailed discussion in similar circumstances.We seek u, solution of (4.35), in the form u := ω0 + ω1 where ω0 is as before and ω1 ∈ Hp(Ω, D) is

yet to be selected. In this scenario, matters are reduced to checking that f = ω0|∂Ω + ω1|∂Ω which isequivalent to asking

f − ω1|∂Ω ∈ Hp(∂Ω, D). (4.38)

To see that there is a choice of ω1 in Hp(Ω, D) which makes (4.38) work, by Theorem 4.3, we need toshow that for any v ∈ Hq(Ω, Dt) ∫

∂Ω〈iσ(D; ν)(f − ω1), v〉 dσ = 0. (4.39)

However, this is an immediate consequence of (4.36) and the integration by parts formula (4.7).Thus, a solution exists and this completes the proof of the first part of the theorem. Finally,

uniqueness in (4.35) is more or less a direct corollary of Theorem 4.1. 2

For later use, we single out the following regularity result.

21

Proposition 4.7 Assume the same hypotheses as in Theorem 3.3 and fix 0 < s < 1, 1 ≤ p ≤ ∞arbitrary. Then, for each u ∈ Hp(Ω, D) ,

u|∂Ω ∈ Bp,ps (∂Ω, E)⇐⇒ u ∈ Bp,p

s+1/p(Ω, E) (4.40)

Proof. Of course, the only new thing left to prove is the left-to-right implication in (4.40). However,this follows by invoking Cauchy’s reproducing formula (4.5) together with (3.21). 2

An important consequence of Theorem 4.6 and Proposition 4.7 is a Hodge type decomposition forD which we now present (a closely related version but for Dirac operators is discussed in the nextsection). To this effect, recall that H1,p

0 (Ω, E) is the space of sections in H1,p(Ω, E) with vanishingboundary trace.

Theorem 4.8 Let E ,F → M , D : E → F be as in Theorem 3.3 and let Ω be an arbitrary Lipschitzsubdomain of M . Then there exists ε = ε(Ω, D) > 0 so that, for any 2− ε < p < 2 + ε,

Lp(Ω,F) = Ker (D∗; Lp(Ω,F))⊕DH1,p0 (Ω, E), (4.41)

where the direct sum is topological (when p = 2 it is in fact orthogonal).

Proof. We begin by observing that

Lp(M,F) = Ker (D∗; Lp(M,F))⊕DH1,p(M, E) (4.42)

holds for any 1 < p < ∞. Indeed, by Theorem 2.1, DH1,p(M, E) is a closed subspace of Lp(M,F)whose (finite) codimension is the same as dim [Ker (D∗; Lp(M,F))]. Note that the space aboveis actually independent of p ∈ (1,∞) and, in fact, included in ∩p∈(1,∞)H

1,p(M,F). In particular,Ker (D∗; Lp(M,F)) ∩DH1,p(M, E) = 0. This proves (4.42).

Going further, for an arbitrary u ∈ Lp(Ω,F) let u ∈ Lp(M,F) denote the extension by zero outsideΩ. Then, (4.42) gives u = w + Dω for some w ∈ Ker (D∗; Lp(M,F)) and ω ∈ H1,p(M, E). Considernext the case when 1 < p ≤ 2 and let

v = ω0 + ω1, ω0 ∈ H1,p(Ω, E), ω1 ∈ Hp(Ω, D), (4.43)

be a solution of Dv = Dω in Ω,

v|∂Ω = 0 on ∂Ω.(4.44)

That this is possible, is ensured by Theorem 4.6. Note that v|∂Ω = 0 forces ω1|∂Ω = −ω0|∂Ω ∈Bp,p

1−1/p(∂Ω, E). Hence, by Proposition 4.7 and the fact that 1 < p ≤ 2, ω1 ∈ Bp,p1 (Ω, E) → H1,p(Ω, E)

so that, ultimately, v ∈ H1,p(Ω, E). Thus, v ∈ H1,p0 (Ω, E) and we have the decomposition

u = w|Ω +Dv in Ω (4.45)

where w|Ω ∈ Ker (D∗, Lp(Ω,F)) and v ∈ H1,p0 (Ω, E), as desired.

Next we turn to the uniqueness part for the decomposition (4.45). To this end, assume that2 − ε < p < ∞ and let v ∈ H1,p

0 (Ω, E) be so that Dv ∈ Ker (D∗, Lp(Ω,F)). It follows that D∗Dv = 0in Ω and, if ε > 0 is small enough, from (a version of) the uniqueness for the Lp-Dirichlet problem forD∗D in Lipschitz domains from [34] we get that v ≡ 0 in Ω. Thus, at this stage, we have existence

22

and uniqueness as long as 2 − ε < p ≤ 2. Finally, the range 2 < p < 2 + ε follows from what we haveproved so far and duality. We omit the details. 2

Let M , D and Ω be as before. In analogy with the classical setting, we define the Szego projection

PD : L2(∂Ω, E) −→ H2(∂Ω, D) → L2(∂Ω, E) (4.46)

as the orthogonal projection onto the closed subspace H2(∂Ω, D) of L2(∂Ω, E). The issue we want tostudy next is whether (4.46) extends to

PD : Lp(∂Ω, E) −→ Hp(∂Ω, D) (4.47)

in a continuous and onto fashion for other values of p ∈ (1,∞). That, in the classical setting when Dis the Cauchy-Riemann operator and Ω is the unit disk in the complex plane, this holds for 1 < p <∞is a famous theorem of M. Riesz which, in fact, is equivalent to the Lp-boundedness of the Hilberttransform on the unit circle. See pp. 151–152 in [18] for more details. Our main result in this respectis as follows.

Theorem 4.9 Let D : E → F be as at the beginning of §4 and let Ω be an arbitrary Lipschitz domainin M . Then there exists ε = ε(D,Ω) > 0 so that PD maps Lp(∂Ω, E) onto Hp(∂Ω, D) for each2− ε < p < 2 + ε.

In particular, with p as above, we have the decomposition

Lp(∂Ω, E) = Hp(∂Ω, D)⊕ iσ(D∗; ν)Hp(∂Ω, D∗), (4.48)

where the direct sum is topological (when p = 2 this is an orthogonal decomposition).

Proof. Eventually symmetrizing the operator D, there is no loss of generality in assuming that E = Fand D = D∗. Furthermore, by perturbing D away from Ω as in Proposition 2.3, we can also assumethat D : H1,2(M, E) → L2(M, E) is invertible. In particular, this allows us to reintroduce the Cauchyoperators (4.14)-(4.15).

Thanks to (4.16) and (4.18), when p = 2 the decomposition (4.48) is simply the statement L2 =(KerT ) ⊕ (ImT ∗) for the (closed-range) operator T := 1

2I + (CD)∗. However, the case when p 6= 2is considerably more subtle and requires a better understanding of the orthogonal projection operator(4.46). To this end, note that the identities

P (12I + CD) = (1

2I + CD), (−12I + CD)P = 0, in L2(∂Ω, E), (4.49)

can be easily justified in light of (4.16) and (4.18). Taking the adjoint of the second equality andsubtracting it from the first yields P (I + CD − (CD)∗) = (1

2I + CD). Next, introduce the boundedoperator

A = AD := CD − (CD)∗ : Lp(∂Ω, E) −→ Lp(∂Ω, E), 1 < p <∞. (4.50)

Then, I + A is accretive and, hence, invertible on L2(∂Ω, E). From general stability results it followsthen that there exists ε > 0 so that

A : Lp(∂Ω, E)→ Lp(∂Ω, E) is invertible for 2− ε < p < 2 + ε. (4.51)

Consequently, we have the Kerzman-Stein type formula

PD = (12I + CD) (I +AD)−1 (4.52)

23

valid in L2(∂Ω, E). This, (4.51) and Theorem 3.1 then show that PD extends as a bounded operatorin Lp(∂Ω, E) for 2− ε < p < 2 + ε and that

PD : Lp(∂Ω, E) −→ Hp(∂Ω, D) is onto for 2− ε < p < 2 + ε. (4.53)

Consider next

Q = QD : L2(∂Ω, E) −→ iσ(D∗; ν)H2(∂Ω, D∗) → L2(∂Ω, E), (4.54)

the complementary orthogonal projection of (4.46). Then, by (4.52), (4.50) and (4.17)

Q = I − P = I − (12I + CD)(I +AD)−1 = (1

2I − (CD)∗)(I +AD)−1

= iσ(D∗; ν)(

12I + CD∗

)[iσ(D∗; ν)]−1(I +AD)−1. (4.55)

This proves that QD extends as a bounded operator on Lp(∂Ω) for 2− ε < p < 2 + ε and that

QD : Lp(∂Ω, E) −→ iσ(D∗; ν)Hp(∂Ω, D∗) is onto for 2− ε < p < 2 + ε. (4.56)

From (4.53) and (4.56), the conclusion regarding (4.48) follows. This finishes the proof of the theorem.2

5 Dirac operators on manifolds with rough boundaries

In this section we explain how Dirac type operators fit in the general framework of § 2 − 4. First, wediscuss some basic facts and terminology relevant for our purpose.

Assume that M is a Riemannian manifold with a C1,1 metric and that the vector bundles E ,F areequipped with C1,1 Hermitian structures. Also, let D be a first order elliptic differential operator as in(2.1)–(2.2) and whose coefficients satisfy

aαβj ∈ C1,1, bαβ ∈ C0,1. (5.1)

We say that D is a Dirac operator if

D∗D has a (real) scalar principal symbol. (5.2)

The reader should be aware that this definition is rather general and is not universally employed in theliterature. It is used in, e.g., [44].

Clearly, if D is Dirac then so is D∗. Now, (5.2) implies that for ξ ∈ T ∗xM\0, x ∈M ,

σ(D∗D; ξ) = g(x, ξ) Idx : Ex → Ex (5.3)

where

g(x, ξ) =∑j,k

gjk(x)ξjξk, ξ ∈ T ∗xM, x ∈M, (5.4)

is a quadratic form. Invoking the ellipticity, positivity and self-adjointness of D∗D we see that (5.4) isstrictly positive definite on T ∗xM . The calculation in (5.3) is independent of the choice of the metrictensor in M (but depends on the Hermitian structures in E and F). Indeed,

g(x, ξ) = |σ(D; ξ)e|2Fx , for any e ∈ Ex with |e|Ex = 1.

24

Thus, as is customary, we may assume that M is equipped with the metric tensor induced by g in(5.4). We make this a standing hypothesis in this section.

In this case, (5.3) reads

σ(D∗D; ξ) = |ξ|2x Idx : Ex → Ex, ξ ∈ T ∗xM\0. (5.5)

Call a Dirac operator D symmetric, if E = F and D = D∗. For D a symmetric Dirac operator,abbreviate

ϑx(ξ) := −iσ(D; ξ), x ∈M, ξ ∈ T ∗xM\0. (5.6)

By convention, we set ϑx(0) = 0 so that ϑ : T ∗M → Hom (E , E) is linear and satisfies

ϑx(ξ)∗ = −ϑx(ζ), ϑx(ξ)2 = −|ξ|2 Idx. (5.7)

The university of the Clifford algebra (cf., e.g., (1.4), p. 4 in [6]) gives that ϑ extends as an algebrahomomorphism

ϑ : C`(M)→ Hom (E , E), (5.8)

where C`(M) := C`(T ∗M, g) is the Clifford (algebra) bundle associated to T ∗M endowed with thequadratic from g. Hence, ϑ is a representation of C`(T ∗M, g) and E becomes a C`(M)-module (to theleft).

On fibers, if a ∈ C`(T ∗xM, g), e ∈ Ex, then the Clifford product a · e ∈ Ex is defined by

a · e := ϑx(a)e ∈ Ex.

Thus, there is a structural correspondence betweenSymmetric Dirac operators

in E →M

and

(left) C`(M)-modulistructures in E →M

. (5.9)

In fact, this correspondence also operates right-to-left. Indeed, if E → M is some C`(M)-module,denote by

m : C0(M,T ∗M ⊗ E)→ C0(M, E) (5.10)

the Clifford multiplication action m(ξ ⊗ u) := ξ · u. Next, equip E with a connection ∇ : C1(M, E)→C0(M,T ∗M ⊗ E) and define D := −im ∇. Hence, D is a first order differential operator and

σ(D; ξ)u = m(ξ ⊗ u) = ξ · u, x ∈M, ξ ∈ T ∗xM\0, (5.11)

since σ(∇; ξ) = iξ⊗Idx. It is possible now to construct a Hermitian metric on E so that 〈m(ξ⊗e), f〉x =〈e,m(ξ ⊗ f)〉x, ∀ ξ ∈ T ∗xM and e, f ∈ Ex. See Lemma 2.2 in [6]. This clearly entails (5.2) for D.

Finally, if ∇ is a so-called Clifford connection it follows that D is also symmetric. See [44, Vol. IIProp. 1.1, p. 246] for a proof of this latter claim, and Proposition 2.5 in [6], p. 16 for a proof of theexistence of such connections. The reader may also consult [6], [4], [35] for a more detailed exposition.

Theorem 5.1 The main results in § 4 (under appropriate smoothness assumptions) are valid for anyDirac operator D on E.

25

Proof. The only thing left for us to check is (3.1). However, this follows from the deep results ofAronszajn [1] (cf. Theorem on p. 235-236 and Remark 3 on p. 248) and Cordes [10] for the secondorder, elliptic differential operators D∗D, DD∗. For this to work, it is of crucial importance that theprincipal symbol is (real and) scalar, a condition automatically ensured in the class of Dirac operators.We also note that, if u ∈ H1,2

loc and D∗Du = 0 (or DD∗u = 0) then u belongs to⋂

1<q<∞H2,q

loc , by

Proposition 2.1 in [34]. 2

Several variants are possible. For example, a related result is the following.

Theorem 5.2 Let E →M be as before and consider a first order, elliptic differential operator D on Ewhose coefficients satisfy appropriate smoothness assumptions. In addition, we assume

D2 has (real) scalar principal symbol. (5.12)

Then the results of § 4 work for D.

Proof. We only need to observe that both D and D∗ have the unique continuation property. In turn,this is seen in the same way as in the previous theorem. 2

In the sequel, we shall call operators satisfying (2.1) and (5.12) of Dirac type.In the second part of this section we first reconsider the problem of the Lp boundedness of the Szego

projections for Dirac type operators in domains which are only slightly smoother than Lipschitz. Thiscontext together with an extra mild assumption on the underlying domain allow for a strengthenedversion of Theorem 4.9 which we now present.

Theorem 5.3 Let D : E → F , Ω be as in Theorem 4.9. Assume, in addition, that the unit conormalν belongs to vmo (∂Ω), the (local version of the) space of functions of vanishing mean oscillations on∂Ω, and that D is of Dirac type.

Then the Szego projection PD extends as a bounded operator from Lp(∂Ω, E) onto Hp(∂Ω, D) foreach 1 < p <∞. As a consequence, for each 1 < p <∞, we have the decomposition

Lp(∂Ω, E) = Hp(∂Ω, D)⊕ iσ(D∗; ν)Hp(∂Ω, D∗), (5.13)

where the direct sum is topological.

Proof. We can, and will, retain the supplementary assumptions made at the beginning of the proofof Theorem 4.9. Specifically, it suffices to treat the case when E = F , D = D∗ and D : H1,2(M, E)→L2(M, E) is invertible. Among other things, these hypotheses allow us to reintroduce the Cauchyoperators (4.14)–(4.15).

Going further, let us suppose for a moment that the operator (4.50) is, in fact, compact on Lp(∂Ω, E)for each 1 < p <∞. Since A∗ = −A it easily follows that I+A is invertible on Lp(∂Ω, E) for 1 < p <∞.With this at hand and proceeding as in the proof of Theorem 4.9, we arrive at the conclusion that(4.53) and (4.56) actually hold for each 1 < p <∞. Thus, (5.13) is valid for the full range 1 < p <∞.

There remains to show that

ν ∈ vmo (∂Ω) =⇒ A is compact in Lp(∂Ω, E), ∀ p ∈ (1,∞). (5.14)

To see this, note that since A =[CDiσ(D; ν) + iσ(D; ν)CD

][iσ(D; ν)]−1, it suffices to show that the

mapping sending f into

26

∫∂Ω〈Θ(x, y), σ(D; ν)(y)f(y)〉 dσ(y) +

∫∂Ω〈σ(D; ν)(x)Θ(x, y), f(y)〉 dσ(y)

=∫∂Ω〈[σ(D; ν)t(y) + σ(D; ν)(x)]Θ(x, y), f(y)〉 dσ(y)

=∫∂Ω〈[σ(D; ν)(x)− σ(Dt; ν)(y)]Θ(x, y), f(y)〉 dσ(y) (5.15)

is compact on Lp(∂Ω, E) for each 1 < p <∞ as long as ν ∈ vmo (∂Ω). This requires a finer analysis ofthe nature of the kernel

[σ(D; ν)(x)− σ(Dt; ν)(y)]Θ(x, y) (5.16)

which we now perform.Our strategy is to work in local coordinates and to decompose this kernel into pieces of two types.

The first type will give rise to commutators between Calderon-Zygmund type singular integral operatorson ∂Ω and operators of pointwise multiplication by vmo (∂Ω) functions. These are known to be compactoperators on Lp(∂Ω) for any 1 < p <∞; cf. the discussion in [39]. The second type will be essentiallythe kernel of a “double layer” singular integral operator. In this case, we relay on a variant of someresults in [19] to conclude that, again, this is a compact operator on Lp(∂Ω, E) for each 1 < p <∞.

The specifics are as follows. Let ψαα be an orthonormal frame for E in some small neighborhoodU of an arbitrary, fixed boundary point. Then

D(uγψγ) =(aµγj ∂ju

γ)ψµ + lower order terms, (5.17)

σ(D, ξ)(uγψγ) = i(aµγj ξju

γ)ψµ. (5.18)

Our assumption that D = D∗ entails

(aµγj )c = −aγµj , ∀µ, γ, j. (5.19)

Next, introduce

Cαγjk := aαβj aβγk − gjkδαγ . (5.20)

Since σ(D2, ξ) = ‖ξ‖2, it follows that Cαγjk ξkξj = 0 for any ξ, ∀α, γ, i.e.

Cαγjk = −Cαγkj , ∀α, γ, j, k. (5.21)

Recall Θ(x, y) from (5.16) and assume that, in local coordinates,

Θ(x, y) = Θαβ(x, y)ψα(x)⊗ ψβ(y). (5.22)

On the other hand, the techniques of [34] adapted to the present case give that the main singularity inΘ(x, y) is contained in Dc

y [e0(x− y, y)ψα(x)⊗ ψα(y)], where

e0(z, y) := Cm

√g(y)

(∑gjk(y)zjzk

)−(m−2)/2. (5.23)

More concretely, for each α, β we have

27

Θαβ(x, y) = −aαβ` (y)∂y`e0(x− y, y) +O(|x− y|−(m−2+ε)), ∀ ε > 0. (5.24)

Now,

σ(D, ν)(x)Θ(x, y) = aµγj (x)νj(x)Θγβ(x, y)ψµ(x)⊗ ψβ(y)

= −aµγj (x)aγβ` (y)νj(x)∂y`e0(x− y, y)ψµ(x)⊗ ψβ(y) +O(|x− y|−(m−2+ε))

= −(gj`(x)δµβ + Cµβj` (x)

)νj(x)∂y`e0(x− y, y)ψµ(x)⊗ ψβ(y)

+O(|x− y|−(m−2+ε)), (5.25)

whereas

−σ(Dt, ν)(y)Θ(x, y) = aγβj (y)νj(y)Θµγ(x, y)ψµ(x)⊗ ψβ(y)

= −aγβj (y)aµγ` (y)νj(y)∂y`e0(x− y, y)ψµ(x)⊗ ψβ(y) +O(|x− y|−(m−2+ε))

= −(gj`(x)δµβ + Cµβ`j (x)

)νj(y)∂y`e0(x− y, y)ψµ(x)⊗ ψβ(y)

+O(|x− y|−(m−2+ε)). (5.26)

Summing up (5.25) and (5.26) yields three types of terms. First, there are weakly singular kernels,i.e. O(|x − y|−(m−2+ε)), which clearly give rise to compact operators on Lp. Second, there are com-mutators of the type [T,Mb] where T is a bounded operator in any weighted Lpω, 1 < p <∞, ω ∈ Ap,Muckenhoupt’s class, and Mb is the (pointwise) multiplication operator with a vmo function. These,again, are compact in Lp(∂Ω) because of rather deep harmonic analysis results; cf. the discussion in[39]. Third, we get (two copies of) the diagonal kernel

gj`(y)νj(y)∂y`e0(x− y, y)ψβ(x)⊗ ψβ(x). (5.27)

To analyze its main singularity, it helps to switch from ν = (νj)j and dσ to n = (nj)j and dσ0, theEuclidean unit normal to ∂Ω and the surface measure induced on ∂Ω by the standard Euclidean metric,respectively. They are related by νj = (gk`nkn`)−1/2nj and dσ =

√g(gk`nkn`)1/2 dσ0, respectively. Let

us also point out that

n ∈ vmo (∂Ω, dσ0)⇐⇒ ν ∈ vmo (∂Ω, dσ). (5.28)

Thus, a direct calculation gives

gj`(y)νj(y)∂y`e0(x− y, y) dσ = Cm〈n, x− y〉 (gjk(y)(xj − yj)(xk − yk))−m/2 dσ0 (5.29)

modulo lower order terms, where 〈·, ·〉 is the ordinary Euclidean inner product. Now, thanks to (5.28),the Euclidean singular integral operator with kernel

〈n, x− y〉 (gjk(y)(xj − yj)(xk − yk))−m/2

is compact. This follows by extending (via spherical harmonics) a similar compactness result aboutthe Euclidean double layer-like potential operators from [19]. The proof of the theorem is thereforecomplete. 2

Next we discuss Lp-based Hodge-Dirac decompositions the context of Dirac type operators in Lips-chitz domains. What is remarkable in this scenario, compared with the situation treated in Theorem 4.8,

28

is that the range of validity for such decompositions is 3/2 − ε < p < 3 + ε, where ε = ε(Ω, D) > 0.Specifically, we have the following.

Theorem 5.4 Let E ,F → M , D : E → F be as at the beginning of this section; in particular, D isassumed to be of Dirac type. Then there exists ε = ε(Ω, D) > 0 so that, for any 3/2− ε < p < 3 + ε,

Lp(Ω,F) = Ker (D∗; Lp(Ω,F))⊕DH1,p0 (Ω, E), (5.30)

where the direct sum is topological. Furthermore, the validity range [3/2, 3] is sharp in the class ofLipschitz domains.

Proof. The crux of the matter is establishing the estimate from below

‖u‖H1,p

0 (Ω,E)≤ C(Ω, D, p)‖D∗Du‖H−1,p(Ω,E) + ‖Comp(u)‖, (5.31)

where 3/2 − ε < p < 3 + ε and ε = ε(Ω, D) > 0. Throughout the proof, Comp will stand for genericcompact operators from H1,p

0 (Ω, E).With D∗D replaced by ∆, the Laplace-Beltrami operator associated with the Riemannian metric

g on M (and assuming that E is trivial) such an estimate has been proved in [41]. Our strategy is toutilize this in conjuction with the identity

D∗D = ∆ + lower order terms (5.32)

valid locally, in open sets over which E is trivial. The rest of the argument consists of patching up localestimates via a suitable partition of unity. More specifically, let (φi)i∈I be a smooth partition of unitysubordinated to some finite open cover (Oi)i∈I of Ω, and select ψi ∈ C∞comp(Oi) such that ψ ≡ 1 onsuppφi, ∀ i ∈ I. Matters can be arranged so that Oi ∩ Ω is a Lipschitz domain for each i ∈ I. Thenthere exists ε > 0 so that if p ∈ (3/2− ε, 3 + ε) the following estimates hold

‖u‖H1,p

0 (Ω,E)≤∑i

‖uφi‖H1,p0 (Oi∩Ω,E)

≤∑i

‖∆(uφi)‖H−1,p(Oi∩Ω,E)

≤∑i

‖D∗D(uφi)‖H−1,p(Oi∩Ω,E) + ‖Comp(u)‖

≤ C(Ω, D, p)‖D∗Du‖H−1,p(Ω,E) + ‖Comp(u)‖. (5.33)

The last inequality is obtained by writing D∗D(uφi) = ψiD∗D(uφi) = ψD∗Du + Comp(u) and then

using the mapping properties of the operator of multiplication by ψi. This concludes the proof of(5.31). Next we aim at showing that

D∗D : H1,p0 (Ω, E) −→ H−1,p(Ω, E) is invertible ∀ p ∈ (3/2− ε, 3 + ε). (5.34)

Consider first the issue of injectivity. When p ≥ 2, this follows from unique continuation. Indeed, ifu ∈ Ker (D∗D,H1,p

0 (Ω, E)), we obtain∫∫

Ω |Du|2 = 0 via integration by parts and, further, Du = 0 inΩ. In turn, with u denoting the extension of u to M by zero outside Ω, this entails Du = 0 on M .Since u vanishes on M \ Ω and D ∈ UCP we finally get that u and, hence, u, must vanish identically.

The case 3/2−ε < p < 2 can be reduced to the previous one via a bootstrap argument. Specifically,if ψ ∈ C∞comp(O) is a smooth cutoff function with small support and u ∈ Ker (D∗D,H1,p

0 (Ω, E)), then,by (5.32), ∆(φu) ∈ Lp(O∩Ω). Since, componentwise, φu ∈ H1,p

0 (O∩Ω), the results in [41] imply thatφu ∈ H1,p+γ

0 (O∩Ω) for some γ > 0. Iterating this scheme sufficiently many times yields u ∈ H1,20 (Ω, E),

29

which is the case already treated. Summarizing, at this stage we have proved that the operator in (5.34)is one-to-one and with closed range for the indicated range of p’s. Dualizing, we see that the sameoperator also has a dense range. With this at hand, (5.34) follows.

Turning now to the actual decomposition (5.30), fix p ∈ (3/2 − ε, 3 + ε) with ε > 0 small, and letf ∈ Lp(Ω,F) be arbitrary. If we now set v := (D∗D)−1(D∗f) ∈ H1,p

0 (Ω, E) and u := f−Dv ∈ Lp(Ω,F),it follows that D∗u = 0 and

f = u+ v, ‖u‖Lp(Ω) + ‖v‖H1,p(Ω) ≤ C‖f‖Lp(Ω). (5.35)

Uniqueness of the above decomposition is already contained in (5.34). The fact that the range [3/2, 3]is sharp in the class of Lipschitz domains follows from the counterexamples in [14]. 2

Our last result in this section is a half-Dirichlet problem for symmetric Dirac operators in Lipschitzdomains. To set the stage, let D : E → E be a symmetric Dirac operator. For an arbitrary, fixedLipschitz subdomain Ω of M define the operators

P± := 12(I ± σ(D; ν)), (5.36)

where recall that ν stands for the outward unit conormal to ∂Ω and I is the identity operator. Whenconsidered on Lp(∂Ω, E), these satisfy (P±)2 = P±, P+ +P− = I, P+P− = P−P+ = 0 and (P±)∗ = P±.Indeed, when p = 2, they become complementary orthogonal projections. Set

Lp±(∂Ω, E) := f ∈ Lp(∂Ω, E); P∓f = 0. (5.37)

Note that

Lp±(∂Ω, E)1<p<∞ is a complex interpolation scale (5.38)

since this is the image of the Lebesgue scale Lp(∂Ω, E)1<p<∞ under common projections.

Lemma 5.5 With the above notation and hypotheses, there exists ε = ε(Ω) > 0 so that∫∂Ω|P+u|p dσ ≈

∫∂Ω|P−u|p dσ (5.39)

for each p ∈ (2− ε, 2 + ε), uniformly in u ∈ Hp(Ω, D).

Proof. Starting from Cauchy’s vanishing formula∫∂Ω〈σ(D; ν)u, uc〉 dσ = 0, (5.40)

valid for any u ∈ H2(Ω, D), one easily derives the identity∫∂Ω〈P±u, uc〉 dσ = 1

2

∫∂Ω|u|2 dσ, ∀u ∈ H2(Ω, D). (5.41)

Now, upon noticing that the left side of (5.40) is ‖P±u‖2L2(∂Ω,E) and that |u|2 = |P+u|2 + |P−u|2 on ∂Ω,we get ∫

∂Ω|P+u|2 dσ =

∫∂Ω|P−u|2 dσ (5.42)

and this, of course, covers the p = 2 version of (5.39). The extension to p ∈ (2 − ε, 2 + ε) for someε = ε(Ω) > 0 then follows from the L2 result, the observation (5.38), plus some general stability resultsfrom [22] (cf. especially the remark on p. 3908). 2

30

Theorem 5.6 Let D be a symmetric Dirac operator on E → M whose coefficients satisfy (2.21) andlet Ω ⊆M be an arbitrary Lipschitz domain.

Then there exists ε = ε(Ω) > 0 so that for any p ∈ (2− ε, 2 + ε) the boundary problems

(BV P±)

u ∈ Hp(Ω, D),

P±(u|∂Ω) = g ∈ Lp±(∂Ω, E),(5.43)

are uniquely solvable. Moreover, there holds

‖u‖Hp(Ω) ≤ κ(Ω, p)‖g‖Lp(∂Ω,E) (5.44)

uniformly in g.

Proof. As before, (cf. the discussion at the beginning of the proof of Theorem 4.9), it suffices to treatthe case when D : H1,2(M, E)→ L2(M, E) is invertible. In particular, this allows us to reintroduce theCauchy operators (4.14)–(4.15). The heart of the matter is proving that there exists ε = ε(Ω) > 0 sothat

P±(λI + CD) are isomorphisms of Lp±(∂Ω, E), (5.45)

for each 2 − ε < p < 2 + ε and λ ∈ R with |λ| ≥ 12 . Accepting (5.45) for a moment, the proof can be

concluded as follows. First, the fact that

u := CD([P±(12I + CD)]−1g) in Ω, (5.46)

solves (5.43) takes care of the existence part and the estimates (5.44). Uniqueness is a simple conse-quence of the estimate contained in Lemma 5.5.

Turning now to the task of establishing (5.45), a few comments are in order. First, let us pointout that (as it will become more apparent shortly) the more general statement, involving λ ∈ R with|λ| ≥ 1

2 , is actually needed even though we only use this result for λ = 12 . Second, thanks to (5.38) and

the stability theory from [22], it suffices to prove (5.45) when p = 2. Third, it is enough to show thatfor each λ ∈ R with |λ| ≥ 1

2 , there holds

‖f‖L2(∂Ω,E) ≤ κ(Ω, λ)‖P±(λI + CD)f‖L2(∂Ω,E), (5.47)

uniformly for functions f ∈ L2±(∂Ω, E). Indeed, the p = 2 version of (5.45) then follows from (5.47) by

letting |λ| → ∞ and using the homotopic invariance of the index.With an eye toward (5.47), fix λ ∈ R with |λ| ≥ 1

2 , f ∈ L2+(∂Ω, E) and define u := CDf in Ω±.

Then, by the jump relations established in §3, we have

P+u|∂Ω± = ±12f + P+CDf = (±1

2 − λ)f + (λI + P+CD)f, (5.48)

and

P−u|∂Ω+ = P−u|∂Ω− = P−CDf. (5.49)

Call the I± the two versions of (5.42) written for Ω+ and Ω−, respectively, and create a new identitywhich, formally, has the form (λ + 1

2)I+ − (λ − 12)I−. From (5.49) it is immediate that the right side

of this new identity is∫∂Ω |P−CDf |2 dσ; in particular, this is ≥ 0. Also, from (5.48) plus some algebra,

the left hand side of this new identity reads

31

−(λ2 − 14)∫∂Ω|f |2 dσ +O

(‖f‖L2(∂Ω,E)‖P+(λI + CD)f‖L2(∂Ω,E)

). (5.50)

From these, (5.47) readily follows when |λ| > 12 .

When λ = ±12 , the argument is simpler (while similar in spirit). The departure point is to write

‖f‖L2(∂Ω,E) ≤ ‖u|∂Ω+‖L2(∂Ω,E)+‖u|∂Ω−‖L2(∂Ω,E) and to note that ‖u|∂Ω±‖L2(∂Ω,E) ≤ ‖P+(u|∂Ω±)‖L2(∂Ω,E).Since u|∂Ω± = (±1

2I +CD)f , the desired conclusion follows. The proof of the theorem is therefore fin-ished. 2

6 Boundary value problems for Hodge-Dirac operators

On a Riemannian manifold M , two basic examples of Dirac type operators are offered by d± δ, whered is the exterior derivative operator and δ its formal adjoint. More generally, one may consider thefamily of (Hodge-Dirac) operators

Dα := d+ αδ, α ∈ R, α 6= 0, (6.1)

in the Grassmann algebra bundle

G = GM := ⊕ml=0ΛlTM, (i.e. GM = C`(M)).

Now, Dα satisfies (5.3) if and only if α = ±1 and is self-adjoint if and only if α = 1. Nonetheless, with∆ := −(dδ + δd) denoting the Hodge-Laplacian on M , the identity

(D∗α)2 = (Dtα)2 = D2

α = −α∆ (6.2)

is valid for any α. As explained in Theorem 5.2, this makes the theory developed in § 2− 4 well-suitedfor the present setting. In particular, for any Lipschitz domain Ω ⊂M , the Hardy type spaces

Hp(Ω, Dα) := u ∈ Lp(Ω,G); Dαu = 0 in Ω, Nu ∈ Lp(∂Ω) (6.3)

enjoy all the properties discussed in § 4.In this section, we are concerned with the “half-Dirichlet” problems for Dα in an arbitrary Lipschitz

subdomain Ω of M . Specifically, we shall consider the problem of prescribing the tangential componentof forms in Hp(Ω, Dα), i.e.

(BV Ptan)

Dαu = 0 in Ω,

Nu ∈ Lp(∂Ω),

ν ∧ u = f ∈ Lpnor(∂Ω,G),

(6.4)

as well as the problem of prescribing the normal component of forms in Hp(Ω, Dα), i.e.

(BV Pnor)

Dαu = 0 in Ω,

Nu ∈ Lp(∂Ω),

ν ∨ u = g ∈ Lptan(∂Ω,G).

(6.5)

Before going any further, let us comment on the notation used so far and make several otherdefinitions. To begin with, recall that ∧, ∨ stand, respectively, for the exterior and the interior productof (differential) forms. Next, we set

Lpnor(∂Ω,G) := f ∈ Lp(∂Ω,G); ν ∧ f = 0 a.e. on ∂Ω

32

and recall from [36], [20], [34] the differential operator d∂ . Specifically, for f ∈ Lpnor(∂Ω,G), we definethe distribution d∂f by requiring that∫

∂Ω〈d∂f, ψ〉 dσ =

∫∂Ω〈f, δψ〉 dσ for any ψ ∈ C1(M,G).

Hereafter, 〈·, ·〉 will denote the ordinary (pointwise) pairing of forms induced by the metric on M . Asin the aforementioned references, we define

Lp,dnor(∂Ω,G) := f ∈ Lpnor(∂Ω,G); d∂f ∈ Lp(∂Ω,G) ,

equipped with the natural norm ‖f‖Lp,dnor(∂Ω,G)

:= ‖f‖Lp(∂Ω,G) + ‖d∂f‖Lp(∂Ω,G). Also, set

Lp,0nor(∂Ω,G) :=f ∈ Lp,dnor(∂Ω,G); d∂f = 0

.

Now, if ∗ stands for the usual Hodge star isomorphism, we set

Lptan(∂Ω,G) := ∗Lpnor(∂Ω,G), Lp,δtan(∂Ω,G) := ∗Lp,dnor(∂Ω,G), Lp,0tan(∂Ω,G) := ∗Lp,0nor(∂Ω,G).

Letting Λl be the collection of l-forms for l ≥ 1 and Λ0 := R, we define πl to be the projection of Gonto Λl (so that Id = ⊕ml=0πl). Then we set

τ :=m∑l=0

(−1)lπl (6.6)

so that τ is an involutive isometry. Now, if f ∈ Lp,δtan(∂Ω,G), define

δ∂f := (−1)m+1 ∗ d∂ ∗ τf. (6.7)

For a more detailed discussion see [34]. Next, we introduce

Hp∧(Ω,G) :=u ∈ C0(Ω,G); du = δu = 0 in Ω, ν ∧ u = 0, Nu ∈ Lp(∂Ω)

, (6.8)

Hp∨(Ω,G) :=u ∈ C0(Ω,G); du = δu = 0 in Ω, ν ∨ u = 0, Nu ∈ Lp(∂Ω)

. (6.9)

It has been shown in [34] that for any Lipschitz domain Ω there exists a small ε = ε(Ω) > 0 so that

Hp∧(Ω,G), Hp∨(Ω,G) are independent of p ∈ (2− ε, 2 + ε). (6.10)

Whenever this is the case, we shall drop the superscript p. Another result of interest from [34] is that,as in the smooth case,

dimH∧(Ω,G) = dimH∨(Ω,G) =m∑l=0

bl(Ω), (6.11)

where bl(Ω) is the l-th Betti number of the Lipschitz domain Ω.One final remark is that we will employ similar notation for the versions of the spaces introduced

above which are obtained by insisting that the forms are ΛlTM -valued for some 0 ≤ l ≤ m (ratherthat being G-valued). Concretely, we shall replace G by ΛlTM and write Lpnor(∂Ω,ΛlTM), etc.

Theorem 6.1 Assume that the Riemann metric tensor on M has C1,1-components and that Ω is aLipschitz subdomain of M . Then there exists ε = ε(Ω) > 0 so that the following assertions are validfor each α ∈ (−∞, 0) ∪ (0, 1] and whenever 2− ε < p < 2 + ε.

33

(1) The problem (BV Ptan) is solvable if and only if f satisfies the compatibility condition

f ∈ ω|∂Ω; ω ∈ H∧(Ω,G). (6.12)

Here . . . is the annihilator in Lq(∂Ω,G) of . . ., for 1p + 1

q = 1. Also, each solution belongs

to Bp,p∨21/p (Ω,ΛlTM).

(2) The space of null-solutions for (BV Ptan) is precisely H∧(Ω,G). In particular, du and δu areuniquely determined (even if u is not).

If (BV Ptan) is solvable and 2− ε < p < 2 + ε then the following regularity results also hold:

(3) N (du) ∈ Lp(∂Ω) (and, hence, N (δu) ∈ Lp(∂Ω)) ⇔ f ∈ Lp,dnor(∂Ω,G). In particular, for u ∈Hp(Ω, Dα),

ν ∧ u ∈ Lp,dnor(∂Ω,G)⇔ ν ∨ u ∈ Lp,δtan(∂Ω,G). (6.13)

Naturally accompanying estimates are valid in each case.

(4) du = 0 (and, hence, δu = 0) ⇔ f ∈ Lp,0nor(∂Ω,G). In particular, for u ∈ Hp(Ω, Dα),

ν ∧ u ∈ Lp,0nor(∂Ω,G)⇔ ν ∨ u ∈ Lp,0tan(∂Ω,G). (6.14)

Finally, similar results are valid for (BV Pnor).

Proof. Let u solve (BV Ptan). Since u ∈ Hp(Ω, Dα) ⇒ u ∈ Bp,p∨21/p (Ω,G), it follows that δu, du ∈⋂

ε>0Bp,p∨2

1/p−1−ε(Ω,G). Furthermore, since ∆(du) = ∆(δu) = 0, Proposition 2.1 in [34] implies that

du, δu ∈⋂ε>0

H1/p+1−ε,ploc (Ω,G). (6.15)

Now fix an arbitrary ω ∈ H∧(Ω,G) and select a suitable sequence of approximating domains Ωj Ω.For each fixed j, the forms ω, u, du, δu have enough regularity in Ωj (cf. the previous discussion) tojustify the following sequence of integrations by parts:

∫∂Ωj

〈νj ∧ u, ω〉 dσj =∫ ∫

Ωj

〈du, ω〉 − 〈u, δω〉 = −α∫ ∫

Ωj

〈δu, ω〉

= −α∫ ∫

Ωj

〈u, dω〉+ α

∫∂Ωj

〈νj ∨ u, ω〉 dσj

= α

∫∂Ωj

〈u, νj ∧ ω〉 dσj (6.16)

Then, using Nu ∈ Lp(∂Ω), Nω ∈ Lq(∂Ω) with 1p + 1

q = 1, one can pass to the limit in (6.16) to obtain∫∂Ω〈f, ω〉 dσ = α

∫∂Ω〈u, ν ∧ ω〉 dσ = 0.

This shows that (6.12) is necessary in order for (BV Ptan) to be solvable. To show that this is alsosufficient, we decompose f = ⊕ml=0fl according to direct summands in G = ⊕ml=0ΛlTM and note that,for each l, fl ∈ Lpnor(∂Ω,ΛlTM). Furthermore, from (6.12),

34

fl ∈ ω|∂Ω; ω ∈ H∧(Ω,ΛlTM), ∀ l ∈ 0, 1, ...,m. (6.17)

The idea now is to solve, for each l ∈ 0, 1, ...,m,

vl ∈ C0(Ω,ΛlTM),

∆lvl = 0 in Ω,

N (vl),N (δvl) ∈ Lp(∂Ω),

ν ∧ vl = 0 on ∂Ω,

ν ∧ δvl = 1αfl ∈ L

pnor(∂Ω,ΛlTM).

(6.18)

From Theorem 5.1 in [34], it is known that (6.18) is solvable if and only if the compatibility condition(6.17) is satisfied. Moreover, the solution also has the property that N (dvl) ∈ Lp(∂Ω), so that

ν ∧ dvl = −d∂(ν ∧ vl) = 0, on ∂Ω. (6.19)

Now, if we set u := (d+ αδ) (∑ml=0 vl) in Ω, it follows that u solves (BV Ptan). This proves (1).

Next, we study the space of null solutions for (BV Ptan). To this end, let u ∈ Hp(Ω, Dα) be so thatν ∧u = 0 and recall the operator C (associated with Dα) as in § 3. Then Cauchy’s reproducing formula(4.5) gives that

u(x) = C(u|∂Ω)(x) =∫∂Ω〈(dy + αδy)E(x, y), (ν · u)(y)〉 dσ(y). (6.20)

Notice that ν · u = ν ∧ u− ν ∨ u = −ν ∨ u, by assumption. Thus,

u = −∫∂Ω〈(dy + αδy)E(x, y), (ν ∨ u)(y)〉 dσ(y). (6.21)

Going to the boundary (nontangentially) and taking ν ∨ ·, it follows that(µI + M

)(ν ∨ u) = − 1

αν ∨∫∂Ω〈dyE(x, y), ν ∨ u〉 dσ, (6.22)

where µ := 1α −

12 and (cf. § 6 in [34])

Mf(x) := ν(x) ∨ p.v.∫∂Ω〈δyE(x, y), f(y)〉 dσ(y), x ∈ ∂Ω. (6.23)

Set g for the right side of (6.22). We claim that g ∈ Lp,δtan(∂Ω,G). Indeed, as in [34],

dyE(x, y) = δxE(x, y) +R(x, y) (6.24)

where the residue R is C1 in a neighborhood of ∂Ω× ∂Ω. Thus,

−α g(x) = νx ∨ δx∫∂Ω〈E(x, y), (ν ∨ u)(y)〉 dσ(y)

+ν ∨∫∂Ω〈R(x, y), (ν ∨ u)(y)〉 dσ(y) =: I + II. (6.25)

Observe now thatI = −δ∂(ν ∨

∫∂Ω〈E, ν ∨ u〉 dσ) ∈ Lp,0tan(∂Ω,G)

35

andδ∂II = −ν ∨

∫∂Ω〈δxR(x, y), (ν ∨ u)(y)〉 dσ(y) ∈ Lp(∂Ω,G).

Hence, I, II ∈ Lp,δtan(∂Ω,G) so that g ∈ Lp,δtan(∂Ω,G) as claimed.At this point, we may invoke Corollary 10.5 in [34] (here it is important that µ ∈ R has |µ| ≥ 1

2) toconclude that ν ∨ u ∈ Lp,δtan(∂Ω,G). Utilizing this back in (6.21) will yield

N (δu) ∈ Lp(∂Ω). (6.26)

More specifically, from (6.21) and (6.24),

u = −∫∂Ω〈E(x, y), δ∂(ν ∨ u)〉 dσ − δx

∫∂Ω〈E(x, y), ν ∨ u〉 dσ −

∫∂Ω〈R(x, y), ν ∨ u〉 dσ (6.27)

so that

δu = −δx∫∂Ω〈E(x, y), δ∂(ν ∨ u)〉 dσ − δx

∫∂Ω〈R(x, y), (ν ∨ u)〉 dσ. (6.28)

This clearly implies (6.26). In turn, (6.26) and the fact that du = −αδu in Ω also give N (du) ∈ Lp(∂Ω).This shows that du, δu ∈ H∧(Ω,G). With this at hand, it is not hard to justify, via a limiting argumentinvolving a suitable sequence Ωj Ω, the following integrations by parts∫ ∫

Ω|du|2 dVol =

∫ ∫Ω〈du, duc〉 dVol =

∫∂Ω〈ν ∧ u, duc〉 dσ = 0. (6.29)

Thus, du = 0 = δu which implies u ∈ H∧(Ω,G). The fact that any element in H∧(Ω,G) is a nullsolution for (BV Ptan) is obvious and this completes the proof of (2).

Next, we tackle the regularity statements, starting with (3). In one direction, clearly N (du) ∈Lp(∂Ω) ⇒ ν ∧ u ∈ Lp,dnor(∂Ω,G) since d∂(ν ∧ u) = −ν ∧ du. Conversely, start with u ∈ Hp(Ω, Dα) sothat ν ∧ u ∈ Lp,dnor(∂Ω,G). Recall (from what we have proved so far) that

u = (d+ αδ)

(m∑l=0

vl

)+ w (6.30)

where vl solves (6.18) for fl := (ν ∧ u)l ∈ Lp,dnor(∂Ω,ΛlTM) and w ∈ H∧(Ω,G). From Theorem 5.1 in[34] it follows that vl has the extra regularity N (dδvl) ∈ Lp(∂Ω). This, (6.30) and the fact that dw = 0give that N (du) ∈ Lp(∂Ω), as desired. Now (3) follows.

Consider now (4). If u ∈ Hp(Ω, Dα) has ν ∧ u ∈ Lp,0nor(∂Ω,G) then, from (3), we have N (du),N (δu) ∈ Lp(∂Ω). Set v := du so that, from what we know,

N (v) ∈ Lp(∂Ω), dv = 0, δv = 0 and ν ∧ v = 0. (6.31)

It follows that v ∈ H∧(Ω,G) so that, in particular, N (du) ∈ L2(∂Ω). This automatically entailsN (δu) ∈ L2(∂Ω). With this at hand, and recalling that dδu = δdu = 0 we may now justify thefollowing identity based on integrations by parts

∫ ∫Ω|du|2 dVol =

∫∂Ω〈ν ∧ u, duc〉 dσ = −α

∫∂Ω〈ν ∧ u, δuc〉 dσ

= −α∫∂Ω〈d∂(ν ∧ u), uc〉 dσ = 0. (6.32)

36

The last equality holds since ν ∧u ∈ L2,0nor(∂Ω,G). This forces du = 0 (and, further, δu = 0), as wanted.

Hence, the proof of (4) is finished.The very last part in the statement of Theorem 6.1 follows in a similar fashion. We leave the details

to the interested reader. This concludes the proof of Theorem 6.1. 2

7 Connections with Maxwell’s equations

In this section we shall introduce another Dirac type operator which is well suited for the analysis ofMaxwell’s equations in Lipschitz domains. The departure point is to embed the Riemannian manifoldM into R ×M so that TM → R ⊕ TM in a canonical fashion. Denote by t the generic variable in Rso that, at the level of complexified Grassmann algebras,

GCM → E := GC

M ⊕C (dt⊗ GCM ) → GC

IR×M . (7.1)

Let us briefly inspect the interplay between the various algebraic structures available. Recall that∧,∨ denote the exterior and interior product, respectively, for differential forms in the Grassmannalgebra GC

IR×M . Also, · stands for the Clifford algebra product in C⊕ GCIR×M . First, there is a natural

conjugation on GIR×M which is compatible with the Clifford multiplication in the sense that u · v = v ·u,and compatible with the metric, meaning

〈u, v〉 = 〈u, v〉 = (uv)0 = (uv)0 = (vu)0 = (vu)0. (7.2)

Here, (. . .)0 denotes the scalar component, i.e. the projection π0 of C ⊕ GCIR×M onto C. Specifically,

recalling the projection operators πl of C⊕ GCIR×M onto Λl, we have

u =m∑l=0

(−1)l(l+1)

2 πlu. (7.3)

Recall the involution τ from (6.6). If α ∈ Λ1 and u ∈ E , then

α · u = α ∧ u− α ∨ u, u · α = α ∧ τu+ α ∨ τu, (7.4)

and

〈α · u, v〉 = 〈u, α · u〉 = −〈u, α · u〉, (7.5)

〈u · α, u〉 = 〈u, v · α〉 = −〈u, v · α〉. (7.6)

Finally, there is a Z2-grading of C⊕ GCIR×M ,

Λod ⊕ Λev :=(⊕l=odd Λl

)⊕(⊕l=even Λl

)(7.7)

which is compatible with the metric (i.e. the sum is direct). Also, note that the Clifford multiplicationby elements from Λ1 switches the parity. In fact, the same applies to the exterior and the interiorproduct with elements from Λ1. In the sequel, we denote by u 7→ uod, u 7→ uev the projection operatorscorresponding to the decomposition (7.7). The usual complex conjugation operator is denoted byu 7→ uc.

Turning now to analysis, we introduce the family of Dirac type operators, indexed by k ∈ C,

IDk := DM + k dt, (7.8)

37

where DM := dM + δM , with dM , δM standing, respectively, for the exterior derivate and co-derivativeoperators in the manifold M . Here dt acts as a Clifford algebra multiplier.

Recall the vector bundle E from (7.1). Clearly,

IDk : C1(M, E)→ C0(M, E) (7.9)

is an elliptic first order differential operator. Some of its most immediate properties are:

ID2k = −∆M − k2, IDk = −IDk, IDc

k = IDkc , IDtk = ID−k, ID∗k = ID−kc , (7.10)

where ∆M := −(dMδM + δMdM ) is the Hodge-Laplacian on M . In particular, IDk is of Dirac type and,hence, the theory in § 2− 4 applies to this operator; cf. also the discussion in § 5. In fact,

σ(IDk; ξ) = i ξ· , ξ ∈ T ∗x\0, x ∈M. (7.11)

As in the previous section, we are interested in the “half-Dirichlet” problem for IDk, i.e. prescribingthe tangential (or normal) components of elements in Hp(Ω, IDk). Here Ω is an arbitrary Lipschitzsubdomain of M , and for 1 < p <∞, k ∈ C,

Hp(Ω, IDk) := u ∈ Lp(Ω, E); IDku = 0 in Ω, Nu ∈ Lp(∂Ω) (7.12)

is the Hardy space naturally associated to IDk. Our main result in this direction is the following.

Theorem 7.1 Let the manifold M be equipped with a C1,1 Riemannian metric and let Ω ⊂ M be anarbitrary Lipschitz domain. Also, recall the Hermitian vector bundle E →M from (7.1) and the Hardyspace Hp(Ω, IDk) from (7.12), k ∈ C, 1 < p <∞.

Then there exist ε = ε(Ω) > 0 and a sequence of real, positive numbers kjj such that:

(1) For each 2− ε < p < 2 + ε and k ∈ C\±kjj the boundary value problem

(BV Ptan)k

u ∈ C0(Ω, E),

IDku = 0 in Ω,

N (u) ∈ Lp(∂Ω),

ν ∧ u = f ∈ Lpnor(∂Ω, E),

(7.13)

has a unique solution. Moreover, the solution satisfies

‖u‖Bp,p∨2

1/p(Ω,E)

+ ‖N (u)‖Lp(∂Ω) ≤ C(p,Ω, k)‖f‖Lp(∂Ω,E). (7.14)

(2) For p and k as before,

N (du) ∈ Lp(∂Ω)⇔ N (δu) ∈ Lp(∂Ω)⇔ f ∈ Lp,dnor(∂Ω, E) (7.15)

plus natural estimates.

(3) For any k ∈ ±kjj the space of null-solutions for (BV Ptan)k is finite dimensional and a solutionexists if and only if the boundary datum satisfies finitely many linear conditions.

Furthermore, the case k = 0 is treated in detail in Theorem 6.1.

38

(4) Similar results are valid for the boundary problem

(BV Pnor)k

v ∈ C0(Ω, E),

IDkv = 0 in Ω,

N (v) ∈ Lp(∂Ω),

ν ∨ v = g ∈ Lptan(∂Ω, E).

(7.16)

Setting the stage for the proof of this theorem, we begin with a series of lemmas, building up tothe Rellich type estimates in Corollary 5.6.

Lemma 7.2 If u ∈ E and α, β ∈ T ∗M ≡ Λ1M , then

|u|2〈α, β〉 = Re 〈α · u, β · uc〉 = Re 〈u · α, uc · β〉. (7.17)

Proof. Based on 〈α, β〉 = −12(α · β + β · α) we may write

|u|2〈α, β〉 = 〈〈α, β〉u, uc〉 = −12〈α · β · u, u

c〉 − 12〈β · α · u, u

c〉

= 12〈β · u, α · u

c〉+ 12〈α · u, β · u

c〉.

Taking the real parts yields the first equality in (7.17).To see the second one, replace u by u in what we have just proved and use the fact that

〈α · u, β · u〉 = 〈α · u, β · u〉 = 〈u · α, u · β〉. (7.18)

The proof of the lemma is complete. 2

Lemma 7.3 Assume that Ω ⊂M is a Lipschitz domain and consider a section u ∈ H2(Ω, IDk), k ∈ C,and a one-form θ ∈ T ∗M . Then

Re∫∂Ω〈ν · u, uc · θ〉 dσ =

∫ ∫ΩO(|u|2) dVol . (7.19)

Proof. Utilizing (7.11), we write

Re∫∂Ω〈ν · u, uc · θ〉 dσ = Re

∫ ∫Ω〈IDku, u

c · θ〉 −∫ ∫

Ω〈u, [ID∗k(u · θ)]

c〉

= Re∫ ∫

Ω〈−(IDku) · θ − ID∗k(u · θ), uc〉

. (7.20)

Since, by assumption, IDku = 0 in Ω we may write −(IDku) · θ − ID∗k(u · θ) as (IDku) · θ − ID∗k(u · θ).The crux of the matter is now that P : u 7→ (IDku) · θ − ID∗k(u · θ) is a zero-order differential operator,as a calculation of symbols shows:

σ(P ; ξ)e = i(ξ · e) · θ − iξ · (e · θ) = 0. (7.21)

Hence, the contribution from this combination of terms is O(|u|2) and the conclusion follows. 2

39

Lemma 7.4 Let Ω ⊂M be Lipschitz. If u ∈ H2(Ω, IDk) and θ ∈ T ∗M , then∫∂Ω|u|2〈θ, ν〉 dσ = Re

∫∂Ω〈(±ν · u+ u · ν), uc · θ〉 dσ +

∫ ∫ΩO (|u|2) dVol . (7.22)

Proof. This is immediate from Lemma 7.2 and Lemma 7.3. 2

Lemma 7.5 Let Ω ⊂M be Lipschitz. Then

‖u‖L2(∂Ω,E) ≈ ‖ν · u± u · ν‖L2(∂Ω,E) modulo ‖u‖L2(Ω,E), (7.23)

uniformly for u ∈ H2(Ω, IDk).

Proof. Choose θ ∈ T ∗M so that 〈θ, ν〉 ≥ c > 0 a.e. on ∂Ω. This is possible since Ω is Lipschitz. Forthis choice, (7.23) is a simple consequence of (7.22) and Cauchy-Schwarz’s inequality. 2

Corollary 7.6 Let Ω be a Lipschitz domain in M . Then

‖N (u)‖L2(∂Ω) ≈ ‖u‖L2(∂Ω,E) ≈ ‖ν ∧ u‖L2(∂Ω,E) ≈ ‖ν ∨ u‖L2(∂Ω,E) modulo ‖u‖L2(Ω,E), (7.24)

uniformly for u ∈ H2(Ω, IDk).

Proof. The first equivalence is part of Theorem 3.1 so we concentrate on the remaining ones. Theimportant observation is that H2

k(Ω, E) splits according to (7.7), i.e.

H2(Ω, IDk) = H2k(Ω,Λod)⊕H2

k(Ω,Λev) (7.25)

where H2k(Ω,Λod) is the space of Λod-valued sections in H2(Ω, IDk), etc. Thus, if u ∈ H2(Ω, IDk) splits

as u = uod + uev, it suffices to prove the last two equivalences in (7.24) for uod and uev separately.Now, from (7.4),

ν · uod = ν ∧ uod − ν ∨ uod, ν · uev = ν ∧ uev − ν ∨ uev (7.26)

anduod · ν = −ν ∧ uod − ν ∨ uod, uev · ν = ν ∧ uev + ν ∨ uev. (7.27)

At this point, everything follows from Lemma 7.5. 2

Denote by Spec (∆) the collection of all complex numbers z so that ∆ − z is not invertible as anoperator from H1,2(M,G) onto H−1,2(M,G). Now, if k ∈ C is so that

k2 /∈ −Spec (∆) (7.28)

then the operator Lk := ∆ + k2 : H1,2(M,G) → H−1,2(M,G) is invertible and we denote by Ek(x, y)the Schwartz kernel of L−1

k . As in § 4 (cf. the discussion in connection with (4.20)–(4.21)), we maydefine Cauchy type operators by setting

Ckf(x) := −∫∂Ω〈IDk,xEk(x, y), ν(y) · f(y)〉 dσ(y), x /∈ ∂Ω, (7.29)

and

Ckf(x) = −p.v.∫∂Ω〈IDk,xEk(x, y), ν(y) · f(y)〉 dσ(y), x ∈ ∂Ω. (7.30)

40

As pointed out in the discussion concerning (4.20), since ID2k satisfies (4.20), the operator

Ck : Lp(∂Ω, E)→ Hp(Ω, IDk) (7.31)

is well-defined, bounded and onto for each 1 < p <∞. In view of (7.31), it is therefore natural to lookfor a solution of (BV Pnor)k in the form

u := Ck(ν ∨ f), f ∈ Lpnor(∂Ω, E). (7.32)

Then, by the results in § 4, u ∈ Hp(Ω, IDk) and

ν ∧ u|∂Ω = ν ∧(

12I + Ck

)(ν ∨ f) =

[12I + ν ∧ Ck(ν ∨ ·)

]f. (7.33)

Our long term goal is to show that

12I + ν ∧ Ck(ν ∨ ·) : Lpnor(∂Ω, E)→ Lpnor(∂Ω, E) (7.34)

is invertible for 2− ε < p < 2 + ε and k ∈ C except for a discrete subset of R.Our proof of (7.34) mimics the main steps used to establish (5.45) but, in this case, the algebra is

considerably more subtle. The idea is to first treat the case p = 2 and show that for any λ ∈ R with|λ| ≥ 1

2 there exists C = C(λ,Ω) > 0 so that

‖f‖L2(∂Ω,E) ≤ C‖ν ∧ (λI + Ck)(ν ∨ f)‖L2(∂Ω,E) + ‖Comp (f)‖, (7.35)

uniformly for f ∈ L2nor(∂Ω, E). Here and elsewhere, Comp will denote compact operators from L2(∂Ω, E)

into normed spaces. To see this, we proceed in two steps.

Step I. Assume λ = ±12 . In this case, it is enough to show that

∥∥∥ν ∧ (12I + Ck

)(ν ∨ f)

∥∥∥L2(∂Ω,E)

≈∥∥∥(ν ∧ (−1

2I + Ck)

(ν ∨ f))∥∥∥

L2(∂Ω,E)

modulo compact operators (7.36)

uniformly for f ∈ L2(∂Ω, E). Indeed, (7.35) follows simply from (7.36) and triangle’s inequality.As for the proof of (7.36), the idea is to apply the estimates in Corollary 7.6 to u := Ck(ν ∨ f) both

for Ω+ := Ω and Ω− := M\Ω. This leads to

∥∥∥ν ∧ (±12I + Ck

)(ν ∨ f)

∥∥∥L2(∂Ω,E)

≈ ‖ν ∨ u|∂Ω±‖L2(∂Ω,E)

modulo compact operators. (7.37)

The important observation now is that, due to (4) in Theorem 3.1 and the fact that f is normal a.e.on ∂Ω,

ν ∨ u|∂Ω+ = ν ∨ u|∂Ω− a.e. on ∂Ω. (7.38)

Utilizing this back into (7.37) yields (7.36). This finishes the proof of Step I.

Step II. Assume |λ| > 12 and fix f ∈ L2

nor(∂Ω, E). The first remark is that it suffices to prove (7.35)for fod and fev separately. This is because ν ∧ (λI +Ck)(ν ∨ ·) is parity preserving so that one can usethe Pythagorean theorem to re-assemble the individual estimates.

41

Note that the splitting L2(∂Ω, E) = L2(∂Ω,Λod)⊕L2(∂Ω,Λev) is compatible with the decompositionL2(∂Ω, E) = L2

tan(∂Ω, E) ⊕ L2nor(∂Ω, E) in the sense that (fnor)ev = (fev)nor, etc. Hence, in our case,

there is no loss of generality in assuming that, say,

f ∈ L2nor(∂Ω,Λod). (7.39)

Going further, let u := Ck(ν ∨ f) in Ω± and notice that, on account of (7.39),

u ∈ H2k(Ω±,Λev). (7.40)

For some smooth θ ∈ T ∗M , chosen so that

essinf 〈θ, ν〉 > 0 on ∂Ω, (7.41)

we write (7.22) in Lemma 7.4 both in Ω+ and Ω−. Call the resulting identities I+ and I−, respectively.Next, we create a new identity, called II, by adding (sidewise)

(λ+ 1

2

)I+ with −

(λ− 1

2

)I−. Let us

analyze the two sides, LHSII and RHSII , of the resulting identity II. To this effect, we first note that

u|∂Ω± =(±1

2I + Ck)

(ν ∨ f) = (λI + Ck)(ν ∨ f) +(−λ± 1

2

)(ν ∨ f) (7.42)

so that

|u|∂Ω± |2 = |(λI + Ck)(ν ∨ f)|2 +(λ∓ 1

2

)2|ν ∨ f |2

+2(−λ± 1

2

)Re 〈ν ∧ (λI + Ck)(ν ∨ f), f c〉

= |(λI + Ck)(ν ∨ f)|2 +∣∣∣λ∓ 1

2

∣∣∣2 |f |2 +R1. (7.43)

Hereafter, Rj , j = 1, 2, . . . , will denote quantities satisfying∫∂Ω|Rj | dσ = O(‖f‖L2(∂Ω,E) · ‖ν ∧ (λI + Ck)(ν ∨ f)‖L2(∂Ω,E)) (7.44)

uniformly in f . Consequently,

LHSII =∫∂Ω|ν ∨ Ck(ν ∨ f)|2〈θ, ν〉 dσ

+[(λ+ 1

2

) (λ− 1

2

)2−(λ− 1

2

) (λ+ 1

2

)2] ∫

∂Ω|f |2〈θ, ν〉 dσ +

∫∂ΩR2 dσ

=∫∂Ω|ν ∨ Ck(ν ∨ f)|2〈θ, ν〉 dσ −

(λ2 − 1

4

) ∫∂Ω|f |2〈θ, ν〉 dσ +

∫∂ΩR2 dσ. (7.45)

Consider now RHSII . Based on (7.40) we may write

ν · u+ u · ν = (ν ∧ u− ν ∨ u) + (ν ∧ u+ ν ∨ u) = 2ν ∧ u. (7.46)

Thus,

(ν · u+ u · ν)|∂Ω± = 2ν ∧ (u|∂Ω±) = 2ν ∧(±1

2I + Ck)

(ν ∨ f)

= 2ν ∧ (λI + Ck) (ν ∨ f) + 2(±1

2 − λ)f. (7.47)

42

Also, 〈uc|∂Ω± · θ, 2ν ∧ (λI + Ck)(ν ∨ f)〉 = R3. Now, u|∂Ω+ − u|∂Ω− = ν ∨ f so that

⟨[2(λ− 1

2

) (λ+ 1

2

)u|∂Ω+ − 2

(λ− 1

2

) (λ+ 1

2

)u|∂Ω+

]c· θ, f

⟩= 2

(λ2 − 1

4

)〈ν ∨ f c, θ · f〉 = −2

(λ2 − 1

4

)〈ν · f c, θ · f〉 = −2

(λ2 − 1

4

)|f |2〈ν, θ〉 (7.48)

by the normality of f and Lemma 7.2.

One final remark is that

(∫ ∫Ω±|u|2 dVol

)1/2

= ‖Comp (f)‖. Indeed, the operator Ck : L2(∂Ω, E)→

L2(Ω, E) is compact, by (2) and (7) in Theorem 3.1. Hence, all in all,

RHSII = −2(λ2 − 1

4

)|f |2〈θ, ν〉+

∫∂ΩR4 dσ + ‖Comp (f)‖2. (7.49)

As a consequence, the entire identity II reads

(λ2 − 1

4

) ∫∂Ω|f |2〈θ, ν〉 dσ +

∫∂Ω|ν ∨ Ck(ν ∨ f)|2〈θ, ν〉 dσ =

∫∂ΩR5 dσ + ‖Comp (f)‖2. (7.50)

On account of (7.41) and (7.44), the estimate (7.35) follows easily from this.As a corollary of (7.35), we have that

ν ∧ (λI + Ck)(ν ∨ ·) : L2nor(∂Ω, E)→ L2

nor(∂Ω, E) is Fredholm

with index zero ∀λ ∈ R, |λ| ≥ 12 and ∀ k as in (7.28). (7.51)

Indeed, as (7.35) shows, the mapping

R\(−1

2 ,12

)3 λ 7→ ν ∧ (λI + Ck)(ν ∨ ·) ∈ L(L2

nor(∂Ω, E))

is a continuous path in the class of semi-Fredholm operators. Since, obviously, the operators becameinvertible for |λ| sufficiently large, the conclusion follows from the homotopic invariance of the index.A remark important in its own right is that the operators

λI + πtanCk, λI + πnorCk, λI + ν ∧ Ck(ν ∨ ·) are Fredholm with

index zero on L2(∂Ω, E)∀λ ∈ R with |λ| ≥ 12 and ∀ k as in (7.28). (7.52)

To see this, take for instance the case of λI + ν ∧ Ck(ν ∨ u). Under the splitting

L2(∂Ω, E) = L2tan(∂Ω, E)⊕ L2

nor(∂Ω, E)

the corresponding matrix realization of this operator is(λI λI

ν ∧ (λI + Ck)(ν ∨ ·) 0

). (7.53)

Now (7.51) applies to yield the desired conclusion. A similar reasoning works for the other two oper-ators. Here πtan := ν ∨ (ν ∧ ·), πnor := ν ∧ (ν ∨ ·) are the orthogonal projections of L2(∂Ω, E) ontoL2

tan(∂Ω, E) and L2nor(∂Ω, E), respectively. Next, we also claim that

43

λI + ν ∧ Ck(ν ∨ ·) : L2,dnor(∂Ω, E)→ L2,d

nor(∂Ω, E) is Fredholm

with index zero ∀λ ∈ R, |λ| ≥ 12 and ∀ k as in (7.28). (7.54)

To see this, for f ∈ L2nor(∂Ω, E) we write

λf + ν ∧ Ck(ν ∨ f) = λf − ν ∧ Ck(ν · f)

= λf + ν ∧∫∂Ω〈δxEk, f〉 dσ

+ν ∧∫∂Ω〈dxEk, f〉 dσ + k dt ν ∧

∫∂Ω〈Ek, f〉 dσ

=: A1 +A2 +A3. (7.55)

Now, A1 = (λI +Nk)f where

Nkf(x) := ν(x) ∧ p.v.∫∂Ω〈δxEk(x, y), f(y)〉 dσ(y), x ∈ ∂Ω, (7.56)

and it has been shown in [34] that λI +Nk satisfies the property described in (7.54). Also, d∂A2 = 0,f 7→ A3 is a compact operator on L2(∂Ω, E) and

A2 = ν ∧∫∂Ω〈δyEk, f〉 dσ = ν ∧

∫∂Ω〈Ek, d∂f〉 dσ.

In particular, the operatorL2,d

nor(∂Ω, E) 3 f 7→ A2 ∈ L2(∂Ω, E)

is compact. Similarly, L2,dnor(∂Ω, E) 3 f 7→ d∂A3 ∈ L2(∂Ω, E) is compact. By combining all the above,

(7.54) follows.The next step in our analysis is to show that, except for a discrete subset of R, the kernel of

12I + ν ∧ Ck(ν ∨ ·) on L2(∂Ω, E) is trivial. To this end, assume that f ∈ L2(∂Ω, E) is sent to zero by12I + ν ∧ Ck(ν ∨ ·) and let u := Ck(ν ∨ f) in Ω±. Then f ∈ L2

nor(∂Ω, E), u ∈ H2(Ω, IDk) and ν ∧ u = 0on ∂Ω.

As in the discussion in the previous paragraph (cf. also the case k = 0 treated in § 6), it can beshown that, in the present case, N (du), N (δu) ∈ L2(∂Ω). Going further,

IDku = 0 ⇒ du+ δu+ k dt · u = 0

⇒ du = −δu− k dt · u and δu = −du− k dt · u. (7.57)

Based on this, we deduce that

∫ ∫Ω|du|2 =

∫ ∫Ω〈du, duc〉 = −

∫ ∫Ω〈δu, duc〉 −

∫ ∫Ω〈k dt · u, duc〉

= −∫ ∫

Ω〈k dt · u, duc〉. (7.58)

Similarly,

44

∫ ∫Ω|δu|2 =

∫ ∫Ω〈δu, δuc〉 = −

∫ ∫Ω〈du, δuc〉 −

∫ ∫Ω〈k dt · u, δuc〉

= −∫ ∫

Ω〈k dt · u, δuc〉. (7.59)

Summing up (7.58) and (7.59) we get∫ ∫Ω|du|2 + |δu|2 = −

∫ ∫Ω〈k dt · u, (δu+ du)c〉 =

∫ ∫Ω|k|2|u|2. (7.60)

We now claim that also ∫ ∫Ω|du|2 + |δu|2 =

∫ ∫Ωk2|u|2. (7.61)

To see this, first observe that

ν ∧ (dt · u) = ν ∧ (dt ∧ u− dt ∨ u)

= −dt ∧ (ν ∧ u) + dt ∨ (ν ∧ u) = −dt · (ν ∧ u) = 0. (7.62)

Going to the boundary in δu = −du− k dt · u = 0 and applying ν ∧ ·, we obtain

ν ∧ δu = d∂(ν ∧ u)− kν ∧ (dt · u) = 0. (7.63)

With this at hand, we now write, via integration by parts

∫ ∫Ω|du|2 + |δu|2 − k2|u|2 =

∫ ∫Ω〈−(∆ + k2)u, uc〉+

∫∂Ω〈u, ν ∧ δuc〉 dσ

−∫∂Ω〈ν ∧ u, duc〉 dσ = 0 (7.64)

where the last equality follows from (7.63), (7.10) and the fact that ν ∧ u = 0 on ∂Ω. This proves(7.61). Combining (7.60) and (7.61) we finally arrive at∫ ∫

Ω(|k|2 − k2)|u|2 dVol = 0. (7.65)

Let us now make the assumption that k /∈ R. It follows from (7.65) that u = 0 in Ω+ and, further,that

ν ∨ u|∂Ω− = ν ∨ u|∂Ω+ = 0.

Hence, in Ω−, u satisfies IDku = 0, ν ∨ u = 0. A similar reasoning to the one above shows that, sincek /∈ R, u must vanish identically in Ω− also. Thus, f = ν ∧ u|∂Ω+ − ν ∧ u|∂Ω− = 0 which, in concertwith (7.51), (7.52) and (7.54), proves that

12I + ν ∧ Ck(ν ∨ ·) is an isomorphism of L2(∂Ω, E), L2

nor(∂Ω, E),

L2nor(∂Ω, E) and L2,d

nor(∂Ω, E) whenever k /∈ R. (7.66)

There remains to analyze the case when k ∈ R. To this effect, fix k0 ∈ C\R and decompose

45

12I + ν ∧ Ck(ν ∨ ·) =

[12I + ν ∧ Ck0(ν ∨ ·)

]−1[I −Rk] (7.67)

where

Rk :=[

12I + ν ∧ Ck0(ν ∨ ·)

][ν ∧ (Ck0 − Ck)(ν ∨ ·)]. (7.68)

Now, from [34], it follows that

C\Spec (∆) 3 k 7→ Rk ∈ L(L2(∂Ω, E)) (7.69)

is an analytic family of compact operators and Rk0 = 0. The analytic Fredholm theorem applies andgives that I−Rk is invertible on L2(∂Ω, E) except at most for a discrete subset U of C\Spec (∆). From(7.66) it then follows that U ⊆ R. Summarizing, the above analysis (carried out in Ω±) shows that

±12I + ν ∧ Ck(ν ∨ ·) is an isomorphism of L2(∂Ω, E), L2

nor(∂Ω, E)

and L2,dnor(∂Ω, E), ∀ k except for a countable subset of R. (7.70)

Emphasizing the boundary condition “ν ∨ u = given on ∂Ω” in place of “ν ∧ u = given on ∂Ω”, leadsto the conclusion that

±12I + ν ∨ Ck(ν ∧ ·) are isomorphisms of L2(∂Ω, E), L2

tan(∂Ω, E)

and L2,δtan(∂Ω, E), ∀ k except for a countable subset of R. (7.71)

There is in fact a more substantial connection between the operators in (7.70) on the one hand, andthe operators in (7.71) on the other hand, than the obvious formal analogy. The point is that, at thelevel of L2(∂Ω, E),

C∗k = ν · C−kc(ν·). (7.72)

This can be proved directly starting from (7.30) and utilizing (7.10) together with the correspondingproperties of Ek(x, y). In turn, (7.72) allows one to conclude that, e.g., for λI + ν ∧ Ck(ν ∨ ·) onLpnor(∂Ω, E), 1 < p <∞, λ ∈ R,

[λI + ν ∧ Ck(ν ∨ ·)]∗ = −ν ∧ [−λI + ν ∨ C−kc(ν ∧ ·)](ν ∨ ·) (7.73)

on Lqnor(∂Ω, E), 1p + 1

q = 1. Next, we aim at showing that there exists ε > 0 so that

(7.70)-(7.71) extend to the Lp-context for |2− p| < ε. (7.74)

To this end, fix k0 ∈ C\R and let ε > 0 be so that (7.70)-(7.71) hold in Lp for 2 − ε < p < 2 + ε andk := k0. That this is possible, is assured by well known stability results; cf, e.g., [22] for a discussion.Thus, since by [34], Ck0 − Ck is compact on the spaces of interest for any k ∈ C\Spec (∆), it followsthat the operators in (7.70)-(7.71) are Fredholm with index zero. Now, generally speaking, if T isFredholm with index zero on Lp(∂Ω), 2 − ε < p < 2 + ε and invertible for p = 2, then T is invertibleon Lp(∂Ω) for each 2− ε < p < 2 + ε. This can be used to finish the proof of (7.74).

Before proceeding any further, there are three remarks which we wish to make at this stage. Thefirst is that, for k as in (7.28) and 2− ε < p < 2 + ε,

46

Im(±1

2I + ν ∧ Ck(ν ∨ ·);Lpnor(∂Ω, E))

= u|∂Ω; u ∈ Hq(Ω∓, ID−k), ν ∧ u = 0, (7.75)

where . . . denotes the annihilator of . . . ⊆ Lq(∂Ω, E) in Lpnor(∂Ω, E), 1p + 1

q = 1. This follows easilyfrom (7.73), and the fact that u = Ck(u|∂Ω) for any u ∈ Hq(Ω±, IDk); cf. § 4. In passing, let us alsonote that the space u ∈ Hq(Ω±, IDk); ν ∧ u = 0 is independent of q ∈ (2− ε, 2 + ε) for ε > 0 small.

The second remark is that, if | Im k| > |Re k|, λ ∈ R, |λ| ≥ 12 , and 2− ε < p < 2 + ε, then

λI + ν ∧ Ck(ν ∨ ·) is invertibleon Lpnor(∂Ω, E) and on Lp,dnor(∂Ω, E). (7.76)

This can be seen by combining the technique in [20] with the results in [34]. We omit the details.Finally, the third remark is that the range 2 − ε < p < 2 + ε for which (7.34) was shown to hold

is, in fact, sharp in the class of all Lipschitz domains. Indeed, counterexamples to the solvability ofthe Lp-Neumann and the Lp-Regularity boundary problems for the Helmholtz in Lipschitz domains forp > 2 (cf. [33] for a related discussion) readily translate into counterexamples to the estimates

‖N (u)‖Lp(∂Ω) ≤ C‖ν ∧ u‖Lp(∂Ω,E), ‖N (u)‖Lp(∂Ω) ≤ C‖ν ∨ u‖Lp(∂Ω,E) (7.77)

for null-solutions u of IDk, for p > 2. In turn, these can be used to show that ±12I + ν ∨ Ck(ν ∧ ·)

and ±12I + ν ∧ Ck(ν ∨ ·) fail to be isomorphisms on Lptan(∂Ω, E) and on Lpnor(∂Ω, E), respectively, for

p > 2 (on appropriate domains). This and (7.73) can now be used to conclude that range of validityfor (7.34) is also sharp when p < 2.

Returning now to the main line of discussion, we are ready to present the

Proof of Theorem 7.1. We tackle part (1). In order to do so, denote by ±kjj the union of allexceptional sets in (7.70)-(7.71), kj ≥ 0. Assuming k /∈ ±kjj and 2− ε < p < 2 + ε, with ε > 0 as in(7.74), a solution for (BV Pnor)k is given by

u := Ck[ν ∨

(12I + ν ∧ Ck(ν ∨ ·)

)−1f

]in Ω. (7.78)

Clearly, (7.14) holds for this u. To see that the solution is unique, start with u ∈ Hp(Ω, IDk)satisfying ν ∧ u = 0 on ∂Ω. Since, by § 4,

u = Ck(u|∂Ω) = Ck(ν ∧ (ν ∨ u)) in Ω, (7.79)

it follows, after going to the boundary and taking ν ∨ · of both sides, that[12I + ν ∨ Ck(ν ∧ ·)

](ν ∨ u) = 0 on ∂Ω. (7.80)

Since k /∈ ±kjj , this in concert with (7.71) imply that ν ∨ u = 0 which, further, forces u = 0 in Ω byvirtue of (7.79). Thus, the solution is unique for k /∈ ±kjj and this completes the proof of (1).

Turning our attention to (2), the first equivalence is clear in view of Nu ∈ Lp(∂Ω) and du+δu+k dt·u = 0 in Ω. Also, N (du) ∈ Lp(∂Ω)⇒ d∂f = d∂(ν ∧ u) = −ν ∧ du ∈ Lp(∂Ω, E) so that f ∈ Lp,dnor(∂Ω, E).Conversely, if f ∈ Lp,dnor(∂Ω, E) then, from (7.70) and (7.78) it follows that

u = Ck(ν ∨ g), where g :=[

12I + ν ∧ Ck(ν ∨ ·)

]−1f ∈ Lp,dnor(∂Ω, E). (7.81)

Thus,

47

u =∫∂Ω〈dxEk(x, y), g(y)〉 dσ(y) +

∫∂Ω〈δxEk(x, y), g(y)〉 dσ(y),

−k∫∂Ω〈Ek(x, y), dt g(y)〉 dσ(y) =: I + II + III. (7.82)

Now, dI = 0, N (dIII) ∈ Lp(∂Ω) are immediate. Finally, the integral kernel of dII is

dxδxEk(x, y) = −δxdxEk(x, y)− (∆x + k2)Ek(x, y) + k2Ek(x, y)

= −δydyEk(x, y) + k2Ek(x, y), (7.83)

so thatdII = −

∫∂Ω〈dyEk(x, y), (d∂g)(y)〉 dσ(y) + k2

∫∂Ω〈Ek(x, y), g(y)〉 dσ(y).

In this writing, it is clear that N (dII) ∈ Lp(∂Ω), as desired. This finishes the proof of (2).Consider next (3), i.e. the case when k ∈ ±kjj so that, in particular, k is real. Let V ≥ 0,

V ∈ C∞(M), suppV ⊆ M\Ω so that ID2k + V is invertible from H1,2(M, E) onto H−1,2(M, E) (cf.

the discussion in [34]). Also, let Ck, Ck be the Cauchy operators constructed as in (4.20)–(4.21) butin connection with the Dirac type operator IDk. Consider a null-solution u of (BV Pnor)k in (7.13).Cauchy’s reproducing formula (4.5) gives

u = Ck(u|∂Ω) = Ck(ν ∧ (ν ∨ u)) in Ω (7.84)

Going to the boundary and taking ν ∨ · implies that[12I + ν ∨ Ck(ν ∧ ·)

](ν ∨ u) = 0 on ∂Ω, (7.85)

i.e. ν∨u ∈ Ker(

12I + ν ∨ Ck(ν ∧ ·); Lptan(∂Ω, E)

). The important observation now is that this kernel is

finite dimensional. Indeed, if follows from [34] that the difference between Ck and Ck0 with k0 ∈ C\Ris a compact operator in Lp(∂Ω, E). Hence, the conclusion follows from (7.71).

In particular, the mapping assigning to each null-solution u of (BV Pnor)k the form ν ∨ u ∈Ker

(12I + ν ∨ Ck(ν ∧ ·); Lptan(∂Ω, E)

)has a finite dimensional range. Since, by (7.84), this correspon-

dence is also one-to-one, we conclude that

dim null solutions of (BV Pnor)k < +∞. (7.86)

Finally, if the boundary datum f belongs to

Im[

12I + ν ∧ Ck(ν ∨ ·); Lpnor(∂Ω, E)

](7.87)

then a solution to (BV Pnor)k can be produced in the form u = Ck(ν ∨ g) for a suitable g ∈ Lpnor(∂Ω, E).Note that, by the previous discussion, the space (7.87) has finite codimension in Lpnor(∂Ω, E). Let usremark that, when k is as in (7.28) and k ∈ ±kjj , then (7.75) offers a better description of thenull-space of (BV Pnor)k. Also, the left side of (7.75) is the “right” space to select boundary data from.

Since (4) follows in a similar manner to (1)-(3), the proof of Theorem 7.1 is finished. 2

An immediate consequence of Theorem 7.1 is the following

48

Corollary 7.7 Let Ω be a Lipschitz subdomain of M and let kjj, 2 − ε < p < 2 + ε be as in thestatement of Theorem 7.1. Then for k /∈ ±kjj, the normal-to-tangential map

NTk : ν ∧ u 7→ ν ∨ u, u ∈ Hp(Ω, IDk), (7.88)

is well-defined, linear and bounded from Lpnor(∂Ω, E) into Lptan(∂Ω, E), as well as from Lp,dnor(∂Ω, E) intoLp,δtan(∂Ω, E). In each case, NTk is an isomorphism, whose inverse is given by NT−1

k : ν ∨ u 7→ ν ∧ u,u ∈ Hp(Ω, IDk), the tangential-to-normal map.

Another related result of interest is recorded below.

Corollary 7.8 Let Ω be a Lipschitz subdomain of M and let kjj, 2 − ε < p < 2 + ε be as in thestatement of Theorem 7.1. Then for k /∈ ±kjj, the operator ν ∧ Ck(ν ∧ ·) is an isomorphism fromLptan(∂Ω, E) onto Lpnor(∂Ω, E) and from Lp,δtan(∂Ω, E) onto Lp,dnor(∂Ω, E). Moreover, a similar set of resultsworks in the case of the operator ν ∨ Ck(ν ∨ ·).

Proof. Writing ν ∧ Ck(ν ∧ ·) = −[−12I + ν ∧ Ck(ν ∨ ·)] NT−1

k , itself a direct consequence of jump-formulas and definitions, and availing ourselves of (7.70) together with Corollary 7.7, the desiredconclusion follows. 2

With the proof of Theorem 7.1 behind us, let us try to rephrase (BV Pnor)k exclusively in termsof differential forms living on M , i.e. GC

M -valued sections, where GCM := ⊕ml=0ΛlTCM . To this effect,

observe that if u = H − i dt ·E where E,H ∈ C0(Ω,GCM ), then IDku = 0 becomes equivalent to having

(E,H) solve (1.13). In particular,

u ∈ Hp(Ω, IDk)⇔u = H − i dt · E, E,H ∈ C0(Ω,GC

M ),

N (E), N (H) ∈ Lp(∂Ω) and (E,H) solves (1.13).(7.89)

Also,

ν ∧ u = h ∈ Lpnor(∂Ω, E)⇔h = g − i dt · f, f, g ∈ Lpnor(∂Ω,GC

M ),

ν ∧ E = f and ν ∧H = g on ∂Ω.(7.90)

Hence, (BV Pnor)k becomes a boundary value problem for the generalized Maxwell system (1.13), i.e.

(BV P nor)k

E,H ∈ C0(Ω,GCM ),

δE + dE − ikH = 0 in Ω,

δH + dH + ikE = 0 in Ω,

N (E), N (H) ∈ Lp(∂Ω),

ν ∧ E = f ∈ Lpnor(∂Ω,GCM ),

ν ∧H = g ∈ Lpnor(∂Ω,GCM ).

(7.91)

This system should be compared with the standard Maxwell system

49

(Maxwell)k

E,H ∈ C0(Ω,GCM ),

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω,

δE = 0, dH = 0 in Ω,

N (E), N (H) ∈ Lp(∂Ω),

ν ∧ E = f ∈ Lpnor(∂Ω,GCM ).

(7.92)

For a treatment of (Maxwell)k see [36], [20] and [34].There are two important things which we would like to point out at this stage. First, as it is

apparent from the ordinary Maxwell system, δE = 0 and dH = 0 in Ω are automatically satisfied whenk 6= 0. Second, (Maxwell)k splits into a direct sum of boundary problems according to the degrees ofthe differential forms involved. Specifically, if for l = 0, 1, . . . ,m, denote by (Maxwelll)k the version of(7.92) in which the first condition has been replaced by E ∈ C0(Ω,ΛlTCM), H ∈ C0(Ω,Λl+1TCM),and the last condition by ν ∧ E = f ∈ Lpnor(∂Ω,Λl+1TCM). Then (E,H) ∈ C0(Ω,GC

M ) ⊕ C0(Ω,GCM )

solves (7.92) if and only if for each l

(El,Hl+1) := (πlE, πl+1H) ∈ C0(Ω,ΛlTCM)⊕ C0(Ω,Λl+1TCM)

solves (Maxwelll)k for the boundary datum fl+1 := πl+1f .Such a phenomenon, however, does not occur for (BV P nor)k. This is due to the more intricate

nature of the PDE in the latter problem. Our aim is to establish conditions under which (7.91) and(7.92) are equivalent. In one direction, the fact that any solution (E,H) of (7.92) also solves (7.91) isequivalent to

f ∈ Lp,dnor(∂Ω,GCM ) and g = −ik−1d∂f ∈ Lp,0nor(∂Ω,GC

M ). (7.93)

Indeed,

g = ν ∧H = −(ik)−1ν ∧ dE = −ik−1d∂(ν ∧ E) = −ik−1d∂f (7.94)

from which the conclusion follows. In fact, as our next theorem shows, granted (7.93), the converseimplication is also true.

Theorem 7.9 Let Ω be a Lipschitz domain in M and let ±kjj, 2 − ε < p < 2 + ε, be as inTheorem 7.1. Then, for k /∈ ±kjj, the boundary problem (BV P nor)k reduces to (Maxwell)k if andonly if the conditions in (7.93) are satisfied.

As a corollary, if 2 − ε < p < 2 + ε, k /∈ ±kjj, f ∈ Lp,dnor(∂Ω,Λl+1TCM) and g = −ik−1d∂f ∈Lp,0nor(∂Ω,Λl+2TCM), then (7.91) is equivalent to (Maxwelll)k. In particular, E and H have homoge-neous degrees l and l + 1, respectively. Also,

‖E‖Bp,p∨2

1/p(Ω,ΛlTCM)

+ ‖H‖Bp,p∨2

1/p(Ω,Λl+1TCM)

≤ C‖f‖Lp,dnor(∂Ω,Λl+1TCM)

. (7.95)

Proof. The only thing left to prove is that if the conditions (7.93) are fulfilled and (E,H) solves(BV P nor)k, then (E,H) also solves (Maxwell)k.

Proceeding backwards, we can start by first producing a solution (E′,H ′) for (Maxwell)k (in thepresent context, this has been done in [34]), so that δE′ = dH ′ = 0, a priori. Then, ν ∧ H ′ =

50

g automatically, as a reasoning analogue to (7.94) shows. It follows that (E′,H ′) is a solution of(BV P nor)k with the given boundary data (f, g) and which satisfies δE′ = dH ′ = 0.

Finally, employing the uniqueness part for (BV P nor)k (which is part of Theorem 7.1), it followsthat (E,H) = (E′,H ′) which yields the desired conclusion. 2

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[2] S. Bell, The Cauchy Transform, Potential Theory and Conformal Mapping, CRC Press, 1992.

[3] J. Bergh and J. Lofstrom, Interpolation spaces. An introduction, Springer-Verlag, 1976.

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Department of MathematicsUniversity of Missouri-ColumbiaMathematical Sciences BuildingColumbia, MO 65211, USAe-mail: marius@math.missouri.edu

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