Triebel–Lizorkin-type spaces on bounded domains

https://doi.org/10.1007/s13163-020-00365-9

Compact embeddings in Besov-type and

Triebel–Lizorkin-type spaces on bounded domains

Helena F. Gonçalves¹ ·Dorothee D. Haroske¹ ·Leszek Skrzypczak²

Received: 7 January 2020 / Accepted: 22 June 2020 / Published online: 16 July 2020

© The Author(s) 2020

Abstract

We study embeddings of Besov-type and Triebel–Lizorkin-type spaces, id_τ:B^sp¹₁^,τ,q¹₁(Ω) → B^sp²₂^,τ,q²₂(Ω)and id_τ:Fp^s1₁^,τ,q¹₁(Ω) → Fp^s2₂^,τ,q²₂(Ω), where Ω ⊂ R^d is a bounded domain, and obtain necessary and sufficient conditions for the compactness of id_τ. Moreover, we characterize its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the cor- responding results for classical Besov and Triebel–Lizorkin spaces as well as for Besov–Morrey and Triebel–Lizorkin–Morrey spaces.

Keywords Besov-type spaces·Triebel-Lizorkin-type spaces·Compact embeddings·Entropy numbers·Approximation numbers

Mathematics Subject Classification 46E35·41A46

1 Introduction

Usually calledsmoothness spaces of Morrey type or, for short,smoothness Morrey spaces, these function spaces are built upon Morrey spacesMu,p(R^d), 0< p≤u<

Helena F. Gonçalves, Dorothee D. Haroske and Leszek Skrzypczak were partially supported by the German Research Foundation (DFG), Grant No. Ha 2794/8-1. Leszek Skrzypczak was also supported by National Science Center, Poland, Grant No. 2013/10/A/ST1/00091.

B

lskrzyp@amu.edu.pl

1 Institute of Mathematics, Friedrich-Schiller-University Jena, 07737 Jena, Germany

2 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Pozna´nskiego 4, 61-614 Poznan, Poland

∞, and attracted some attention in the last decades, motivated firstly by possible appli- cations. They include Besov–Morrey spacesNu^s,p,q(R^d), Triebel–Lizorkin–Morrey spaces E_u^s_,p,q(R^d), 0 < p ≤ u < ∞, 0 < q ≤ ∞,s ∈ R, Besov-type spaces B^sp^,τ,q(R^d)and Triebel–Lizorkin-type spaces F^sp^,τ,q(R^d), 0 < p < ∞, 0 <q ≤ ∞, τ ≥0,s∈R.

The classical Morrey spacesMu,p, 0< p ≤ u < ∞, were introduced by Mor- rey [22] and are part of a wider class of Morrey–Campanato spaces, cf. [23]. They can be seen as a complement toLpspaces, sinceMp,p(R^d)=Lp(R^d).

The Besov–Morrey spacesNu^s,p,q(R^d)were introduced by Kozono and Yamazaki [19]

and used by them and later on by Mazzucato [21] in the study of Navier–Stokes equa- tions. In [34], Tang and Xu introduced the corresponding Triebel–Lizorkin–Morrey spaces Eu^s,p,q(R^d), thanks to establishing the Morrey version of Fefferman–Stein vector-valued inequality. Some properties of these spaces including their wavelet char- acterizations were later described in the papers by Sawano [27,28], Sawano and Tanaka [29,30] and Rosenthal [26]. Recently, some limiting embedding properties of these spaces were investigated in a series of papers [13–16].

Another class of generalizations, the Besov-type spaceB^sp^,τ,q(R^d)and the Triebel–

Lizorkin-type spaceFp^s,τ,q(R^d)were introduced in [45]. Their homogeneous versions were originally investigated by El Baraka [8–10] and by Yuan and Yang [40,41]. There are also some applications in partial differential equations for spaces of typeB^sp^,τ,q(R^d) andFp^s,τ,q(R^d), such as (fractional) Navier–Stokes equations, cf. [20].

Although the above scales are defined in different ways, they share some prop- erties and are related to each other by a number of embeddings and coincidences.

For instance, they both include the classical spaces of type B^s_p,q(R^d)andF_p^s_,q(R^d) as special cases. We refer to our papers mentioned above, to the recently published papers [43,44], but in particular to the fine surveys [32,33] by Sickel.

There is still a third approach, due to Triebel, who introduced and studied in [38]

local spaces and in [39] hybrid spaces, together with their use in heat equations and Navier–Stokes equations. However, since the hybrid spaces coincide with appropri- ately chosen spaces of type B^sp^,τ,q(R^d)or Fp^s,τ,q(R^d), respectively, cf. [46], we do not have to deal with them separately now.

In this paper we investigate the compactness of the embeddings of the spaces B^sp^,τ,q(Ω)andFp^s,τ,q(Ω), whereΩ ⊂R^dis a bounded domain, i.e. a bounded open set inR^d. In particular, our first goal is to find necessary and sufficient conditions for the compactness of the embeddings

id_τ:A^s_p¹₁^,τ_,q¹₁(Ω) → A^s_p²₂^,τ_,q²₂(Ω), (1.1) where A=Bor A=F, cf. Theorem3.2. Here we prove that id_τ is compact if, and only if,

s₁−s₂ d >max

τ2− 1

p2

+− τ1− 1

p1

+, 1

p1−τ1−min 1

p2−τ2, 1

p2(1−p₁τ1)+

,

(1.2)

where we use the notationa₊ := max{a,0}. At this point, this work can be seen as a counterpart of the papers [14–16], where we studied the compactness of the corresponding embeddings of the spacesNu^s,p,qandEu^s,p,q.

Usually one would start by studying the continuity of such embeddings and later proceed to the compactness. Here we do it differently and start by dealing with the compactness. Our technique relies basically on embeddings. Since for compactness one always has strict inequalities, like condition (1.2), one can always have further embeddings in between the considered spaces. Therefore, we take advantage of the relations between this scale, the smoothness Morrey spacesNu^s,p,qandEu^s,p,qand the classical spaces of type B^s_p,q and F_p^s_,q, and use the corresponding results for these spaces to obtain our main result.

Afterwards we qualify the compactness of id_τ in (1.1) by means of entropy and approximation numbers. In the recent works [16,17], we characterised entropy and approximation numbers of the embedding

id_A:A^su¹₁,p₁,q₁(Ω) →A^su²₂,p₂,q₂(Ω),

withA=N orA=E. However, to the best of our knowledge, apart from a result obtained in [43] for approximation numbers when the target space isL_∞, nothing is known on this matter for embeddings between spaces of typeA^sp^,τ,q. Here we contribute a little more to the development of this topic, establishing some partial counterparts of the results proved in [17].

This paper is organized as follows. In Sect.2we present and collect some basic facts about smoothness Morrey spaces, onR^d and on bounded domainsΩ ⊂ R^d, and introduce the notions of entropy and approximation numbers. In Sect.3we are concerned with the compactness of the above-described embeddings of Besov-type and Triebel–Lizorkin-type spaces on bounded domains. We also prove an extension of the results obtained in [14] for the scaleN_u^s_,p,qto the cases when pi =ui = ∞, i = 1,2. Moreover, we collect some immediate consequences of the main result, when we consider particular source and/or target spaces. In Sect. 4 we end up by characterizing entropy and approximation numbers of the embedding id_τ in (1.1), collecting also some special cases.

2 Preliminaries

First we fix some notation. ByNwe denote theset of natural numbers, byN0the set N∪ {0}, and byZ^dtheset of all lattice points inR^dhaving integer components. For a ∈ R, leta₊ := max{a,0}. All unimportant positive constants will be denoted by C, occasionally with subscripts. By the notation A B, we mean that there exists a positive constantC such that A ≤ C B, whereas the symbol A ∼ B stands for A B A. We denote by B(x,r) := {y ∈ R^d: |x−y| < r}the ball centred at x ∈ R^dwith radiusr >0, and| · |denotes the Lebesgue measure when applied to measurable subsets ofR^d.

Given two (quasi-)Banach spaces X andY, we writeX → Y ifX ⊂Y and the natural embedding ofXintoY is continuous.

2.1 Smoothness spaces of Morrey type onR^d

LetS(R^d)be the set of allSchwartz functionsonR^d, endowed with the usual topology, and denote byS(R^d)itstopological dual, namely, the space of all bounded linear functionals on S(R^d)endowed with the weak ∗-topology. For all f ∈ S(R^d) or S(R^d), we use f to denote itsFourier transform, and f^∨for its inverse. LetQbe the collection of all dyadic cubes inR^d, namely, Q := {Qj,k := 2^−j([0,1)^d + k):j ∈ Z,k ∈ Z^d}. The symbol(Q)denotes the side-length of the cube Q and

jQ := −log₂(Q).

Letϕ0, ϕ∈S(R^d)be such that

suppϕ0⊂ {ξ ∈R^d: |ξ| ≤2}, |ϕ0(ξ)| ≥Cif|ξ| ≤5/3 (2.1) and

suppϕ⊂ {ξ ∈R^d:1/2≤ |ξ| ≤2} and |ϕ(ξ)| ≥Cif 3/5≤ |ξ| ≤5/3, (2.2) where C is a positive constant. In what follows, for allϕ ∈ S(R^d) and j ∈ N, ϕj(·):=2^{j d}ϕ(2^j·).

Definition 2.1 Lets ∈R,τ ∈ [0,∞),q ∈(0,∞]andϕ0,ϕ ∈S(R^d)be as in (2.1) and (2.2), respectively.

(i) Let p∈(0,∞]. TheBesov-type space B^sp^,τ,q(R^d)is defined to be the collection of all f ∈S(R^d)such that

f |B^s_p^,τ_,q(R^d) := sup

P∈Q

1

|P|^τ

⎧⎪

⎨

⎪⎩ ∞ j=max{j_P,0}

2^{j sq}

⎡

⎣

P

|ϕj ∗ f(x)|^pdx

⎤

⎦

q p

⎫⎪

⎬

⎪⎭

1 q

<∞

with the usual modifications made in case ofp= ∞and/orq = ∞.

(ii) Let p ∈ (0,∞). TheTriebel–Lizorkin-type space Fp^s,τ,q(R^d)is defined to be the collection of all f ∈S(R^d)such that

f |F_p^s,τ_,q(R^d) := sup

P∈Q

1

|P|^τ

⎧⎪

⎨

⎪⎩

P

⎡

⎣ ^∞

j=max{j_P,0}

2^{j sq}|ϕj∗ f(x)|^q

⎤

⎦

p q

dx

⎫⎪

⎬

⎪⎭

1 p

<∞

with the usual modification made in case ofq = ∞.

Remark 2.2 These spaces were introduced in [45]. To some extent the scale of Nikol’skij–Besov type spacesB^sp^,τ,q(R^d)had already been studied in [8–10].

We shall collect some features of these spaces below, but introduce first another scale of smoothness spaces of Morrey type. Recall first that theMorrey spaceMu,p(R^d),

0 < p ≤ u < ∞, is defined to be the set of all locally p-integrable functions f ∈L^loc_p (R^d)such that

f |Mu,p(R^d) := sup

x∈R^d,R>0

R^du−dp

B(x,R)|f(y)|^pdy ¹

p < ∞.

Remark 2.3 The spacesMu,p(R^d)are quasi-Banach spaces (Banach spaces for p ≥ 1). They originated from Morrey’s study on PDE (see [22]) and are part of the wider class of Morrey–Campanato spaces; cf. [23]. They can be considered as a complement toLpspaces. As a matter of fact,Mp,p(R^d)=Lp(R^d)withp∈(0,∞). To extend this relation, we putM∞,∞(R^d)=L_∞(R^d). One can easily see thatMu,p(R^d)= {0} foru <p, and that for 0<p2≤ p1≤u <∞,

Lu(R^d)=Mu,u(R^d) →Mu,p1(R^d) →Mu,p2(R^d). (2.3) In an analogous way, one can define the spacesM_∞,p(R^d),p∈(0,∞), but using the Lebesgue differentiation theorem, one can easily prove thatM_∞,p(R^d)=L_∞(R^d).

Next we recall the definition of the other scale of smoothness spaces of Morrey type we deal with in this paper.

Definition 2.4 Let 0< p ≤u <∞or p =u = ∞. Letq ∈ (0,∞],s ∈Randϕ0, ϕ∈S(R^d)be as in (2.1) and (2.2), respectively.

(i) TheBesov–Morrey spaceN_u^s_,p,q(R^d)is defined to be the set of all distributions f ∈S(R^d)such that

f |N_u^s_,p,q(R^d):=

_∞

j=0

2^{j sq}ϕj ∗ f |Mu,p(R^d)^q1/q

<∞ (2.4)

with the usual modification made in case ofq = ∞.

(ii) Letu ∈(0,∞). TheTriebel–Lizorkin–Morrey spaceEu^s,p,q(R^d)is defined to be the set of all distributions f ∈S(R^d)such that

f |Eu^s,p,q(R^d):=

_∞

j=0

2^{j sq}|(ϕj∗ f)(·)|^q 1/q

|Mu,p(R^d)

<∞ (2.5) with the usual modification made in case ofq = ∞.

Remark 2.5 Besov–Morrey spaces were introduced by Kozono and Yamazaki [19].

They studied semi-linear heat equations and Navier–Stokes equations with initial data belonging to Besov–Morrey spaces. The investigations were continued by Mazzucato [21], where one can find the atomic decomposition of the spaces. The Triebel–

Lizorkin–Morrey spaces were later introduced by Tang and Xu [34]. We follow the ideas of Tang and Xu [34], where a somewhat different definition is proposed. The

ideas were further developed by Sawano and Tanaka [27–30]. The most systematic and general approach to the spaces of this type can be found in the monograph [45]

or in the survey papers by Sickel [32,33].

Remark 2.6 Note that foru= porτ =0 we re-obtain the usual Besov and Triebel–

Lizorkin spaces:

N^sp,p,q(R^d)=B^s_p,q(R^d)=B^s_p^,_,⁰_q(R^d) (2.6) and

E^s_p,p,q(R^d)=F^s_p,q(R^d)=F^s_p^,_,⁰_q(R^d), (2.7) whereB^s_p,q(R^d)andF_p^s_,q(R^d)denote the classical Besov spaces and Triebel–Lizorkin spaces, respectively. There exists extensive literature on such spaces; we refer, in particular, to the series of monographs [35–37] for a comprehensive treatment.

ConventionWe adopt the nowadays usual custom to writeA^s_p,qinstead ofB^s_p,qorF_p^s_,q, A^sp^,τ,qinstead ofB^sp^,τ,qorF^sp^,τ,q, andA^su,p,qinstead ofNu^s,p,qorEu^s,p,q, respectively, when both scales of spaces are meant simultaneously in some context.

We collect some basic properties of the scalesA^sp^,τ,q(R^d)andA^su,p,q(R^d). The spaces A^sp^,τ,q(R^d)andA^su,p,q(R^d)are independent of the particular choices ofϕ0,ϕappearing in their definitions. They are quasi-Banach spaces (Banach spaces for p,q≥1), and S(R^d) → A^su,p,q(R^d),A^sp^,τ,q(R^d) → S(R^d). In case ofτ <0 oru < pwe have

A^sp^,τ,q(R^d)=A^su,p,q(R^d)= {0}.

Next we recall some basic embeddings results needed in the sequel. We refer to the references given above. For the spaces A^sp^,τ,q(R^d)it is known that

A^s_p^+ε,τ_,r (R^d) → A^s_p^,τ_,q(R^d) if ε∈(0,∞), r,q ∈(0,∞], (2.8) and

A^s_p^,τ_,q1(R^d) → A^s_p^,τ_,q2(R^d) if q1≤q2, (2.9) as well as

B^s_p^,τ_,min{p,q}(R^d) → F_p^s,τ_,q(R^d) → B^s_p^,τ_,max{p,q}(R^d), (2.10) which directly extends the well-known classical case from τ = 0 toτ ∈ [0,∞), p∈(0,∞),q ∈(0,∞]ands∈R. Moreover, it is known from [45, Proposition 2.6]

that

A^s_p^,τ_,q(R^d) → B^s+d(τ−

1 p)

∞,∞ (R^d). (2.11)

The following remarkable feature was proved in [42].

Proposition 2.7 Let s ∈ R, τ ∈ [0,∞)and p,q ∈ (0,∞] (with p < ∞ in the F -case). If eitherτ > ¹_p orτ = ¹_pand q= ∞, then A^sp^,τ,q(R^d)=B^s+d(τ−

1 p)

∞,∞ (R^d).

As for the scaleA^s_u,p,q(R^d)the counterparts to (2.8)–(2.10) read as

A^su^+ε,p,r(R^d) →A^su,p,q(R^d) if ε >0, r∈(0,∞], (2.12)

and

A^su,p,q1(R^d) →A^su,p,q2(R^d) if q1≤q2. (2.13) However, there also exist some differences. Sawano proved in [27] that, fors∈Rand 0<p<u <∞,

N_u^s_,p,min{p,q}(R^d) → Eu^s,p,q(R^d) → Nu^s,p,∞(R^d), (2.14) where, for the latter embedding,r= ∞cannot be improved—unlike in case ofu =p [see (2.10) withτ =0]. More precisely,

Eu^s,p,q(R^d) →Nu^s,p,r(R^d) if, and only if, r= ∞oru= pandr≥max{p,q}.

On the other hand, Mazzucato has shown in [21, Proposition 4.1] that E_u⁰_,p,2(R^d)=Mu,p(R^d), 1< p≤u <∞, in particular,

E⁰p,p,2(R^d)=Lp(R^d)=F_p⁰_,2(R^d), p∈(1,∞). (2.15) Remark 2.8 We obtained a lot more embedding results within the scales of spaces A^s_u,p,q(R^d)andA^sp^,τ,q(R^d), respectively, in [14,15,43,44], but will recall some of them in detail below as far as needed for our argument. We turn to the relation between the two scales of smoothness Morrey spaces. Lets,u,pandqbe as in Definition2.4and τ ∈ [0,∞). It is known from [45, Corollary 3.3, p. 64] that

Nu^s,p,q(R^d) →B^s_p^,τ_,q(R^d) with τ = 1 p −1

u. (2.16)

Moreover, the above embedding is proper ifτ >0 andq <∞. Ifτ =0 orq = ∞, then both spaces coincide with each other, in particular,

Nu^s,p,∞(R^d)=B^s,

1 p−¹u

p,∞ (R^d). (2.17)

As for theF-spaces, if 0≤τ <1/p, then F^s_p^,τ_,q(R^d)=Eu^s,p,q(R^d) with τ = 1

p −1

u, 0< p≤u <∞; (2.18) cf. [45, Corollary 3.3, p. 63]. Moreover, ifp∈(0,∞)andq ∈(0,∞], then

F^s,

1 p

p,q (R^d)=F_∞,^s _q(R^d)=B^s,

1 q

q,q(R^d); (2.19)

cf. [32, Propositions 3.4 and 3.5] and [33, Remark 10].

For later use we recall the definition of the space bmo(R^d), i.e., the local (non- homogeneous) space of functions of bounded mean oscillation, consisting of all locally integrable functions f ∈ L^loc₁ (R^d)satisfying that

f |bmo(R^d):= sup

|Q|≤1

1

|Q|

Q

|f(x)− fQ|dx+ sup

|Q|>1

1

|Q|

Q

|f(x)|dx<∞,

whereQappearing in the above definition runs over all cubes inR^d, and fQdenotes the mean value of f with respect to Q, namely, fQ := _|Q¹_|

Q f(x) dx, cf. [35, 2.2.2(viii)]. Hence the above result (2.19) implies, in particular,

bmo(R^d)=F_∞,⁰ ₂(R^d)=F⁰_p^,_,2^1/p(R^d), 0<p<∞. (2.20) Remark 2.9 In contrast to this approach, Triebel followed the original Morrey- Campanato ideas to develop local spacesL^rA^s_p,q(R^d)in [38], and so-called ‘hybrid’

spaces L^rA^s_p,q(R^d) in [39], where 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and

−^d_p ≤r <∞. This construction is based on wavelet decompositions and also com- bines local and global elements as in Definitions2.1and2.4. However, Triebel proved in [39, Chapter 3] that

L^rA^s_p,q(R^d)=A^s_p^,τ_,q(R^d), τ = 1 p + r

d, (2.21)

in all admitted cases. Therefore we do not have to deal with these spaces separately in the sequel.

2.2 Spaces on domains

We assume that Ω is a bounded domain in R^d. We consider smoothness Morrey spaces onΩdefined by restriction. LetD(Ω)be the set of all infinitely differentiable functions supported inΩand denote byD(Ω)its dual. IfΩis aC^∞domain, then we are able to define the extension operator ext:D(Ω)→S(R^d), cf. [31], the restriction operator re:S(R^d)→D(Ω)can be defined naturally as an adjoint operator

re(f), ϕ = f,ext(ϕ), f ∈S(R^d), whereϕ ∈D(Ω). We will write f|Ω =re(f).

Definition 2.10 Lets∈Randq ∈(0,∞].

(i) Let 0 < p ≤ u < ∞or p =u = ∞(withu <∞in case ofA = E). Then A^su,p,q(Ω)is defined by

A^su,p,q(Ω):=

f ∈D(Ω): f =g|Ω for someg∈A^su,p,q(R^d)

endowed with the quasi-norm f |A^s_u,p,q(Ω):=inf

g|A^s_u,p,q(R^d): f =g|_Ω, g ∈A^s_u,p,q(R^d) . (ii) Letτ ∈ [0,∞)andp∈(0,∞](withp<∞in case ofA=F). Then A^sp^,τ,q(Ω)

is defined by

A^s_p^,τ_,q(Ω):=

f ∈D(Ω): f =g|_Ω for someg∈ A^s_p^,τ_,q(R^d) endowed with the quasi-norm

f | A^s_p^,τ_,q(Ω):=inf

g| A^s_p^,τ_,q(R^d):f =g|Ω, g∈ A^s_p^,τ_,q(R^d) . Remark 2.11 The spacesA^su,p,q(Ω)andA^sp^,τ,q(Ω)are quasi-Banach spaces (Banach spaces for p,q ≥ 1). When u = p or τ = 0 we re-obtain the usual Besov and Triebel–Lizorkin spaces defined on bounded domains. Several properties of the spaces A^s_u,p,q(Ω), including the extension property, were studied in [31]. As for the spaces A^sp^,τ,q(Ω)we also refer to [45, Section 6.4.2]. In particular, if the domain is smooth then, according to [45, Theorem 6.13], there exists a linear and bounded extension operator

ext_τ:A^s_p^,τ_,q(Ω)→ A^s_p^,τ_,q(R^d), where 1≤ p<∞,0<q ≤ ∞,s∈R, τ ≥0, (2.22) such that

re◦ext_τ =id in A^s_p^,τ_,q(Ω), (2.23) where re:A^sp^,τ,q(R^d)→ A^sp^,τ,q(Ω)is the restriction operator as above.

Several types of embeddings related to these scales were already considered for bounded smooth domains. For instance, embeddings within the scale of spaces A^su,p,q(Ω)as well as to classical spaces likeC(Ω)or Lr(Ω)were investigated in [14,15]. In [12] we studied the question under what assumptions these spaces con- sist of regular distributions only. Moreover, in [43] we considered the approximation numbers of some special compact embedding ofA^sp^,τ,q(Ω)intoL_∞(Ω).

Remark 2.12 Let us mention that we have the counterparts of many continuous embed- dings stated in the previous subsection for spaces onR^d when dealing with spaces restricted to bounded domains. This concerns, in particular, the elementary embed- dings and coincidences (2.8)–(2.10), Proposition2.7and (2.12)–(2.18).

For a matter of completion, we finish this subsection by giving the definition of bmo(Ω), that we will use later on. As previously, this is done by restriction, that is, bmo(Ω)is defined as being the space of all restrictions toΩof functions in bmo(R^d), equipped with the norm

f |bmo(Ω) :=inf

g|bmo(R^d):f =g|_Ω, g∈bmo(R^d) .

2.3 Entropy numbers

As explained in the beginning already, our main concern in this paper is to character- ize the compactness of embeddings in further detail. Therefore we briefly recall the concepts of entropy and approximation numbers.

Definition 2.13 Let XandY be two complex (quasi-) Banach spaces,k∈Nand let T ∈L(X,Y)be a linear and continuous operator fromXintoY.

(i) Thekth entropy number ek(T)ofT is the infimum of all numbersε >0 such that there exist 2^k−1balls inYof radiusεwhich cover the imageT BX of the unit ball BX = {x∈ X: x|X ≤1}.

(ii) Thekth approximation number ak(T)ofT is defined by

ak(T)=inf{T −S:S ∈L(X,Y),rankS <k}, k∈N. (2.24) Remark 2.14 For details and properties of entropy and approximation numbers we refer to [3,4,18,25] (restricted to the case of Banach spaces), and [7] for some exten- sions to quasi-Banach spaces. Among other features we only want to mention the multiplicativity of entropy numbers: let X,Y,Z be complex (quasi-) Banach spaces andT1∈L(X,Y),T2∈L(Y,Z). Then

ek₁+k₂−1(T2◦T1)≤ek₁(T1)ek₂(T2), k1,k2∈N. (2.25) Note that one has in general limk→∞ek(T)= 0 if, and only if,T is compact. The last equivalence justifies the saying that entropy numbers measure ‘how compact’ an operator acts. This is one reason to study the asymptotic behavior of entropy numbers (that is, their decay) for compact operators in detail.

Approximation numbers share many of the basic features of entropy numbers, but are different in some respect. They can—unlike entropy numbers—be regarded as specials-numbers, a concept introduced by Pietsch [24, Section 11]. Of special importance is the close connection of both concepts, entropy numbers as well as approximation numbers, with spectral theory, in particular, the estimate of eigenvalues.

We refer to the monographs [3,4,7,18,25] for further details.

Remark 2.15 We recall what is well-known in the case of the embedding

idA:A^s_p¹_1,q1(Ω)→ A^s_p²_2,q2(Ω),

where −∞ < s2 ≤ s1 < ∞, 0 < p1,p2 ≤ ∞ (p1,p2 < ∞ in the F-case), 0<q1,q2≤ ∞, and the spacesA^s_p,q(Ω)are defined by restriction. Let

δ =s1−s2−d 1

p1− 1 p2

, δ₊=s1−s2−d 1

p1− 1 p2

+. (2.26)

Then idAis compact whenδ+>0; cf. [7, (2.5.1/10)]. In this situation Edmunds and Triebel proved in [5,6] (see also [7, Theorem 3.3.3/2]) that

ek(idA)∼k^−s1−ds2, k∈N, (2.27) where s1 ≥ s2, 0 < p1,p2 ≤ ∞(p1,p2 < ∞in the F-case), 0 < q1,q2 ≤ ∞, andδ₊ > 0. It was originally proved there for smooth domains, but the extension to arbitrary bounded domains is also covered by [37, Theorem 1.92]. In the case of approximation numbers the situation is more complicated; the result of Edmunds and Triebel for smooth domains in [7, Theorem 3.3.4], partly improved by Caetano [2], reads as

ak(idA) ∼ k^−δd+−κ, k∈N, (2.28) with

κ=

min{p₁,p2}

2 −1

+·min δ

d, 1

min{p₁,p2}

, (2.29)

whereδis given by (2.26) andp₁ denotes the conjugate ofp1defined by _p¹

1+_p¹

1 =1

if 1≤ p1≤ ∞and p₁ = ∞if 0< p1<1. The above asymptotic result is almost complete now, apart from the restrictions that(p1,p2)= (1,∞)or _d^δ = _min{p¹

1,p2}

when 0 < p1 <2 < p2 ≤ ∞. Note thatκ =0 unless p1 <2 < p2, andδ ≥ δ₊ withδ =δ+if p1≤ p2.

3 Compact embeddings

First we recall our compactness result as obtained in [14] (forA=N) and [15] (for A=E), with a supplement related to arbitrary bounded domains proved in [17]. We shall heavily rely on this result in our argument below.

ConventionHere and in the sequel we shall understand_u^pi

i =1 in case ofpi =ui = ∞, i =1,2.

Theorem 3.1 Let si ∈R,0<qi ≤ ∞,0<pi ≤ui <∞, or, in the case ofN-spaces, allow also pi =ui = ∞, i=1,2. Then the embedding

id_A:A^su¹₁,p₁,q₁(Ω) →A^su²₂,p₂,q₂(Ω) (3.1) is compact if, and only if, the following condition holds:

s1−s2

d >max

0, 1 u1− 1

u2,p1

u1

1 p1 − 1

p2

. (3.2)

In particular, if p1=u1= ∞andA^su¹1,p1,q1 =Nu^s11,p1,q1, thenid_Agiven by(3.1) is compact if, and only if, s1>s2. If p2 =u2= ∞andA^su²₂,p₂,q₂ =Nu^s2₂,p₂,q₂, then id_Ais compact if, and only if,^s1−_d^s2 >_u¹₁.