Generalized Dirac Operators on Nonsmooth Manifolds and Maxwell s Equations

Transcription

1 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 subdomains of 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 of basic results from complex function theory in this general setting. For example, we study Plemelj- Calderón-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 Szegö projections and the corresponding L p -decompositions. Our approach relies on an extension of the classical Calderón- Zygmund theory of singular integral operators which allows one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell s equations, the main novelty is the treatment of the corresponding electro-magnetic boundary 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 operator whose square is the wave operator, Dirac type operators have become of central importance in many branches 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 elliptic systems of the first order generalizing the classical Cauchy-Riemann system give rise to a natural, rich function theory. The general aim of this paper is to develop such a function theory for a general Dirac operator D on a manifold M under minimal smoothness assumptions. A special emphasis is placed on studying Hardy type spaces associated with D, H p (, D) := {u; Du = 0 in, N (u) L p ()}, (1.1) in an arbitrary Lipschitz subdomain of M; here N (u) stands for the nontangential maximal function of 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 the complex plane, this topic is classical and a great deal of information is known; cf. the excellent accounts in [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 higher dimensions is the Dirac operator e j j within the context of a Clifford algebra generated by the Supported in part by NSF grant DMS # Mathematics Subject Classification. Primary 31C12, 42B20, 35F15, 42B30; Secondary 58G20, 42B25, 78A25. Key words. Dirac operators, Hardy spaces, Maxwell s equations, Lipschitz domains. 1

2 (anticommuting) imaginary units {e j } j. Hardy spaces in Lipschitz domains of R n associated with such operators have been studied in [29], [17], [35]. In the present paper we continue this line of research and take the next natural step by considering generalized 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 unique continuation property assumption. As is well known, Dirac operators naturally associated with Clifford algebra structures automatically satisfy (1.2). Thus, from this perspective, the primary role of Clifford algebras is to provide natural examples of operators D for which (1.2) holds. On the other hand, each operator 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 decomposition theorems which we now proceed to describe. Let be a Lipschitz domain in the manifold M and set + :=, := M \. Also, consider the boundary Hardy spaces H p ±(, D) := {u ; u H p ( ±, D)}. As is well known, when D :=, the Cauchy-Riemann operator and M := C, the Plemelj- Calderón decomposition L p () = H p (, D) H p +(, D), 1 < p <, (1.3) plays a basic role in complex and harmonic analysis. In particular, (1.3) is equivalent to the L p - boundedness of the classical Cauchy singular integral operator on the Lipschitz curve ; cf. the discussion in [32]. A natural question is the extent to which (1.3) remains valid when D is a generalized Dirac operator and the interface a Lipschitz submanifold (of codimension one) of M. We shall prove that (1.3) continues to hold in this general context but modulo finite dimensional spaces. Specifically, ( H p (, D), H+(, p D) ) is a Fredholm pair (cf. [23]) and Index ( H p (, D), H p +(, D) ) = Index D. (1.4) In particular, the index of D on M can be read off data living on. See Theorem 4.4 for a complete statement; this result has been announced in [38]. Let us point out that a version of this theorem in the smooth context has first been proved (cf. [6]) via techniques which do not work in the context of nonsmooth domains. At the heart of the matter is the fact that, in the presence of boundary irregularities, the relevant operators are longer pseudodifferential and only belong to the class of singular integrals. Another important consequence of (1.4) is that the transmission boundary problem (T BV P ) { u+ H p ( +, D), u H p (, D), u + u = f L p () is Fredholm solvable and its index is precisely Index D. In particular, the index of (T BV P ) is independent of the particular Lipschitz domain (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 arbitrary Lipschitz domain in M and denote by ν the outward unit normal to. Then there exists ɛ = ɛ(, D) > 0 so that (1.5) L p () = H p (, D) iσ(d ; ν)h p (, D ) (1.6) 2

3 for any 2 ɛ < p < 2 + ɛ, where the sum is direct and topological. Moreover, when σ(dd ; ξ) is scalar and ν vmo (), the class of functions of vanishing mean oscillations on, then (1.6) is valid for any 1 < p <. As explained in 4, the decomposition (1.6) is essentially equivalent to the statement that the orthogonal projection of L 2 () onto H 2 (, D), the so-called Szegö projection, extends to a bounded operator on L p (). In particular, it is illuminating to point out that (1.6) contains Riesz s classical estimate + + c n e inθ L C p c n e inθ L, 1 < p <, (1.7) n=0 p ( π,π) n= p ( π,π) as a very special case. We prove (1.6) from the analysis of a Kerzman-Stein type formula which we deduce in the present context (cf., e.g., [2] for more on this in smoother settings). We also consider L p -based Hodge type decompositions for Dirac type operators in Lipschitz domains. Specifically, we show that there exists ɛ = ɛ(, D) > 0 so that, for any 2 ɛ < p < 2 + ɛ, L p () = Ker (D ; L p ()) D H 1,p 0 (), (1.8) where the direct sum is topological. Furthermore, if D is actually Dirac (i.e. D D has scalar principal symbol), then (1.8) is shown to hold for the larger range 3/2 ɛ < p < 3 + ɛ. A key ingredient in the latter result is the recent solution of the Poisson problem for the Laplace-Beltrami operator in Lipschitz domains from [41]. In the second part of the paper we consider boundary problems for Dirac type operators in Lipschitz domains. 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 M and with P ± standing for 1 2 (I ± σ(d; ν)), I being the identity operator, we discuss in 5 the following boundary problem: (BV P ± ) { u H p (, D), P ± (u ) prescribed in L p (). The trace u is taken in the pointwise (nontangential) sense; this is meaningful a.e. on. At the present time there is no analogue of the concept of regular elliptic problem in the nonsmooth setting but, broadly speaking, each such problem is teated via ad-hoc methods. Here we identify the right spaces and operators for the method of layer potentials to apply. Next, we consider what we call the Maxwell-Dirac operator (1.9) ID k := 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 is to initiate a detailed investigation of boundary problems of the type (BV P ) { u H p (, ID k ), u tan or u nor prescribed in L p (). (1.11) Here, again, is an arbitrary Lipschitz subdomain of M and u tan, u nor are, respectively, the tangential and the normal component of u on. It is therefore natural to think of (1.11) as half-dirichlet problems for ID k. 3

4 When all structures involved are smooth and solutions are sought in a sufficiently regular class of functions, 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 changes when 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 pseudodifferential operators. One notable difficulty in the case we are interested, i.e. Lipschitz boundaries, metric tensors with a very limited amount of smoothness, is the absence of an algebra structure and the lack of a symbolic calculus within the class of general singular integral operators. Our approach utilizes an array of tools from harmonic analysis which have been successful in the treatment of second-order, constant coefficient elliptic boundary problems in Lipschitz domains of the Euclidean space. See [25] and the references cited there for a survey of the state of the art in this field up 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 variable first-order elliptic systems, is a natural continuation of this work. A basic goal of this program is to achieve an elliptization of (the non-coercive) Maxwell s equations (Maxwell) { de ikh = 0 in, δh + ike = 0 in, (1.12) with N (E), N (H) L p () and E tan or E nor prescribed in L p (), by embedding (1.12) into a larger, half-dirichlet problem for a suitable Dirac type operator. Indeed, as explained in last part of our paper, (1.12) can be understood as a particular manifestation of the half-dirichlet problem for Maxwell-Dirac operator (1.10). In this case, under the identification u = H i dt E, (1.11) becomes ( Generalized Maxwell ) E, H C 0 (, G C M ), δe + de ikh = 0 in, δh + dh + ike = 0 in, (1.13) with N (E), N (H) L p () and E tan, H tan or E nor, H nor prescribed in L p (), and we give necessary and sufficient conditions on the boundary data guaranteeing that (1.11) and (1.13) are equivalent. Of course, for this result to have practical value, we first need to give a thorough solution to (1.11) to begin with. See Theorem 7.1 for this. Following the work in the three-dimensional case in [37], the boundary problem (1.12) has been first solved on Lipschitz domains in [36], [20], [34] via integral equation methods. The philosophy of the approach in these papers is to reduce (1.12) to boundary problems for the (perturbed) Hodge-Laplacian with absolute and relative boundary conditions. In particular, one transforms it into a second order PDE. Here we provide an alternative approach based on working directly with the more general, first order system (1.13). Ultimately, implementing this idea requires understanding connections between Dirac operators and Maxwell s equations in non-smooth domains. In the flat, Euclidean setting and for constant coefficient operators, this direction of research has been initiated in [31] and [30]. A basic ingredient in [31] is a Rellich type estimate to the effect that u tan L 2 () u nor L 2 (), (1.14) for two-sided monogenic functions in, i.e. elements in H 2 (, ID k ) which are also annihilated by the action of ID k to the right. A step forward in the direction of dealing with one-sided Clifford modules 4

5 (as in the case of manifolds) was taken in [30] where the monogenicity assumptions were relaxed. In the present paper we continue this program by showing that (1.14) holds for any u H 2 (, ID k ) and arbitrary Lipschitz subdomain of a Riemannian manifold. In somewhat greater detail, the organization of the paper is as follows. Section 2 contains a discussion of the global invertibility properties of D and Laplacians associated with D. A function theory associated with a general, first-order, variable coefficient elliptic system, with a special emphasis on Hardy type spaces and Cauchy like operators in Lipschitz domains is developed in 3-4. How Dirac type operators fit in this general framework makes the subject of Section 5. Boundary value problems for Hodge-Dirac operators of the form d+αδ, α R, in Lipschitz subdomains of Riemannian manifolds are studied in Section 6. The first part of Section 7 is reserved for a similar discussion, this time pertaining to the Maxwell-Dirac operator (1.10). Finally, in the second part of this section we elaborate on the connections between the half-dirichlet problem for the perturbed Dirac operator (1.10) and the Maxwell system (1.12). Acknowledgments. I am grateful to Alan McIntosh for sharing his ideas with me and for the many spirited conversations we have had during his visit at UMC in the Fall of His constructive suggestions led to several improvements in 7. I also thank Bernhelm Booß-Bavnbek for his insightful comments, David Calderbank for giving me a copy of his thesis [8] and Michael Taylor for an inspiring discussion during his visit at UMC in the Fall of Inverting generalized Dirac operators and Laplacians Let M be a compact, boundaryless, smooth, orientable manifold, of real dimension m. E, F M smooth vector bundles and Consider D : C 1 (M, E) Meas (M, F) (2.1) a first order, elliptic (i.e. with an invertible symbol) differential operator mapping C 1 sections of E into measurable sections of F. Assume that, in local coordinates, Du = a αβ j j u β f α + b αβ u β f α, (2.2) where (e β ) β, (f α ) α are local frames for E and F, respectively, and u = u β e β, with a αβ j C γ, γ > 0, and b αβ L r, r > m. (2.3) The first order of business is to study the global action of the operator D on the manifold M. In this context, we shall prove that D is invertible modulo finite dimensional spaces, i.e. it is a Fredholm operator. To state the main result in this direction, denote by H s,p the usual scale of Sobolev spaces. Also, recall the index r from (2.3). Theorem 2.1 Granted (2.3), the operator D : H 1,p (M, E) L p (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: In particular, if γ > 1, then u H τ,p, Du L q u H 1,q. (2.5) 5

6 u L p and Du = 0 = u C α for some α > 0. (2.6) In the smooth context, this is well known. The primary interest in this result stems from the low regularity 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 subsequent pseudodifferential operators properly supported. Then, via a partition of unity, we can write with D # OP C S 1 1,δ, D = D # + D b + B (2.7) D b OP C γ S 1 γδ 1,δ, B L (M, Hom(E, F)) (2.8) for some 0 < δ < 1. Let E OP C S 1 1,δ be a two-sided parametric for the elliptic operator D#. Then ED = I + ED b + EB modulo a smoothing operator. (2.9) Now, if 1 γ < s < 1 + γ(1 δ), it follows that D b : H s,p H s 1+γδ,p. (2.10) See [7], [27] and [43] for a discussion of mapping properties of pseudodifferential operators whose symbols have a limited amount of smoothness. In particular, for s as before, ED b : H s,p H s+γδ,p. (2.11) Hence, ED b is a compact operator from H 1,p into itself. Going further, assume p m and observe that EB : H 1,p H 1,p factors as H 1,p L p B L t E H 1,t H 1,p (2.12) where 1 p := 1 p 1 m, 1 t := 1 p + 1 r, and the first inclusion is the usual Sobolev embedding. Note that r > m entails t > p so that the last inclusion is well-defined and compact. The case p > m is similar and requires r > p. Hence, in any event, D in (2.4) has a quasi-inverse to the left. Similarly, DE = I + D b E + BE modulo smoothing operators, and the composition is compact. Also, if p m, then where p < p, 1 p = 1 p 1 m and 1 t = 1 p L p E H 1,p Db H γδ,p L p (2.13) L p E H 1,p H 1,p L p B L t L p (2.14) + 1 r. Since the first inclusion is compact, so is the entire composition. When p > m it follows that q =, t = r and the last inclusion holds if r > p. This proves the first part of the theorem. With regard to (2.5), we have from (2.9) u = E(Du) ED b u EBu, mod C. (2.15) 6

7 Now, u H τ,p with τ + γ > 1 implies ED b u H τ+γδ,p. If p m τ we see, as in (2.12) but starting with H τ,p in place of H 1,p, that EBu H 1,t with 1 t = 1 p τ m + 1 r. If p > m τ, then EBu H1,r. Also, Du L q E(Du) H 1,q. Hence, u H 1,q + H τ+δγ,p + H 1,t 0, δ<1 where t 0 := t if p m τ, and t 0 := r otherwise. Since r > max{m, q} (by our hypotheses), this is an improvement over the original regularity assumption on u and the procedure can be iterated sufficiently many times to yield (2.5). Next, we take up the task of studying the kernel of D in (proper) Lipschitz subdomains of M when no boundary conditions are imposed. As we shall see momentarily, this leads to a natural concept of Hardy spaces associated with D. Before we do so, however, we make the following definition. The operator D is said to have the unique continuation property (abbreviated UCP henceforth) if u H 1,2 (M, E), Du = 0 on M u 0 or supp u = M. (2.16) If this holds, we simply write D UCP. Assume that the Riemannian metric on M has H 2,r coefficients for some r > m = dim M, i.e. g H 2,r (M, Hom(T M T M, R)), (2.17) and denote by d Vol the corresponding volume element on M. Also, equip E and F with H 2,r Hermitian structures. Let D be as in (2.1)-(2.2). From now on, strengthen (2.3) to a αβ j H 2,r, b αβ H 1,r for some r > m. (2.18) An important observation is that, under the current assumptions, the coefficients of D, the formal adjoint of D, also satisfy (2.18). Fix V C (M), a scalar, positive, non-identically zero function with M\ supp V and consider the second-order differential operator Locally, if u = u β e β, L := DD + V. (2.19) Lu = j a αβ jk ku β f α + b αβ j j u β f α + j (c αβ j u β )f α + d αβ u β f α. (2.20) Then L is a formally self-adjoint, strongly elliptic operator whose coefficients satisfy a αβ jk C1+γ, b αβ j, c αβ j H 1,r, d αβ L r, (2.21) for some γ > 0, r > m. Proposition 2.2 With the above notation and hypotheses, we have In a similar fashion, if D UCP L : H 1,2 (M, F) H 1,2 (M, F) is invertible. (2.22) L := D D + V (2.23) 7

8 then D UCP L : H 1,2 (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 H 1,2 is so that Lu = 0, it follows that 0 = Lu, u = ( D u 2 + V u 2 ) dvol. (2.25) M Hence, D u = 0 in M and u = 0 in supp V. 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. Next we tackle the issue of invertibility for D itself under the additional hypothesis that D is formally 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 in M. There exists a smoothing operator P on M whose integral kernel is supported in O O and so that is an invertible operator. Proof. Consider the mapping Φ : C comp(o) D P : H 1,2 (M, E) L 2 (M, E) (2.26) [ Ker (D : H 1,2 (M, E) L 2 (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 : H 1,2 (M, E) L 2 (M, E)) is finite dimensional, one can select a finite dimensional subspace V C comp(o) so that Φ : V [ Ker (D : H 1,2 (M, E) L 2 (M, E))] is an isomorphism. (2.28) Pick now P : L 2 (M, E) L 2 (M, E) to be the orthogonal projection onto V. There remains to show that the operator (2.26) is invertible. To this end, observe first that, since P is of finite rank, (2.26) is Fredholm, thanks to Theorem 2.1. Next we prove that it is also one-to-one. Indeed, this follows more or less directly from the fact that V Im D = {0} (2.29) since, as (2.28) implies, Ker P Ker D = {0}. In turn, (2.29) is a simple consequence of (2.28) and Green s formula (note that the selfadjointness of D is used here). This concludes the proof of the injectivity of the operator (2.26). Passing to adjoint, we get that D P : L 2 (M, E) H 1,2 (M, E) is onto. (2.30) Now, if f L 2 (M, E), it follows from (2.30) that there exists u L 2 (M, E) so that (D P )u = f. Thus, since P is smoothing, Du = f + P u L 2 (M, E) and, further, u H 1,2 (M, E), by the regularity result in Theorem 2.1. This proves that the operator (2.26) is surjective. 8

9 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 elliptic differential operator D satisfying (3.1) a generalized Dirac operator. It is interesting to point out in this connection that, according to an old conjecture of L. Schwartz, the two conditions in (3.1) are in fact equivalent (in the smooth context). In order to continue, we need some notation. Let M be a fix Lipschitz domain. This means that, in appropriate local coordinates, is locally described by graphs of (Euclidean) Lipschitz functions. Denote by dσ the surface measure on and by ν T M the unit (outward) conormal to. We set + :=, := M \, and let ± be the nontangential boundary trace operators on ±. That is, u ± (x) := lim u(y), x, (3.2) y γ ± (x) where γ ± (x) ± are appropriate nontangential approach regions. Finally, we let N stand for the nontangential maximal operator defined for sections u defined in + or by setting N u(x) := sup { u(y) ; y γ ± (x)}, x, (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 supp V =. For a fixed such V, we consider: E the Schwartz kernel of L 1, E D (M M, F F), (3.4) Ẽ the Schwartz kernel of L 1, Ẽ D (M M, E E), (3.5) Γ(x, y) := (D x Id y )E(x, y), Γ D (M M, E F), (3.6) Γ(x, y) := (Id x D y )Ẽ(x, y), Γ D (M M, E F). (3.7) In (3.7), Du := [Du c ] 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 L p boundedness and jump relations) to the ordinary Cauchy operator on Lipschitz curves of the complex plane as discussed in [9]. For now, if f : E is an arbitrary section set: Cf(x) := Cf(x) := Cf(x) := p.v. Γ(x, y), iσ(d; ν(y))f(y) y dσ(y) for x /, Γ(x, y), iσ(d; ν(y))f(y) y dσ(y) for x /, Γ(x, y), iσ(d, ν(y))f(y) y dσ(y) for x, 9

10 Cf(x) := p.v. Γ(x, y), iσ(d; ν(y))f(y) y dσ(y) for x. Here p.v. indicates that the integral is taken in the Cauchy principal value sense, i.e. by removing geodesic balls. See [39] and [34] for a discussion. Also, σ(d; ξ) stands for the principal symbol of D at ξ Tx M \ 0, x M, and i = 1. While the operators C, C (and C, C) are, in many respects, similar, there are some important differences. Most notably, generic elements in the range of C satisfy a first order PDE in, whereas those in the range of C satisfy a second order PDE in. Another way of understanding C is as follows. Note that the integral kernel of C is iσ(d t ; ν(y)) D y Ẽ(x, y). The first order differential operator σ(d t ; ν) D can be thought of as a natural conormal derivative for L = D D + V and, hence, C can be thought 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, this extends results from the Euclidean context and for standard constant coefficient Dirac operators in [9], [24], [17] and [35]. Theorem 3.1 Let be a Lipschitz subdomain of M. With the above notation and hypotheses, the following are true: (1) C and C are bounded operators on L p (, E) for 1 < p < ; (2) N (Cf) L p () C f L p (,E) uniformly for f L p (, E), and we have DCf = 0 in ; (3) N ( Cf) L p () C f L p (,E) uniformly for f L p (, E) and L c Cf = 0 in ; (4) Cf ± = (± 1 2 I + C)f and Cf ± = (± 1 2 I + C)f for any f L p (, E), 1 < p < (Plemelj jump 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 (D x Id y )Γ(x, y) = δ x (y) for x. (3.8) Hence, D(Cf) = 0 in. The estimate N (Cf) L p () C f L p (,E) is also a direct consequence of Theorem 2.9 in [34]. The point (3) follows similarly. As for (4), the general jump formulas from [34] give Cf ± (x) = 1 2 i σ(d ; ν(x))σ(l; ν(x)) 1 iσ(d; ν(x))f(x) + Cf(x), for any f L p (, E), 1 < p < and a.e. x. Upon noticing that σ(d ; ν)σ(l; ν) 1 σ(d; ν) = σ(d ; ν)σ(d ; ν) 1 σ(d; ν) 1 σ(d; ν) = Id, the first part of (4) follows. The case of C is handled analogously. In a series of theorems below we address the issue of the global interior regularity for the operator C. This is going to be measured either on the Sobolev spaces H s,p or the Besov spaces Bs p,q. A good reference for the latter class (considered on or on ) is [21]. To state our first result recall that a b := max {a, b}. 10

11 Theorem 3.2 Let 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 is well defined and bounded. C : L p (, E) B p,p 2 1/p (, E) (3.9) Proof. Assume that the metric tensor g is given locally by g jk dx j dx k and set (g jk ) j,k := [(g jk ) j,k ] 1, g := det [(g jk ) j,k ]. As in 2 of [34], locally we decompose Ẽ(x, y) = 1 } {e 0 (y, x y) + e 1 (x, y), (3.10) g(y) where e 0 (y, x y) is the Schwartz kernel of the classical pseudodifferential operator E 0 (D, x) ØPC 1+µ S 2 cl, some µ > 0, whose symbol is [ E 0 (ξ, y) := ( j,k,γ ] 1, a αγ j (y)a βγ k (y)c ξ j ξ k ) αβ (3.11) and e 1 (x, y) is a residual term. Thanks to (2.3), in local coordinates the latter satisfies e 1 (x, y) x y 2 + x e 1 (x, y) x y + y e 1 (x, y) x y + x y e 1 (x, y) C ɛ x y (m 1+ɛ), ɛ > 0. (3.12) The decomposition (3.10), at the level of kernels, naturally induces a splitting C = C 0 + C 1. That C 1 maps L p (, E) into B p,p 2 1/p (, E) for each 1 < p < is elementary and follows from (3.12). As for C 0, the following result from [41] applies. Let q(d, x) ØPC 0 Scl 1 have an odd principal symbol. Then the integral operator whose kernel is the Schwartz kernel of q(d, x) maps L p (, E) into B p,p 2 1/p (, E) for each 1 < p <. We continue by discussing the action of C on the Sobolev scale H 1,p (, E), 1 < p <. Theorem 3.3 Let be an arbitrary Lipschitz subdomain of M. Also, assume that the coefficients of the operator D and the metric tensors are C 1+γ for some γ > 0. Then, for each 1 < p <, there exists C = C(p, ) > 0 so that N ( Cf) L p () C f H 1,p (,E) (3.13) for any f H 1,p (, E). In particular, the operator C : H 1,p (, E) H 1,p (, E) is well-defined and bounded. Proof. We first present a proof which works in the case when the coefficients of D and the metric tensors are C. Then we indicate how this can be modified in the case of structures exhibiting a limited amount of smoothness. To this end, with an eye on (3.13), fix f H 1,p (, E) for some 1 < p <. Also, recall the local description of D in (2.2). Working in local coordinates and with orthonormal frames we see that the α-component of Cf is, modulo lower order terms, given by a µβ Σ j (y) c a µγ (y)n k(y) k ( ) ẽ αβ (x, y) f γ (y) g(y) dσ Σ (y) (3.14) y j 11

12 where the superscript c denotes complex conjugation. Hereafter, the summation convention is tacitly used. Also, R m is the image of in local coordinates, dσ Σ the area element on inherited from the Riemannian metric, n = (n k ) k is the unit normal to with respect to the Euclidean metric, (ẽ αβ ) α,β are the entries in Ẽ(x, y), and dvol = g dx, where dx is the ordinary Lebesgue measure in R m. Before we proceed with the main arguments, we make an important observation to the effect that, for any 1 < p < and j {1,..., m}, ( Ẽ x (x, y) + Ẽ ) (x, y) y j x j ( Ẽ and y (x, y) + Ẽ ) (x, y) y j x j are kernels which yield bounded operators on L p (Σ). (3.15) Indeed, take for instance the first expression in (3.15). With [, ] standing for the usual commutator bracket, this is the Schwartz kernel of x [ L 1, / x i ]. Now, if p(x, ξ) Scl 2 is the principal symbol of L 1 and if {p 1, p 2 } := ξj p 1 xj p 2 xj p 1 ξj p 2 denotes the Poisson bracket, then the principal symbol of [ L 1, / x i ] OPScl 2 is (cf., e.g., [44], Vol. 2, pp. 13) i {ξ j, p(x, ξ)} = i p x j (x, ξ). (3.16) Since p x j (x, ξ) S 2 cl 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 / x s, 1 s m, to (3.14). In this regard, there are two cases to study. First, when / x s hits the lower order terms, the highest singularity comes from terms of the form xs ẽ αβ (x, y). The contribution from these kernels can be handled directly by the theory developed in [34]; the conclusion is that the corresponding integral operators are bounded on L p (Σ) for any 1 < p <. Second, there is the case when / x s hits the first part of the expression (3.14). This time, the main singularities are contained in terms of the form xs yj ẽ αβ (x, y). In the sequel, we find it convenient to replace these by ys yj ẽ αβ (x, y). By (3.15), this can be arranged modulo operators bounded on L p (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 j (y) c a µγ k (y)n k(y) ( ) ẽαβ (x, y) f γ (y) y s y j ( = a µβ j (y) c a µγ k (y) n k (y) n s (y) ) ( ) ẽ αβ (x, y) f γ (y) y s y k y j ( ) +a µβ j (y) c a µγ k (y)n s(y) ẽ αβ (x, y) f γ (y) =: I s + II s. (3.17) y k y j a µβ Observe that I s contains a tangential derivative. Hence, when I s is integrated against Σ dσ Σ, this tangential derivative can be passed on to the other factors. The resulting terms obviously obey the type of estimate we are after since we assume that f belongs to H 1,p (Σ). There remains II s. In order to treat this term, we shall used the PDE satisfied by Ẽ. The point is that II s = ( L t yẽ(x, y)) αγ n s(y)f γ (y) + {residual terms}. (3.18) 12

13 The source of main singularities in the residual terms is y Ẽ(x, y) and, once again by [34], the integral operators with kernels of this type can be controlled in the desired fashion. Finally, the fact that L t yẽ(x, y) = 0 for x y takes care of II s. This concludes the proof of (3.13). The claim about the boundedness of C on H 1,p (, 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 C 1+γ, we first remark that (3.15) is satisfied by the most singular part of Ẽ in the decomposition (3.10). As for the contribution from e 1 (x, y), matters are readily reduced (cf. also (3.34), (3.36) below) to analyzing integrals of the type T f(x) := e1 (x, y)[f(y) f(x)]dσ(y), where f is a scalar-valued function in H 1,p () and e 1 stands for two arbitrary derivatives on e 1. In this context, the key estimate, provided by the analysis in [34], is that e 1 (x, y) C ɛ x y (n 1+ɛ) for any ɛ > 0. Based on this, an elementary interpolation argument gives that T : B p,p θ () L p () for any 1 p and θ > 0. The desired conclusion now follows easily since H 1,p () B p,p θ () for 1 < p < and θ (0, 1). 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 is well defined and bounded. Also, for 1 < p, q < and 0 < s < 1 C : H s,p (, E) B p,p 2 s+1/p (, E) (3.19) are well defined and bounded. C : Bs p,q (, E) B p,q s+1/p (, E), C : B p,q s (, E) Bs p,q (, E) (3.20) Proof. It only remains to treat the case s = 1; the full result then follows from this, Theorem 3.2 and complex interpolation (cf. [3]). The problem localizes and, working in local coordinates, it suffices to show that xl C sends H 1,p () boundedly into B p,p 2 1/p () for each 1 < p <, l = 1,..., m. Taking into account (3.14), (3.15) (with appropriate modifications when dealing with structures which are noly C 1+γ ) and the identity (3.17) this can be proved much as we did for Theorem 3.3. Now, (3.19) and real interpolation gives (3.20) as long as 1 < p, q <, 0 < s < 1. 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 and 0 < s < 1, the operator is well defined and bounded. In particular, is well defined and bounded for each 0 < s < 1 and 1 p. C : Bs p,p (, E) B p,p s+1/p (, E) (3.21) C : Bs p,p (, E) Bs p,p (, E) (3.22) Proof. Our strategy is to treat the end-point cases p = 1 and p = separately and then use interpolation in order to cover the whole range 1 p. To this effect, assume first that p = 1 and observe that the problem at hand localizes. Also, recall from [16] that each f Bs 1 () has an atomic decomposition of the form 13

14 f = j λ j a j, (λ j ) j l 1, a j B 1 s ()-atom, f B 1 s () j λ j. (3.23) In (3.23), each B 1 s ()-atom a satisfies supp a S r, tan a L () r s m, (3.24) where S r is a surface ball on of radius r > 0. We shall now proceed to analyze the action of C on an individual B 1 s ()-atom f. Our aim is to show that there exists C(s, ) > 0, independent of f so that dist (, ) 1 s 2 Cf L 1 () + Cf L 1 () + Cf L 1 () C(s, ). (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 (, ) 1 s 2 Cf to L 1 (). First, based on the identity (3.17), locally we can write Cf(x) = k 0 (x y, y), tan f(y) dσ(y) + k 1 (x y, y), f(y) dσ(y) =: A 0 + A 1, (3.26) where k 0 (x y, y), k 1 (x y, y) are kernels which behave similarly to Schwartz kernels of pseudodifferential operators in ØPC Scl 1 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 into sections over ) whose kernel k(x, y) satisfies i x j yk(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 (, ) s 1+µ+τ 1+µ Kf L 1 () + µ Kf L 1 () + Kf L 1 () C(s, ) f (B s ()) (3.28) for µ = 0, 1,..., N 1. This is a slight generalization of a lemma in [41] which, in turn, extends some Euclidean estimates from [14]. Two key observations which allow us to use this result in the present context are as follows. First, (3.27) with N = 2, τ = 1, holds for k 0 (x y, y) and, second, tan f (B s ()) C(s, ) < (3.29) is valid uniformly for f Bs 1 ()-atom. These take care of the contribution coming from A 0 (cf. (3.26)) in 2 Cf in the context of (3.25). As for the contribution of A1, note that dist (x, ) x k 1 (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 exceed a (fixed) multiple of Mf( x), where M is the Hardy-Littlewood maximal operator on and x is the projection of x onto (along a suitable smooth transversal direction). Consequently, the contribution of A 1 (cf. (3.26)) to dist (x, )1 s 2 Cf(x) d Vol(x) can be controlled by 14

15 ( diam () 0 )( ) t s dt (Mf)( x) dσ( x), (3.31) where t plays the role of dist (x, ). Now, since B 1 s () L τ(s) () with τ(s) > 1 (in fact, 1/τ(s) = 1 s/(m 1)), the boundedness of M on L τ(s) () can be used to obtain the desired conclusion. This concludes the proof of the fact that dist (, ) 1 s 2 Cf L 1 () C(s, ). (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 (, ) 1 s Cf L () + Cf L () C(s, ) f B s (), (3.33) uniformly for f B s (). To this end, assume that f has support in U, a small open subset of M, and that {ψ α } α is a frame of E over U. Fix an arbitrary x 0 U and denote by x 0 its projection on along some smooth, a priori fixed transversal field. In particular, d := dist (x 0, ) dist ( x 0, ), uniformly in x 0. Going further, let θ C c (U) be a cut off function which is identically one near x 0 and, if f = α f αψ α in U, introduce f(x) := θ(x)( α f α( x 0 ))ψ α (x) for x U. Clearly, f agrees with f at x 0 and f Lip C f L. With an eye toward (3.33) we write Cf(x 0 ) = C(f f)(x 0 ) + C f(x 0 ) =: I + II. (3.34) To treat I, for a large constant C, split the domain of integration into {y : dist (y, x 0 ) Cd} and {y : dist (y, x 0 ) > Cd}. In the first resulting integral majorize the kernel of C by Cd m, while in the second one by C dist (y, x 0 ) m. That this works, is guaranteed by the expansion (3.10) and the estimates that e 0, e 1 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, y, on L (). A crude estimate gives T f(x) C log (dist (x, )) f L (), x, (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 () and seek a bound of the order of φ Lip(). The idea is to integrate by parts in Cφ and write this as φ plus a Newtonian potential. That is, Cφ(x) = Γ(x, y), iσ(d; ν(y))f(y) y dσ(y) = φ(x) 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 the proof of (3.21) corresponding to p =, s (0, 1). 15

16 Theorem 3.6 Retain the same hypotheses as in Theorem < s < 1, the operator Then, for each 1 < p < and is well defined and bounded. C : Bs p,p (, E) L p s+1/p (, E) (3.37) 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) between this 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 B p,p s+1/p () Lp s+1/p (); see for this Lemma 4.5 and Proposition 4.4 in [41]. The proof is finished. In closing, we would like to point out that all our results about C in this section continue to hold under 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 L p maximum principle, (a sharp form of) Cauchy s vanishing formula, decomposition theorems and (global) regularity issues. Through the section, will denote an arbitrary Lipschitz domain in M. In relation to D, an elliptic first order differential operator as in (2.1) (2.2) and so that (2.21), (3.1) are satisfied, we introduce some Hardy type spaces. Specifically, for 0 < p, we set H p (, D) := {u L p (, E); Du = 0 in and N u L p ()} (4.1) and equip it with the norm u H p (,D) := N u L p (). As such, H p (, D) becomes a Banach space if 1 < p. Theorem 4.1 Let D and be as above. Then, for each 1 < p <, there holds H p (, D) B p,p 2 1/p (). (4.2) Also, for any u H p (, E) there exists u in the nontangential pointwise sense and u L p (,E) N u L p (), (4.3) uniformly for u H p (, D) (the L p -version of the maximum principle). In particular, the boundary version Hardy spaces H(, D) := {u ; u H p (, D)} (4.4) are well defined, closed subspaces of L p (, E) for each 1 < p <. Proof. The crux of the matter is Cauchy s reproducing formula u = C(u ) in, (4.5) valid for any u H p (, D) which is also continuous up to and including the boundary of. Once this is available, the extra hypothesis of continuity can be eliminated via a routine limiting argument. 16

17 Then, the existence of u 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 representation formula to the effect that u(x) = C(u ) + Γ(x, y), (Du)(y) y dvol(y), x, (4.6) for any u C 1 (, E). Indeed, (4.6) follows based on Id x D y Γ(x, y) = δx (y) and Du, v dvol = u, D t v dvol 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 smooth approximating domains and the boundary behavior theory which we established for functions belonging to Hardy spaces. Going further, (4.6), the fact that Cf plus a limiting argument show, much as in the Euclidean setting (cf., e.g., [35]) that u exists a.e. on and u = C(u ) in for any u H p (, D). This finishes the proof. We now introduce Hardy spaces whose elements are more regular functions. Specifically, set H p 1 (, D) := {u Hp (, D); N ( u) L p ()}. (4.8) Theorem 4.2 Let D be as in Theorem 3.3. Then, for any 1 < p <, and u H p (, D) has u H 1,p (, E) u H p 1 (, D) (4.9) Moreover, for any 1 < p <, N ( u) L p () u H 1,p (,E). (4.10) H p 1 (, D) Bp,p 2 1+1/p (). (4.11) Proof. The left-to-right implication in (4.9) together with the inequality in (4.10) follow directly from (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. We next present a sharp form of Cauchy s vanishing formula. To state this result, we let [...] stand for the annihilator of [...] under the pairing (u, v) = u, vc dσ. Theorem 4.3 Let D, be as at the beginning of 4 and fix 1 < p, q <, two conjugate exponents. Then the mapping is an isomorphism. Φ : H p (, D ) [H q (, D)], Φ(u) := iσ(d ; ν)u, (4.12) Proof. That Φ is well-defined is immediate from Theorem 4.1 and an integration by parts. Also, note that 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 in assuming that E = F and D = D, the adjoint of D. This is seen by symmetrizing the operator D, i.e. by working with 17

18 ID := ( 0 D D 0 ), on F E, (4.13) since the original claim can be ultimately recovered from the corresponding one for ID. Next, we produce an invertible perturbation D of D. More specifically, there exists a symmetric, smoothing operator P L supported away from so that D := D P is invertible, say, from H 1,2 (M, E) onto L 2 (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 integral operators C D f(x) := Θ(x, y), iσ(d; ν)(y)f(y) y dσ(y), x, (4.14) C D f(x) := p.v. Θ(x, y), iσ(d; ν)(y)f(y) y dσ(y), x. (4.15) The point is that C D satisfies properties similar to those enjoyed by the Cauchy operators in Theorem 3.1. As a consequence, we have ( Im 1 2 I + C D; L p (, E)) = H p (, D) = Ker ( ) 1 2 I + C D; L p (, E), (4.16) for any 1 < p <. One final property we want to single out is that the adjoint of C D acting on L p (, E) is (C D ) = iσ(d ; ν)c D [iσ(d ; ν)] 1 on L q (, E), 1/p + 1/q = 1. (4.17) Here C D is the singular integral operator constructed analogously to C D but with D replacing D. In particular, (4.17) remains valid with D replaced by D. Thus, from (4.17) and the second equality in (4.16), we have ( Ker 1 2 I + (C D) ; L p (, E)) ( = Ker 1 2 I iσ(d ; ν)c D [iσ(d ; ν)] 1 ; L p (, E)) ( = iσ(d ; ν)ker 1 2 I C D ; Lp (, E)) = iσ(d ; ν)h p (, D ). (4.18) Returning to the study of Φ in (4.12), we may write Hence Φ in (4.12) is onto also. ( ) Im Φ = iσ(d ; ν)h p (, D ) = Ker ( 1 2 I + C D) ; L p (, E) [ ( )] = Im 1 2 I + C D; L q (, E) = [H q (, E)]. (4.19) The Cauchy reproducing formula (4.5) involves the operator C which does not map all L p (, E) into H p (, D), as C does. An important case when matters can be arranged so that C = C occurs for E = F, D 2 = (D t ) 2, D 2 : H 1,2 (M, E) H 1,2 (M, E) invertible. (4.20) Note that (4.20) is automatically satisfied when D = D t and the kernel Ker (D : H 1,2 (M, E) L 2 (M, E)) is trivial. 18

19 The idea is that, granted (4.20), L(= D 2 ) is invertible from H 1,2 (M, E) into H 1,2 (M, E) and we may base the construction of the Cauchy operators C, C on the kernels Γ := (D x Id y )E(x, y) and Γ := (Id x D t y)e(x, y), (4.21) where E is the Schwartz kernel of L 1. That these kernels (and, hence, the associated) operators coincide, is a consequence of the commutation identity DL 1 = L 1 D read at the level of Schwartz kernels. In this case, C 2 = 1 4I and, if we set then the Plemelj-Calderón type decomposition H p ±(, D) := {u ; u H p ( ±, D)} (4.22) L p (, E) = H p +(, E) H p (, D) (4.23) is valid for any Lipschitz domain 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 definition. Recall from [23] that (A, B) is called a Fredholm pair for the Banach space X if A, B are closed subspaces of X so that dim (A B) < and dim (X/(A + B)) <. In this case, one defines Index (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]. To state 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 arbitrary Lipschitz subdomain of M. Then the Hardy spaces ( H p (, D), H p +(, D) ) are a Fredholm pair for L p (, E) and for each 1 < p <. Index ( H p (, D), H p +(, D) ) = Index ( ) D : H 1,p (M, E) L p (M, F) (4.24) The point is that in the general case the analogue of (4.23) is valid only modulo finite dimensional linear spaces. A version of Theorem 4.4 corresponding to p = 2 in the smooth case has been first conjectured 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 that ± supp V ± =. Also, set L ± := DD + V ±. As before, it follows that L ± : H 1,2 (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) := (Dx Id y )E ± (x, y), Γ ± D (M M, E F) and the Cauchy type operators acting on arbitrary sections f : E by C ± f(x) := Γ ± (x, y), iσ(d; ν(y))f(y) y dσ(y), for x /, (4.25) and C ± f(x) := p.v. Γ ± (x, y), iσ(d)(y, ν(y))f(y) y dσ(y), for x. (4.26) There are several properties of these operators which are going to be of importance for us in the sequel. First, C ± : L p (, E) H p ( ±, D) are well defined and bounded for each 1 < p <. Second, for all f L p (, E), 19

20 C + f ± = (± 1 2 I + C +)f, C f ± = (± 1 2 I + C )f a.e. on, (4.27) and, third, One immediate conclusion is that Im C ± are bounded on L p (, E), 1 < p <. (4.28) ( ) ± 1 2 I + C ±; L p (, E) H±(, p D), 1 < p <. (4.29) Going further, a key observation is that the main singularity in Γ ± (x, y), as described in (3.10), is independent of V ±. In particular, if m := dim M, then Γ + (x, y) Γ (x, y) = O( x y (m 2+ɛ) ) for any ɛ > 0. Hence, the integral operator K := C + C is compact from L p (, E) into itself, 1 < p <. Now, since I + K = ( 1 2 I + C +) ( 1 2 I + C ), it follows from (4.29) that Im (I + K; L p (, E)) H p +(, D) + H p (, D). (4.30) Consequently, since I + K is Fredholm, it follows that H p +(, D) + H p (, D) is closed and has finite codimension in L p (, E). Now, if f H p (, D) H p +(, D), then there exist (unique, by Theorem 4.1) functions u ± H p ( ±, D) so that u + = f = u. Define u f L p (M, E) by setting u f := u ± in ± so that u f Ker ( D : H 1,p (M, E) L p (M, F) ). Consequently, the assignment f u f is linear, well-defined and, by Theorem 4.1, one-to-one from H p (, D) H p +(, D) into the space Ker ( D : H 1,p (M, E) L p (M, F) ). Invoking Theorem 2.1, it is clear that this is also onto (note that (2.21) implies (2.3) with γ > 1 and r = ). Hence, dim ( H (, p D) H+(, p D) ) ( ) = dim Ker D : H 1,p (M, E) L p (M, F) <. (4.31) At this point, the first part in Theorem 4.4 follows. There remains (4.24) to which we now turn. A combination of Theorem 2.1, (4.31) and the fact that the mapping (4.12) is an isomorphism, allows us to write dim Ker (D ; L q (M, F)) = dim Ker (D ; H 1,q (M, F)) = dim ( H (, q D ) H+(, q D ) ) = dim ( H p (, D) H p +(, D) ) = dim ( H p (, D) + H p +(, D) ) (4.32) where 1/p + 1/q = 1. Now, the index formula (4.24) follows from this, the fact that H (, p D) + H+(, p D) is closed in L p (, E) and (4.31). This completes the proof of the theorem. A natural, direct consequence of Theorem 4.4 is the following. Corollary 4.5 With the previous notation and hypotheses, the transmission problem (T BV P ) { u H p ( +, D), v H p (, D), u v = f L p (, E), (4.33) is Fredholm solvable for any 1 < p < and its index is the same as the index of the Fredholm pair ( H p (, D), H p +(, D) ) in L p (, E). 20