The Banach envelopes of Besov and Triebel-Lizorkinspaces and applications to partial differential equations

Osvaldo MendezDepartment of Mathematics

University of Missouri-ColumbiaColumbia, MO 65211

Marius Mitrea∗

Department of MathematicsUniversity of Missouri-Columbia

Columbia, MO 65211

September 21, 2001

Abstract

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that

Bs,qp (Rn) = B

s−n(

1p−1),1

1 (Rn), F s,qp (Rn) = Bs−n(

1p−1),1

1 (Rn),

if 0 < p < 1, 0 < q < 1, s ∈ R, while

Bs,qp (Rn) = B

s−n(

1p−1),q

1 (Rn), F s,qp (Rn) = Bs−n(

1p−1),1

1 (Rn),

if 0 < p < 1, 1 ≤ q < +∞, s ∈ R, and

Bs,qp (Rn) = Bs,1

p (Rn), F s,qp (Rn) = F s,1p (Rn),

if 1 ≤ p < +∞, 0 < q < 1, s ∈ R.Applications to questions regarding the global interior regularity of solutions to

Poisson type problems for the three-dimensional Lame system in Lipschitz domainsare presented.

1 Introduction

The Banach envelope of a quasi-Banach space X (whose dual separates points) is the minimal“enlargement” of X to a Banach space X. In particular,

the inclusion ι : X → X is continuous with dense range. (1.1)

∗Author was supported in part by NSF grant DMS-9870018.1991Mathematics Subject Classification. Primary 46E35, 42B20, 46A16; Secondary 35J55, 47G10, 73C02.Key words: Banach envelope, Besov and Triebel-Lizorkin spaces, Lipschitz domains, layer potentials, bound-ary problems.

1

Applying “hat” has good functorial properties such as preserving the linearity, boundednessand the quality of being an isomorphism for operators between quasi-Banach spaces. As aconsequence, (

X)∗

= X∗. (1.2)

The main result in the first part of this paper is a simple, practical criterion for computingthe Banach envelope of a quasi-Banach space X. The idea is that (1.1)–(1.2) identify theBanach space X uniquely (up to an isomorphism). When applied to the scale of Besov andTriebel-Lizorkin spaces this result yields the following formulas.

Theorem 1.1. For s ∈ R, 0 < p < 1 and 0 < q < 1 we have

Bs,qp (Rn) = F s,q

p (Rn) = Bs−n( 1

p−1),1

1 (Rn) = Fs−n( 1

p−1),1

1 (Rn). (1.3)

If 0 < p < 1 and 1 ≤ q <∞, then

Bs,qp (Rn) = B

s−n( 1p−1),q

1 (Rn), F s,qp (Rn) = B

s−n( 1p−1),1

1 (Rn) = Fs−n( 1

p−1),1

1 (Rn). (1.4)

Finally, in the case when s ∈ R, 1 ≤ p < +∞ and 0 < q < 1 we have

Bs,qp (Rn) = Bs,1

p (Rn), F s,qp (Rn) = F s,1

p (Rn). (1.5)

Our primary motivation for establishing these results stems from problems arising at theinterface between harmonic analysis and PDE’s. A paradigm very useful for applications isdiscussed below. To state it, for each 0 < p < +∞ we set

p ∧ 1 := min p, 1, p ∨ 1 := max p, 1, p′ := (max 0, 1− 1/p)−1. (1.6)

Theorem 1.2. Let T be a linear and bounded operator from F s,qp (Rn) into itself for each

(p, q, s) in a neighborhood of a point (p0, q0, s0) ∈ (0,∞)× (0,∞)× R and assume that T isan isomorphism of F s0,q0

p0(Rn) (alternatively, assume that this operator is Fredholm, onto, or

has finite dimensional cokernel).Then there exists a neighborhood U of (p0, q0, s0) so that:

(i) T extends to an isomorphism of Fs−n( 1

p∧1−1),r

p∨1 (Rn) onto itself for each (p, q, s) ∈ U(respectively, is Fredholm of the same index, onto, or has finite dimensional cokernel).Here r = r(p, q) := q if p, q ≥ 1, and 1 otherwise;

(ii) T ∗ extends to an isomorphism of F−s+n( 1

p∧1−1),r′

p′ (Rn) onto itself for each (p, q, s) ∈ U .

The proof is conceptually simple so that we can outline the main steps here; details areprovided in subsequent sections.

2

Proof. There are three basic ingredients in the proof. The first one is a result from [29] (cf.Corollary 4.8 for the version relevant for us here) to the effect that, for a linear, boundedoperator on some complex interpolation scale of quasi-Banach spaces, the property of beingan isomorphism is stable with respect to the scale parameter (inside the scale). Notice that,in order for this to work in the present context, we need to show that F s,q

p (Rn), s ∈ R,0 < p, q ≤ +∞, is a complex interpolation scale of quasi-Banach spaces (see §4 for theprecise definition of our method of complex interpolation and for a proof of this result).

The second ingredient is that properties such as being an isomorphism, Fredholm, onto,or having a finite dimensional cokernel, are preserved by applying “hat” (cf. Theorem 2.2).Finally, the third ingredient is the actual identification of the Banach envelopes of the quasi-Banach spaces in the Triebel-Lizorkin class. For this see Theorem 1.1.

In order to be more specific, let us assume, for instance, that p0 < 1 and that T isan isomorphism of F s0,q0

p0(Rn). Then, repeated applications of the aforementioned stabil-

ity/preservation results give that there exists a neighborhood U := A×B ×C of (p0, q0, s0)in (0,∞)× (0,∞)×R so that T extends to an isomorphism of F s,q

p (Rn) onto itself for each(p, q, s) ∈ U . Invoking (1.3) and the functorial properties of “hat”, the conclusion is that

T extends to an isomorphism of Fs−n( 1

p−1),1

1 (Rn) for each (p, s) ∈ A × C. This proves thep0 < 1 case in (i). A similar pattern works in the remaining cases as well. Now, (ii) is aconsequence of (i) and duality.

On the scale of Besov spaces we have a similar result (whose proof closely parallels thatof Theorem 1.2).

Theorem 1.3. Let T be a linear and bounded operator from Bs,qp (Rn), into itself for each

(p, q, s) in a neighborhood of a point (p0, q0, s0) in (0,∞)× (0,∞)×R and assume that T isan isomorphism of Bs0,q0

p0(Rn) (alternatively, assume that this operator is Fredholm, onto, or

has finite dimensional cokernel).Then there exists a neighborhood V of (p0, q0, s0) so that:

(i) T extends to an isomorphism of Bs−n( 1

p∧1−1),q∨1

p∨1 (Rn) onto itself for each (p, q, s) ∈ Vwith s 6= s0 (respectively, is Fredholm of the same index, onto, or has finite dimensionalcokernel);

(ii) T ∗ extends to an isomorphism of B−s+n( 1

p∧1−1),q′

p′ (Rn) onto itself for each (p, q, s) ∈ Vwith s 6= s0.

Let us point out that other variations on this theme are possible. For example, some of thescale parameters may be kept fixed, the operator T may depend analytically on a parameter,and we can replace the Euclidean space Rn by the boundary of a Lipschitz domain (providedp0, q0 are sufficiently close to 1 and s ∈ (0, 1)).

It should be remarked that there are several notable predecessors to our stability typeresults. The general philosophy is that certain functional analytic properties of an operatorT acting on complex interpolation scale Xθθ are preserved under small changes in thescale parameter. For the Banach space setting, see [40], [6], and [47]. More recently, thistheory has been refined to allow scales of quasi-Banach spaces in [29]. What is new in

3

Theorems 1.2–1.3 is the fact that, while our hypotheses are made on T acting on Xθθ,the conclusions refer to T acting on a different scale, i.e. Xθθ. In this connection, it isuseful to recall that while “hat” leaves Banach spaces unchanged, it fundamentally affectsquasi-Banach spaces. This is well illustrated by Theorem 1.1, where the Banach envelopesof all the quasi-Banach spaces in the Besov and Triebel-Lizorkin scales are identified for thefirst time. For the applications we have in mind, this is particularly useful.

In the second part of this paper we explain how (appropriate versions of) Theorems 1.2–1.3 can be applied to PDE’s in nonsmooth domains. Below, we elaborate more on this point.A natural approach to the Lp-treatment of elliptic boundary value problems in Lipschitz do-mains, originally developed in [12] and subsequently used successfully in other importantcircumstances, requires considering the mapping properties of certain boundary integral op-erators at the level of “atomic” Hardy spaces (which are quasi-Banach for p < 1). Consider,for instance, the case of the Dirichlet problem with regular data for the three-dimensionalLame system in a Lipschitz domain Ω. In this context, the atomic estimates of B. Dahlbergand C. Kenig from [13] imply that 1

2I + K, the boundary double layer potential operator for

the Lame system, is invertible on Hp1 (∂Ω), the (local) Hardy type space spanned by regular

atoms, if p = 1. Since Hp1 (∂Ω) turns out to be the (local) Triebel-Lizorkin space F 1,2

p (∂Ω),

it follows from Theorem 1.2 that 12I + K is also an isomorphism of B1−s,1

1 (∂Ω) for eachs ∈ (0, s0) with s0 = s0(∂Ω) > 0 small.

In §6 we develop these ideas and establish sharp invertibility results for Lame layerpotentials in Lipschitz subdomains of R3. Then, based on these results, we present a completeanalysis of Poisson type problems for the Lame system in Lipschitz subdomains of R3.Finally, we indicate how these results can be applied to the problem of determining themapping properties of the square root of the Lame system in Lipschitz subdomains of R3.

Lately, there has been a growing interest in these topics and similar or related programsfor scalar elliptic PDE’s have been carried out in [25], [18], [1], [35], [2], [8]. In the presentpaper we initiate the study of such issues for systems of PDE’s. We want to point out thatour methods are rather general and work (almost axiomatically) whenever “atomic” esti-mates have been established. These ideas have been already used in [35] for layer potentialsassociated with the Laplace-Beltrami operator in a Lipschitz subdomain of a Riemannianmanifold.

The organization of the paper is as follows. Section 2 contains functional analysis pre-requisites. In Section 3 we recall the definition and some of the relevant properties of Besovand Triebel-Lizorkin spaces. Section 4 contains a discussion of the complex method of in-terpolation in the context of A-convex quasi-Banach spaces. Here we prove that F s,q

p (Rn)and Bs,q

p (Rn), s ∈ R, 0 < p, q < +∞, are interpolation scales of quasi-Banach spaces fora “natural” complex interpolation method. In the quasi-Banach range, i.e. when p < 1 orq < 1, this seems new. Theorem 1.1 is then proved in Section 5. Applications to PDE’s innon-smooth domains are discussed in Section 6. Among other things, here we formulate andsolve Poisson type problems for the Lame system in arbitrary bounded Lipschitz domains inR

3. Finally, some counterexamples are discussed in §7.

Acknowledgments. It is a pleasure to thank Nigel Kalton and Mike Frazier for the manyconversations we have had on related themes and for some important suggestions and bib-liographical references which have been incorporated in the text. We also thank the referee

4

for the careful reading of the manuscript and some useful comments.

2 Quasi-Banach spaces and Banach envelopes

A quasi-norm on a vector space X is a non-negative positive-homogeneous, real-valued func-tion ‖ · ‖ on X satisfying ‖x‖ = 0 ⇔ x = 0 and for which there exists κ > 0 so that‖x + y‖ ≤ κ(‖x‖ + ‖y‖) for all x, y ∈ X. The best constant κ(≥ 1) satisfying this lastinequality is called the modulus of concavity of X. The quasi-normed space (X, ‖ · ‖) is saidto be topologically p-convex (or p-normable) for some 0 < p ≤ 1 if there exists an equivalentquasi-norm ‖ · ‖∗ and C > 0 so that

‖x1 + x2 + ...+ xn‖p∗ ≤ C(‖x1‖p∗ + ‖x2‖p∗ + ...+ ‖xn‖p∗) (2.1)

for any finite collection x1, x2, ..., xn ∈ X. According to a theorem of Aoki and Rolewicz([30, Theorem 1.3, p.7]), (X, ‖ · ‖) is topologically p-convex if p := (1 + log2κ)−1, where κ isthe modulus of concavity of X.

Any quasi-norm on X induces a locally bounded topological vector space structure onX and, conversely, any such topological structure is induced by any of the equivalent quasi-norms

‖x‖B := infr > 0; x/r ∈ B, x ∈ X, (2.2)

where B is a bounded neighborhood of the origin in X. See [30]. A quasi-Banach space isa locally bounded topological vector space X which is complete with respect to the quasi-norms (2.2). If we prefer to stress the choice of a particular quasi-norm, we shall simplywrite (X, ‖ · ‖).

When s ∈ R and either 0 < p < 1 or 0 < q < 1, the Besov and Triebel-Lizorkin spaces,Bs,qp (Rn) and F s,q

p (Rn), respectively, (cf. the discussion in the subsequent sections) are someof the most familiar examples of quasi-Banach spaces.

Let (X, ‖ · ‖) be a quasi-Banach space and fix B a bounded neighborhood of the originin X. Also, denote by co(B) the convex hull of B in X. Then, if the dual of X separatespoints, the Minkowski functional

‖x‖c := ‖x‖co(B) = infr > 0; x/r ∈ co(B) (2.3)

defines a norm on X. Moreover, a different choice of B generates an equivalent norm. Define(X, ‖ ·‖c), the Banach envelope of (X, ‖ ·‖), to be the completion of X in the norm ‖ ·‖c (see[30]). When no reference to the norm is necessary, we simply write X in place of (X, ‖ · ‖c).Note that

X → X continuously and densely. (2.4)

Also, if X is Banach, then X = X.Consider next two quasi-Banach spaces, (Xi, ‖ · ‖Xi), i = 1, 2, and denote by L(X1, X2)

the collection of all linear, continuous maps from X1 into X2. Clearly, for each boundedneighborhood B of the origin in X1,

5

‖T‖L(X1,X2) := supx∈B‖Tx‖X2 , T ∈ L(X1, X2), (2.5)

defines a quasi-norm on L(X1, X2). As expected, other choices of B yield equivalent quasi-norms. In the end, L(X1, X2) is a quasi-Banach space when furnished with (2.5). Moreover,when X2 is actually Banach then so is L(X1, X2).

A simple but useful observation is contained below.

Proposition 2.1. Let (Xi, ‖·‖Xi), i = 1, 2, be quasi-Banach spaces. Then any T ∈ L(X1, X2)extends to an operator T ∈ L(X1, X2) and the mapping

L(X1, X2) 3 T 7→ T ∈ L(X1, X2) (2.6)

is continuous and injective.In particular, for any quasi-Banach space X and any Banach space Y ,

L(X,Y ) = L(X, Y ). (2.7)

Proof. The verification of the first part is straightforward; cf. also [18]. This also gives theleft-to-right inclusion in (2.7). The opposite one is a consequence of the fact that T |X ∈L(X, Y ), for each T ∈ L(X, Y ).

Theorem 2.2. Let (Xi, ‖ · ‖Xi), i = 1, 2, be quasi-Banach spaces. Then the following aretrue.

(i) If T ∈ L(X1, X2) is an isomorphism then so is T ∈ L(X1, X2).

(ii) If T ∈ L(X1, X2) is onto then so is T ∈ L(X1, X2).

(iii) If T ∈ L(X1, X2) is compact then so is T ∈ L(X1, X2).

(iv) If T ∈ L(X1, X2) is Fredholm then so is T ∈ L(X1, X2). Furthermore,

index (T ;X1, X2) = index (T ; X1, X2). (2.8)

(v) If T ∈ L(X1, X2) is has a finite dimensional cokernel then the same holds for T ∈L(X1, X2).

Proof. For (i), apply Proposition 2.1 to T and T−1 in order to obtain (T )−1 = (T−1).Consider next (ii). Let y ∈ X2 with ‖y‖X2

≤ 1; then there exists y ∈ co (BX2) such that‖y − y‖X2

≤ 1/2. Thus, y =∑λjyj with yj ∈ BX2 , 0 ≤ λj ≤ 1,

∑λj = 1. By hypothesis,

there exist κ := κ(X1, X2, T ) < ∞ and xj ∈ X1 so that ‖xj‖X1 ≤ κ and Txj = yj for each

j. If we set x :=∑λjxj ∈ X1 then ‖x‖X1

≤ κ and Tx = y. In particular, ‖y − T x‖ ≤ 1/2.

This suffices to conclude that T ∈ L(X1, X2) is actually onto; cf., e.g., Lemma 2.4 in [29].Going further, (iii) follows more or less directly from the fact that T commutes with

the operation of taking the convex hull and definitions. In turn, (iii) readily implies thatFredholmness is preserved under “hat”. Hence, to finish the proof of (iv), we are left with

6

proving (2.8). To this end, let E be a finite dimensional subset of X2 so that E⊕T (X1) = X2

and denote by F the kernel of T ∈ L(X1, X2). Then introduce the linear operator

T ′ : X1/F ⊕ E → X2, T ′([x], y) := Tx+ y, (2.9)

which clearly, is an isomorphism. Thus, by (i), it remains an isomorphism after applying the“hat” to the spaces in (2.9). Since “hat” commutes with ⊕, /, and leaves E,F invariant (cf.Theorem 2.4 below), it follows that

T ′ : X1/F ⊕ E → X2 is an isomorphism. (2.10)

This easily gives (2.8). Finally, with the aid of (ii), (v) is proved in a similar fashion to (iv);we leave the details to the interested reader.

In order to continue, we need to explain one more thing. Specifically, if (Xi, ‖ · ‖Xi),i = 1, 2, are quasi-Banach spaces so that X1 ∩ X2 is dense in (Xi, ‖ · ‖Xi), i = 1, 2, we saythat (X1, ‖ · ‖X1)∗ → (X2, ‖ · ‖X2)∗ if any Λ ∈ (X1, ‖ · ‖X1)∗ has the property that Λ|X1∩X2

extends to some element Λ ∈ (X2, ‖ · ‖X2)∗ with

‖Λ‖(X2,‖·‖X2)∗ ≤ C‖Λ‖(X1,‖·‖X1

)∗ (2.11)

for some C > 0 independent of Λ. Also, we say that (X1, ‖ · ‖X1)∗ = (X2, ‖ · ‖X2)∗ if(X1, ‖ · ‖X1)∗ → (X2, ‖ · ‖X2)∗ and (X2, ‖ · ‖X2)∗ → (X1, ‖ · ‖X1)∗.

Corollary 2.3. For any quasi-Banach space X, there holds (X)∗ = X∗.

Proof. It follows directly from Proposition 2.1.

We now turn to the main result of this section which can be thought of as a converseto Corollary 2.3 in the class of Banach spaces satisfying (2.4). See also [30] for a differentcriterion.

Theorem 2.4. Let (X, ‖ · ‖X) be a quasi-Banach space and (Y, ‖ · ‖Y ) be a Banach space.

Then (X, ‖ · ‖X) = (Y, ‖ · ‖Y ) if and only if the following two conditions are fulfilled:

(i) the inclusion map ι : (X, ‖·‖X) → (Y, ‖·‖Y ) is well-defined, continuous and with denseimage;

(ii) (X, ‖ · ‖X)∗ = (Y, ‖ · ‖Y )∗.

Proof. The necessity is contained in Corollary 2.3. As for sufficiency, since the completionof X with respect to ‖ · ‖Y and ‖ · ‖c (cf. (2.3)) is Y and X, respectively, we only need toprove that

‖ · ‖c ≈ ‖ · ‖Y on X. (2.12)

To see this, we use the Hahn-Banach theorem in order to find, for each fixed x ∈ X, afunctional Λ = Λx,

Λ ∈ (X, ‖ · ‖c)∗ = (X, ‖ · ‖X)∗ = (Y, ‖ · ‖Y )∗ (2.13)

7

with ‖Λ‖(X,‖·‖c)∗ ≤ 1 and |Λ(x)| = ‖x‖c. Thus, ‖Λ‖(Y,‖·‖Y )∗ ≤ C so that

‖x‖c ≤ ‖Λ‖(Y,‖·‖Y )∗‖x‖Y ≤ C‖x‖Y (2.14)

for some C > 0 independent of x ∈ X.Conversely, there exists Λ′ = Λ′x, Λ′ ∈ (Y, ‖ · ‖Y )∗ = (X, ‖ · ‖X)∗ = (X, ‖ · ‖c)∗ so that

‖Λ′‖(Y,‖·‖Y )∗ ≤ 1 and |Λ′(x)| = ‖x‖Y . Hence, ‖x‖Y ≤ ‖Λ′‖(X1‖·‖c)∗‖x‖c ≤ C ′‖x‖c with C ′ > 0independent of x ∈ X. This proves (2.12) and finishes the proof of the theorem.

To exemplify these ideas, let us give a short proof of the well known fact that ˆp = `1 for0 < p ≤ 1. Indeed, since for this range of indices (`p)∗ = `∞ ([16]), Theorem 2.4 applies andyields the desired conclusion. Other examples of specific calculations of Banach envelopesare offered in the context of Besov and Triebel-Lizorkin spaces in subsequent sections.

For later reference, let us note here one more criterion useful in the calculation of Banachenvelopes.

Proposition 2.5. Let Xi, i = 1, 2 be two quasi-Banach spaces and Yi, i = 1, 2, be twoBanach spaces so that the inclusions Xi → Yi are well-defined and continuous. Next, letT ∈ L(Y1, Y2) be onto and so that T |X1 : X1 → X2 is well-defined, continuous and has aretract (i.e. a continuous inverse to the right).

Then, if X1 = Y1 we also have X2 = Y2.

Proof. The proof is very simple. Using the fact that T has a continuous inverse to theright and the functorial properties of “hat”, we see that T : X1 → X2 is onto. Thus, byhypotheses, X2 = ImT = Y2.

3 Besov and Triebel-Lizorkin spaces

In this section we present the definitions and a summary of basic results for Besov andTriebel-Lizorkin spaces. In the interest of brevity, we shall only develop those aspects whichare relevant for us in the sequel. More detailed accounts can be found in, e.g., [19], [20], [21],[22], [42], [43], [39], [37].

Let Φ ∈ S(Rn), ϕ ∈ S(Rn) be two Schwartz functions such that:

1. supp F(Φ) ⊆ ξ ∈ Rn; |ξ| ≤ 2 and |F(Φ)(ξ)| ≥ c > 0 uniformly for |ξ| ≤ 53,

2. suppF(ϕ) ⊆ ξ ∈ Rn; 12≤ |ξ| ≤ 2 and |F(ϕ)(ξ)| ≥ c > 0 uniformly for 3

5≤ |ξ| ≤ 5

3,

3.∑

i∈Z |F(ϕ)(2iξ)|2 = 1, if ξ 6= 0.

Here, F(f) denotes the Fourier transform of f . Set ϕi(x) := 2inϕ(2ix), i ∈ Z. For s ∈ Rand 0 < p, q ≤ +∞ the (inhomogeneous) Besov spaces are defined as follows:

Bs,qp (Rn) :=

f ∈ S ′(Rn); ‖f‖Bs,qp (Rn) := ‖Φ ∗ f‖Lp +

( ∞∑i=1

(2is‖ϕi ∗ f‖Lp)q) 1q< +∞

, (3.1)

8

whereas, for s ∈ R, 0 < p < +∞ and 0 < q ≤ +∞, the (inhomogeneous) Triebel-Lizorkinspaces are defined as

F s,qp (Rn) :=

f ∈ S ′(Rn); ‖f‖F s,qp (Rn) := ‖Φ∗f‖Lp+

∥∥∥( ∞∑i=1

(2is|ϕi∗f |)q) 1q∥∥∥Lp< +∞

. (3.2)

There is also an appropriate version of (3.2) corresponding to p = +∞; see, e.g., [39].As is well-known, (3.1)–(3.2) are Banach spaces for p, q ≥ 1, but only quasi-Banach

when either p < 1 of q < 1. Note that the “diagonals” of the two scales coincide, i.e.Bs,pp (Rn) = F s,p

p (Rn) for 0 < p ≤ +∞ and s ∈ R. There are also homogeneous versions

of these spaces, denoted by Bs,qp (Rn) and F s,q

p (Rn), which are obtained by dropping therequirement that ‖Φ ∗ f‖Lp < +∞ and extending the sum in i to the range i ∈ Z.

Following [20], we also introduce a discrete version of the Triebel-Lizorkin scale of spacesby defining f s,qp , for s ∈ R, 0 < p < +∞ and 0 < q ≤ +∞ as the collection of all sequencess = sQQ indexed by dyadic cubes Q ⊂ Rn such that

‖s‖fs,qp :=∥∥∥( ∑

Q dyadic

(|Q|−1/2−s/n|sQ|χQ)q)1/q∥∥∥

Lp< +∞, (3.3)

where χQ is the characteristic function of Q and |Q| is the (Euclidean) volume of Q. Thereis also an appropriate version of this definition when p = +∞. Specifically,

‖s‖fs,q∞ := supP dyadic

(1

|P |

∫P

∑Q⊂P

(|Q|−1/2−s/n|sQ|χQ(x))q dx

)1/q

. (3.4)

Observe that Jα(sQQ) := |Q|−α/nsQQ is an isomorphism between f s,qp and f s+α,qp for

each α ∈ R. Also, as is well known, each f s,qp is a quasi-Banach lattice. The inhomogeneoussequence space f s,qp is defined as in (3.3) except that, this time, one insists on (indexing thesequence and) performing the sum only over dyadic cubes Q ⊂ Rn with |Q| ≤ 1. Theseenjoy similar properties as their homogeneous counterparts.

Next, as in §5 of [22], the sequence spaces bs,qp associated with the Besov scale are intro-duced, for s ∈ R, and 0 < p, q ≤ +∞, as the collection of all sequences s = sQQ, indexedby dyadic cubes Q ⊂ Rn satisfying

‖s‖bs,qp :=∑ν∈Z

(∥∥∥ ∑|Q|=2−nν

|Q|−1/2−s/n|sQ|χQ∥∥∥Lp

)q1/q

=∑ν∈Z

( ∑|Q|=2−nν

(|Q|−1/2−s/n+1/p|sQ|)p)q/p1/q

< +∞. (3.5)

Again, the inhomogeneous spaces bs,qp are defined similarly except that the sum only involvescubes Q satisfying |Q| ≤ 1. They are all quasi-Banach spaces for the indicated ranges ofindices.

9

According to Theorem 7.20 in [22], the continuous versions of the spaces introducedabove are actually isomorphic to their respective discrete versions via an wavelet transform.Nonetheless, the occasional advantage of working with the sequence spaces is that they arequasi-Banach lattices. In the sequel, we shall make repeated use (sometimes tacitly) of thisobservation.

In the next several theorems we collect some basic properties of Besov and Triebel-Lizorkin spaces (cf., e.g., [42], [39] and the references therein).

Theorem 3.1 (Embeddings). For 0 < p0 ≤ p1 ≤ +∞, s0, s1 ∈ R, 0 < q0 ≤ q1 ≤ +∞with s0 − n

p0= s1 − n

p1, the inclusion

Bs0,q0p0

(Rn) → Bs1,q1p1

(Rn) (3.6)

is continuous with dense range. Moreover, the same holds for the inclusion

F s0,q0p0

(Rn) → F s1,q1p1

(Rn), (3.7)

provided 0 < p0 < p1 < +∞, 0 < q0, q1 ≤ +∞ and s0 − np0

= s1 − np1

.

Theorem 3.2 (Duality). For s ∈ R, 0 < p ≤ 1 and 0 < q < +∞, one has

(Bs,qp (Rn))∗ = B

−s+n( 1p−1),q′

∞ (Rn) (3.8)

and

(F s,qp (Rn))∗ = F

−s+n( 1p−1),∞

∞ (Rn) = B−s+n( 1

p−1),∞

∞ (Rn). (3.9)

Also, for 1 ≤ p < +∞, 0 < q < +∞ and s ∈ R,

(Bs,qp (Rn))∗ = B−s,q

′

p′ (Rn) (3.10)

and

(F s,qp (Rn))∗ = F−s,q

′

p′ (Rn). (3.11)

Since (3.11) with 0 < q < 1 does not seem to be covered in the current literature, wepresent a short proof based on Proposition 2.5 and a result from [46] to the effect that thecorresponding duality statement holds at the level of sequence spaces. More specifically, ifT denotes the inverse ϕ-transform of Frazier-Jawerth ([20]), then

T : f s,qp (Rn)→ F s,qp (Rn) (3.12)

is well-defined, linear, bounded and has a continuous right inverse for any s ∈ R, 0 < p <+∞, 0 < q ≤ +∞; see Theorem 12.2 in [20]. Given the aforementioned result from [46],Theorem 2.4 gives that

f s,qp (Rn) = f s,1p (Rn), if 1 ≤ p < +∞, 0 < q ≤ 1, s ∈ R. (3.13)

10

Thus, the hypotheses of Proposition 2.5 are verified and, hence,

F s,qp (Rn) = F s,1

p (Rn), for 1 ≤ p < +∞, 0 < q ≤ 1, s ∈ R. (3.14)

In particular, we get that F s,qp (Rn) and F s,1

p (Rn) have the same dual for the ranges of indices

specified in (3.14). Now, the fact that (F s,1p (Rn))∗ = F−s,∞p′ (Rn) is known; see Remark 5.14,

p. 80 in [20] and p. 20 in [39].To state the next result, let P denote the collection of all polynomials in Rn.

Theorem 3.3 (Lifting Property). For f ∈ S ′(Rn)/P and s ∈ R, 0 < p, q ≤ +∞, the twoassertions below are equivalent:

(i) f ∈ F s,qp (Rn);

(ii) ∂kf ∈ F s−1,qp (Rn) for each 1 ≤ k ≤ n,

with the obvious equivalence of norms. Also, for each tempered distribution f ∈ S ′(Rn), thefollowing three statements are equivalent:

(iii) f ∈ F s,qp (Rn);

(iv) f ∈ Lp(Rn) ∩ F s,qp (Rn);

(v) f ∈ F s−1,qp (Rn) and ∂kf ∈ F s−1,q

p (Rn) for each 1 ≤ k ≤ n,

with the corresponding equivalence of norms.

As is well known, the scales F s,qp (Rn), Bs,q

p (Rn) encompass many of the more familiar,classical spaces. For later reference, let us recall here that:

Bs,∞∞ (Rn) = Cs(Rn), 0 < s 6= integer, (3.15)

F s,2p (Rn) = Lps(R

n), 1 < p < +∞, s ∈ R, (3.16)

F 0,2p (Rn) = Hp(Rn), 0 < p ≤ 1, (3.17)

F 0,2p (Rn) = Hp(Rn), 0 < p ≤ 1, (3.18)

F 0,2∞ (Rn) = BMO(Rn). (3.19)

Hereafter, for 0 < p ≤ 1 we set

Hp(Rn) :=f =

∑j≥0

λjaj ∈ S ′(Rn)/P ; (λj)j ∈ `p, aj is an Lp1 − atom for Hp, (3.20)

and equipped it with the natural quasi-norm. Recall that in the case when nn+1

< p ≤ 1 and

max1, p < p1 < +∞ an Lp1-atom for Hp is a function satisfying supp a ⊆ Q for some Qcube in Rn and so that

11

∫Rn

a = 0, ‖a‖Lp1 ≤ |Q|1p1− 1p , (3.21)

(recall that |Q| denotes the Euclidean volume of Q). For other (smaller) values of p see, e.g.,[32]. Also, Hp(Rn), the local Hardy space, is defined as in (3.20) except for the followingdifference. This time, all atoms are supported in cubes Q satisfying |Q| ≤ 1 and, moreover,those atoms supported in cubes Q with |Q| = 1 are no longer required to have vanishingmoments.

In the last part of this section, we discus one more such identification. This involvesHardy-like spaces spanned by “regular” atoms, a class which arises naturally in PDE’s. Tobe specific let n

n+1< p ≤ 1 and max1, p < p1 < +∞. Call A an Lp1-atom for Hp

1 if thereexists some cube Q in Rn so that

suppA ⊆ Q, ‖∇A‖Lp1 ≤ |Q|1p1− 1p . (3.22)

Then, inspired by [12], for nn+1

< p ≤ 1 set

Hp1 (Rn) :=

f =

∑j≥0

λjAj ∈ S ′(Rn)/R; (λj)j ∈ `p, Aj is an Lp1 − atom for Hp1

(3.23)

and equip it with the natural quasi-norm. To be more pedantic, we should have writtenHp,p1

1 (Rn), but it turns out that different choices of p1 (so that max1, p < p1 < +∞) yieldisomorphic spaces.

Proposition 3.4. For each nn+1

< p ≤ 1 we have the identification

Hp1 (Rn) = F 1,2

p (Rn). (3.24)

Proof. From [42], [39], [20] and (3.18),

f ∈ F 1,2p (Rn)⇔ f ∈ S ′(Rn)/R and ∇f ∈ F 0,2

p (Rn) = Hp(Rn) (3.25)

plus equivalences of norms. Since ∇A ∈ Hp(Rn) for any A which is an Lp1-atom for Hp1

(note that the vanishing moment condition is automatically satisfied), and ‖∇A‖Hp ≤ Cwith C independent of A, it follows that

∇ : Hp1 (Rn)→ Hp(Rn) is well-defined and bounded. (3.26)

Now (3.25) and (3.26) prove the left-to-right inclusion in (3.24). As for the opposite inclusion,Theorem 7.4 in [20] shows that it is enough to check that, if 1 < p1 < +∞, then any atomA for F 1,2

p (Rn) is also a fixed multiple of an Lp1-atom for Hp1 . To this end, recall from [20]

that an atom for F 1,2p (Rn) satisfies

(i) suppA ⊆ J , dyadic cube in Rn;

(ii) A =∑

Q⊆J sQaQ, where (aQ)Q are smooth atoms for F 1,2p (Rn) and the sequence of

coefficients s = sQQ satisfies ‖s‖f1,2p≤ |J |−1/p.

12

Recall that f s,qp , introduced at the beginning of this section, stands for the discrete Littlewood-Paley spaces from [20] (cf. also [21]). Then, so we claim,

‖∇A‖Lp1 ≤ C‖A‖F 1,2p1≤ C‖s‖f1,2

p1≤ C|J |

1p1 ‖s‖f1,2

∞≤ C|J |

1p1− 1p . (3.27)

Indeed, the first inequality above follows by (a version of) (3.25) and (3.16), the second oneby Theorem 4.1, p. 60 in [20], the third one by the inequality displayed in Remark 7.3, p. 89of [20], and the last one by the properties of A listed above. The estimate (3.27) proves theright-to-left inclusion in (3.24) and finishes the proof of the proposition.

We also introduce Hp1 (Rn), the inhomogeneous version of the space Hp

1 (Rn) by consid-ering only cubes of side-length ≤ 1 in the definition of Hp

1 (Rn). We have

Proposition 3.5. For nn+1

< p ≤ 1, there holds

Hp1 (Rn) = F 1,2

p (Rn). (3.28)

Proof. The right-to-left inclusion can be proved along the same lines as the correspondinghomogeneous identification. Here, the only novelty is the limitation on the size of the supportof atoms. As for the left-to-right inclusion, we shall need the fact that

‖ · ‖Lp + ‖ · ‖F 1,2p

(3.29)

is an equivalent norm on F 1,2p (Rn). See the theorem on p. 98 of [43] (and Proposition 3, p. 95

in [39]) for this. Now, since

‖f + g‖pF s,qp≤ ‖f‖p

F s,qp+ ‖g‖p

F s,qp, s ∈ R, 0 < p ≤ 1, p ≤ q ≤ +∞, (3.30)

plus a similar p-triangle inequality for ‖ · ‖Lp , it suffices to show that, for some fixedmax 1, p < p1 < +∞, the collection of all Lp1-atoms for Hp

1 is a bounded subset ofF 1,2p (Rn). Let A be such an atom. Note that ‖A‖Lp ≤ C, with C > 0 independent of A,

because A is supported in a cube Q of side-length ≤ 1 and satisfies ‖A‖Lp1 ≤ |Q|1/n+1/p1−1/p.Also, by Proposition 3.4, ‖A‖F 1,2

p≤ C since, clearly, A ∈ Hp

1 (Rn). The desired conclusionfollows.

Let us now briefly explain how the scales F s,qp (Rn), Bs,q

p (Rn) can be adapted to subdo-mains of Rn. For a more detailed discussion the interested reader may also consult Chapter4 in [44] and [25]. A domain Ω ⊆ Rn is called Lipschitz if its boundary ∂Ω is locally given bygraphs of Lipschitz functions; see, e.g., [36] for more on this. For a fixed, bounded Lipschitzdomain Ω ⊆ Rn and s > 0, 0 < p, q ≤ +∞, define Bs,q

p (Ω), F s,qp (Ω) as the collections of

restrictions to Ω of functions from Bs,qp (Rn) and F s,q

p (Rn), respectively. These are endowedwith the natural metrics. An important subclass is that of potential spaces on Ω, i.e.

Lpα(Ω) = Fα,2p (Ω), 1 < p < +∞, α ≥ 0. (3.31)

If α < 0, 1 < p < +∞, 1p

+ 1q

= 1, then a distribution f ∈ D′(Ω) belongs to the space Lpα(Ω)if

‖f‖Lpα(Ω) := sup|〈f, ϕ〉|; ϕ ∈ C∞c (Ω), ‖ϕ‖Lq−α(Rn) ≤ 1

< +∞. (3.32)

13

Also, for any α ∈ R we set Lpα,0(Ω) := f ∈ Lpα(Rn); supp f ⊆ Ω. Let us also point outthat Bs,q

p (∂Ω), F s,qp (∂Ω) and Hp

1 (∂Ω) can also be introduced by localizing and pulling-backthe corresponding scales from R

n−1.It is not hard to see that with this definition we can obtain atomic decompositions for the

spaces of Besov and Triebel-Lizorkin on ∂Ω (at least for s > 0) similar to those in [19], wherethe corresponding atoms on ∂Ω are defined in the obvious way (cf. [18]). In the propositionbelow, we single out a particular instance (which is the analogue of Proposition 3.5).

Proposition 3.6. For n−1n< p < 1 and a bounded Lipschitz domain Ω ⊆ Rn, we have

Hp1 (∂Ω) = F 1,2

p (∂Ω)

=f ∈ Lip (∂Ω)′; f =

∑i≥0

λiai, (λi)i ∈ `p, ai Lp1 − atom for Hp1 (∂Ω)

.

(3.33)

Finally, we present a further characterization of Hp1 (∂Ω).

Proposition 3.7. Let Ω ⊆ Rn be a bounded Lipschitz domain. Then, for n−1n

< p < 1, wehave

Hp1 (∂Ω) = B

1−n( 1p−1),1

1 (∂Ω) ∩f ∈ Lip (∂Ω)′; ∇tanf ∈ Hp(∂Ω)

, (3.34)

where ∇tan denotes the tangential gradient on ∂Ω.

Proof. The left to right inclusion follows by observing that the atoms for Hp1 (∂Ω) are the

same as those for B1−n( 1

p−1),1

1 (∂Ω) and that the tangential gradient operator maps Hp1 (∂Ω)

boundedly into Hp(∂Ω) (as it can be seen at the level of atoms). To prove the right toleft inclusion we observe that by the classical embedding results (recalled in the first partof this section) any distribution in the right-hand side belongs to Lr(∂Ω) for some r > 1.As is well known, this latter space embeds further into Hp(∂Ω). Thus, if f belongs to thespace in the right side of (3.34), it follows that ψf ∈ Hp(∂Ω) and ∇tan(ψf) ∈ Hp(∂Ω) forany compactly supported Lipschitz function ψ on ∂Ω. Assuming that ψ has small support,pulling everything to Rn−1 and invoking the lifting theorem, gives that ψf ∈ Hp

1 (∂Ω). Hence,f ∈ Hp

1 (∂Ω), as desired.

4 Interpolation of Besov and Triebel-Lizorkin spaces

In this section, we shall be concerned with the interpolation of Besov and Triebel-Lizorkinspaces for the full range of indices. For the real method, complete results are available andthe reader is referred to, e.g., [42], [43], [20], [17] and [39] for details. Here we shall focus onthe complex method for the Besov and Triebel-Lizorkin classes; we elaborate on this below.

Let us first recall that if X is a quasi-Banach space and Ω ⊆ C is an open subset of thecomplex plane then f : Ω → X is called analytic if for each z0 ∈ Ω there exists r > 0 suchthat there is a power series expansion f(z) =

∑∞n=0 xnz

n, xn ∈ X, converging uniformly for|z−z0| < r. This definition of analyticity is actually necessary since complex differentiability

14

turns out to be too weak a notion; see [27], [28], [45] for a more detailed account of the theoryof quasi-Banach valued analytic functions.

Going further, if (X0, X1) is an interpolation couple of quasi-Banach spaces, we defineF , the class of admissible functions, as the collection of all bounded, analytic functionsf : z ∈ C; 0 < Re z < 1 → X0 +X1 which extend continuously to the closure of the stripsuch that the traces t 7→ f(j + it) are bounded continuous functions into Xj, j = 0, 1.

Following Calderon’s original definition for Banach spaces, it is natural to try to endowF with the quasi-norm

‖f‖F := max

supt∈R‖f(it)‖X0 , sup

t∈R‖f(1 + it)‖X1

. (4.1)

Then one sets [X0, X1]θ := x ∈ X0 + X1; x = f(θ) for some f ∈ F, 0 < θ < 1, andequip it with the quasi-norm ‖x‖[X0,X1]θ := inf ‖f‖F ; f ∈ F , f(θ) = x, for x ∈ [X0, X1]θ.Note that the modulus of concavity of [X0, X1]θ can be controlled in terms of the moduli ofconcavity for Xj, j = 0, 1. Also, if T is a linear bounded operator mapping Xj into itself,j = 0, 1, then T is clearly bounded on X0 + X1 and is therefore an interpolating operator(i.e. composition with T is a bounded endomorphism of F).

Although natural, the above version of complex interpolation is not, however, usuallyemployed in the existing literature. In order to avoid discussing analytic functions withvalues in quasi-Banach spaces, a popular variant is to replace F by the subspace generatedby functions having finite-dimensional ranges. Originating in [38], this is used in many placesin the literature such as [24], [23] and [9].

Compared to the case of Banach spaces, some immediate difficulties occur in the generalquasi-Banach context. Most notably, first F and then the intermediate spaces [X0, X1]θ,are not necessarily complete. As is well known, this is due to the possible failure of theMaximum Modulus Principle in quasi-Banach spaces. However, there exists a distinguishedsubclass of quasi-Banach spaces, called A-convex (analytically convex) in [28], in which theMaximum Modulus Principle is nonetheless valid.

A quasi-Banach space (X, ‖ · ‖X) is called A-convex if there is a constant C so that forevery polynomial P : C → X we have ‖P (0)‖X ≤ C max|z|=1 ‖P (z)‖X . For our purposes,the following result from [28], [15] is going to be of basic importance.

Theorem 4.1. For a quasi-Banach space (X, ‖ · ‖X) the following conditions are equivalent:

(i) X is A-convex;

(ii) X has an equivalent quasi-norm ‖ · ‖ which is plurisubharmonic, i.e.

1

2π

∫ 2π

0

‖x+ eiθy‖ dθ ≥ ‖x‖, for any x, y ∈ X;

(iii) X has an equivalent quasi-norm ‖ · ‖ so that log ‖ · ‖ is plurisubharmonic;

(iv) X has an equivalent quasi-norm ‖ · ‖ so that ‖ · ‖p is plurisubharmonic for some 0 <p < +∞;

15

(v) X has an equivalent quasi-norm ‖ · ‖ so that ‖ · ‖p is plurisubharmonic for each 0 <p < +∞;

(vi) there exists C so that max ‖f(z)‖X ; 0 < Re z < 1 ≤ Cmax ‖f(z)‖X ; Re z = 0, 1for any analytic function f : z; 0 < Re z < 1 → X which is continuous and boundedon z; 0 ≤ Re z ≤ 1.

Clearly, any Banach space is A-convex. Other examples of A-convex quasi-Banach spacescan be manufactured by means of the following simple observation, which closely parallelsProposition 2.3 in [15].

Lemma 4.2. Assume that (Ω,Σ, µ) is a measure space, 0 < p ≤ +∞ and that (X, ‖ · ‖X)is an A-convex quasi-Banach space. Then Lp(Ω, X), the space of X-valued functions whichare p-th power integrable on Ω, is an A-convex quasi-Banach space.

Proof. Assume first that 0 < p < +∞, Then, by Theorem 4.1, there exists an equivalentquasi-norm ‖·‖ on X so that ‖·‖p is plurisubharmonic. It follows that for any f, g ∈ Lp(Ω, X)and each fixed ω ∈ Ω, the function uω(z) := ‖f(ω) + zg(ω)‖p is subharmonic. Hence, so isU(z) :=

∫Ωuω(z) dµ(ω) =

∫Ω‖f(ω) + zg(ω)‖p dµ(ω) = ‖f + zg‖pLp(Ω,X). Now, the desired

conclusion follows from Theorem 4.1.When p = +∞, we simply write

1

2π

∫ 2π

0

supω∈Ω‖f(ω) + eiθg(ω)‖X dθ ≥ sup

ω∈Ω

( 1

2π

∫ 2π

0

‖f(ω) + eiθg(ω)‖X dθ)

≥ supω∈Ω‖f(ω)‖X . (4.2)

The proof of the lemma is finished.

The relevance of this lemma for the applications we have in mind is brought forward by thefollowing.

Proposition 4.3. For any s ∈ R, 0 < p, q ≤ +∞, the spaces F s,qp (Rn) and Bs,q

p (Rn) are A-convex. The same is true for their homogeneous versions and for the associated (homogeneousand inhomogeneous) discrete spaces.

Proof. This is a consequence of Lemma 4.2 and the way the quasi-norms in (3.1), (3.2) and(3.3) are defined.

Next we discuss another criterion for A-convexity valid for quasi-Banach lattices of func-tions. To this effect, let (Ω,Σ, µ) be a σ−finite measure space and let L0 be the collectionof all complex-valued, µ-measurable functions on Ω. Recall that a quasi-Banach functionspace X on (Ω,Σ, µ) is an order-ideal in the space L0 containing a strictly positive function,equipped with a quasi-norm ‖ · ‖X so that (X, ‖ · ‖X) is complete, and if f ∈ X and g ∈ L0

with |g| ≤ |f | a.e. then g ∈ X with ‖g‖X ≤ ‖f‖X .Going further, a quasi-Banach lattice of functions (X, ‖ · ‖X) is called lattice r-convex if∥∥∥( m∑

j=1

|fj|r)1/r∥∥∥

X≤( m∑j=1

‖fj‖rX)1/r

(4.3)

16

for any finite family fj1≤j≤m of functions from X (see, e.g., [26]; cf. also [33], Vol. II). Thisimplies that the space

[X]r :=f measurable; |f |1/r ∈ X

, normed by ‖f‖[X]r := ‖ |f |1/r‖rX , (4.4)

is a Banach function space, called the r-convexification of X (cf. also [33], Vol. II, pp. 53-54).The theorem below is due to N. Kalton ([26], [28]).

Theorem 4.4. Let X be a (complex) quasi-Banach lattice of functions and denote by κ itsmodulus of concavity. Then the following assertions are equivalent:

(i) X is A-convex;

(ii) X is lattice r-convex for some r > 0;

(ii) X is lattice r-convex for each 0 < r < (1 + log2κ)−1.

Proof. This follows directly from Theorem 4.4 in [28] and Theorem 2.2 in [26] provided Xsatisfies an upper p-estimate with p := (1+log2κ)−1. That is, for some equivalent quasi-norm‖ · ‖ and some constant C > 0,

‖ |x1| ∨ ... ∨ |xn| ‖p ≤ Cn∑j=1

‖xj‖p (4.5)

for any finite collection x1, ..., xn ∈ X. However, this is a simple consequence of the fact thatin our case |x1| ∨ ... ∨ |xn| ≤ |x1|+ ...+ |xn| and the Aoki-Rolewicz theorem (recalled at thebeginning of §2).

Next, consider two quasi-Banach lattices of functions (Xj, ‖·‖Xj), j = 0, 1. The Calderon

product X1−θ0 Xθ

1 , where 0 < θ < 1, is the function space defined by the quasi-norm

‖f‖ := inf‖f0‖1−θ

X0‖f1‖θX1

; |f | ≤ |f0|1−θ|f1|θ, fj ∈ Xj, j = 0, 1. (4.6)

A simple yet important feature for us here is that the Calderon product “commutes” withthe process of convexification. More concretely, if X0, X1 are as above and, in addition, X0,X1 are also lattice r-convex for some r > 0, it is straightforward to check that

[X1−θ0 Xθ

1 ]r = ([X0]r)1−θ([X1]r)θ, ∀ θ ∈ (0, 1), (4.7)

in the sense of equivalence of quasi-norms.

Lemma 4.5. For each s ∈ R, 0 < p, q ≤ +∞ and 0 < r < min p, q, the spaces f s,qp and

bs,qp are lattice r-convex. Moreover,

[f s,qp ]r = f s′,q′

p′ , [bs,qp ]r = bs′,q′

p′ , (4.8)

where the indices are related by

p′ = p/r, q′ = q/r, s′ = r(s+ n/2)− n/2. (4.9)

Similar results are valid for the inhomogeneous sequence spaces.

17

Proof. It is trivial to check that if r > 0 then |sQ|1/rQ ∈ f s,qp ⇐⇒ sQQ ∈ f s′,q′

p′ , where

the primed indices are as in (4.9). The assertions made about the scale f s,qp are clear from

this. The case of bs,qp is virtually identical.

Note that we can also use the above lemma together with Theorem 4.4 in order to givean alternative proof of the fact that the sequence spaces f s,qp , bs,qp are A-convex.

Proposition 4.6. For s0, s1 ∈ R, 0 < p0, p1, q0, q1 ≤ +∞, 0 < θ < 1, 1/p := (1 − θ)/p0 +θ/p1, 1/q := (1− θ)/q0 + θ/q1 and s := (1− θ)s0 + θs1, there holds

f s,qp =(f s0,q0p0

)1−θ(f s1,q1p1

)θ. (4.10)

If, in addition, s0 6= s1, then also

bs,qp =(bs0,q0p0

)1−θ(bs1,q1p1

)θ. (4.11)

Finally, similar results are valid for the inhomogeneous sequence spaces.

Proof. The Calderon product on the f s,qp scale has been computed in Theorem 8.2 of [20].However, a proof of (4.11) does not seem to be readily available in the literature; we includeone here.

Our first observation is that (4.11) is valid if, in addition, p0, p1, q0, q1 ≥ 1, in which caseall spaces involved are actually Banach. Indeed, in this situation, (4.11) is a consequence ofCalderon’s formula (allowing one to identify the Calderon’s product with the intermediatespaces obtained via complex interpolation ([4])) and the fact that

Bs,qp (Rn) =

[Bs0,q0p0

(Rn), Bs1,q1p1

(Rn)]θ, (4.12)

under the current assumptions on the indices. For (4.12) see Theorem 6.4.5 in [3].Turning now to the general case, all we need to do is to utilize (4.7), for some r > 0

sufficiently small, in order to reduce matters to the Banach case (i.e. when all integrabilityindices are ≥ 1). Since this has been treated before, and since convexification commutes withthe Calderon product, the desired conclusion follows easily. The proof of the proposition istherefore finished.

Returning now to the task of discussing the complex method for an interpolation coupleof quasi-Banach spaces X0, X1, let us make the additional assumption that X0 + X1 is A-convex. This entails

sup‖f(z)‖X0+X1 ; 0 < Re z < 1

≤ C‖f‖F , (4.13)

uniformly for f ∈ F . With this at hand, it is easy to correct all the aforementioned defi-ciencies of the complex method. Furthermore, the axioms of the abstract stability theorydeveloped in [29] are satisfies and, in particular, the following result holds.

Proposition 4.7. Let X0, X1 be an interpolation couple of quasi-Banach spaces and assumethat X0 + X1 is A-convex. Also, let T : Xj → Xj be a linear and bounded operator forj = 0, 1, and assume that T is an isomorphism of [X0, X1]θ0 for some 0 < θ0 < 1.

18

Then there exists ε > 0, which depends only on θ0, the moduli of concavity of Xj, j = 0, 1,‖T‖L(Xj), j = 0, 1, and ‖T−1‖L([X0,X1]θ0 ), so that T is also an isomorphism of [X0, X1]θ foreach 0 < θ < 1 satisfying |θ − θ0| < ε.

A version better suited for the applications we discuss in this paper is recorded below.

Corollary 4.8. Assume that the family of A-convex quasi-Banach spaces Xs,qp , indexed

by triplets (s, q, p) in some open connected subset U of R × (0,∞) × (0,∞), is a complexinterpolation scale. That is, for any (s0, q0, p0) and (s1, q1, p1) ∈ U , there holds[

Xs0,q0p0

, Xs1,q1p1

]θ

= Xs,qp , ∀ θ ∈ (0, 1), (4.14)

where s := (1− θ)s0 + θs1, 1/p := (1− θ)/p0 + θ/p1, 1/q := (1− θ)/q0 + θ/q1. Also, supposethat T is a linear operator mapping each Xs,q

p boundedly into itself and so that there exists

(s∗, q∗, p∗) ∈ U for which T is an isomorphism when considered from Xs∗,q∗

p∗ onto itself.Then there exists an open neighborhood V of (s∗, q∗, p∗) in U so that T continues to be

an isomorphism of Xs,qp for each (s, q, p) ∈ V.

Proof. Fix a small cube Q in the (s, 1/q, 1/p) space which is centered at (s∗, 1/q∗, 1/p∗). Itis not difficult to check that there exists a positive, finite constant C so that ‖T‖L(Xs,q

p ) ≤ Cand so that the modulus of concavity of Xs,q

p is ≤ C for each (s, 1/q, 1/p) ∈ Q. Indeed, Ccan be taken to depend only on the action of T on X

sj ,qjpj , where (sj, 1/qj, 1/pj), j = 1, 2, ..., 8

are the vertices of Q.Next, apply Proposition 4.7 to T on each segment L ⊂ Q which passes through the point

(s∗, 1/q∗, 1/p∗) and whose endpoints belong to ∂Q. The crux of the matter is the fact thatε = ε(L) > 0 provided by Proposition 4.7 is constructive and it can be checked that thereexists c > 0 so that ε(L) ≥ c > 0, uniformly in L. From this, the conclusion follows.

It has been pointed out in [29] that the complex method described above gives the resultpredicted by the Calderon formula for nice pairs of function spaces. A specific result, buildingon earlier work in [23], and which has been proved in [29], is as follows.

Proposition 4.9. Let Ω be a Polish space, µ a σ-finite Borel measure on Ω, and let X0, X1

be a pair of quasi-Banach function spaces on (Ω, µ).Then, if both X0 and X1 are A-convex and separable, it follows that X0 +X1 is A-convex

and [X0, X1]θ = X1−θ0 Xθ

1 .

As observed in [29], the hypothesis of separability in this case is equivalent to σ-ordercontinuity. For a general quasi-normed space X, this property asserts that a non-negative,non-increasing sequence of functions in X which converges a.e. to zero also converges to zeroin the quasi-norm topology of X (cf., e.g., [33], Vol. II). An equivalent reformulation is thatif g ∈ X and |fn| ≤ |g| for all n and fn → f a.e. then ‖fn−f‖X → 0. For us, it is of interestto also note a result, proved in Theorem 1.29 of [10], to the effect that

one of the lattices X0, X1

is σ-order continuous=⇒ X1−θ

0 Xθ1 is σ-order continuous

for each θ ∈ (0, 1).(4.15)

Before we return to Besov and Triebel-Lizorkin spaces, we would like to include animportant comment here.

19

Remark. For the applications we have in mind (i.e. sequence spaces), Proposition 4.9continues to hold in the case when just one of the two quasi-Banach lattices X0, X1 isseparable. Indeed, in [29], the separability hypotheses on Xj, j = 0, 1, was used to ensurethat if f0 ∈ X0 and f1 ∈ X1 then the function z 7→ |f0|1−z|f1|z is admissible (i.e. belongs toF). In fact, the one property which is not immediate is its continuity on the closure of thestrip 0 < Re z < 1. Nonetheless, this issue can be handled as follows.

If g := |f0|1−θ|f1|θ ∈ X1−θ0 Xθ

1 for fj ∈ Xj and 0 < θ < 1, then gχE = |f0χE|1−θ|f1χE|θfor any E ⊂ Ω. In particular, if E is finite and χE stands for the characteristic functionof E then, clearly, z 7→ |f0χE|1−z|f1χE|z is admissible. Thus, as in [29], gχE ∈ [X0, X1]θand ‖gχE‖[X0,X1]θ ≤ C‖gχE‖X1−θ

0 Xθ1. Consider now En Ω, a nested family of finite sets

exhausting E (which can be arranged if the Xj’s are sequence spaces). Replacing E byEj \ Ek and using the fact that, by (4.15), gχEn → g in X1−θ

0 Xθ1 , ultimately gives that

gχEnn is Cauchy in [X0, X1]θ The same argument further yields that gχEnn convergesto g in X0 + X1. Hence, g ∈ [X0, X1]θ and ‖g‖[X0,X1]θ ≤ C‖g‖X1−θ

0 Xθ1. From this point on,

one proceeds as in the proof of Theorem 3.4 in [29].

After these preparations, we are finally ready to present the main result of this section.

Theorem 4.10 (Complex Interpolation). Let s0, s1 ∈ R, 0 < p0, p1 ≤ +∞, 0 < q0, q1 ≤+∞, 0 < θ < 1, s = (1− θ) s0 + θs1 and 1/p := (1− θ)/p0 + θ/p1, 1/q := (1− θ)/q0 + θ/q1.Also, suppose that either p0 + q0 < +∞ or p1 + q1 < +∞. Then[

F s0,q0p0

(Rn), F s1,q1p1

(Rn)]θ

= F s,qp (Rn). (4.16)

Similar results are valid for discrete and inhomogeneous Triebel-Lizorkin spaces.Furthermore, if in addition s0 6= s1, then also[

Bs0,q0p0

(Rn), Bs1,q1p1

(Rn)]θ

= Bs,qp (Rn). (4.17)

Once again, analogous results are valid for discrete and inhomogeneous Besov spaces.

Before presenting the proof, a few comments are in order here. First, this result seemsnew only when min p0, q0, p1, q1 < 1; for the other case see, e.g., Corollary 8.3 in [20] andTheorem 6.4.5 in [3].

Second, a result which formally resembles ours has been proved in §2.4.7 of [42]. Nonethe-less, it should be pointed out that the complex method utilized there is different from oursand, more importantly, does not seem suited for the applications we have in mind. In par-ticular, it is not known whether it has the so-called interpolation property (i.e., preservationof the boundedness of linear operators); see the comment at the beginning of §2.4.8 in [42].

Third, (4.16) contains the complex interpolation of Hardy spaces, i.e. for each 0 < θ < 1and 0 < p0, p1 < +∞,[

Hp0(Rn), Hp1(Rn)]θ

= Hp(Rn), where 1/p := (1− θ)/p0 + θ/p1. (4.18)

It also contains complex interpolation between Hardy spaces and BMO, i.e.

[Hp0(Rn),BMO(Rn)

]θ

= Hp(Rn), 0 < θ < 1, 0 < p0 < +∞, 1/p := (1− θ)/p0. (4.19)

20

The same formulas, but for different methods of complex interpolation, have been obtained in[5], [24], [9], [23]. Clearly, (4.16)-(4.17) contain several other particular cases of independentinterest; we leave their formulation to the interested reader.

Proof. For the Triebel-Lizorkin class, we shall only treat the case of the scale f s,pq as the

rest follows from this by well known methods. For example, F s,pq can be identified with the

discrete scale via the wavelet transform by taking a common unconditional wavelet basis (cf.Theorem 7.20 in [22]). This identification is independent of the indices of the spaces involvedand, hence, amenable to interpolation.

The point is that Proposition 4.3 and the extra lattice structure on the discrete spacesallow us to use Proposition 4.9 (cf. also the remark following its statement) in order toobtain [

f s0,q0p0, f s1,q1p1

]θ

= f s,qp (4.20)

for the indices specified in the statement of the theorem. Then (4.10) can be used to producethe desired conclusion.

Finally, thanks to Proposition 4.6, a similar argument works in the case of Besov spaces.This completes the proof of the theorem.

In closing, let us note that natural versions of the results in this section continue to holdif the underlying Euclidean space is replaced by the boundary of a Lipschitz domain Ω. Inparticular, as expected, F s,q

p (∂Ω) and Bs,qp (∂Ω) continue to be complex interpolation scales

for appropriate ranges of indices. See also [25, §2 and p. 200] for a related discussion.

5 Banach envelopes of Besov and Triebel-Lizorkin spaces

The main aim in this section is to identify the Banach envelopes of all quasi-Banach spaceson the scales of Besov and Triebel-Lizorkin spaces. We debut with the

Proof of Theorem 1.1. By Theorem 3.1, the quasi-Banach space F s,qp (Rn) is continuously

and densely included in the Banach space Fs−n( 1

p−1),1

1 (Rn). Also, by Theorem 3.2, thesetwo spaces have the same dual. Then Theorem 2.4 proves “half” of (1.3). The proof of theremaining half is similar. Going further, note that the second equality in (1.5) has actuallybeen obtained in (3.14). Parenthetically, let us note that, granted the duality result (3.11),plus the fact that F s,q

p (Rn) → F s,1p (Rn), Theorem 2.4 applies once again and yields the same

result. The rest of the cases are handled analogously; we omit the details.

Corollary 5.1. Forn

n+ 1< p < 1, one has

Hp1 (Rn) = B

1−n( 1p−1),1

1 (Rn), (5.1)

Hp(Rn) = B−n( 1

p−1),1

1 (Rn). (5.2)

Proof. This follows from Theorem 1.1, (3.18) and Proposition 3.5.

21

In conjunction with Proposition 2.1 and the remark preceding it, the above results yieldthe following.

Corollary 5.2. Let 0 < p, q < +∞ and s ∈ R. Then any bounded linear operator T from

F s,qp (Rn) into itself has a natural bounded extension T acting from B

s−n( 1p−1),1

1 (Rn) into itself,if p < 1, and from F s,q∧1

p (Rn) into itself, if p ≥ 1.Furthermore, any linear and bounded operator T from Bs,q

p (Rn) into itself has a bounded

extension to Bs−n( 1

p∧1−1),q∨1

p∨1 (Rn).

Other variants are, obviously, possible. Here we only want to point out that similarresults can then be lifted to ∂Ω, the boundary of a bounded Lipschitz domain Ω in Rn. Let

us denote by B−(n−1)( 1

p−1),1

1 (∂Ω) the `1-span of Hp(∂Ω) atoms, i.e.,

B−(n−1)( 1

p−1),1

1 (∂Ω) :=f ∈ Lip (∂Ω)′; f =

∑i≥0

λiai, (λi)i ∈ `1, ai Hp(∂Ω)− atom

. (5.3)

Then we have the following.

Corollary 5.3. Let Ω be a bounded Lipschitz domain in Rn. For n−1n< p < 1, we have

Hp1 (∂Ω) = B

1−(n−1)( 1p−1),1

1 (∂Ω), (5.4)

Hp(∂Ω) = B−(n−1)( 1

p−1),1

1 (∂Ω). (5.5)

Proof. The identity (5.4) follows much as (5.1). Also, because of the atomic characterizations,

Hp(∂Ω) → B−(n−1)( 1

p−1),1

1 (∂Ω) continuously and densely. Also, the fact that(B−(n−1)( 1

p−1),1

1 (∂Ω))∗

=(Hp(∂Ω)

)∗= B

(n−1)( 1p−1),∞

∞ (∂Ω) (5.6)

can be verified directly starting from (5.3). Then Theorem 2.4 gives (5.5).

In closing, it should be pointed out that the space introduced in (5.3) interpolates natu-rally with the scale Bs,r

q (∂Ω), for −1 < s < 0 and 1 < r, q < +∞. This can be checked byusing (5.6) together with the duality theorem for interpolation.

6 Applications to PDE’s in non-smooth domains

Here we discuss certain applications to elliptic PDE’s in Lipschitz domains with a specialemphasis on the three-dimensional Lame system. We want to stress that our methodsare rather flexible (they are independent of positivity and apply indiscriminately to singleequations and systems), and can be used in other cases of interest as well.

Let us now formulate two classical boundary problems for the Lame system of linearelastostatics on a Lipschitz domain Ω in R3:

22

L~u := µ∆~u+ (λ+ µ)∇(div ~u), in Ω, (6.1)

where µ > 0 and λ > −23µ are the so-called Lame constants. Here ~u = (u1, u2, u3); however,

when clear from the context, our notation will not always indicate the scalar or vector natureof functions and spaces involved. Also, in this section we deviate (slightly) from the notationused so far and denote Bs,p

p (Ω), Bs,pp (∂Ω) by Bp

s (Ω) and Bps (∂Ω), respectively.

More precisely, for 1 < p, q < +∞, 1/p+ 1/q = 1 and 0 < s < 1, we consider the Poissonproblem with Dirichlet boundary conditions

L~u = f ∈ Lps+ 1

p−2

(Ω),

Tr ~u = g ∈ Bps (∂Ω),

~u ∈ Lps+ 1

p

(Ω),

(6.2)

where Tr stands for the trace map on ∂Ω, and the Poisson problem with Neumann-typeboundary conditions

L~u = f ∈ Lq1q−s−1,0

(Ω),

∂~u

∂ν= g ∈ Bq

−s(∂Ω),

~u ∈ Lq1−s+ 1

q

(Ω).

(6.3)

In the classical setting,

∂~u

∂ν:=(λ(div ~u)n+ µ[∇~u+∇~ut]n

)∣∣∣∣∂Ω

(6.4)

is the traction (n is the unit normal to ∂Ω and the superscript t indicates transposition; inthis case, of the 3×3 matrix ∇~u = (∂iuj)i,j). The sense in which (6.4) should be interpretedin the present context is as the distribution in Bq

−s(∂Ω) =(Bps (∂Ω)

)∗, given by

⟨∂~u∂ν, φ⟩

:=〈f , φ〉+ µ〈∇~u+∇~ut,∇φ+∇φt〉

+ λ〈div ~u , div φ〉, ∀φ ∈ Bps (∂Ω),

(6.5)

where φ ∈ Lps+ 1

p

(Ω) is an extension (in the trace sense) of φ. The last two pairings in the

right side of (6.5) are well defined since ∇φ ∈ Lps+ 1

p−1

(Ω), ~u ∈ Lq−s+ 1q

(Ω) and

Lps−1+ 1

p

(Ω) =(Lq−s+ 1

q

(Ω))∗, Lq−s+ 1

q

(Ω) =(Lps−1+ 1

p

(Ω))∗, (6.6)

for 0 < s < 1 and 1 < p, q < +∞ with 1p

+ 1q

= 1. It is not difficult to check that

the definition (6.5) is correct, agrees with (6.4) when ~u is sufficiently smooth, and satisfiesnatural estimates. Let us point out that, even if the notation does not reflect it, (6.5) dependsstrongly on the choice of f = L~u as an element in Lq1

q−s−1,0

(Ω).

23

Going further, recall the Kelvin matrix Γ = (Γi,j)i,j of fundamental solutions for thesystem of elastostatics, where for 1 ≤ i, j ≤ 3,

Γi,j(X) :=3

8π

(1

µ+

1

2µ+ λ

)δi,j|X|

+3

8π

(1

µ− 1

2µ+ λ

)XiXj

|X|3, X ∈ R3. (6.7)

Here, δi,j is the Kronecker symbol; see, e.g., [DKV]. For vector valued densities on ∂Ω, thesingle (elastic) layer potential operator is defined by

S~f(X) :=

∫∂Ω

Γ(X −Q) ~f(Q) dσ(Q), X ∈ R3, (6.8)

where dσ stands for the canonical surface measure on ∂Ω. Also, the double (elastic) layerpotential operator is given by

D~f(X) :=

∫∂Ω

(∂

∂ν(Q)Γ(X −Q)

)t~f(Q) dσ(Q), X ∈ Ω. (6.9)

Here the operator ∂/∂ν applies to each column of the matrix Γ.Finally, we record the corresponding jump relation

limX→P, X∈γ±(P )

D~f(X) = ±12~f(P ) + p.v.

∫∂Ω

(∂

∂ν(Q)Γ(X −Q)

)t~f(Q) dσ(Q)

=:(±1

2I + K

)~f(P ), P ∈ ∂Ω,

(6.10)

and

limX→P, X∈γ±(P )

∂S~f

∂ν(X) =

(∓1

2I + K∗

)~f(P ), P ∈ ∂Ω, (6.11)

where K∗ denote the formal adjoint of K. In (6.10)-(6.11), γ±(P ) are suitable nontangentialapproach regions with vertex at P ∈ ∂Ω, which are contained in Ω+ := Ω and Ω− := R3 \ Ω,respectively. See, e.g., [DKV] for more details.

Our first result collects the main properties of layer potentials for the Lame systemconsidered on scales of Sobolev-Besov spaces. Even though this is stated only for domainsin the three dimensional Euclidean space, virtually the same holds true in Rn with n ≥ 3.

Theorem 6.1. Let Ω be a bounded Lipschitz domain in R3. For any 1 ≤ p ≤ +∞ and0 < s < 1, the single layer potential S is a bounded linear map from Bp

−s(∂Ω) into Bp

1+ 1p−s(Ω).

In fact, if 1 < p < +∞, then S is a bounded linear map from Bp−s(∂Ω) into Bp

1+ 1p−s(Ω) ∩

Lp1+ 1

p−s(Ω), i.e.

max

‖Sf‖Bp

1+ 1p−s

(Ω) , ‖Sf‖Lp1+ 1

p−s(Ω)

≤ C‖f‖Bp−s(∂Ω) (6.12)

24

uniformly for f ∈ Bp−s(∂Ω). In particular, if we also denote by S the composition between Tr

and (6.8), then this operator maps Bp−s(∂Ω) boundedly into Bp

1−s(∂Ω) for each 1 ≤ p ≤ +∞and 0 < s < 1.

Furthermore, for 1 ≤ p ≤ +∞ and 0 < s < 1, the operator D extends to a bounded linearmap from Bp

s (∂Ω) into Bp

s+ 1p

(Ω). In fact, if 1 < p < +∞, then D also extends as a bounded

linear map from Bps (∂Ω) into Bp

s+ 1p

(Ω)∩Lps+ 1

p

(Ω). In other words, for any 1 < p < +∞ and

0 < s < 1, the estimate

max

‖Df‖Lp

s+ 1p

(Ω) , ‖Df‖Bps+ 1

p(Ω)

≤ C‖f‖Bps (∂Ω) (6.13)

holds uniformly for f ∈ Bps (∂Ω). Finally, Tr D = 1

2I + K so that, in particular, K is well

defined and bounded on Bps (∂Ω) for any 1 ≤ p ≤ +∞, 0 < s < 1.

This can be proved much the same way as in the case of scalar (harmonic) potentials; see[18] and [35] for this latter case.

Next, let Ψ be the six-dimensional linear space of vector-valued functions ψ in R3 satis-fying

∂iψj + ∂jψ

i = 0, 1 ≤ i, j ≤ 3, (6.14)

and note that each ψ ∈ Ψ is a null-solution of L. Also, for a space X (∂Ω) ⊆(Lip (∂Ω)

)∗,

set

X (∂Ω)Ψ := u ∈ X (∂Ω); 〈u, ψ|∂Ω〉 = 0, ∀ψ ∈ Ψ. (6.15)

If X (Ω) ⊆ L1(Ω) is a space of functions in Ω then X (Ω)Ψ is defined analogously.

Theorem 6.2. Let Ω be a bounded Lipschitz domain in R3. There exists ε ∈ (0, 1] with thefollowing significance. Let 1 ≤ p ≤ +∞ and 0 < s < 1 be so that one of the conditions(I)–(III) below are satisfied:

(I) :2

1 + ε< p <

2

1− εand 0 < s < 1; (6.16)

(II) : 1 ≤ p <2

1 + εand

2

p− 1− ε < s < 1; (6.17)

(III) :2

1− ε< p ≤ +∞ and 0 < s <

2

p+ ε. (6.18)

Also, let 1 ≤ q ≤ +∞ denote the conjugate exponent of p. Then the operators listed beloware invertible:

1. 12I + K : Bp

s (∂Ω) −→ Bps (∂Ω);

2. 12I + K∗ : Bq

−s(∂Ω) −→ Bq−s(∂Ω);

3. S : Bq−s(∂Ω) −→ Bq

1−s(∂Ω).

25

4. ±12I + K : Bp

s (∂Ω)/Ψ −→ Bps (∂Ω)/Ψ;

5. ±12I + K∗ : Bq

−s(∂Ω)Ψ −→ Bq−s(∂Ω)Ψ;

6. S : Bq−s(∂Ω)Ψ −→ Bq

1−s(∂Ω)/Ψ.

For each 0 ≤ ε ≤ 1 consider the region Rε ⊆ R2 which is the interior of the hexagonOABCDE, where O = (0, 0), A = (ε, 0), B = (1, 1−ε

2), C = (1, 1), D = (1 − ε, 1),

E = (0, 1+ε2

). Then the “invertibility” region described in (6.16)–(6.18) simply says that(s, 1/p) belongs to Rε or, possibly, to the (open) segments OA, CD. Note that the regionencompassed by the parallelogram with vertices at (0, 0), (1, 1

2), (1, 1) and (0, 1

2) is common

for all Lipschitz domains, and that Rε can be thought as an enhancement of it.

Proof. To begin with, note that the operators

S :Hp(∂Ω) −→ Hp1 (∂Ω), (6.19)

12I + K :Hp

1 (∂Ω) −→ Hp1 (∂Ω), (6.20)

12I + K∗ :Hp(∂Ω) −→ Hp(∂Ω) (6.21)

are well-defined and bounded for any 2/3 < p ≤ 1 (in fact, appropriate versions hold forp > 1 also). This follows from [7] and the discussion in the last part of §3. Next, we claimthat there exists ε = ε(Ω) > 0 so that the operators (6.19)–(6.21) are isomorphisms for1− ε < p ≤ 1. Indeed, when p = 1 this follows as in [12], granted the results of [13]. ThenTheorem 1.2, applied in concert with Corollary 5.3, allows us to conclude that

S :B1−s(∂Ω) −→ B1

1−s(∂Ω), (6.22)

12I + K :B1

1−s(∂Ω) −→ B11−s(∂Ω), (6.23)

12I + K∗ :B1

−s(∂Ω) −→ B1−s(∂Ω) (6.24)

are isomorphisms for s ∈ (0, s0), with s0 = s0(Ω) > 0 small.Note that, as far as the operator 1

2I + K is concerned, this covers the segment CD on

the boundary of Rε. Interpolation between (6.20), the dual of (6.21) on the one hand, andthe L2-results in [14] on the other hand, yields that 1

2I + K is an isomorphism on Lp(∂Ω)

for 2 − ε < p < +∞ and Lp1(∂Ω) for 1 < p < 2 + ε, thus covering the segments OE andBC. Finally, the fact that 1

2I + K is an isomorphism of B∞s (∂Ω) for s > 0 small, follows

from (6.21) (or (6.24)) and duality; this has been first observed in [29]. This takes care ofthe segment OA and the full region is then covered by interpolation.

The remaining operators can be treated via the same pattern and this finishes the proofof the theorem.

Our next theorem deals with the case of the Poisson problem with Dirichlet boundaryconditions for the three dimensional Lame system.

26

Theorem 6.3. Let Ω be a bounded Lipschitz domain in R3. Then there exists ε = ε(Ω) > 0with the following property. If p ∈ (1,∞) and s ∈ (0, 1) are such that one of the conditions(I)–(III) in (6.16)–(6.18) are satisfied, then the Poisson problem with Dirichlet boundaryconditions (6.2) has a unique solution.

Moreover, there exists C > 0 depending only on Ω, p, s, such that the solution satisfiesthe estimate

‖~u‖Lps+ 1

p(Ω) ≤ C‖f‖Lp1

p+s−2(Ω) + C‖g‖Bps (∂Ω). (6.25)

Finally, similar results are valid when all functions involved are taken from scales ofBesov spaces.

Proof. Let us first deal with the existence part. To this end, let f ∈ Lps+1/p−2(R3) be a

compactly supported extension of f (whose norm is controlled by that of f). Also, letF (X) :=

∫R3 Γ(X − Y )f(Y ) dY , X ∈ Ω, be the elastostatic Newtonian potential of f . It

follows that F ∈ Lps+1/p(Ω) and

‖F‖Lps+1/p

(Ω) ≤ C‖f‖Lps+1/p−2

(Ω) (6.26)

for some C > 0 independent of f . Subtracting F from ~u, we see that matters can be reducedto the case when f = 0. In this latter situation, a solution is given by ~u := D

((1

2I + K)−1g

)in Ω. Note that by Theorem 6.1, this solution satisfies the estimate (6.25) (with f = 0). Insummary, this and (6.26) show that there exists a solution ~u of (6.2) which satisfies (6.25).

As for uniqueness, the departure point is to write down the Green type integral repre-sentation formula

~v = D(Tr~v)− S(∂~v∂ν

), (6.27)

valid for each ~v ∈ Lrτ (Ω), with r > 1 and τ > 1/r, satisfying L~v = 0 in Ω. Thus, if~u ∈ Lp

s+ 1p

(Ω) solves the homogeneous version of (6.2), then taking ∂/∂ν of both sides of

(6.27) readily gives that (12I + K∗)(∂~u

∂ν) = 0. The important thing is that the region Rε is

invariant to the transformation (s, 1p) 7→ (1 − s, 1 − 1

p) and that ∂~u

∂ν∈ Bp

s−1(∂Ω). Thus, on

account of Theorem 6.2, ∂~u∂ν

= 0 also. Utilizing this back in (6.27) yields ~u ≡ 0 in Ω, asdesired.

Next, we discuss the Poisson problem with Neumann boundary conditions.

Theorem 6.4. Let Ω be a bounded Lipschitz domain in R3. There exists ε = ε(Ω) > 0having the following property. Suppose that p ∈ (1,∞), s ∈ (0, 1) are such that one of theconditions (I)–(III) in (6.16)–(6.18) is satisfied and denote by q the conjugate exponent of p.

Then the Poisson problem for the Lame system with Neumann boundary condition (6.3),subject to the (necessary) compatibility condition

〈f, ψ〉 = 〈g, ψ〉, ∀ψ ∈ Ψ, (6.28)

has a unique, modulo Ψ, solution. Moreover, there exists a positive constant C which dependsonly on Ω, p, s, such that

27

‖~u‖Lq1−s+ 1

q(Ω)/Ψ ≤ C‖f‖Lq1

q−s−1,0(Ω) + C‖g‖Bq−s(∂Ω). (6.29)

Once again, similar results are valid when all functions involved are considered on scalesof Besov spaces.

Proof. Granted Theorems 6.1–6.2, the proof of this result closely parallels that of Theo-rem 6.3, and we omit the details.

For A := (aαβij )1≤α,β,i,j≤3 with real entries consider the second order differential operator

(L~u)α :=∑β

∑i,j

aαβij ∂i∂juβ, ∀α. (6.30)

Assume that A satisfy the Legendre-Hadamard ellipticity condition∑α,β

∑i,j

aαβij ξiξjηαηβ ≥ C|ξ|2|η|2, ∀ ξ, η ∈ R3, (6.31)

the symmetry condition

aαβij = aβαji , 1 ≤ α, β, i, j ≤ 3 (6.32)

and the positive semi-definiteness condition⟨Aζ, ζ〉 :=

∑α,β

∑i,j

aαβij ζαi ζ

βj ≥ 0, ∀ ζ = (ζαi )α,i ∈ R9. (6.33)

Then there exist two (unique) nonpositive, self-adjoint operators LD, LN so that

〈LD~u,~v〉 = −∫

Ω

⟨A∇~u,∇~v

⟩, ∀ ~u,~v ∈ L2

1,0(Ω), (6.34)

and

〈LN~u,~v〉 = −∫

Ω

⟨A∇~u,∇~v

⟩, ∀ ~u,~v ∈ L2

1(Ω), (6.35)

respectively. It should be noted that a certain peculiarity occurs for systems (as opposed tosingle equations). Specifically, while the choice of the matrix A (used in the writing of thedifferential operator L) is immaterial for LD (as is seen, e.g., from (6.34) and Plancherel’sformula), different choices of A may, in turn, lead to different operators LN .

When the above procedure is applied to the Lame system (6.1) viewed as in (6.30) for

aαβij := µ(δi,jδα,β + δi,βδj,α) + λδi,αδj,β, (6.36)

we denote the corresponding Dirichlet and Neumann Lame operators by LD and LN . Also,the (unique) nonnegative, self-adjoints square-roots of the negatives of these operators aredenoted by

√−LD and

√−LN . Alternative descriptions of these square-roots (as unbounded

operators on L2(Ω)) can be given in terms of the (square-roots of the) eigenvalues of −LDand −LN .

28

Note that the choice A := (aαβij )α,β,i,j with aαβij as in (6.36) leads to the traction conormalderivative (6.4). Also, for the same choice of A,

〈A∇~u,∇~u〉 = λ|div ~u|2 + µ2|∇~u+∇~ut|2 ≥ C|∇~u+∇~ut|2, (6.37)

where the last inequality follows from the elementary estimate |div ~u|2 ≤ 34|∇~u+∇~uT |2, plus

our assumptions on the Lame constants. Now, Korn’s inequality (cf., e.g., [36]) to the effectthat

‖~u‖L21(Ω) ≤ C‖∇~u+∇~ut‖L2(Ω) (6.38)

for any ~u ∈ L21(Ω) such that∫

Ω

〈~u, ψ〉+ 〈∇~u,∇ψ〉 = 0, ∀ψ ∈ Ψ, (6.39)

together with elementary functional analysis allow us to conclude that√−LN : L2

1(Ω)/Ψ −→ L2(Ω)Ψ is an isomorphism. (6.40)

Consider next√−LD. Since the choice of the tensor A (used to write the Lame system

in the form (6.30)) given by

aαβij := µδi,jδα,β +(λ− 2µ2

3µ+ λ

)δi,αδi,β +

µ(µ+ λ)

3µ+ λδi,βδj,α (6.41)

(leading to the so-called pseudo-stress conormal derivative) yields

〈A∇~u,∇~u〉 ≥ C|∇~u|2, (6.42)

simple functional analysis gives that√−LD : L2

1,0(Ω) −→ L2(Ω) is an isomorphism. (6.43)

Starting from these observation and relying on Theorems 6.3–6.4 and the Littlewood-Paley theory from [41], the next corollary can be proved via interpolation much as in thescalar case; cf. [25] and [34].

Corollary 6.5. Assume that Ω be a bounded Lipschitz domain in R3. There exists ε =ε(Ω) > 0 so that

(√−LD)−1 : Lp(Ω) −→ Lp1,0(Ω) (6.44)

and

(√−LN)−1 : Lp(Ω)/Ψ −→ Lp1(Ω)Ψ (6.45)

are isomorphisms for 1 < p < 3 + ε. Furthermore, the operators√−LD : Lp1,0(Ω) −→ Lp(Ω) (6.46)

29

and √−LN : Lp1(Ω) −→ Lp(Ω) (6.47)

are bounded for 1 < p < +∞.

Our last result deals with mapping properties of the Green and Neumann functions forthe Lame system in Lipschitz domains. This builds on the work in [11] and [25], wherethe Green function for the Laplacian has been considered, [34] dealing with the NeumannLaplacian, and [35] where more general variable coefficient (scalar) operators are treated.

Corollary 6.6. Assume that Ω is a bounded Lipschitz domain in R3, and consider G(X,Y )and N(X, Y ), the Green and Neumann functions of the Lame system in Ω. Also, denote byG and N the operators sending f , respectively, to

∫ΩG(·, Y )f(Y ) dY and

∫ΩN(·, Y )f(Y ) dY .

Then, for some ε = ε(Ω) > 0,

‖∇Gf‖Lq(Ω) ≤ C‖f‖Lp(Ω), f ∈ Lp(Ω), (6.48)

and

‖∇Ng‖Lq(Ω) ≤ C‖g‖Lp(Ω), g ∈ Lp(Ω)Ψ, (6.49)

provided 1 < p < 32

+ ε and 1q

:= 1p− 1

3(note that this entails q ∈ (1, 3 + ε)).

Proof. Consider first (6.48). The desired conclusion follows from the observation that G =(−LD)−1 = (

√−LD)−2, together with

(√−LD)−1 : Lp(Ω) −→ Lp1,0(Ω) → Lq(Ω), (

√−LD)−1 : Lq(Ω) −→ Lq1(Ω), (6.50)

under the given hypotheses on p and q, by (6.12). The proof of (6.49) is very similar.

In closing, let us point out that, since all major ingredients have been taken care of,one can also carry out an analysis of the mapping properties of the complex powers of theoperators

√−LD and

√−LN parallel to the study of the Dirichlet and Neumann Laplacians

from [34], [35].

7 Some counterexamples

For an arbitrary bounded Lipschitz domain Ω ⊆ R3 and 1 < p < +∞, consider the Poissonproblem for the Lame system with homogeneous Dirichlet boundary conditions:

~u ∈ Lp1,0(Ω),

L~u = f ∈ Lp−1(Ω).(7.1)

From Theorem 6.3 this is uniquely solvable with natural estimates for any 32− ε < p < 3 + ε

for some ε = ε(Ω) > 0. We aim at proving that the range p ∈ [32, 3] is sharp in the class of

all Lipschitz domains.

30

To this end, let

T :(L2

1,0(Ω))∗

= L2−1(Ω) −→ L2

1,0(Ω) (7.2)

be the solution operator for the problem (7.1), mapping f into ~u. Clearly, in the context of(7.2), T is well defined, linear and bounded. Moreover, by Green’s formula, T also satisfies

〈f1, T f2〉 = 〈f2, T f1〉, f1, f2 ∈ L2−1(Ω), (7.3)

for the natural pairing between functionals in(L2

1,0(Ω))∗

and elements in L21,0(Ω). Thus,

whenever T extends as a bounded mapping of Lp−1(Ω) into Lp1,0(Ω), then T also extends asa bounded mapping of Lq−1(Ω) into Lq1,0(Ω), where q is the conjugate exponent of p.

On the other hand, given q > 3, there exist a bounded (cone-like) Lipschitz domain Ω inR

3 and a field ~u such that

~u ∈ L21,0(Ω), L~u ∈ C∞(Ω), ~u /∈ Lq1(Ω). (7.4)

This can be seen by appealing to the results in [31] where the singularities of null-solutionsof the Lame system near the vertex of a cone have been analyzed. More specifically, forφ ∈ (0, π), there exists an infinite cone-like domain in Ωφ ⊂ R3 (whose aperture is 2φ) and~u0 ∈ L2

1,0(Ωφ, loc) so that |∇~u0(X)| ∼ |X|λ(φ)−1 for X near the vertex (assumed to be theorigin of R3), where the parameter λ(φ) ∈ (0, 1] satisfies λ(φ) 0 as φ π. Truncating ~u0

yields a filed ~u with the properties listed in (7.4). In the light of our discussion, this impliesthe failure of the well-posedness of (7.1) for p /∈ [3

2, 3].

Going further, we have the following theorem.

Theorem 7.1. The range of validity for the Theorem 6.3 in the class of Lipschitz domainsis optimal.

Proof. By considerations of interpolation, the optimal range of solvability for the problem(6.3) is a convex subset of the unit square [−1, 0] × [0, 1] which contains the parallelogramwith vertices at (−1, 0), (0, 1

2), (0, 1), (−1, 1

2), and whose trace on the main diagonal of this

square coincides (by the above discussion) with the segment joining (−13, 1

3) and (−2

3, 2

3).

From this, the conclusion follows by elementary geometrical considerations.

An immediate consequence of this and Theorem 6.2 is the following.

Theorem 7.2. The range of validity for the Theorem 6.2 in the class of Lipschitz domainsis optimal.

References

[1] V. Adolfsson and J. Pipher, The inhomogeneous Dirichlet problem for ∆2 in Lipschitzdomains, J. Funct. Anal., 159 (1998), 137–190.

[2] P. Auscher and Ph. Tchamitchian, On the square root of the Laplacian on Lipschitzdomains, prepublication du LAMFA et CNRS, (1998).

31

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

[4] A. P. Calderon, Intermediate spaces and interpolation, the complex method, StudiaMath., 24 (1964), 113–190.

[5] A. P. Calderon and A. Torchinsky, Parabolic maximal functions associated with a distri-bution, II, Adv. in Math., 24 (1977), 101–171.

[6] W. Cao and Y. Sagher, Stability of Fredholm properties on interpolation scales, Ark. forMath., 28 (1990), 249–258

[7] R. Coifman, A. McIntosh, and Y. Meyer, L’integrale de Cauchy definit un operateurborne sur L2 pour les courbes Lipschitziennes, Ann. Math., 116 (1982), 361–388.

[8] T. Coulhon and X. T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. of Amer. Math.Soc., 351 (1999), 1151–1169.

[9] M. Cwikel, M. Milman and Y. Sagher, Complex interpolation of some quasi-Banachspaces, J. Funct. Anal., 65 (1986), 339–347.

[10] M. Cwikel and P. Nilsson, Interpolation of weighted Banach lattices, Technion PreprintSeries No. MT-834.

[11] B. Dahlberg, Lq estimates for Green potentials in Lipschitz domains, Math. Scand., 44(1979), 149-170.

[12] B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in Lp for Laplace’sequation in Lipschitz domains, Annals of Math., 125 (1987), 437–465.

[13] B. Dahlberg and C. Kenig, Lq-Estimates for the three-dimensional system of elastostaticson Lipschitz domains, Lecture Notes in Pure and Applied Math., Vol. 122, C. Sadoskyed., 1990, 621–634.

[14] B. Dahlberg, C. Kenig, and G. Verchota, Boundary value problems for the system ofelastostatics on Lipschitz domains, Duke Math. J., 57 (1988), 795–818.

[15] W. J. Davis, D. J. H. Garling and N. Tomczak-Jaegermann, The complex convexity ofquasi-normed linear spaces, J. Funct. Anal., 55 (1984), 110–150.

[16] M. M. Day, The spaces Lp with 0 < p < 1, Bull. Amer. Math. Soc., 46 (1940), 816–823.

[17] R. A. DeVore and V. A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc.,Vol. 305 No. 1 (1988), 397–414.

[18] E. Fabes, O. Mendez and M. Mitrea, Boundary layers on Sobolev-Besov spaces andPoisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998),323–368.

[19] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34(1985), 777–799.

32

[20] M. Frazier and B. Jawerth, A discrete transform and decompositions of distributionspaces, J. Funct. Anal., Vol. 93 No. 1 (1990), 34–170.

[21] M. Frazier and B. Jawerth, Applications of the ϕ and wavelet transforms to the theoryof function spaces, “Wavelets and Their Applications” Ruskai et. al. eds., Jones andBartlett, Boston 1992, 377–417.

[22] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of FunctionSpaces, CBMS Regional Conference Series in Mathematics, Vol. 79, AMS, Providence,RI, 1991.

[23] M. Gomez and M. Milman, Complex interpolation of Hp-spaces on product domains,Ann. Math. Pura Appl., 155 (1989), 103–115.

[24] J. Janson and P. W. Jones, Interpolation between Hp spaces: The complex method, J.Funct. Anal., 48 (1982), 58–80.

[25] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J.Funct. Anal., 130 (1995), 161–219.

[26] N. J. Kalton, Convexity conditions for non-locally convex lattices, Glasgow Math. J., 25(1984), 141–152.

[27] N. J. Kalton, Analytic functions in non-locally convex spaces, Studia Math., 83 (1986),275–303.

[28] N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math., 84(1986), 297–324.

[29] N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spacesand applications, Trans. Amer. Math. Soc., Vol. 350, No. 10 (1998), 3903–3922.

[30] N. Kalton, N. Peck and J. Roberts, An F space sampler, London Mathematical Society,LNS No. 89, 1984.

[31] V. A. Kozlov, V. G. Maz’ya and C. Schwab, On singularities of solutions of the displace-ment problem of linear elasticity near the vertex of a cone, Arch. Rational Mech. Anal.,119 (1992), 197–227.

[32] R. H. Latter, A decomposition of Hp(Rn) in terms of atoms, Studia Math., 62 (1978),92–101.

[33] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. I-II, Springer-Verlag,Berlin Heidelberg, New York, 1977.

[34] O. Mendez and M. Mitrea, Complex powers of the Neumann Laplacian in Lipschitz do-mains, to appear in Math. Nachrichten, (2000).

33

[35] M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian man-ifolds: Sobolev-Besov spaces and the Poisson problem, to appear in J. Funct. Anal.,(2000).

[36] J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Academia,Prague, 1967.

[37] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Series, 1976.

[38] N. M. Riviere, Interpolation Theory in s-Banach spaces, Dissertation, Univ. of Chicago,Chicago, IL, 1966.

[39] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, andNonlinear Partial Differential Operators, de Gruyter, Berlin, New York, 1996.

[40] I. Y. Sneiberg, Spectral properties of linear operators in interpolation families of Banachspaces, Mat. Issled., 9 (1974), 214–229.

[41] E. M. Stein, Topics in Harmonic Analysis Related to Littlewood-Paley Theory, Annalsof Math. Studies, Vol. 63, Princeton Univ. Press, Princeton, NJ, 1970.

[42] H. Triebel, Theory of Function Spaces, Birkhauser, Berlin, 1983.

[43] H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, Vol. 84,Birkhauser Verlag, Basel, 1992.

[44] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd edition,Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

[45] P. Turpin, Convexites dans les espaces vectoriels topologiques generaux, DissertationesMath. No. 131, 1974.

[46] I. E. Verbitsky, Imbedding and multiplier theorems for discrete Littlewood-Paley spaces,Pacific J. Math., Vol. 176 No. 2 (1996), 529–556.

[47] A. T. Vignati and M. Vignati, Spectral theory and complex interpolation, J. Funct. Anal.,80 (1988), 387–397.

———————————————–

mendez@math.missouri.edumarius@math.missouri.edu

34