The Poisson Problem for the Exterior Derivative Operator with Dirichlet Boundary Condition on Nonsmooth Domains

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The Poisson Problem for the Exterior Derivative Operator with Dirichlet Boundary Condition on Nonsmooth Domains
Dorina Mitrea, Marius Mitrea and Sylvie Monniaux *

Introduction
In this paper we study the boundary value problem du = f in Ω, Tr u = g on ∂Ω, (1.1) where Ω is a given Lipschitz subdomain of a manifold M , d is the exterior derivative operator, and f , g are given differential forms in Ω and on ∂Ω, respectively. The goal is to find a natural functional analytic framework where (1.1) has a solution u whose regularity is consistent with that of the data f , g, and which satisfies a natural estimate. As such, two scales inherently lend themselves for the task at hand, namely, B p,q s , the scale of Besov spaces, and F p,q s , the scale of Triebel-Lizorkin spaces (cf. §2.2 for definitions). Since most of the time we shall work with both these scales, we shall often write A p,q s , A ∈ {B, F }, (with the obvious interpretation) as a way of referring to them simultaneously.
There are two types of issues associated with the problem (1.1), i.e., of analytical nature (such as those stemming from the low regularity assumptions on the domain and the compatibility conditions the data must satisfy), and of topological nature (since the fact that every closed form is exact entails that certain Betti numbers vanish). Our main results with regard to the solvability of (1.1) fall under two categories. In the case when the smoothness of the datum f is low, we have the following (precise definitions are given in §2; here we only want to point out that ν stands for the outward unit conormal to ∂Ω): Theorem 1.1 Let Ω be a Lipschitz subdomain of the smooth, compact, boundaryless manifold M , and fix 1 < p, q < ∞, −1 + 1/p < s < 1/p. Then for each 0 ≤ ℓ ≤ n − 1 the following two statements are equivalent. (ii) There exists a finite constant C > 0 with the following significance. For any differential form f ∈ A p,q s (Ω, Λ ℓ ) and any (1.2) subject to the (necessary) compatibility conditions there exists u ∈ A p,q s+1 (Ω, Λ ℓ−1 ) such that du = f in Ω, Tr u = g on ∂Ω, (1.4) and for which (1.5) Finally, corresponding to ℓ = n, we have the following conclusion. There exists a finite constant C > 0 such that for any f ∈ A p,q s (Ω, Λ n ) and any g as in (1.2) with ℓ = n, subject to the compatibility conditions (1.6) where χ E is the characteristic function of a set E, V M stands for the volume element on M and {Ω j } 1≤j≤b 0 (Ω) are the connected components of Ω, there exists u ∈ A p,q s+1 (Ω, Λ n−1 ) satisfying (1.4) and (1.5) with ℓ = n.
When the smoothness of the datum f (and, hence, that of the solution u) is larger than what has been considered so far, the ordinary trace operator alone is no longer adequate in describing the nature of u on ∂Ω. Hence, the very formulation of the problem has to be changed in order to reflect this novel aspect. Specifically, we have the following result (for simplicity, stated here for Euclidean Lipschitz domains): Theorem 1.2 Let Ω be an arbitrary bounded Lipschitz domain in R n and assume that 1 < p, q < ∞, k ∈ N and −1 + 1/p < s − k < 1/p. Furthermore, suppose that either A = B and p = q or A = F and q = 2. Then, for each ℓ ∈ {0, 1, ..., n − 1}, the following two statements are equivalent. (ii) There exists a finite constant C > 0 with the following significance. The boundary value problem      du = f ∈ A p,q s (Ω, Λ ℓ ) in Ω, u ∈ A p,q s+1 (Ω, Λ ℓ−1 ), Tr [∂ α u] = g α ∈ B p,p s+1−k−1/p (∂Ω, Λ ℓ−1 ) on ∂Ω, ∀ α : |α| ≤ k, is solvable if and only if the following compatibility conditions are satisfied (below, {e j } 1≤j≤n is the standard orthonormal basis in R n ): and Tr [∂ α f ] = n j=1 g α+e j ∧ dx j , ∀ α : |α| ≤ k − 1. (1.8) Furthermore, granted (1.8), the solution u can be chosen to satisfy Finally, in the case ℓ = n, the boundary problem (1.7) has a solution which, in addition, satisfies (1.9) if and only if    (ν j ∂ k − ν k ∂ j )g α = ν j g α+e k − ν k g α+e j , ∀ α : |α| ≤ k − 1, ∀ j, k ∈ {1, ..., n}, Ω j f, dx 1 ∧ · · · ∧ dx n dx = ∂Ω j ν ∧ g (0,...,0) , dx 1 ∧ · · · ∧ dx n dσ, 1 ≤ j ≤ b 0 (Ω). (1.10) Of course, when M is equipped with a (smooth) metric tensor and with δ and ∨ denoting the adjoint of d and the interior product of forms, respectively, there are natural dual versions of the above theorems corresponding to a formal application of the Hodge star isomorphism. In the case of Theorem 1.1, the dual statement reads as follows.
(ii) For any differential form f ∈ A p,q s (Ω, Λ ℓ−1 ) and any g belonging to B p,q there exists u ∈ A p,q s+1 (Ω, Λ ℓ ) such that δu = f in Ω, Tr u = g on ∂Ω, (1.12) and such that the estimate naturally associated with (1.12) holds. Finally, corresponding to the case ℓ = 1, the following conclusion is valid. There exists a finite constant C > 0 such that for any f ∈ A p,q s (Ω) and any g belonging to B p,q f, χ Ω j = g, χ ∂Ω j ν , for each 1 ≤ j ≤ b 0 (Ω), (1.13) there exists u ∈ A p,q s+1 (Ω, Λ 1 ) which solves (1.12) and which satisfies the estimate naturally associated with this problem.
As for the Hodge dual version of Theorem 1.2, below we restrict ourselves to the case of vector fields (leaving the formulation of the full statement to the interested reader).

Corollary 1.4
Let Ω be a bounded Lipschitz domain in R n and assume that 1 < p, q < ∞, k ∈ N and −1 + 1/p < s − k < 1/p. Also, suppose that either A = B and p = q or A = F and q = 2. Then the boundary value problem (1.14) is

solvable (in which case the solution obeys natural estimates) if and only if
and (1.15) The above results provide a fairly complete picture of the solvability of the Poisson problem, equipped with a Dirichlet boundary condition, for the exterior derivative operator (and its adjoint) in Lipschitz domains, when the smoothness of the solution, as well as data, is measured on Besov and Triebel-Lizorkin spaces. As regards the latter scale, of particular interest is the case when q = 2, corresponding to Sobolev (potential) spaces.
In the case when Ω has a smooth boundary, (1.1) can eventually be reduced to an elliptic problem for which standard techniques apply; this approach is carried out by G. Schwarz in §3.3 of his monograph [43]; cf. also [44]. Nonetheless, for a number of applications, it is important to allow ∂Ω to only be minimally smooth, in the sense of E. Stein (cf. [46]).
The particular case of Corollary 1.3 when ℓ = 1 and Ω is a connected, bounded, Lipschitz domain in R n , has received a lot of attention in the literature. This is due, in part, to the fact that the Poisson boundary value problem for the divergence equation, i.e., div u = f in Ω, Tr u = g on ∂Ω, (1. 16) arises quite often in applications of physical interest. In this setting, u typically models the displacement field in the equations of elasticity, or the velocity field in the hydrodynamics. In fact, it was precisely its usefulness in the context of the Navier-Stokes equations that gave us the impetus to undertake a systematic study of the problem (1.16) and carry out a thorough study of the regularity properties of solution on scales of Besov-Triebel-Lizorkin spaces in Lipschitz domains; cf. [38].
One of the earliest references in which (1.16) is treated in non-smooth domains is J. Nečas' book [39]. In Lemma 7.1 of Chapter 3 of that monograph, the case when Ω is Lipschitz, p = 2 and s = 0 is treated via an approach which relies on duality (i.e., by studying the mapping properties of the gradient operator).
A different approach, which makes extensive use of the mapping properties of singular integral operators of Calderón-Zygmund type, was developed by M.E. Bogovski in the 80's. In [3], for a bounded, connected Lipschitz domain in R n , the author constructs an integral operator J , mapping scalar functions to vector fields, and with the following additional properties: (1.1)), then u := J ℓ f has du = f . This is strongly reminiscent of the classical Poincaré lemma and, indeed, our definition of the operators J ℓ has, as starting point, an elegant construction going back to the seminal work of E. Cartan. Cartan's solution of Poincaré's lemma in an Euclidean domain Ω which is star-like with respect to the origin, involves an explicit construction which requires integrating over rays emerging from 0 ∈ Ω. Since in the present work we are naturally led to considering differential forms with discontinuous coefficients, this construction is no longer suitable in its original inception, but a certain averaged version of it will do. Remarkably, while these averaged Cartan-like operators fail to be local in the sense of (1.20), it is their adjoints who satisfy (1.20). Conjugating these adjoints with the Hodge star-isomorphism finally yields a family of integral operators which are smoothing of order one and which satisfy (1.20)-(1.21). This interpretation helps put Bogovski's construction in the proper historical perspective while, at the same time, de-mystifies some of its more unusual features. The above discussion pertains to the local aspect of the work carried out in this paper. Passing to global results is then done by invoking the powerful abstract machinery of De Rham theory. As a result, a trade-mark feature which most of our main results inherit is that certain topological characteristics of the underlying domain (in our case, the vanishing of Betti numbers) can be described in purely analytical terms (i.e., well-posedness of certain boundary value problems). It is this combination of techniques from seemingly unrelated fields we consider to be our main contribution to the problem at hand.
Let us now survey further work in connection with the problems studied here. In [2], D.N. Arnold, L.R. Scott and M. Vogelius proved higher-order regularity results for (1.16) in the case when Ω is a polygonal domain in R 2 , and their main results are covered by our Corollary 1.4. When Ω is a contractible, bounded, three-dimensional, Euclidean Lipschitz domain, the problem (1.1) corresponding to f ∈ L 2 (Ω, R 3 ) (i.e., a differential form of degree one) and g = 0, has been solved by Z. Lou and A. McIntosh in [29]. The approach employed by these authors consists of reducing this PDE to a scalar problem and, as mentioned on page 1493 of [29], cannot be adapted to case when the data are higher-degree differential forms. In our Theorem 1.1 we have successfully dealt with this issue.
That the problem (1.16) formulated in a bounded, Lipschitz domain Ω ⊂ R n has a solution u ∈ C 0 (Ω, R n ) ∩ W 1,n (Ω, R n ) whenever g = 0 and f ∈ L n (Ω) satisfies Ω f dx = 0, is a fairly recent, deep result due to J. Bourgain and H. Brezis [5]. A peculiarity of the problem considered in this context is that the solution operator cannot be chosen to be linear. Shortly thereafter, a new approach to (1.16) for f ∈ L p (Ω), 1 < p < ∞, Ω f = 0, and g = 0 in bounded, Lipschitz subdomains of R n has been developed by J. Bourgain and H. Brezis in [6]. In the same paper, these authors also study the limiting cases p = 1 and p = ∞, for which they produce intricate counterexamples to the solvability of (1.1) in W 1,p (Ω, R n ) even when Ω is an n-dimensional torus (in which scenario, the boundary condition is void).

Geometrical preliminaries
Let M be a smooth, compact, oriented manifold of real dimension n, equipped with a smooth metric tensor, j,k g jk dx j ⊗ dx k . Denote by T M and T * M the tangent and cotangent bundles to M , respectively. Occasionally, we shall identify T * M ≡ Λ 1 canonically, via the metric. Set Λ ℓ for the ℓ-th exterior power of T M . Sections in this latter vector bundle are ℓ-differential forms. The Hermitian structure on T M extends naturally to T * M := Λ 1 and, further, to Λ ℓ . We denote by ·, · the corresponding (pointwise) inner product. The volume form on M , V M , is the unique unitary, positively oriented differential form of maximal degree on M . In local coordinates, V M := [det (g jk )] 1/2 dx 1 ∧ dx 2 ∧ ... ∧ dx n . In the sequel, we denote by dλ M the Borelian measure induced by the volume form V M on M , i.e., dλ M = [det (g jk )] 1/2 dx 1 dx 2 ...dx n in local coordinates.

(2.2)
Let d stand for the (exterior) derivative operator and denote by δ its formal adjoint (with respect to the metric introduced above). For further reference some basic properties of these objects are summarized below.
Let Ω be a Lipschitz subdomain of M . That is, ∂Ω can be described in appropriate local coordinates by means of graphs of Lipschitz functions. Then the unit conormal ν ∈ T * M is defined a.e., with respect to the surface measure dσ, on ∂Ω. For any two sufficiently well-behaved differential forms (of compatible degrees) u, w we then have the integration by parts formula We conclude with a brief discussion of a number of notational conventions used throughout the paper. We denote by Z the ring of integers and by N = {1, 2, ...} the subset of Z consisting of positive numbers. Also, we set N o := N ∪ {0}. By C k (Ω), k ∈ N o ∪ {∞}, we shall denote the space of functions of class C k in Ω, and by C ∞ c (Ω) the subspace of C ∞ (Ω) consisting of compactly supported functions. When viewed as a topological space, the latter is equipped with the usual inductive limit topology and its dual, i.e. the space of distributions in Ω, is denoted by D ′ (Ω) := C ∞ c (Ω) ′ . Also, we set C k (Ω, Λ ℓ ) := C k (Ω) ⊗ Λ ℓ , etc. Finally, we would like to alert the reader that, besides denoting the pointwise inner product of forms, ·, · is also used as a duality bracket between a topological space and its dual (in each case, the spaces in question should be clear from the context).

Review of smoothness spaces
We start by defining the Besov and Triebel-Lizorkin scales in R n . The classical Littlewood-Paley definition of Triebel-Lizorkin and Besov spaces (see, for example, [41]) has the following form. Consider a family of functions {ζ j } ∞ j=0 in the Schwartz class with the following additional properties: (i) there exist positive constants C 1 , C 2 , C 3 such that Then, with F denoting the Fourier transform in R n , for s ∈ R and 0 < q ≤ ∞, 0 < p < ∞ the Triebel-Lizorkin spaces are defined as where S ′ (R n ) stands for the space of tempered distributions in R n . For 0 < p ≤ ∞, the Besov spaces are defined as As is well-known, the following embeddings hold

8)
A p,q 1 s 1 (R n ) ֒→ A p,q 2 s 2 (R n ) if s 1 > s 2 and p, q 1 , q 2 are arbitrary, (2.9) and for each p, q, s, with equivalence of norms. Next, the class A p,q s (M ), 1 < p, q < ∞, s ∈ R, is obtained by lifting the Euclidean scale A p,q s (R n ) to M via a C ∞ partition of unity and pull-back. Given an arbitrary open subset Ω of M , we denote by R Ω f ∈ D ′ (Ω) the restriction of a distribution f on M to Ω. For 0 < p, q ≤ ∞ and s ∈ R we then set (2.11) The convention we make in (2.11) is that either A = F and p < ∞ or A = B, corresponding to, respectively, the definition of Triebel-Lizorkin and Besov spaces in Ω.
Two other types of function spaces which will play an important role for us later on are as follows. First, for 0 < p, q ≤ ∞, s ∈ R, we set A p,q s,0 (Ω) := {f ∈ A p,q s (M ) : supp f ⊆ Ω}, f A p,q s,0 (Ω) := f A p,q s (M ) , f ∈ A p,q s,0 (Ω), (2.12) where, as usual, either A = F and p < ∞ or A = B. Thus, B p,q s,0 (Ω), F p,q s,0 (Ω) are closed subspaces of B p,q s,0 (M ) and F p,q s,0 (M ), respectively. Second, for 0 < p, q ≤ ∞ and s ∈ R, we introduce (where, as before, A = F and p < ∞ or A = B). For further reference, it is worth singling out the scale of Sobolev (potential) spaces defined for 1 < p < ∞, s ∈ R, as For the remainder of this subsection we assume that Ω is a Lipschitz subdomain of M . In this case, according to [42], there exists a universal linear extension operator. More specifically, we have

Proposition 2.2
If Ω is a Lipschitz subdomain of M , then there exists a linear operator E mapping C ∞ c (Ω) into distributions on M , and such that for any 0 < p, q ≤ ∞ and s ∈ R, boundedly, and Other properties of interest are summarized in the propositions below.
where tilde denotes extension by zero outside Ω.
Proposition 2.6 If 1 < p, q < ∞ and s ∈ R then R Ω , the operator of restriction to Ω maps in a linear, bounded and onto fashion. Moreover, if −1 + 1/p < s then R Ω in (2.26) is also one-to-one, hence an isomorphism. In this latter case, its inverse is the operator of extension by zero outside Ω. In particular, this allows the identification Another family of spaces which are goint to play an important role in our work is where, as usual, Proposition 2.7 For every 1 < p, q < ∞ and s ∈ R, continuously. Furthermore, where, as before, tilde denotes the extension by zero outside Ω.
Turning to spaces defined on Lipschitz boundaries, assume 1 < p, q < ∞, 0 < s < 1, and that Ω is the unbounded region in R n lying above the graph of a Lipschitz function ϕ : R n−1 → R. We then define B p,q s (∂Ω) as the space of locally integrable functions g for which the assignment . In particular, with dσ denoting the area element on ∂Ω, it can be shown that The above definition then readily adapts to the case of a Lipschitz subdomain of the manifold M , via a standard partition of unity argument. Having defined Besov spaces on ∂Ω with a positive, sub-unitary amount of smoothness, we then set Next, recall (cf. [23]) that the trace operators are well-defined, bounded and onto if 1 < p, q < ∞ and 1 p < s < 1 + 1 p . They also have a common bounded right-inverse The nature of some of the problems addressed in this paper requires that we work with Besov spaces (defined on Lipschitz boundaries) which exhibit a higher order of smoothness (than considered in (2.37)). Following [34], we now make the following definition.

Definition 2.9
Let Ω be a bounded Lipschitz domain in R n . For p ∈ (1, ∞), k ∈ N and s ∈ (0, 1), define the (higher order) Besov spaceḂ p,p k−1+s (∂Ω) as the collection of all familieṡ Of course, when k = 1, condition (2.42) simply becomes (2.37). The trace theory summarized in (2.39)-(2.40) has a natural analogue in the context of higher smoothness spaces. More specifically, the following holds. Proposition 2.10 Consider a bounded Lipschitz domain Ω in R n , and let 1 < p, q < ∞, 1/p < s < 1 + 1/p and k ∈ N. Furthermore, suppose that either A = B and q = p or A = F and q = 2. In this context, define the higher order trace operator by setting where the traces in the right-hand side are taken in the sense of (2.39). Then (2.43)-(2.44) is a well-defined, linear, bounded operator, which is onto and whose kernel is given by That is,

Moreover, the trace operator (2.43)-(2.44) has a bounded, linear right-inverse, i.e., there exists a linear, continuous operator
This is a version of a result proved in [34]. Related results have been proved by A. Jonsson and H. Wallin in [23] (where the authors have dealt with more general sets than Lipschitz domains). We conclude our review with one more equivalent characterization of the spaceḂ p,p k−1+s (∂Ω), also proved in [34]. To state it, let {e j } j denote the canonical orthonormal basis in R n and set ν = (ν 1 , ..., ν n ) for the outward unit normal to Ω ⊂ R n .

Differential forms with Besov and Triebel-Lizorkin coefficients
In this paper we shall work with certain nonstandard smoothness spaces which are naturally adapted to the type of differential operators we intend to study. Specifically, if Ω is an open subset of M and if X is a space of distributions in Ω, we introduce equipped with the natural graph norms. Throughout the paper, all derivatives are taken in the sense of distributions.
Next, inspired by (2.3), for each u ∈ D ℓ (d; A p,q s (Ω)) we can define ν ∧ u as a functional on ∂Ω by setting whenever Tr Ψ = ψ in one of the following two scenarios: It follows (2.46) and (2.23), (2.38) that the operator is well-defined and bounded. Similarly, if u ∈ D ℓ (δ; A p,q s (Ω)), we can then define ν ∨ u as a functional by setting whenever Tr Φ = ϕ in one of the following two scenarios: Much as before, it follows that the operator is well-defined, linear and bounded. The ranges of the operators (2.53), (2.55) are denoted by respectively. These spaces are equipped with the natural "infimum" norms. It follows that the operator is well-defined, linear and bounded. Similarly, we define the operator which, once again, is well-defined, linear and bounded. We conclude by discussing a useful approximation result.

Lemma 2.12
Let Ω be a Lipschitz subdomain of M and assume that 1 < p, q < ∞.
Proof. Given a differential form u, we remark that all three approximation properties we seek to prove are both local in nature and stable under pull-back. Hence, in all three cases, there is no loss of generality in assuming that Ω is a bounded Lipschitz domain in R n and that there exists an open, upright, truncated, circular cone Γ, centered at the origin of R n , such that Assuming that this is the case, we pick two scalar functions ϕ ± ∈ C ∞ c (±Γ) with ϕ ± = 1 and set ϕ ± ε := ε −n ϕ ± (· ε −1 ) for each ε > 0, sufficiently small. After this preamble, we are ready to proceed with the proof of (i). Thus, assuming that u is as above, we let w ∈ A p,q s (R n , Λ ℓ ) be compactly supported and such that R Ω (w) = u. Then, so we claim, the sequence u ε : vanishes on Ω and, hence, concluding the proof of claim (i).
Next, if u is as in (ii), the fact that ν ∧ u = 0 on ∂Ω and Proposition 2.6 give that u ∈ D ℓ (d; A p,q s,0 (Ω)) and dũ = du. Thus, in this case, the sequence of differential forms Finally, if u is as in (iii), then the same type of reasoning applies though, this time, dũ = du is justified slightly differently. More specifically, matters are readily reduced to checking that (here δ is the formal adjoint of d with respect to the Euclidean metric). To see this, we may invoke (i) and select u ε ∈ A p,q since Tr u ε → Tr u = 0 in, say, L p (∂Ω, Λ ℓ ) as ε → 0 + . This justifies (2.66) and finishes the proof of the lemma.

Singular homology and sheaf theory
For a topological space X , we set H ℓ sing (X ; R) for the ℓ-th singular homology group of X over the reals, ℓ = 0, 1, ... (cf., e.g., [31]). Then b ℓ (X ), the ℓ-th Betti number of X , is defined as the dimension of H ℓ sing (X ; R). As is well known, b ℓ (X ), ℓ = 0, 1, ..., are topological invariants of X . In fact, b 0 (X ) is simply the number of connected components of X . The most important case for us is when X is a Lipschitz subdomain Ω of the manifold M .
Next, we include a brief synopsis of some basic terminology together with some fundamental results from sheaf theory. Recall that a sheaf F on a topological space X is a double In order to lighten notation, for each ω ∈ F(U ) and any V ⊆ U open, we may write ω| V in place of ρ U V (ω). By virtue of (2.67), this is without loss of information.
3. For each U , open subset of X , any open covering {U i } i∈I of U , and any family {ω i } i∈I , ω i ∈ F(U i ), satisfying the compatibility condition there exists a unique section ω ∈ F(U ) such that ω| U i = ω i for any i ∈ I.
Given two sheaves F, G over X , a sheaf homomorphism ϑ : F → G is a collection of vector space homomorphisms {ϑ(U ) : F(U ) → G(U )} U ⊆X , indexed by open subsets of X , which commute (in a natural way) with the restriction mappings. We define supp (ϑ) as the smallest closed set outside of which ϑ is the null sheaf homeomorphism.
A sheaf F over X is said to be fine if for each open, locally finite cover {U i } i∈I of X there exists a family of sheaf homomorphisms ϑ i : F → F, i ∈ I, such that Next, assume that F 0 , F 1 , ... are sheaves over the topological space X and that, for ℓ = 0, 1, ..., the mappings ϑ ℓ : F ℓ → F ℓ+1 are sheaf homomorphisms. Then is called an exact complex provided the following two conditions are true: Then the sheaf of locally constant functions on X is given by Recall that for any reasonable topological space X one associates H ℓ sing (X ; R), the classical ℓ-th singular homology group of X over the reals; see [31]. In this connection, we shall make use of a deep theorem of De Rham which we present below in an abstract form, well suited for our purposes.

Mapping properties of singular integral operators
For 0 ≤ δ, ρ ≤ 1 , m ∈ R, let S m ρ,δ be the class of symbols consisting of all functions p ∈ C ∞ (R n × R n ) such that for each pair of multi-indices β, γ there exists a constant C β,γ such that uniformly for (x, ξ) ∈ R n × R n . For p ∈ S m ρ,δ we define the pseudodifferential operator p(x, D) by and denote by OP S m ρ,δ the class {p(x, D) : p ∈ S m ρ,δ }. The following is a consequence of Theorem 6.2.2 on p. 258 of [50] (cf. also Remark. 3 on p. 257 of [50]). Proposition 3.1 Let m ∈ R, 0 ≤ δ < 1 and fix s ∈ R, 1 < p, q < ∞, arbitrary. Then any T ∈ OP S m 1,δ induces a bounded, linear operator An immediate consequence of the above result, which is going to be useful for us here is recorded separately.
Corollary 3.2 Assume that m ∈ R, m > −n, and

4)
where, for each γ ∈ N n o ,

viewed below as multiplication operators). Then
is a bounded operator.
We now turn our attention to mapping properties of operators given by singular integrals. The result that suits our purposes reads as follows. Theorem 3.3 Let 1 < p, q < ∞ and s ∈ R be arbitrary and assume that is a function which satisfies: (i) for some N = N (n, p, q, s) ∈ N sufficiently large, for all multi-indices β, γ ∈ N n o such that |β| + |γ| ≤ N (where S n−1 denotes the unit sphere in R n ); , then it is also assumed that Then, if T is defined as (with the integral taken in the principal value sense when m = 0), it follows that for each φ, ψ ∈ C ∞ c (R n ) the operator is bounded.
Prior to presenting the proof of the above theorem, we record a useful, well-known result (cf., e.g., p. 73 in [46]). Lemma 3.4 There exists a sequence {h j } j∈No , h j ∈ N and there exist homogeneous polynomials {Y hj } j∈No, 1≤h≤h j in R n of degree j ∈ N o which are harmonic in R n and whose restrictions to S n−1 form an orthonormal basis for L 2 (S n−1 ). In addition for each j ∈ N o there holds

13)
where ∆ S n−1 is the Laplace-Beltrami operator on S n−1 . Finally, fix −n ≤ m ≤ 0. Then for each j ∈ N o and 1 ≤ h ≤ h j ,

Proof of Theorem 3.3.
We first consider the case when −n < m < 0. With k(x, z) as in the statement of Theorem 3.3 we may write, making use of Lemma 3.4, where a hj (x) := In particular, it is standard to deduce from (3.13) and (3.8) that, for some sufficiently large N ∈ N, Thus, Thanks to (3.15), Corollary 3.2 applies and allows us to conclude that if 1 < p, q ≤ ∞ and s ∈ R are arbitrary, and if φ, ψ ∈ C ∞ c (R n ), then for some C and M depending only on n, p, q, s. Now, the fact that the operator (3.12) is bounded follows from this and (3.18). When m = 0 we proceed in a similar fashion, the sole difference being that, in this case, by virtue of (3.10). Finally, there remains to consider the case when m = −n. In this scenario, we first notice that by (3.8), the fact that k(x, λz) = k(x, z), integrations by parts and duality, it is relatively straightforward to show that, for each choice of the cutoff functions φ, ψ ∈ C ∞ c (R n ), for each k ∈ Z with |k| ≤ M . Here, M is a constant which can be as large as desired by ensuring that N (introduced in connection with (3.8)) is sufficiently large. By induction on θ (see below), we shall now show that, for any 1 < p, q < ∞ and s ∈ R, any operator T satisfying the current assumptions and any φ, ψ ∈ C ∞ c (R n ), for each θ ∈ {0, 1, ..., n}, provided N is large enough. To prove (3.24) when θ = 0, given 1 < p, q < ∞ and s ∈ R, pick k ∈ Z such that s ∈ (k, k + 1) and such that (3.23) holds.
Having proved (3.12) (when m = −n) for the F -scale, the corresponding statement for the B-scale can be deduced from this and real interpolation. This concludes the proof of Theorem 3.3. 2 Our next goal is to prove similar mapping properties for a local version of the operator (3.11). This portion of our analysis only requires knowing that, for some m ∈ R, is a bounded operator. We shall therefore assume that this is the case and, given a bounded Lipschitz domain Ω in R n , define where · and R Ω stand, respectively, for the extension by zero outside Ω, and the restriction to Ω of distributions in R n . Thus, T Ω maps C ∞ c (Ω) to (C ∞ c (Ω)) * and we aim at establishing mapping properties for this operator on Besov and Triebel-Lizorkin scales in Ω. A preliminary result in this regard is as follows. Proof. For each p, q ∈ (1, ∞), s > 1 p − 1, and any f ∈ C ∞ c (Ω), we may write s,z (Ω) .

(3.29)
Indeed, equality (1) is a consequence of the way T Ω has been defined. Inequality (2) is due to the boundedness of the restriction operator R Ω , whereas inequality (3) comes from the assumption (3.27). Going further, equality (4) is due to the fact that the norm in A p,q s,0 (Ω) is inherited from the one in A p,q s (R n ). Finally, inequality (5)  Proof. For f ∈ C ∞ c (Ω), we claim that In order to justify this, for any g ∈ C ∞ c (Ω), we write where all pairings are to be understood in the sense of distributions. Indeed, equality (1) is a consequence of the definition of the adjoint of T Ω , whereas equality (2) is based on the definition of T Ω . Next, equality (3) follows from the way R Ω acts on distributions, while equality (4) is simply the definition of the adjoint of T . Finally, equality (5) is once again based on the definition of R Ω . Since, by duality, T * satisfies the same properties as T , Proposition 3.5 applies and the desired conclusion follows from the representation (3.30). 2 Theorem 3.7 Let p, q ∈ (1, ∞), s ∈ R, and denote by p ′ , q ′ the conjugate exponents of p, q, i.e. 1 p + 1 p ′ = 1 and 1 q + 1 q ′ = 1. Then the operator is bounded.
Proof. From Theorem 3.7 we know that T Ω maps (A p ′ ,q ′ −s (Ω)) * boundedly into A p,q s−m (Ω) for all p, q ∈ (1, ∞) and s ∈ R. Thanks to (2.33) and (3.36), we then obtain that T Ω maps (A p ′ ,q ′ −s (Ω)) * into the closure of C ∞ c (Ω) in A p,q s−m (Ω) and, by definition, the latter space is precisely

Local theory: distinguished homotopy operators
In this section we shall construct a class of homotopy operators which allow us to prove some Poincaré type results (pertaining to the fact that closed forms are locally exact) while keeping precise track of the smoothness of the differential forms involved. Our main result in this regard is the theorem below, whose proof occupies the bulk of this section.

2)
and which have the following additional properties.
(2) There exists θ ∈ C ∞ c (Ω) such that for any ℓ-form u with coefficients distributions in (4.5) or s > 1/p and u ∈ D ℓ (d; A p,q s,z (Ω)). (4.8) Proof. Given the local nature of the results and their invariance under pull-back, it suffices to work under the assumption that M = R n (equipped with the standard Euclidean metric) and that Ω is a bounded Lipschitz domain which is star-like with respect to some (Euclidean) ball B ⊂ Ω. Assume that this is the case and bring in the classical Cartan homotopy operator, which we now recall. Specifically, if ℓ ∈ {1, ..., n} and y ∈ B is fixed, define for each ℓ-differential form u = j 1 <···<j ℓ u j 1 ...j ℓ dx j 1 ∧ · · · ∧ dx j ℓ (where, as customary, 'hat' indicates that the symbol underneath is omitted). A straightforward calculation then shows that (4.10) See, e.g., Theorem 4.11 in [45], or [48] for more details. We intend to work with differential forms whose coefficients are not necessarily continuous and, hence, need to alter the definition (4.9) as to avoid integrating over thin sets. One way to achieve this is to average the definition (4.9) with respect to y ∈ B. Concretely, for some fixed function θ ∈ C ∞ c (B) with θ = 1, we introduce for each 1 ≤ ℓ ≤ n where u ∈ C 1 (Ω, Λ ℓ ). Above, x − y is identified with n j=1 (x j − y j )dx j and ∨ stands for the interior product of forms in R n . Then the homotopy property (4.10) is further inherited by the new family of operators. More specifically, For reasons which will become clear in a moment, we find it convenient to consider K t ℓ , the transpose (in the sense of distributions) of (4.11), meaning A straightforward calculation (based on a couple of changes of variables) shows that x ∈ Ω, (4.14) whenever u ∈ C ∞ c (Ω, Λ ℓ−1 ). One most notable feature of the operator (4.13) is that supp (K t ℓ u) is a subset of {λx + (1 − λ)y : x ∈ supp u, y ∈B, 0 ≤ λ ≤ 1}. In particular, since Ω is assumed to be starlike with respect to the ball B, we may conclude that Going further, we note that the dual of (4.12) then becomes (4.16) where, in the current context, δ denotes the formal transpose of d with respect to the (flat) Euclidean metric in R n . Let us point out that the case ℓ = 0 of (4.16) has also been derived in [3].
With 'star' denoting the standard Hodge isomorphism in R n , we now introduce another operator of interest to us, i.e.
Taking (4) in Proposition 2.1 into account, it easily follows from (4.17) and (4.16 (4.18) and Our next goal is to study the mapping properties of the operators J ℓ . To this end, it clearly suffices to analyze scalar integral operators of the form x ∈ Ω, (4.20) where 1 ≤ j ≤ n, 1 ≤ ℓ ≤ n. To get started, let us first write T ℓ,j f in the form where is the integral kernel of T ℓ,j . Expanding (1+τ ) n−ℓ−1 via the Binomial Theorem and changing variables so that z re-scales to a unit vector eventually shows that k ℓ,j (x, z) can be written as a linear combination of terms of the form Next, observe that each such kernel satisfies the uniform estimate and the homogeneity condition k j,i (x, λz) = λ −n+i+1 k j,i (x, z), for each λ > 0. In particular, Theorem 3.8 guarantees that the integral operator with kernel k j,i (x, x − y) (Ω) for each p, q ∈ (1, ∞) and each s ∈ R. Thus, all in all, the operator (4.3) is bounded. In fact, a similar argument yields (4.4) is bounded as well.
Remark I. As a corollary of (4.3)-(4.4) and Proposition 2.4, given any 1 < p, q < ∞, the operators are well-defined, linear and bounded for each 1 ≤ ℓ ≤ n. In fact, (4.25) self-improves to Remark II. An inspection of the above proof shows the following. If D is an open subset of Ω such that Ω\D is also star-like with respect to the ball B, then supp (K ℓ u) ⊂ D whenever

Remark III.
If Ω is a bounded, open subset of R n which is star-like with respect to a ball B ⊂ Ω then, necessarily, Ω is a Lipschitz domain. See p. 17 in [33].

29
In this section we compute the dimension of the so-called relative cohomology groups of a Lipschitz domain Ω ⊂ M , for the exterior derivative operator considered in the context of Besov-Triebel-Lizorkin spaces. Our approach consists of two steps. In a first stage, we employ Theorem 2.13 for the complex associated with d on the scale A p,q s , in which scenario, no boundary conditions are involved. In a subsequent step, boundary conditions are brought into play in a natural fashion, by dualizing the results obtained in step one.
To set the stage, we first recall an elementary, abstract result. For any Banach space X, we denote by X * its dual. Also, if V ⊆ X is a closed subspace of X, we set for the annihilator of V (relative to X).
Lemma 5.1 Let X be a Banach space and let 0 ⊆ W ⊆ V ⊆ X be closed subspaces of X. Then The proof is elementary and is left to the interested reader. The special cases V = X and W = 0 are, in fact, well-known; cf., e.g., p. 86 in [40].
Consider next the family of unbounded operators with domains D ℓ (d; A p,q s (Ω)) and which act according to d ℓ (u) := du for each differential form u ∈ D ℓ (d; A p,q s (Ω)). The first order of business is to identify the dual of (5.3), assuming that M is equipped with a (smooth) Riemannian metric.

Lemma 5.2
Let Ω be a Lipschitz domain, 1 < p < ∞, and fix s < 1 p with s = −1 + 1 p . Also, let 1 < p ′ < ∞ be such that 1/p + 1/p ′ = 1. Then, for each 0 ≤ ℓ ≤ n, the adjoint of the operator (5.3) is and and which acts according to d * ℓ u = δu for each u in the domain of d * ℓ .

Proposition 5.3
Let Ω be a Lipschitz domain and fix 1 < p, q < ∞, s < 1/p. Then, in the context of (5.3), where, as usual R O∩Ω∩Wx denotes the operator of restriction (in the sense of distributions) to the open set O ∩ Ω ∩ W x , etc. Next, we set L ℓ (U ) := {u ∈ A p,q s,loc (U, Λ ℓ ) : du ∈ A p,q s,loc (U, Λ ℓ+1 )} (5.13) so that L ℓ := (L ℓ (U )) U , indexed by open subsets (in the relative topology) of Ω, becomes a sheaf on the compact topological space Ω when equipped with the family of restriction operators (5.14) Note that (5.14) is meaningful in the sense that if Going further, since d ℓ+1 • d ℓ = 0, the family {d ℓ } ℓ≥0 yields the complex Here LCFΩ stands for the sheaf of germs of locally constant functions on Ω, and ι is the natural inclusion operator. Since each A p,q s,loc (U, Λ ℓ ) is stable under multiplication by smooth, compactly supported functions, a partition of unity argument shows that (5.15) provides a fine resolution of the sheaf LCFΩ.
Next, we claim that, in fact, the complex (5.15) is exact. As explained in §2.4, checking this comes down to verifying the following property. Fix an index ℓ ∈ {1, ..., n}, a point ∩ Ω, a differential form v ∈ A p,q s,loc (V, Λ ℓ−1 ) for which R W ∩Ω (u) = dv. To this end, we note that the membership of u to A p,q s,loc (U, Λ ℓ ) entails, by definition, the existence of an open neighborhood W of x o with the property that W ∩ Ω ⊂ U and such that R W ∩Ω (u) ∈ A p,q s (W ∩ Ω, Λ ℓ ). In addition, there is no loss of generality in assuming that W ∩ Ω is small and, when viewed in appropriate local coordinates, it becomes a Lipschitz domain which is starlike with respect to a ball (in the Euclidean geometry). Assuming that this is the case, we denote by K ℓ the family of operators constructed as in Theorem 4.1 but in connection with the Lipschitz domain W ∩ Ω. In particular, since d R W ∩Ω (u) = 0, the representation (4.5) yields R W ∩Ω (u) = dv where v := K ℓ (R W ∩Ω (u)). Moreover, v ∈ A p,q s+1 (W ∩ Ω, Λ ℓ−1 ) ֒→ A p,q s (W ∩ Ω, Λ ℓ−1 ) by (4.26), (2.9), and since there exists w ∈ A p,q s (M, Λ ℓ−1 ) such that v = R W ∩Ω (w) we may ultimately conclude that v ∈ A p,q s,loc (W ∩ Ω, Λ ℓ−1 ), thus finishing the proof the fact that the complex (5.15) is exact. The analysis carried out so far shows that the De Rham theory (cf. Theorem 2.13) applies, and it remains to identifying the cohomology groups associated with the complex (5.15). Concretely, (5.10) follows as soon as we prove that In turn, (5.16) is an easy consequence of Turning our attention to (5.17) we note that, in one direction, if u ∈ A p,q s,loc (Ω, Λ ℓ ) then, from the definition of this space, there exist a finite, open cover {W i } i∈I of Ω along with Hence, if {ξ i } i∈I is a smooth partition of unity subordinate to this cover, it follows that i∈I ξ i w i ∈ A p,q s (M, Λ ℓ ) and u = R Ω i∈I ξ i w i ∈ A p,q s (Ω, Λ ℓ ), as desired. Conversely, if u ∈ A p,q s (Ω, Λ ℓ ) then, by definition, there exists w ∈ A p,q s (M, Λ ℓ ) such that u = R Ω (w). From this, we see that u ∈ A p,q s,loc (Ω, Λ ℓ ), justifying (5.17). This completes the proof of (5.10). Finally, (5.11) is a direct consequence of definitions. 2 Returning to the unbounded operator (5.3), we can now formulate the following Let Ω be a Lipschitz domain and recall the unbounded operators d ℓ introduced in (5.3) and (5.8) for −1 ≤ ℓ ≤ n. Then, if 1 < p, q < ∞ and s < 1/p, these operators have closed ranges, and the same is true for their adjoints. Consequently, Proof. Assume 0 ≤ ℓ ≤ n. The first claim follows from (5.10), the fact the the singular homology groups of Ω have finite dimension and a general functional analytic result to the effect that if T : X → Y is a closed, unbounded operator between two Banach spaces, with the property that Im T , the image of T , has finite codimension in Y , then Im T is a closed subspace of Y . That the adjoint of the operator (5.3) has also a closed range is a consequence of what we have proved so far and the version of Banach's closed range theorem corresponding to closed, densely defined, unbounded operators. See Theorem 5.13 on p. 234 Kato's book. This theorem also gives (5.18). Finally, the case ℓ = −1 is elementary and the proof of the corollary is complete. 2 Proposition 5.5 Assume that Ω is a Lipschitz domain and that 1 < p, q < ∞. Then for each ℓ = 1, ..., n, we have that dim {u ∈ D ℓ (δ; A p,q s (Ω)) : δu = 0 and ν ∨ u = 0} {δω : ω ∈ D ℓ+1 (δ; A p,q s (Ω)) and ν ∨ ω = 0} = b ℓ (Ω) (5.19) if − 1 + 1 p < s < 1 p , Furthermore, corresponding to the case ℓ = 0 we have

21)
and Proof. Assume first that 1 ≤ ℓ ≤ n. Based on Lemma 5.1, Corollary 5.4 and Proposition 5.3, we may write Moreover, corresponding to the case ℓ = n we have Proof. This is an immediate consequence of Proposition 5.5 and Hodge theory; cf. Proposition 2.1. 2 Parenthetically, we record a related, useful result.

Proposition 5.7
Let Ω be an arbitrary open subdomain of M . Then for each ℓ = 1, ..., n, we have that Proof. This is a consequence of results in §2.4, §3. 2

The proofs of the main results
We debut by stating and proving a weaker version of Theorem 1.1.

Proposition 6.1
Let Ω be a Lipschitz domain and fix 1 < p, q < ∞, −1 + 1/p < s < 1/p. Then, for each 0 ≤ ℓ ≤ n − 1, the following are equivalent: (i) the (n − ℓ)-th Betti number of Ω vanishes, i.e. b n−ℓ (Ω) = 0; (ii) there exists a finite constant C > 0 such that for any f ∈ A p,q s (Ω, Λ ℓ ) with df = 0 and ν ∧ f = 0, there exists u ∈ A p,q s (Ω, Λ ℓ−1 ) with du = f , ν ∧ u = 0, and such that Corresponding to ℓ = n, we have the following statement. There exists a finite constant C > 0 such that for any f ∈ A p,q s (Ω, Λ n ) with f, χ Ω j V M = 0, 1 ≤ j ≤ b 0 (Ω), there exists u ∈ A p,q s (Ω, Λ n−1 ) with du = f , ν ∧ u = 0, and such that is onto. When the space intervening in (6.3) are equipped with natural norms (graph norm for the space on the left; the space on the right is simply viewed as a closed subspace of A p,q s (Ω, Λ ℓ )), this operator becomes bounded also. Then the desired conclusion (in particular, the estimate (6.1)) follows from the Open Mapping Theorem.
The last part of the proposition is a consequence of (5.27). 2 We now turn to the as wanted.
By the local theory developed in §4, for each j there exits u j ∈ A p,q s+1,z (O j ∩ Ω, Λ ℓ ) such that Indeed, if the operators J ℓ are as in Theorem 4.1 (with Ω replaced by O j ∩ Ω), we may take u j := J ℓ F j . Then the properties (6.8) follow from (4.6)-(4.7) and (4.25). Going further, we recall that tilde denotes the extension by zero operator (cf. §2.2) and note that u j ∈ A p,q s+1,0 (Ω, Λ ℓ−1 ) satisfies Finally, u := v + G ∈ A p,q s+1 (Ω, Λ ℓ−1 ), solves (1.4) and obeys (1.5). To deal with the last part in the theorem, we note that there is no loss of generality in assuming that Ω is connected and g = 0 (the latter reduction is ensured by reasoning as before). Then Proposition 6.1 yields some v ∈ A p,q s (Ω, Λ n−1 ) such that dv = f , ν ∧ v = 0 and v A p,q s (Ω,Λ n−1 ) ≤ C f A p,q s (Ω,Λ n ) . Let {O j } 1≤j≤N be a finite, open covering of Ω such that each O j ∩ Ω is contained in a coordinate patch and becomes a Lipschitz domain which is star-like with respect to a ball, when viewed as a subset of the Euclidean space. Also, fix {ϕ j } j a smooth partition of unity such that supp ϕ j ⊆ O j for 1 ≤ j ≤ N . Finally, set f j := R O j ∩Ω (d(ϕ j v)) ∈ A p,q s (O j ∩ Ω, Λ n ) and notice that ν ∧ f j = −d ∂ (ν ∧ (ϕ j v)) = 0 on ∂(O j ∩ Ω). Next, since δV M = δ( * 1) = − * d1 = 0, formula (2.3) gives that, for each j, Having established (6.13), consider the operators J ℓ from Theorem 4.1 with Ω replaced by O j ∩ Ω. In view of (6.13), the last identity in (4.6) then allows us to write (Ω,Λ n ) for each j. Consequently, the differential form u := R Ω j u j belongs to A p,q s+1,z (Ω, Λ n−1 ) satisfies du = f , as well as (Ω, Λ ℓ−1 ) then Tr [∂ α v] = g α for each multi-index α of length at most k. In particular, the differential form F := f − dv ∈ A p,q s (Ω, Λ ℓ ) (6.14) satisfies dF = 0 in Ω and, for each multi-index α with |α| ≤ k − 1, g α+e j ∧ dx j = 0. (6.15) Consequently, F ∈ A p,q s,z (Ω, Λ ℓ ) by (2.46). Thus, (5.26) and the current assumptions imply that there exists w ∈ A p,q s+1,z (Ω, Λ ℓ ) such that dw = F in Ω, plus a naturally accompanying estimate. It follows that u := v + w solves (1.7) and, in addition, it obeys (1.9).
Conversely, assume that (ii) in the statement of the theorem holds. By taking g α = 0 for every multi-index α of length ≤ k − 1, it follows that for any f ∈ A p,q s,z (Ω, Λ ℓ ) with df = 0 there exists u ∈ A p,q s+1,z (Ω, Λ ℓ ) with du = f . Thus, by (5.26), b n−ℓ (Ω) = 0, as desired. To treat the last part in the statement of the theorem, corresponding to the case when ℓ = n, we note that, on the one hand, it is straightforward to check that the conditions in (1.10) are indeed necessary for the solvability of (1.7) when ℓ = n.
On the other hand, assuming that the compatibility conditions (1.10) are verified, we can constructġ, v, F , as before. Then, with V R n := dx 1 ∧ · · · ∧ dx n denoting the Euclidean volume form, for each j = 1, ..., b 0 (Ω), we may write thanks to the fact that δV R n = 0 and the second condition in (1.10). With this in hand, (5.28) gives that there exists w ∈ A p,q s+1,z (Ω, Λ n−1 ) satisfying dw = F in Ω and a natural estimate. Thus, u := v + w is the desired solution.

38
We start by recording some useful particular cases of Theorems 1.1-1.2.
Proof. In the case when b n−ℓ (Ω) = 0, (7.1)-(7.2) follow directly from the fact that the boundary value problems dealt with in Theorems 1.1-1.2 are solvable (with zero boundary data). Also, the converse implication is a consequence of Corollary 5.6. Finally, the second part of the proposition follows from the first, after an application of the Hodge star-isomorphism. Proof. This follows immediately from Proposition 7.1 and (2.30). 2
We next discuss a lifting result on Besov and Triebel-Lizorkin spaces on (Euclidean) Lipschitz domains.
Second, when ℓ = 1, the scalar function v appearing in (7.15) actually belongs to L p s+1 (Ω), as a simple application of Proposition 7.6 shows. Third, when Ω is a bounded, three-dimensional, Euclidean domain with a C 2 boundary and when ℓ = 1, the above Hodge decomposition result has been proved by R. Griesinger in [20]. On p. 245 of that paper the author asks whether the higher dimensional version of (7.15) holds, an issue addressed by our proposition above.