Higher degree layer potentials for non-smooth domains with arbitrary topology

Transcription

1 Higher degree layer potentials for non-smooth domains with arbitrary topology Dorina Mitrea and Marius Mitrea 2 Introduction It has long been understood that there are intimate connections between the scalar harmonic layer potentials associated with some domain Ω and the topology of Ω. The same is true for the vector-valued operators naturally associated with problems arising in static electromagnetism and hydrodynamics. However, the intrinsic logic of the theory itself dictates the consideration of layer potentials associated with differential forms of arbitrary degree rather than just those for scalar harmonic functions or vector fields. Addressing this higher order case, which brings together several seemingly unrelated theories is the main goal of the present paper. For maximum applicability it is important to replace the Euclidean space with a Riemannian manifold and to allow the underlying domain to have an irregular boundary; here we shall work with the class of Lipschitz domains. At the level of scalar-valued functions in Lipschitz domains with arbitrary topology in the flat Euclidean setting, a systematic study has been carried out in MiD2]. An attempt to extend this to higher degree differential forms, while still retaining the flat Euclidean setting is in MiD]. In this connection, see also MM]. The present work, which continues this line of research, relies on the recent advances in MMT] where, however, the main emphasis is on boundary problems for the Hodge-Laplacian. Thus, we are able to revisit fundamental problems, having to do with boundary integral operators, originating in the classical theory of harmonic integrals initiated by W. V. D. Hodge and, in the smooth context, studied at length by a large number of authors, including, D. C. Spencer, P. R. Garabedian, G. F. Duff, J. J. Kohn and C. Miranda among others. See, for instance, Ho], Du2], KS], GS], Mir]. Our goal is to prove the effectiveness of tools from harmonic analysis, especially those associated with Calderón s program in such a context. An approach involving similar ideas has been extremely successful in dealing with homogeneous, constant-coefficient operators in the flat, Euclidean setting cf. DK], DKV], FKV], Fa], FJR], MMP], Ve]). Partially supported by a UMC Research Board grant 2 Partially supported by NSF grant DMS Mathematics Subject Classification. Primary 42B20, 3B0, 58A0; Secondary 45F5, 58G20 Key words: layer potentials, Lipschitz domains, harmonic differential forms, Betti numbers. Typeset by AMS-TEX

2 2 D. MITREA AND M. MITREA Our main theorems are too long to be recalled here but, for the purpose of this introduction, we would like to single out a result which is significant for the theory developed in this paper. In order to be specific, assume that the smooth, compact, connected, oriented Riemannian manifold M is a homology sphere so that on l-forms has a trivial kernel modulo constants if l = 0, m). Denote by Γ l x, y) the Schwartz kernel of and introduce the family of layer potentials.) M l fx) := p.v. νx) δ x Γ l x, y), fy) dσy), x. Here Ω M is a Lipschitz domain and ν, dσ are, respectively, the outward unit conormal and the surface area on. Also, δ is the formal adjoint of d, the exterior derivative operator, stands for the interior product of forms and f L 2 tan, Λ l T M), the space of tangential l-form with square integrable coefficients on. Finally, p.v. indicates that the integral is taken in the principal value sense, i.e. by removing geodesic balls with respect to some smooth background metric. It is both illuminating and rewarding to note that, in the flat Euclidean setting, the family of singular integral operators {M l } 0 l m encompasses both the classical harmonic double layer potential operator.2) Kfx) := p.v. ω m y x, νy) fy) dσy), x, x y m ω m is the area of the unit sphere in R m ) as well as its formal adjoint K. This happens for extremal values of l. More concretely, if l = 0, then.3) M 0 = K. Also, if l = m and with denoting the Hodge star operator,.4) ) m ν M m )ν ) = K. In R 3, the same family also encompasses the so-called magnetostatic integral operator defined by ).5) Mf := p.v. νx) curl x fy) 4π x y dσy), x, for tangential vector-valued densities f : R 3 ; cf. CK], CK2], MMP]. That is, when l = and m = 3,.6) M = M. The fundamental role played by the topology of the underlying domain is exhibited by the basic equalities.7) dim Ker 2 I + M l; L 2 tan, Λ l T M) ) = b m l Ω),

3 GENERALIZED LAYER POTENTIALS 3.8) dim Ker 2 I + M l; L 2 tan, Λ l T M) ) = b l Ω). Here and elsewhere, b l Ω) := dim Hsing l Ω; R) is the l-th Betti number of Ω. A fundamental feature of.7).8) is that they relate purely analytical objects the singular integral operators in the left hand sides) to purely topological ones the Betti numbers in the right hand side). Much of the theory developed in this paper appears to be new even when all structures involved are smooth. A fundamental difficulty in the Lipschitz case as opposed to the C case) is that the operators.) are no longer weakly singular when acting on tangential forms). In particular, ± 2 I + M l is no longer of the form identity +compact but, nonetheless, remains Fredholm with index zero on L p tan, Λ l T M) for 2 ε < p < 2 + ε where ε = εω) > 0. This range is sharp in the class of Lipschitz domains if one allows forms of arbitrary degrees. However, if is C then all the results of the present paper extend to the full range < p <. For certain results in this latter context and for the flat Euclidean setting see Mir], SS]. The layout of the paper is as follows. In Section 2 we recall a number of basic definitions and set up the notation used throughout the paper. Section 3 contains the statements of all our main results. Finally, in Section 4, we present the proofs of the results from 3. Acknowledgments. We would like to thank Dmitry Kahvinson and Erhard Meister for calling to our attention the references Dy-2], GK] and Kr], Pi-3], We], respectively. 2 Basic definitions and notation Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and denote by T M, T M its tangent and cotangent bundles, respectively. We also assume that M is equipped with a Riemannian metric g = j,k g jkdx j dx k so that 2.) the Riemannian metric tensor g has coefficients in C,. Also, we let dvol stand for the corresponding volume form on M. Sections of Λ l T M, the l-th exterior power of the tangent bundle, consist of l-differential forms. The Hermitian structure in the fibers on T M extends naturally to T M and, further, to Λ l T M. We will make considerable use of the Hodge star operator, which can be characterized as the unique vector bundle morphism : Λ l T M Λ m l T M such that 2.2) u u) = u 2 dvol.

4 4 D. MITREA AND M. MITREA Here we regard dvol as an m-form on M, making use of the orientation we assume M has. If stands for the usual exterior product of forms, we define the interior product between a -form α and an l-form u by setting 2.3) α u := ) l )m α u). Next, recall the exterior) derivative operator d = m j= x j dx j and its formal adjoint with respect to the metric described above), the co-differential operator δ. In particular, if Ω is a Lipschitz subdomain of M with outward unit conormal ν defined a.e. on with respect to the surface measure dσ) and u C Ω, Λ l T M), v C Ω, Λ l+ T M), then 2.4) Ω du, v dvol Ω u, δv dvol = ν u, v dσ = u, ν v dσ. Let us also note that if ι : M stands for the canonical inclusion and the superscript # indicates pull-back, then 2.5) ν u, v dσ = ι # u v) for any forms u, v of degree l and l +, respectively. The Hodge Laplacian is then defined as 2.6) := dδ + δd). For further reference as well as for the convenience of the reader, some basic, elementary properties of these objects are summarized in the following lemma. Lemma 2.. For arbitrary one-forms α, β, and any l-form u, m l)-form v, and l + )-form w, the following are true: ) u = ) lm l) u; 2) u, v = ) lm l) u, v and u, v = u, v ; 3) α α u) = 0 and α α u) = 0; 4) α β u) + β α u) = α, β u; 5) α u, w = u, α w ; 6) α u) = ) l α u) and α u) = ) l α u). Moreover, if α is normalized such that α, α =, then also: 7) u = α α u) + α α u); 8) α α u) = α u and α α u) = α u. Finally, 9) d d = 0, δ δ = 0, d = d, δ = δ; 0) δ = ) ml+)+ d and δ = ) l d, δ = ) l+ d on l-forms.

5 GENERALIZED LAYER POTENTIALS 5 Fix Ω an arbitrary Lipschitz domain in M with nonempty boundary and recall that ν denotes the unit outward conormal. Let 2.7) f tan := ν ν f), f nor := ν ν f) stand, respectively, for the tangential and normal components of f : Λ l T M, and define 2.8) L p tan, Λ l T M) := {f L p, Λ l T M); f nor = 0}, L p nor, Λ l T M) := {f L p, Λ l T M); f tan = 0}, < p <, l = 0,,..., m. A measurable section f : Λ l T M is called tangential if ν f = 0 a.e. on, and normal if ν f = 0 on. Going further, an l-form f L p tan, Λ l T M), p, is said to have its boundary exterior) co-derivative in L p if there exists an l )-form in L p, Λ l T M), which we denote by δ f, so that 2.9) dψ, f dσ = ψ, δ f dσ for any ψ C M, Λ l T M). We set 2.0) L p,δ tan, Λ l T M) := { f L p tan, Λ l T M); δ f L p, Λ l T M) }, and equip it with the natural norm 2.) f L p,δ tan,λl T M) := f L p,λ l T M) + δ f L p,λ l T M). It is easy to check that δ is a local operator, i.e. supp δ f) supp f for any form f in Ltan, p,δ Λ l T M) and, hence, in the smooth context, δ is a first order differential operator. However, the coefficients would involve second order derivatives of the transition functions between local charts in the manifold and, hence, such a description is no longer possible in the Lipschitz case. It is precisely for circumventing this difficulty that we resort to the distributional definition above. In particular, introducing the closed) subspace L p,0 tan, Λ l T M) of L p,δ tan, Λ l T M) by { } 2.2) L p,0 tan, Λ l T M) := f L p,δ tan, Λ l T M); δ f = 0, it follows that 2.3) δ : L p,δ tan, Λ l T M) L p,0 tan, Λ l T M) is well-defined and bounded.

6 6 D. MITREA AND M. MITREA For f L p nor, Λ l T M), we define the distribution d f by requiring that 2.4) Set δψ, f dσ = ψ, d f dσ for any ψ C M, Λ l+ T M). 2.5) L p,d nor, Λ l T M) := { f L p nor, Λ l T M); d f L p, Λ l+ T M) }, equipped with the natural norm 2.6) f L p,d nor,λ l T M) := f L p,λ l T M) + d f L p,λ l+ T M). Analogously to 2.2), if we set 2.7) L p,0 nor, Λ l T M) := { f L p,d nor, Λ l T M); d f = 0 }, then 2.8) d : L p,d nor, Λ l T M) L p,0 nor, Λ l+ T M) is well-defined and bounded. A discussion pertaining to the relationship between d and d, the intrinsic exterior differential operator of, can be found in Appendix A of MMT]. Here we only want to point out that for each f C 0, Λ l T M) we have 2.9) f L p,0 nor, Λ l T M) ν f = 0 and d ι # f) = 0. Along the paper, all restrictions to the boundary of are taken in the pointwise nontangential sense. That is, 2.20) u x) := lim uy), x, y γx), y x where γx) Ω is an appropriate nontangential approach region. Also, N is going to denote the nontangential maximal operator defined on some form u in Ω by 2.2) N ux) := sup { uy) ; y γx)}, x. Following K. Kodaira Ko]), we shall call the forms of class C which are annihilated both by d and δ harmonic fields. There are several classes of harmonic fields which are going to be of importance for us. For < p < and 0 l m we introduce 2.22) H l,p Ω) := { u C Ω, Λ l T M); N u) L p ), du = 0, δu = 0 in Ω },

7 GENERALIZED LAYER POTENTIALS 7 and 2.23) H l,p Ω) := { u H l,p Ω); ν u = 0 }, 2.24) H l,p Ω) := { w H l,p Ω); ν w = 0 }. As proved in MMT], the spaces 2.23) 2.24) are actually independent of p provided 2 ε < p < 2 + ε for some ε = εω) > 0. In such cases, we shall occasionally drop the superscript p and simply write H Ω), l H Ω). l Also, 2.25) dim H l,p Ω) = b l Ω), dim H l,p Ω) = b m l Ω), where b l Ω) stands for the l-th Betti number of Ω. At this stage we make the standing assumption that 2.26) M is a homology sphere, so that all singular homology groups H q sing M; R) vanish for 0 < q < m. Recall that under these conditions, for each 0 < l < m, the operator : H,2 M, Λ l T M) H,2 M, Λ l T M) has an inverse,, whose Schwartz kernel, Γ l x, y), is a symmetric double form of bidegree l, l). For l = 0, m, we need to make a slight adjustment. In order to be more specific, assume that, e.g., l = 0. Then : H,2 M)/R) H,2 M)/R) has an inverse, whose Schwartz kernel Γ 0 x, y) is a distribution defined modulo constants. In the sequel, we shall assume that the normalization M Γ 0x, ) dvol = M Γ 0, y) dvol = 0 is automatically enforced. The case l = m is similar. A few basic properties of the family {Γ l } l are as follows. First, since and commute, 2.27) x y Γ l x, y) = Γ m l x, y), where the subscripts indicate the variables in which the corresponding Hodge star-) operators are acting. Since d, δ and commutes, at the level of Schwartz kernels we get 2.28) δ x Γ l+ x, y)) = d y Γ l x, y)), d x Γ l x, y)) = δ y Γ l+ x, y)). Next, let Ω be a Lipschitz domain in M and denote by S l the single layer potential operator on with kernel Γ l x, y), i.e., 2.29) S l fx) := Γ l x, y), fy) dσy), x M \,

8 8 D. MITREA AND M. MITREA where f L p, Λ l T M). Note that S l f = 0 in M\. Also, set S l f := S l f. Going further, let us introduce the principal value singular integral operators 2.30) M l fx) := p.v. νx) d x Γ l x, y), fy) dσy), x, and 2.3) N l fx) := p.v. νx) δ x Γ l x, y), fy) dσy), x. Here, p.v.... is taken in the sense of removing geodesic balls with respect to some smooth background metric); see MT] for more details. These operators are the higher degree analogue of the so-called magnetostatic and electrostatic operators arising in scattering theory in R 3 cf., e.g., CK], CK2]). Finally,...] will denote the annihilator of the subspace...]. 3 Statement of main results We retain the assumptions made in 2; in particular, we assume that 2.) and 2.26) hold. Some of the main properties of M l, N l, S l, as well as some related operators, are summarized in the next eight theorems. Theorem 3.. Let Ω M be a Lipschitz domain and let 0 l m. Then there exists ε > 0 such that if 2 ε < p < 2 + ε, then 3.) S l : L p, Λ l T M) H,p, Λ l T M) is an isomorphism. For the rest of the paper, given some Lipschitz domain Ω M, we use the notation Ω + := Ω and Ω := M \ Ω. Theorem 3.2. Let Ω be a bounded Lipschitz domain in M and 0 l m. Then there exists ε = εω) > 0 so that for each 2 ε < p < 2+ε, the following statements are valid: ) The kernels of the operators ± 2 I+M l on each of the spaces L p tan, Λ l T M), L p,δ tan, Λ l T M), L p,0 tan, Λ l T M), coincide and are independent of p. In fact, 3.2) Ker ± 2 I + M l; L p tan, Λ l T M) ) = { ν u ± ; u H l+ Ω ± ) }. In particular, 3.3) dim Ker 2 I + M l; L p tan, Λ l T M) ) = b m l Ω)

9 GENERALIZED LAYER POTENTIALS 9 and 3.4) dim Ker 2 I + M l; L p tan, Λ l T M) ) = b l Ω). Also, 3.5) Image ± 2 I + M l; L p tan, Λ l T M) ) = L p tan, Λ l T M) { u ; u H l Ω ) }. Similar descriptions are valid for the images of ± 2 I + M l when acting on the spaces L p,δ tan, Λ l T M) and Ltan, p,0 Λ l T M). Furthermore, the following operators are isomorphisms on the indicated spaces: ] 2) ± 2 I + M l acting on δ L p,δ tan, Λ l+ T M). 3) ± 2 I + M l acting on the quotient spaces 3.6) L p,δ tan, Λ l T M) L p,0 tan, Λ l T M), p Ltan, Λ l T M) L p,0 tan, Λ l T M). 4) ± 2 I + M l acting on the quotient spaces 3.7) L p tan, Λ l T M) Ker ± 2 I + M l), L p,δ tan, Λ l T M) Ker ± 2 I + M l), L p,0 tan, Λ l T M) Ker ± 2 I + M l). 5) ± 2 I + M l acting on ν L q,0 nor, Λ l+ T M) ] L p tan, Λ l T M), where p + q =. 6) ν S l ± 2 I + M l ) δ ν S l+ )ν S l )] acting from 3.8) Ker ± 2 I + N l) onto Ker ± 2 I + M l ), ] where ± 2 I + M l ) are considered on the space δ Ltan, p,δ Λ l T M). Before stating our next result, a few comments are in order here. First, as this theorem shows, there are natural obstructions to inverting the boundary layer potential operators ± 2 I+M l on L p tan, Λ l T M), L p,δ tan, Λ l T M), L p,0 tan, Λ l T M); these are expressed in the form of the non-vanishing of certain singular homology groups of the underlying domain. Second, in the smooth context, the space 3.9) L p,0 tan, Λ l T M) {u ; u H l Ω)},

10 0 D. MITREA AND M. MITREA which appeared in the last part of ) above, can be described in more classical terms. In order to be specific, recall that if C then ω C, Λ l T ) is called an admissible tangential boundary value if it is closed in and has zero period on all l-cycles of which bound in Ω. See Tu], Du]. The remark we wish to make is that, if C and f is smooth and belongs to 3.9) then, so we claim, ι # f is an admissible tangential boundary value. This can be seen from 2.9) and a theorem of G. De Rham, as formulated in VII, p. 59 of Mir]; we leave the details to the interested reader. Third, as expected, there are many applications of Theorem 3.2 to boundary value problems involving harmonic differential forms; for more on this and related topics the interested reader is referred to, e.g., Du-2], Kr], Ho], MMT], Pi- 3], Sch], Tu], We] and the references therein. Here we only want to illustrate this idea by considering the following basic example. In Tu], Du], the problem of existence of a harmonic field with a prescribed admissible tangential boundary value in a smooth domain has been stated and solved. Uniqueness can be ensured by prescribing periods or by making the additional requirement that the form is also closed. In the context of Lipschitz domains, this problem translates into u C Ω, Λ l+ T M), δu = 0 in Ω, du = 0 in Ω, 3.0) BV P ) l N u) L p ), ν u = f L p, Λ l T M), u closed. The last condition is to be interpreted as the equivalent of u Im d, where d is consider here as an unbounded operator on L p Ω)). Formulated as such, the problem BV P ) l has a solution if and only if f belongs to the space 3.9) and the solution is unique in this case. The solution can be expressed as 3.) u = ds l 2 I + M l) f ), in Ω. Next, we discuss the Hodge dual of Theorem 3.2. Theorem 3.3. Let Ω be a bounded Lipschitz domain in M and 0 l m. Then there exists ε = εω) > 0 so that for each 2 ε < p < 2+ε, the following statements are valid: ) The kernels of the operators ± 2 I+N l on each of the spaces L p nor, Λ l T M), L p,d nor, Λ l T M), Lnor, p,0 Λ l T M), coincide and are independent of p. In fact, 3.2) Ker ± 2 I + N l; L p nor, Λ l T M) ) = { ν u ; u H l Ω ) }.

11 GENERALIZED LAYER POTENTIALS In particular, 3.3) dim Ker 2 I + N l; L p nor, Λ l T M) ) = b m l Ω) and 3.4) dim Ker 2 I + N l; L p nor, Λ l T M) ) = b l Ω). Also, 3.5) Image ± 2 I + N l; L p nor, Λ l T M) ) = L p nor, Λ l T M) { u ± ; u H l Ω ± ) }. Similar descriptions are valid for the images of ± 2 I + N l when acting on the spaces L p,d nor, Λ l T M) and L p,0 nor, Λ l T M). Furthermore, the following operators are isomorphisms on the indicated spaces: 2) ± 2 I + N l acting on d L p,d nor, Λ l T M) ]. 3) ± 2 I + N l acting on the quotient spaces 3.6) L p,d nor, Λ l T M) L p,0 nor, Λ l T M) and on Lp nor, Λ l T M) L p,0 nor, Λ l T M). 4) ± 2 I + N l acting on the quotient spaces 3.7) L p nor, Λ l T M) Ker ± 2 I + N l), L p,d nor, Λ l T M) Ker ± 2 I + N l) Lp,0, and nor, Λ l T M) Ker ± 2 I + N l). 5) ± 2 I + N l acting on p + q =. ν L q,0 tan, Λ l T M)] L p nor, Λ l T M), where 6) ν S l ± 2 I + N l+) d ν S l+ )ν S l )] acting from 3.8) Ker ± 2 I + M l) onto Ker ± 2 I + N l+), where ± 2 I + N l+) are considered on the space d L p,d nor, Λ l T M) ]. Theorem 3.4. Let Ω M be a Lipschitz domain. Then there exists ε > 0, such that if 2 ε < p < 2 + ε and 0 l m, the following operators are isomorphisms: 3.9) ± 2 I + M l on { u ; u H l Ω ) } L p tan, Λ l T M) and 3.20) ± 2 I + N l on { w ± ; u H l Ω ± ) } L p nor, Λ l T M).

12 2 D. MITREA AND M. MITREA Theorem 3.5. Let Ω M be a Lipschitz domain. Then there exists ε = εω) > 0 so that if 0 l m and 2 ε < p < 2 + ε, the following hold true: ] ) Image ν δs l ; L p,0 nor, Λ l T M)) = δ L p,δ tan, Λ l T M). In particular, there holds dim Coker ν δs l : L p,0 nor, Λ l T M) L p,0 tan, Λ l 2 T M)) = b l 2 ); 2) Ker ν δs l ) = Ker 2 I + N ) l Ker 2 I + N l) when all the operators under discussion are acting on the space L p,0 nor, Λ l T M). In particular, there holds dim Ker ν δs l ; L p,0 nor, Λ l T M)) = b l ); 3) Image ν ds l ; L p,0 tan, Λ l T M)) = d L p,d nor, Λ l+ T M) ]. In particular, there holds dim Coker ν ds l : Ltan, p,0 Λ l T M) L p,0 nor, Λ l+2 T M)) = b l+ ). 4) Ker ν ds l ) = Ker 2 I + M ) l Ker 2 I + M l), when all the operators considered here are acting on the space L p,0 tan, Λ l T M). In particular, there holds dim Ker ν ds l ; L p,0 tan, Λ l T M)) = b l ); Theorem 3.6. Let Ω M be a Lipschitz domain. Then there exists ε = εω) > 0 so that if 0 l m and 2 ε < p < 2+ε, the following operators are isomorphisms between the indicated spaces: ) ν δs l : d L p,d nor, Λ l T M) ] ] δ L p,δ tan, Λ l T M) ; 2) ν ds l : δ L p,δ tan, Λ l+ T M) ] d L p,d nor, Λ l+ T M)) ] ; ] 3) ν S l : δ L p,δ tan, Λ l+ T M) Lp,d nor, Λ l+ T M) Lnor, p,0 Λ l+ T M) ; 4) ν S l : d L p,d nor, Λ l T M) ] Lp,δ tan, Λ l T M) Ltan, p,0 Λ l T M). Theorem 3.7. Let Ω M be a Lipschitz domain. Then there exists ε = εω) > 0 such that for each 0 l m and 2 ε < p < 2 + ε, the following are true: ) The operator 3.2) ν S l : Ltan, p,0 Λ l T M) Lp,d nor, Λ l+ T M) L p,0 nor, Λ l+ T M) is onto and the dimension of its kernel is b l ). In fact, 3.22) ν S l : L p,0 tan, Λ l T M) Ker 2 I + M ) l Ker 2 I + M ) Lp,d nor, Λ l+ T M) l L p,0 nor, Λ l+ T M) is an isomorphism.

13 GENERALIZED LAYER POTENTIALS 3 2) The operator 3.23) ν S l : L p,0 nor, Λ l T M) Lp,δ tan, Λ l T M) L p,0 tan, Λ l T M) is onto and the dimension of its kernel is b l ). In fact, 3.24) ν S l : L p,0 Ker 2 I + N l is an isomorphism. nor, Λ l T M) ) Ker 2 I + N l ) Lp,δ tan, Λ l T M) L p,0 tan, Λ l T M) To state our next theorem, recall that for a linear and bounded operator T : X X, σ p T ; X) stands for the point spectrum of T, i.e. the collection of all z C so that zi T is not injective. Theorem 3.8. Let Ω M be a Lipschitz domain. Then there exists ε = εω) > 0 so that for each p > 2 ε and 0 l m, we have 3.25) σ p M l ; L p,δ tan, Λ l T M)) 2, 2 ], σ p N l ; L p,d nor, Λ l T M)) 2, 2 ]. Let us now define 3.26) H l,p ) := { u ; u C 0 Ω, Λ l T M), N u) L p ), du = 0, δu = 0 in Ω }, and 3.27) K l,p ) := { u ν ; u C Ω, Λ l T M), where 3.28) ν := ν d ν δ N u), N du), N δu) L p ), u = 0 in Ω}, is the conormal derivative associated naturally with dδ + δd) =. Theorem 3.9. Let Ω be a Lipschitz domain in M. Then there exists ε = εω) > 0 so that if 0 l m, 3.29) H l,q ) ] = K l,p ), K l,q ) ] = H l,p ), for each 2 ε < p, q < 2 + ε with /p + /q =. In particular, 3.30) L 2, Λ l T M) = H l,2 ) K l,2 ), where the direct sum is also orthogonal. Next we discuss two corollaries of this theorem which have independent interest. The first one is a significant extension of Theorem 4, p. 83 in Dy2] cf. also Dy]).

14 4 D. MITREA AND M. MITREA Corollary 3.0. Let Ω be a Lipschitz domain in M. Then there exists ε = εω) > 0 such that if 2 ε < p, q < 2+ε, /p+/q =, for f L p, Λ l T M) the following statements are equivalent: 3.3) 3.32) ) There exists u H l,p Ω) so that u = f; 2) For each ω C Ω, Λ l T M) so that ω = 0, N ω), N dω), N δω) L q ), there holds ω ν, f dσ = 0; 3) For each v C Ω, Λ l T M), w C Ω, Λ l+ T M) so that N v), N w) L q ), dv = δw and such that v, w exist a.e., there holds ν v ν w, f dσ = 0; 4) Same as 3) except that, in addition, δv = 0, v = 0 and dw = 0, w = 0 in short, v, w satisfy the generalized Cauchy-Riemann equations). Our second corollary deals with the so-called annihilator problem; cf. GK] for some partial results in smooth subdomains of the flat Euclidean space. This is, in fact, a dual statement to Corollary 3.0. Corollary 3.. Let Ω be a Lipschitz domain in M. Then there exists ε = εω) > 0 with the following significance. Suppose that 2 ε < p, q < 2 + ε, /p + /q = and that f L p, Λ l T M). Then the following conditions are equivalent: 3.33) ) For each u H l,q Ω) there holds f, u dσ = 0; 2) There exist v C 0 Ω, Λ l T M), w C 0 Ω, Λ l+ T M) so that N v), N w) L p ), dv = δw and such that v, w exist a.e., with the property that 3.34) ν v = f nor, ν w = f tan ; 3) Same as 2) except that, in addition, δv = 0, v = 0 and dw = 0, w = 0 i.e., v, w satisfy the generalized Cauchy-Riemann equations); 4) There exists u C Ω, Λ l T M) so that u = 0, N u), N du), N δu) L p ) for which u ν = f. Our next result extends Theorem 2 in Dy2] to Lipschitz subdomains of Riemannian manifolds. As such, it answers the conjecture made on p. 67 of Dy2].

15 GENERALIZED LAYER POTENTIALS 5 Proposition 3.2. Let Ω be a Lipschitz domain in M and assume that 0 < l < m and < p <. Then 3.35) H l,p + ) H l,p ) = ν L p,0 nor, Λ l T M) ] ] ν L p,0 tan, Λ l T M), where the direct sums are also topological. In the case when l {0, m}, the direct sum in the left side should be replaced by +. In connection with the conormal derivative 3.28), we define the double layer potential operators Γl D l fx) := x, y), fy) dσy) ν y 3.36) =δs l+ ν f)x) ds l ν f)x), x /, and 3.37) Γl K l fx) :=p.v. x, y), fy) dσy) ν y =δs l+ ν f)x) ds l ν f)x), x, 3.38) K l f := S lf ν = ν ds lf ν δs l f, for f L p, Λ l T M), < p <. Then Kl is the formal adjoint of K l. Also, for f L p, Λ l T M), < p <, the following jump relations hold: 3.39) S l f) ν ± = 2 I + ) K l f, D l f ± = ± 2 I + K l) f. Finally, from the work in MiM], the following estimates are valid: 3.40) N D l f)) L p ) C f L p,λ l T M), D l f C α Ω,Λ l T M) C f Cα,Λ l T M), K l f C α,λ l T M) C f C α,λ l T M), K l f H,p,Λ l T M) C f H,p,Λ l T M), Kl f H,p,Λ l T M) C f H,p,Λ l T M), for each < p < and 0 < α <. Nonetheless, the operators ± 2 I+K l, ± 2 I+K l are not Fredholm on Lp, Λ l T M). This is a direct consequence of the results described in the next proposition.

16 6 D. MITREA AND M. MITREA Proposition 3.3. Let Ω be a bounded Lipschitz domain in M and fix some 0 l m. Then there exists ε = εω) > 0, such that for every 2 ε < p < 2 + ε 3.4) Image 2 I + K l ; L p, Λ l T M) ) = K l,p ± ) and 3.42) Ker 2 I + K l; L p, Λ l T M) ) = H l,p ± ). In particular, the operators ± 2 I + K l, ± 2 I + K l on L p, Λ l T M) have closed images. We conclude this section with an analogue of Proposition 3.2. Proposition 3.4. Let Ω be a bounded Lipschitz domain in M and 0 l m. Then there exists ε = εω) > 0 such that for every 2 ε < p < 2 + ε, 3.43) K l,p + ) + K l,p ) = L p, Λ l T M) and, if /p + /q =, 3.44) K l,p + ) K l,p ) = where the annihilator is taken in L p, Λ l T M). ν L q,0 tan, Λ l T M) ν L q,0 nor, Λ l T M)], In closing, let us point out that in the case when is in fact C then all the results in this section are valid in L p for the full range < p <. 4 Proofs of main results In the first part of this section we collect several theorems from MMT] which are going to be of importance for us in the sequel. Theorem 4.. Let Ω M be a Lipschitz domain. Then for each < p < we have 4.) N S l f) L p ), N ds l f) L p ), N δs l f) L p ) C f L p,λ l T M), uniformly for f L p, Λ l T M). Also, with boundary traces taken in the pointwise nontangential sense, the following jump-relations are valid: 4.2) ds l f = 2 ν f + ds lf, δs l f ± = ± 2 ν f + δs lf ± a.e. on for each f L p, Λ l T M) and < p <. Moreover, 4.3) δs l f = S l δ f), ds l g = S l+ d g), f L p,δ tan, Λ l T M), g L p,d nor, Λ l T M).

17 GENERALIZED LAYER POTENTIALS 7 Theorem 4.2. For each l = 0,,..., m, < p <, the operators M l, N l are bounded from L p, Λ l T M) into L p tan, Λ l T M) and from L p, Λ l T M) into L p nor, Λ l T M), respectively. Also, the adjoint of M l acting on L p tan, Λ l T M) is the operator Ml t acting on L q tan, Λ l T M), with /p + /q =, given by 4.4) M t l = ν N l+ ν ), and 4.5) M l = N m l and M l = N m l on l-forms. Finally, M l, N l are well-defined, bounded from L p,δ tan, Λ l T M) and L p,d nor, Λ l T M), respectively, into themselves and 4.6) δ M l f = M l δ f), d N l g = N l+ d g), f L p,δ tan, Λ l T M), g L p,d nor, Λ l T M). Parenthetically, let us point out that we could have defined M 0 via 4.7) M 0 fx) := p.v. νx) δ y Γ x, y), fy) dσy), which is a formula suggested by 4.4). x, Theorem 4.3. Let Ω M be a Lipschitz domain. There exists ε = εω) > 0 so that for each l {0,,..., m}, each 2 ε < p < 2 + ε, and each λ R with λ 2, 4.8) the operators λi + M l are Fredholm with index zero on the spaces L p,δ tan, Λ l T M), L p,0 tan, Λ l T M) and L p tan, Λ l T M) and 4.9) the operators λi + N l are Fredholm with index zero on the spaces L p,d nor, Λ l T M), L p,0 nor, Λ l T M) and L p nor, Λ l T M). In the sequel, we shall need a Green type formula for a form u C 0 Ω, Λ l T M) satisfying u = 0 in Ω and such that N u), N du), N δu) L p ) for some < p <. As proved in MMT], MiD], in this case u, du, δu have non-tangential boundary traces in L p ) and 4.0) u = ds l ν u) + δs l+ ν u) + S l ν δu) S l ν du) in Ω.

18 8 D. MITREA AND M. MITREA For further reference, let us point out that, granted 2.26), it follows that for any Lipschitz domain Ω M we have the following identities relating the Betti numbers of Ω ±, : 4.) b l Ω ) = b m l Ω + ) and 4.2) b l ) = b l Ω) + b m l Ω). Turning now to the task of proving the results stated in 3 we debut with the Proof of Theorem 3.. From the work in 3 of MMT] we have that there exists ε > 0 such that if 2 ε < p < 2 + ε the operator 3.) is Fredholm with index zero. Therefore, it suffices to prove that it is also one-to-one. To this end, let f L p, Λ l T M) be such that S l f = 0 and set u := S l f in Ω ±. Then u is harmonic in Ω ±, N u) L p ), and u ± = 0. Using the uniqueness result for the Dirichlet problem in Lipschitz domains in Dahlberg s sense) proved in 4 of MMT], we can conclude that u = 0 in Ω ±. In turn, based on jump relations 4.2) and Lemma 2., the latter implies f = 0. With this the injectivity of S l is proved, and the proof of Theorem 3. is completed. Proof of Theorems 3.2 and 3.3. First we deal with ) in Theorem 3.2. To this end, the first order of business is to show that there exists ε > 0 so that if 2 ε < p < 2+ε then ) ) Ker ± 2 I + M l; L p,δ tan, Λ l T M) = Ker ± 2 I + M l; L p,0 tan, Λ l T M) 4.3) Partial results in this direction are 4.4) = {ν u ± ; u H l+ Ω ± )}. Ker ± 2 I + M l; L p tan, Λ l T M) ) Ker ± 2 I + M l; L 2 tan, Λ l T M) ), Ker ) ) ± 2 I + M l; L p,δ tan, Λ l T M) Ker ± 2 I + M l; L 2,δ tan, Λ l T M), for some ε > 0 and each 2 ε < p < 2 + ε, 0 l m. These are clear when p 2. When p < 2, let ε > 0 so that Theorem 4.3 above and Theorem 2.9 in KM] hold true. Based on these, the first inclusion in 4.4) follows. Finally, the second inclusion follows from this and the fact that the family of operators M l commutes with δ cf. 4.6)). With 4.4) at hand, since the last set in 4.3) does not depend on p and, as is seen from Green s formula 4.0), it is contained in the first two sets in 4.3)), it suffices to prove 4.3) for p = 2 which we shall assume next. Let the mappings 4.5) Φ ± : Ker ± 2 I + M l; L 2,δ tan, Λ l T M) ) H l+ Ω ± )

19 GENERALIZED LAYER POTENTIALS 9 and ) 4.6) Ψ ± : H l+ Ω ± ) Ker ± 2 I + M l; L 2,0 tan, Λ l T M), be defined by Φ ± f) := ds l f for each f L p,δ tan, Λ l T M) with ± 2 I + M l) f = 0, and Ψ ± w) := ν w ± for each w H l+ Ω ± ), respectively. We claim that these mappings are well-defined, linear, bounded and inverse to each other. To this end, fix an arbitrary f as above and set u := ds l f. It is then immediate that u C Ω ±, Λ l+ T M), N u), N δu) L 2 ), du = 0, u = 0 in M \ and ν u = ± 2 I + M l) f = 0. Using integration by parts as in 2.4), we see that δu = 0 in Ω. In particular, 0 = δu = δds l f = ±dδs l f = ±ds l δ f) in Ω. The latter and 4.2) then imply that 0 = ν δu ± ) = ν ds l δ f)) = ν ds l δ f)) ±. Using 2.4) again, we obtain that δu = 0 also in Ω ±. Hence, in order to conclude that Φ ± are well-defined, we are left with checking that ν u ± = 0. One more integration by parts gives that 4.7) u, ū = S l f, δū + S l f, ν ū dσ = 0. Ω Ω Thus, u = 0 in Ω and 0 = ν u = ν u ±. This concludes the proof of the fact that Φ ± are well-defined. Their boundedness is contained in Theorem 4., while linearity is immediate. Next, turning attention to the mappings 4.6), notice that if w H l+ Ω ± ) then δ ν w ± ) = ν δw) ± = 0. Hence, ν w ± L 2,0 tan, Λ l T M). If we now write the Green type formula 4.0) for w it follows that 4.8) w = ds l ν w ± ) in Ω ±. Taking the non-tangential limits to the boundary of the two sides in 4.8) and then applying ν to the resulting identity, we obtain based on 4.2)) that 4.9) ν w ± = 2 ν w ± ) M l ν w ± ). Clearly, this amounts to ± 2 I + M l)ψ ± w) = 0 which shows that Ψ ± are well defined. Once again, the boundedness of Ψ ± follows from Theorem 4., while the linearity is immediate. To continue, we consider the inclusion mapping 4.20) ) ) ι : Ker ± 2 I + M l; L 2,0 tan, Λ l T M) Ker ± 2 I + M l; L 2,δ tan, Λ l T M). Fix w and f arbitrary forms in the domains of Ψ ± and Φ ±, respectively. Then, 4.2) Φ ± ιψ ± w))) = Φ ± ν w ± ) = ds l ν w ± ) = w,

20 20 D. MITREA AND M. MITREA and 4.22) ιψ ± Φ ± f))) = Ψ ± ds l f) = ν ds l f) ± ) = 2 I M l) f = f. Now 4.2) 4.22) prove that ι is a surjection, and that Φ ± and Ψ ± are inverse to each other. Thus, 4.3) follows. With 4.3) at hand, we now proceed to show that ) 4.23) Ker ± 2 I + M l; L p,δ tan, Λ l T M) = Ker ± 2 I + M l; L p tan, Λ l T M) ). Indeed, this is a consequence of Theorem 4.3 and the lemma below whose proof is given in MiD]). Lemma 4.4. Let X, Y be two Banach spaces such that the inclusion Y X is continuous with dense range. Furthermore, let T be an operator which is welldefined and Fredholm both on X and on Y, and assume that index T ; X) = index T ; Y ). Then Ker T ; X) = Ker T ; Y ). Summing up, 4.3), 4.23) and 2.25) prove 3.2) 3.4) in part ) Theorem 3.2. Also, 3.2) 3.4) in Theorem 3.3 are direct consequences of what we have proved so far, Hodge duality and 4.5). At this point we would like to digress for a moment and point out that the above reasoning can be also adapted to the case when the underlying manifold is the Euclidean space R m. What is new in this context is the fact that Ω := R m \ Ω is no longer relatively compact. Nonetheless, in this case, Theorem in MiD] can be used to obtain the same conclusion as before. The description of the images of ± 2 I + M l in 3.5) is obtained from Theorem 4.2 by using the information we have about their kernels. More precisely, for ε > 0 as above and 2 ε < p, q < 2 + ε, /p + /q =, 4.24) Image ± 2 I + M l; L p tan, Λ l T M) ) = Ker ± 2 I + M l) t ; Ltan, q Λ l T M) )] L p tan, Λ l T M) = Ker ν ± 2 I + N ) l+ ν ; L q tan, Λ l T M) )] L p tan, Λ l T M) = ν Ker ± 2 I + N l+; L q nor, Λ l+ T M) ))] L p tan, Λ l T M) = { ν ν u ); u H l ) } L p tan, Λ l T M) = { u ; u H l ) } L p tan, Λ l T M). With this, ) in Theorem 3.2 is completely proved.

21 GENERALIZED LAYER POTENTIALS 2 Next we show that the operators in 2) 4) in Theorem 3.2 are isomorphisms. In MMT] we have proved that, for each < p <, 4.25) dim Lp,0 tan, Λ l T M) δ L p,δ tan, Λ l+ T M) ] = b m l ) < +. ] In particular, the space δ L p,δ tan, Λ l+ T M), which is the image of the bounded operator 2.3) at the level of l + forms), is closed in L p,0 tan, Λ l T M). See, e.g., Proposition 4.4.8, p. 67 in Va]. Also, from 4.6) we get that the mappings λi + M l are well defined from the space δ L p,δ tan, Λ l T M)] into δ L p,δ tan, Λ l T M)] for any λ R with λ 2. The property of being bounded from below modulo compact operators is stable to taking the restriction of an operator to some invariant subspace. Thus, from Theorem 4.3 and the invariance of the index to homotopic transformations, we see that ± 2 I + M l are Fredholm operators with index zero on δ L p,δ tan, Λ l T M)]. Hence, in order to conclude that they are isomorphisms it suffices to check that they are also one-to-one. Moreover, by Lemma 4.4, it suffices to do so when p = 2 only. To this end, let f L 2,δ tan, Λ l T M) such that ± 2 I + M l)δ f) = 0. We set u := ds l δ f). Then, du = 0, δu = ds l δ δ f) = 0 in M \ and ν u = 0 on we have used 4.3) and 4.2)). Integrating by parts gives 4.26) u 2 = S l δ f), δū + S l δ f), ν ū dσ = 0. Ω Ω Hence, u = 0 in Ω, which in turn implies that ν u ± = 0. With this at hand, since u = dδs l f = δds l f, some more integrations by parts give 4.27) u 2 = Ω ± ds l f, dū + Ω ± ds l f, ν ū dσ = 0. ± Thus, u = 0 in Ω ±, and furthermore, δ f = ν u ν u + = 0. This finishes the proof of the fact that the operators in 2) of Theorem 3.2 are isomorphisms. Since M l maps each of the spaces Ltan, p Λ l T M), Ltan, p,δ Λ l T M), L p,0 tan, Λ l T M), boundedly into itself, it follows that the operators in 3) of Theorem 3.2 are welldefined and bounded. This is also clear for the operators in 4). Also, Theorem 4.3 and general Fredholm theory imply that they are Fredholm with index zero. Hence we are left with showing that they are also injective. In order to continue, let us recall a regularity result. Specifically, the implication 4.28) f L p tan, Λ l T M), λi + M l )f L p,δ tan, Λ l T M) f L p,δ tan, Λ l T M),

22 22 D. MITREA AND M. MITREA is valid for λ R, with λ 2, and 2 ε < p < 2+ε cf. Corollary 0.5 in MMT]). Now the injectivity of the operators in 3) is a consequence of 4.28), 4.6) and the injectivity of the operators in 2) at the level of l forms). As far as the injectivity of the operators in 4) of Theorem 3.2 is concerned, suppose that f L p tan, Λ l T M) is such that ± 2 I + M l)f Ker ± 2 I + M l) L p,0 tan, Λ l T M), i.e. 4.29) ± 2 I + M l)± 2 I + M l)f = 0. Next, for u := ds l f, f L p,0 tan, Λ l T M), we write 4.0) in Ω ± and take the corresponding nontangential boundary traces on. Applying ν to the both sides of the resulting identity gives that 4.30) 2 I + M ) l ± 2 I + M ) l = ν δsl+2 )ν ds l ) on L p,0 tan, Λ l T M). Combining 4.30) and 4.29) we conclude that 4.3) ± 2 I + M l) f = I + ± 2 I + M l )) ± 2 I + M ) ] l f δ L p,δ tan, Λ l T M). Since the operators in 2) are isomorphisms, from 4.29) and 4.3) we obtain that ± 2 I + M l)f = 0 which, in turn, completes the proof of the injectivity of the operators in 4). Given what we have proved up to this point, Hodge-duality and 4.5), imply that the statements ) 4) in Theorem 3.3 are also true. Based on these, we may conclude that ± 2 I + M l) t = ν ± 2 I + N l+)ν ) is an isomorphism L p of tan, Λ l T M) ν Lnor, p,0. The fact that the operators in 5) of Theorem 3.2 Λ l+ T M) are isomorphisms now follows from this and some standard functional analysis arguments. Furthermore, using Hodge duality, the same can be said about the operators in 5) of Theorem 3.3. Finally, we are left with proving that the operators in 6) of Theorems 3.2 and 3.3 are isomorphisms. We will prove only 6) in Theorem 3.3, since 6) in Theorem 3.2 is proved similarly. To this end, let g Ker ± 2 I + M l) Ltan, p,0 Λ l T M) and set u := S l g in M \. Then, ν du) = 0 on and δu = 0 in M \. The Green identity 4.0) written for this u in Ω becomes: 4.32) S l g = ±ds l ν S l g) δs l+ ν S l g). By taking the non-tangential limits of the two sides of 4.32) on and then applying ν we arrive at 4.33) 2 I + N l+) ν Sl g) = ν ds l )ν S l g).

23 GENERALIZED LAYER POTENTIALS 23 This shows that the mappings in 6) of Theorem 3.3 are well-defined. Clearly the operators in 6) are linear and because dim Ker ± 2 I + M l) = dim Ker ± 2 I + N l+ ) < +, in order to conclude that they are isomorphisms we only have to prove that they are injective. Let f L 2,0 tan, Λ l T M) be such that ν S l f = ± 2 I+N l+) ν ds l )ν S l f)]. Then, using the fact that ν ds l = d ν S l ), it follows that ν S l f = d g for some g L 2,d nor, Λ l T M). Furthermore, δds l f = ds l δ f) = 0 so integrating by parts gives that 4.34) Ω± ds l f 2 = Sl f, δds l f ± ν Sl f, ds l f) ± dσ Ω ± = d g, ds l f) ± dσ = g, δdsl f) ± dσ = 0. Thus, ds l f = 0 in Ω ± and, further, f = ν ds l f) ν ds l f) + = 0. This completes the proof of the injectivity of the operators in 6) of Theorem 3.3 and, with it, the proof of Theorems Proof of Theorem 3.4. Let ε > 0 be such that the results of Theorems hold true. First we observe that for 2 ε < p < 2 + ε, the mapping 4.35) ν : Ltan, 2 Λ l T M) ν Ker ± 2 I + N l+) L2 nor, Λ l+ T M) Ker ± 2 I + N l+) is an isomorphism, with inverse ν. Combining this with 4) in Theorem 3.3 we arrive at the fact that ν ± 2 I+N L p l+)ν are isomorphisms on tan, Λ l T M) ν Ker ± 2 I + N l+). This, in concert with 4.4) and simple functional analysis, allows us to conclude that ± 2 I + M l are isomorphisms on 4.36) ν Ker ± 2 I + N l+; L p nor, Λ l+ T M) )] L p tan, Λ l T M) = { u ; u H l Ω ) } L p tan, Λ l T M), where for the last equality we have used 3.2). The fact the operators in 3.20) are also isomorphisms can be proved along the same lines and we omit the details. Proof of Theorem 3.5. Let ε > 0 be such that the results of Theorems hold true. Based on the identity ν δs l+2 = δ ν S l+2 ), we see that 4.37) Image ν δs l+2 ; L p,0 nor, Λ l+2 T M) ) ] δ L p,δ tan, Λ l+ T M),

24 24 D. MITREA AND M. MITREA hence, 4.38) ) dim Coker ν δs l+2 : L p,0 nor, Λ l+2 T M) L p,0 tan, Λ l T M) dim Lp,0 tan, Λ l T M) ] δ L p,δ tan, Λ l+ T M) = b m l ) = b l ), where the next to the last equality is 4.25), while the last one follows from Poincaré duality. On the other hand, 4.30) and 2) in Theorem 3.2 imply that the operator 4.39) ν δs l+2 : d L p,d nor, Λ l+ T M) ] ] δ Ltan, p,δ Λ l+ T M) is onto. The latter, shows that the opposite inequality to 4.38) also holds true, so that 4.40) dim Coker ν δs l+2 : L p,0 nor, Λ l+2 T M) L p,0 tan, Λ l T M)) = b l ), and the inclusion 4.37) is in fact equality. This proves ) in Theorem 3.5. Next, we concentrate on proving 2). Starting with identity 4.0) for u := δs l+2 f, f L p,0 nor, Λ l+2 T M), we obtain much in the spirit of 4.30)) that 4.4) ν ds l )ν δs l+2 ) = 2 I + N l+2) 2 I + N l+2), on L p,0 nor, Λ l+2 T M). Therefore, dim Ker ν δs l+2 ; L p,0 nor 4.42) ) dim Ker 2 I + N l+2; Lnor p,0 = b l+ Ω) + b m l 2 Ω) = b l+ ), ) + dim Ker 2 I + N l+2; L p,0 ) nor where for the first equality we have used 3.3) 3.4). inequality to 4.42) also holds true since, so we claim, However, the opposite 4.43) Ker ± 2 I + N l+2; L p,0 ) nor Ker ν δsl+2 ; L p,0 nor). Assuming 4.43) for the moment, we see that in fact 4.42) holds true with equality. As a byproduct, we also have that 4.44) Ker ν δs l ) = Ker 2 I + N l ) Ker 2 I + N l),

25 GENERALIZED LAYER POTENTIALS 25 when the operators are acting on L p,0 nor, Λ l T M). This completes the proof of 2), modulo that of 4.43) which we treat next. Let f L p,0 nor, Λ l+2 T M) be such that 2 I + N l+2)f = 0 the case 2 I + N l+2 )f = 0 is treated along the same lines). Thus 0 = ν δs l+2 ) 2 I + N l+2)f = 2 I + M l)ν δs l+2 )f], where the second equality is obtained by writing 4.0) for u := δs l+2 f and proceeding much as in the proof of 4.30). Now 2) in Theorem 3.2 implies that ν δs l+2 )f = 0, i.e., f Ker ν δs l+2 ; Lnor) p,0. This proves the inclusion 4.43). Finally, the statements regarding the operator ν ds l can be proved analogously. We omit the details. Proof of Theorem 3.6. Let ε > 0 be such that Theorems are valid. Then, from the proof of Theorem 3.5 we have that the operators ) 2) are onto. Next, we show that they are one-to-one. First, let f L p,d nor, Λ l T M) be such that ν δs l d f) = 0. From the ontoness of the operator 2), we know that there exists g Ltan, p,δ Λ l T M) with ν ds l 2 δ g) = d f. Hence, ν δs l )ν ds l 2 δ g)) = 0. Combining this with 4.30) and 2) in Theorem 3.2, we can conclude that δ g = 0. Thus, d f = 0 and the operator ) is injective. Similarly, one can prove that ν ds l is one-to-one, and hence, an isomorphism, between the spaces indicated in the statement of the theorem. The rest of the theorem follows from what we have proved so far and the properties of the operators d and δ. More precisely, combining the identity d ν S l ) = ν ds l on L p,δ tan, Λ l T M) with the fact that the operator 4.45) d : Lp,d nor, Λ l+ T M) Lnor, p,0 Λ l+ T M) d L p,d nor, Λ l+ T M) ] together with the operator in 2) are isomorphisms, we arrive at the conclusion that the operator in 3) is an isomorphism. The operator in 4) can be handled in the same way. The proof of the theorem is finished. Proof of Theorem 3.7. Let ε > 0 be such that Theorems are valid. The ontoness of the operator in ) follows from 3) in Theorem 3.6. Moreover, from the proof of Theorem 3.5 we see that 4.46) Ker ν S l ; L p,0 tan) = Ker ν ds l ; L p,0 tan) = Ker 2 I + M l) Ker 2 I + M l). Note that the dimension of this last space is b m l Ω)+b l Ω) = b l ), as desired. Finally, the results regarding ν S l can be proved similarly. Proof of Theorem 3.8. Let ε > 0 be sufficiently small. Note that it suffices to only prove the first inclusion in 3.25), since the second one is an immediate consequence

26 26 D. MITREA AND M. MITREA of this and Hodge duality. To proceed, first we prove a partial result to the effect that 4.47) σ p M l ; L p,0 tan, Λ l T M)) 2, 2 ]. To this end, let z C be so that there exists f L p,0 tan, Λ l T M), f = 0, satisfying zi M l )f = 0. Set u := ds l f in Ω ± and note that δu = 0 and u B p,p /p Ω) L2 Ω) if p is sufficiently close to 2 where p = max{p, 2}). In turn, these force 4.48) ± ν u, S l f dσ = ± u 2 =: λ ± 0, ). Ω ± Notice that we cannot have λ + = λ = 0 since this would imply u = 0 in Ω ± and, further, f = ν u + ν u = 0, a contradiction. Since ν u ± = 2 I + M l)f = z 2 )f, by assumptions, 4.48) entails λ +/λ = 2 z)/ 2 + z). Thus, after some simple algebra, we get that 4.49) z = 2 as desired, finishing the proof of 4.47). λ λ + ) λ + λ + ) 2, 2 ], Now, the fact that the family {M l } l commutes with δ, readily implies 4.50) ) σ p M l ; L p,δ tan, Λ l T M) ) ) σ p M l ; Ltan, p,0 Λ l T M) σp M l ; L p,0 tan, Λ l T M). Thus, the desired conclusion follow from 4.47) and 4.50). Proof of Theorem 3.9. If g H l,q ) ] then, by Theorem 5.2 in MMT], there exists u C Ω, Λ l T M) so that u = 0, N u), N du), N δu) L p ) and u ν = g. Thus, g Kl,p ) and this proves the left-to-right inclusion of the first equality in 3.29). Conversely, if g = u ν for some u C Ω, Λ l T M) so that u = 0 in Ω and N u), N du), N δu) L p ) then, for each ω H l,q Ω), 4.5) g, ω dσ = u ν = ± =0. Ω, ω dσ u, ω ± Ω du, δω ± Ω δu, dω

27 GENERALIZED LAYER POTENTIALS 27 This finishes the proof of the first equality in 3.29). To see the second one, all we need to do is to check that H l,p ) is closed in L p, Λ l T M). However, if u H l,p Ω) then Green s integral representation formula 4.0) gives 4.52) u = ds l ν u) + δs l ν u) in Ω, so that 4.53) N u) L p ) u L p,λ l T M). The desired result follows from this; the proof of the theorem is thus finished. Proof of Corollary 3.0. That ) implies 3) is seen via simple integrations by parts. Also, 3) 4) is obvious, while 4) 2) follows by setting v := δω and w := dω. Finally, 2) ) is a consequence of Theorem 3.9. Proof of Corollary 3.. That ) implies 4) is a consequence of Theorem 3.9. Next, 3) 2) is obvious, while 2) ) is seen via integrations by parts. Finally, 4) 3) follows by setting v := δu and w := du. Proof of Proposition 3.2. That both H l,p + ) and H l,p ) are included in the right side of 3.35) is clear. Conversely, let f ν L p,0 nor, Λ l T M) ] ] ν L p,0 tan, Λ l T M) and set 4.54) ω ± := δs l ν f) ds l ν f) in Ω ±. Note that N ω ± ) L p ) and 4.55) N ω ± ) L p ) C f L p,λ l T M). Also, 4.56) dω ± = dδs l ν f) = δds l ν f) = δs l d ν f)) = 0 in Ω ±, and 4.57) δω ± = δds l ν f) = dδs l ν f) = ds l δ ν f)) = 0 in Ω ±. Thus, ω ± H l,p Ω) and ω + + ω = f. Based on these observations we may conclude that 4.58) H l,p + ) + H l,p ) = ν L p,0 nor, Λ l T M) ] ] ν Ltan, p,0 Λ l T M). The fact that H l,p + ) H l,p ) = 0 for 0 < l < m is a consequence of the fact that M is a homology sphere. Finally, the fact that the sum in the left side of 3.35) is topological follows from 4.55).

28 28 D. MITREA AND M. MITREA Proof of Proposition 3.3. Let ε > 0 be such that all the results stated before hold true. Then the left-to-right inclusion in 3.4) is immediate from the definition of K l,p ± ). In order to prove the opposite inclusion, let g K l,p ± ). Theorem 3.9 gives that g H l,q ± ) ], where p + q =. Hence, g nor ] H l. Ω) ± Invoking Theorem 3.3 we infer that ± 2 I + N l)f = g nor for some f L p nor, Λ l T M). Therefore, if we set u := S l f, then 4.59) K l,p ± ) g + u ν ± = g tan + ν ds l f L p tan, Λ l T M). Making now use of Lemma 2.4 in MMT], we may conclude that 4.60) g + u ν L p,0 tan, Λ l T M) H l ] Ω) ±. ± Hence, ) in Theorem 3.2 guarantees the existence of h L p,0 tan, Λ l T M) such that 4.6) g + u ν = ν ds l h) ± = S lh ± ν ± ; recall that δs l h = 0 for any h Ltan, p,0 Λ l T M)). Summing up, we have proved that g = ν S lh f)) ± = 2 I + K l )h f). With this, the proof of Proposition 3.3 is completed. Proof of Proposition 3.4. The fact that 3.43) holds is a consequence of the jumprelations 3.39). With this at hand, 3.44) follows with the aid of 3.29) and Proposition 3.2 in this connection see also p in Ru]). References CK] D. Colton and R. Kress, Integral equation methods in scattering theory, Wiley, New York, 983. CK2] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, Springer-Verlag, 992. DK] B. Dahlberg and C. Kenig, Hardy spaces and the L p Neumann problem for Laplace s equation in a Lipschitz domain, Annals of Math., ), DKV] B. Dahlberg, C. Kenig, and G. Verchota, Boundary value problems for the system of elastostatics on Lipschitz domains, Duke Math. J., ),

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