Compact embeddings in Besov-type and Triebel–Lizorkin-type spaces on bounded domains

We study embeddings of Besov-type and Triebel–Lizorkin-type spaces,idτ:Bp1,q1s1,τ1(Ω)↪Bp2,q2s2,τ2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {id}}_\tau {:}\,{B}_{p_1,q_1}^{s_1,\tau _1}(\varOmega )\,\hookrightarrow \,{B}_{p_2,q_2}^{s_2,\tau _2}(\varOmega )$$\end{document} and idτ:Fp1,q1s1,τ1(Ω)↪Fp2,q2s2,τ2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {id}}_\tau {:}\,{F}_{p_1,q_1}^{s_1,\tau _1}(\varOmega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\varOmega ) $$\end{document}, where Ω⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subset {{\mathbb R}^d}$$\end{document} is a bounded domain, and obtain necessary and sufficient conditions for the compactness of idτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {id}}_\tau $$\end{document}. Moreover, we characterize its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the corresponding results for classical Besov and Triebel–Lizorkin spaces as well as for Besov–Morrey and Triebel–Lizorkin–Morrey spaces.

The classical Morrey spaces M u, p , 0 < p ≤ u < ∞, were introduced by Morrey [22] and are part of a wider class of Morrey-Campanato spaces, cf. [23]. They can be seen as a complement to L p spaces, since M p, p (R d ) = L p (R d ).
The Besov-Morrey spaces N s u, 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 equations. In [34], Tang and Xu introduced the corresponding Triebel-Lizorkin-Morrey spaces E s u, p,q (R d ), thanks to establishing the Morrey version of Fefferman-Stein vector-valued inequality. Some properties of these spaces including their wavelet characterizations 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][14][15][16].
Another class of generalizations, the Besov-type space B s,τ p,q (R d ) and the Triebel-Lizorkin-type space F s,τ p,q (R d ) were introduced in [45]. Their homogeneous versions were originally investigated by El Baraka [8][9][10] and by Yuan and Yang [40,41]. There are also some applications in partial differential equations for spaces of type B s,τ p,q (R d ) and F s,τ p,q (R d ), such as (fractional) Navier-Stokes equations, cf. [20]. Although the above scales are defined in different ways, they share some properties 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 ) and F s p,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 appropriately chosen spaces of type B s,τ p,q (R d ) or F s,τ p,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 s,τ p,q (Ω) and F s,τ p,q (Ω), where Ω ⊂ R d is a bounded domain, i.e. a bounded open set in R d . In particular, our first goal is to find necessary and sufficient conditions for the compactness of the embeddings id τ : A s 1 ,τ 1 p 1 ,q 1 (Ω) → A s 2 ,τ 2 p 2 ,q 2 (Ω), (1.1) where A = B or A = F, cf. Theorem 3.2. Here we prove that id τ is compact if, and only if, where we use the notation a + := max{a, 0}. At this point, this work can be seen as a counterpart of the papers [14][15][16], where we studied the compactness of the corresponding embeddings of the spaces N s u, p,q and E s u, 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 spaces N s u, p,q and E s u, p,q and the classical spaces of type B s p,q and F s p,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 However, to the best of our knowledge, apart from a result obtained in [43] for approximation numbers when the target space is L ∞ , nothing is known on this matter for embeddings between spaces of type A s,τ p,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. 2 we present and collect some basic facts about smoothness Morrey spaces, on R d and on bounded domains Ω ⊂ R d , and introduce the notions of entropy and approximation numbers. In Sect. 3 we 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 scale N s u, p,q to the cases when p i = u i = ∞, 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.

Preliminaries
First we fix some notation. By N we denote the set of natural numbers, by N 0 the set N ∪ {0}, and by Z d the set of all lattice points in R d having integer components. For a ∈ R, let a + := 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 constant C 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 d with radius r > 0, and | · | denotes the Lebesgue measure when applied to measurable subsets of R d .
Given two (quasi-)Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding of X into Y is continuous.

Smoothness spaces of Morrey type on R d
Let S(R d ) be the set of all Schwartz functions on R d , endowed with the usual topology, and denote by S (R d ) its topological 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 its Fourier transform, and f ∨ for its inverse. Let Q be the collection of all dyadic cubes in R d , namely, Q : The symbol (Q) denotes the side-length of the cube Q and j Q := − log 2 (Q).
with the usual modifications made in case of p = ∞ and/or q = ∞.
with the usual modification made in case of q = ∞.
We shall collect some features of these spaces below, but introduce first another scale of smoothness spaces of Morrey type. Recall first that the Morrey space M u, p (R d ),

Remark 2.3
The spaces M u, 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 to L p spaces. As a matter of fact, for u < p, and that for 0 < p 2 ≤ p 1 ≤ u < ∞, In an analogous way, one can define the spaces M ∞, p (R d ), p ∈ (0, ∞), but using the Lebesgue differentiation theorem, one can easily prove that Next we recall the definition of the other scale of smoothness spaces of Morrey type we deal with in this paper.
with the usual modification made in case of q = ∞.
with the usual modification made in case of q = ∞.

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][28][29][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 for u = p or τ = 0 we re-obtain the usual Besov and Triebel-Lizorkin spaces: are independent of the particular choices of ϕ 0 , ϕ appearing in their definitions. They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), and . Next we recall some basic embeddings results needed in the sequel. We refer to the references given above. For the spaces A s,τ p,q (R d ) it is known that and as well as which directly extends the well-known classical case from (2.11) The following remarkable feature was proved in [42].
As for the scale A s u, p,q (R d ) the counterparts to (2.8)-(2.10) read as and However, there also exist some differences. Sawano proved in [27] that, for s ∈ R and 0 < p < u < ∞, where, for the latter embedding, r = ∞ cannot be improved-unlike in case of u = p [see (2.10) with τ = 0]. More precisely, Remark 2. 8 We obtained a lot more embedding results within the scales of spaces [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. Let s, u, p and q be as in Definition 2.4 and τ ∈ [0, ∞). It is known from [45, Corollary 3.3, p. 64] that Moreover, the above embedding is proper if τ > 0 and q < ∞. If τ = 0 or q = ∞, then both spaces coincide with each other, in particular, For later use we recall the definition of the space bmo(R d ), i.e., the local (nonhomogeneous) space of functions of bounded mean oscillation, consisting of all locally integrable functions f ∈ L loc where Q appearing in the above definition runs over all cubes in R d , and f Q denotes the mean value of f with respect to Q, namely,

Remark 2.9
In contrast to this approach, Triebel followed the original Morrey-Campanato ideas to develop local spaces L r A s p,q (R d ) in [38], and so-called 'hybrid' This construction is based on wavelet decompositions and also combines local and global elements as in Definitions 2.1 and 2.4. However, Triebel proved in [39,Chapter 3] that in all admitted cases. Therefore we do not have to deal with these spaces separately in the sequel.

Spaces on domains
We assume that Ω is a bounded domain in R d . We consider smoothness Morrey spaces on Ω defined by restriction. Let D(Ω) be the set of all infinitely differentiable functions supported in Ω and denote by D (Ω) its dual. If Ω is a C ∞ domain, then we are able to define the extension operator ext : can be defined naturally as an adjoint operator where ϕ ∈ D(Ω). We will write f | Ω = re( f ).

Remark 2.11
The spaces A s u, p,q (Ω) and A s,τ p,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 s,τ p,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 where re: A s,τ p,q (R d ) → A s,τ p,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 s u, p,q (Ω) as well as to classical spaces like C(Ω) or L r (Ω) were investigated in [14,15]. In [12] we studied the question under what assumptions these spaces consist of regular distributions only. Moreover, in [43] we considered the approximation numbers of some special compact embedding of A s,τ p,q (Ω) into L ∞ (Ω). 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 to Ω of functions in bmo(R d ), equipped with the norm

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

Definition 2.13
Let X and Y be two complex (quasi-) Banach spaces, k ∈ N and let T ∈ L(X , Y ) be a linear and continuous operator from X into Y .
(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 extensions 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 and Note that one has in general lim k→∞ e k (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 special s-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 0 < q 1 , q 2 ≤ ∞, and the spaces A s p,q (Ω) are defined by restriction. Let and δ + > 0. It was originally proved there for smooth domains, but the extension to arbitrary bounded domains is also covered by [ where δ is given by (2.26) and p 1 denotes the conjugate of p 1 defined by 1 The above asymptotic result is almost complete now, apart from the restrictions that ( p 1 ,

Compact embeddings
First we recall our compactness result as obtained in [14] (for A = 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. Convention Here and in the sequel we shall understand p i u i = 1 in case of p i = u i = ∞, i = 1, 2.
Theorem 3.1 Let s i ∈ R, 0 < q i ≤ ∞, 0 < p i ≤ u i < ∞, or, in the case of N -spaces, allow also p i = u i = ∞, i = 1, 2. Then the embedding is compact if, and only if, the following condition holds: In particular, if p 1 = u 1 = ∞ and A s 1 u 1 , p 1 ,q 1 = N s 1 u 1 , p 1 ,q 1 , then id A given by (3.1) is compact if, and only if, s 1 > s 2 . If p 2 = u 2 = ∞ and A s 2 u 2 , p 2 ,q 2 = N s 2 u 2 , p 2 ,q 2 , then id A is compact if, and only if, Proof The cases when 0 < p i ≤ u i < ∞, i = 1, 2, were proved in [14,15] for A = N and A = E respectively. So we are left with the cases p 1 = u 1 = ∞ or p 2 = u 2 = ∞. At first, let us consider the case when p 1 = u 1 = ∞ and A s 1 u 1 , p 1 ,q 1 = N s 1 u 1 , p 1 ,q 1 . If s 1 − s 2 > 0, the compactness of id A follows from as the first embedding is compact when s 1 − s 2 > 0. Now we assume that id A is compact. We have where the last embedding was proved in [14,15]. Then, the compactness of the first embedding implies the compactness of the embedding between the outer spaces, which in turn implies s 1 − s 2 > 0. Now let p 2 = u 2 = ∞, A s 2 u 2 , p 2 ,q 2 = N s 2 u 2 , p 2 ,q 2 and s 1 −s 2 d > 1 u 1 . As the case p 1 = u 1 = ∞ (when A = N ) is already covered by our preceding observation, we may further assume that 0 < p 1 ≤ u 1 < ∞. By a straightforward extension of our continuity result in [14, Theorem 3.1] (to the cases when p i = u i = ∞ for i = 1 or i = 2) we have the continuous embedding Moreover, in case of s 1 − d u 1 > s 2 , it is well-known that the embedding is compact. Thus N s 1 u 1 , p 1 ,q 1 (Ω) → N s 2 ∞,∞,q 2 (Ω) compactly for any q 1 , q 2 ∈ (0, ∞]. The compactness of E s 1 u 1 , p 1 ,q 1 (Ω) → N s 2 ∞,∞,q 2 (Ω) is then a consequence of (2.14). The necessity follows from the following chain of embeddings in the same way as above. Finally we apply (2.14) for the case A s 1 Now we give the counterpart of Theorem 3.1 for Besov-type and Triebel-Lizorkintype spaces. For convenience we use some abbreviation for the following expression, which plays an essential role in the sequel. So let us denote (3.8) is compact if, and only if, the following condition holds: Proof We shall use sharp embeddings and identities like (2.11), Proposition 2.7 and (2.18) (all adapted to spaces restricted to the domain Ω, recall Remark 2.12) together with our previous result Theorem 3.1 several times. Therefore we shall always distinguish below between the cases τ i < 1 p i and τ i ≥ 1 p i (with some additional restrictions on q i occasionally), i = 1, 2. For that reason it seems convenient to reformulate condition (3.10) according to these cases; that is, the goal is to prove that id τ given by (3.9) is compact if, and only if, Step 1. Let us assume that τ 2 ≥ 1 p 2 with q 2 = ∞ if τ 2 = 1 p 2 . First we prove the sufficiency of (3.11) for the compactness of id τ , that is, we assume now Then the embedding (2.11) and Proposition 2.7 yield (3.13) and the embedding between the Besov spaces is compact. So id τ is compact. Now we turn to the necessity of (3.12) for the compactness of id τ . So we assume that id τ given by (3.9) is compact. Let first τ 1 ≥ 1 p 1 with q 1 = ∞ if τ 1 = 1 p 1 . Then by Proposition 2.7 we obtain which results in a compact embedding between the outer Besov spaces. This is wellknown to imply (3.12) as desired. A similar argument works for p 1 = ∞, τ 1 = 0 and q 1 < ∞, since then we have .

Substep 2.1
If also 0 ≤ τ 1 < 1 p 1 , then (3.10) reads as For the F-spaces, the result immediately follows from coincidence (2.18) and Theorem 3.1. Note that in this case (3.2) coincides with the last line in (3.11) in view of 1  (3.17) and the embedding between the Besov-Morrey spaces is compact. Therefore id τ is compact.
We now turn to the necessity and assume that id τ is compact. Then where we have used again (2.16), (2.9) and the coincidence (2.17). In view of Theorem 3.1 this leads to the desired condition (3.16).

Substep 2.2
An analogous argument works for τ 1 ≥ 1 p 1 with q 1 = ∞ if τ 1 = 1 p 1 , cf. Proposition 2.7. This time (3.10) coincides with the second line of (3.11). So for the sufficiency we assume that Then Proposition 2.7, the embedding (2.16) and the coincidence (2.18) give (3.20) Then, by Theorem 3.1, id τ is compact. Conversely, let us assume that id τ is compact. We benefit from Proposition 2.7 and the coincidences (2.17) and (2.18) to obtain (3.21) In case of A = F, the last embedding is true due to the elementary embeddings (2.14). Therefore, the compactness of id τ and Theorem 3.1 lead to the desired condition

Substep 2.3
Assume finally τ 1 = 1 p 1 with q 1 < ∞. Note that in this case, due to the middle line of (3.11), the condition (3.10) reads as s 1 > s 2 . We apply (2.11) to obtain Since B s 1 ∞,∞ (Ω) = N s 1 ∞,∞,∞ (Ω), in view of Theorem 3.1 the second embedding is compact for s 1 > s 2 , and the last embedding is a consequence of (2.16) and (2.18), respectively. Conversely, if id τ is compact in this case, then we can argue as follows.
where we used (2.17) and (2.18) in the last equality and (2.14) in the last embedding. In view of Theorem 3.1 this leads to s 1 > s 2 as required.
Step 3 It remains to deal with τ 2 = 1 p 2 and q 2 < ∞. In that case (3.11) always reads as s 1 − s 2 > d( 1 Substep 3.1 Assume first τ 1 < 1 p 1 and let 1 u 1 = 1 p 1 − τ 1 . Then by elementary embeddings and the coincidences (2.17) and (2.18), (3.24) and the embedding of the outer spaces is compact for any q 0 , since the second embedding is compact by Theorem 3.1 with (3.10). The last embedding is continuous where we apply [44,Theorem 2.5]. If A = B, we put q 0 = q 2 and the argument is complete, while in case of A = F we choose q 0 ≤ min{ p 2 , q 2 } and finally use the continuous embedding into F s 2 ,τ 2 p 2 ,q 2 (Ω) due to (2.10). On the other hand, [44,Corollaries 5.2,5.9] and (2.11) ensure (3.25) such that the compactness of id τ implies s 1 − d u 1 > s 2 by the well-known classical results. This proves the necessity of the condition.
We can argue in the same way as above. Let Then by the identities in Proposition 2.7 and (2.19), 26) where the first embedding is compact for s 1 + d(τ 1 − 1 p 1 ) > s 0 and the last embedding is continuous for s 0 > s 2 by (2.8) and (2.10). Hence id τ is compact. Conversely, the compactness of id τ implies where we used (2.11) and Proposition 2.7. But the resulting compactness of the outer embedding of Besov spaces leads to the desired condition s 1 + d(τ 1 − 1 p 1 ) > s 2 . Substep 3.3 Let finally τ 1 = 1 p 1 with q 1 < ∞. So we are in the double-limiting case and need to show the compactness of id τ if, and only if, s 1 > s 2 . The sufficiency can be obtained via where we use [44,Corollary 5.2] in the last embedding in case of A = B, extended by the same argument as above to A = F via (2.10). The second embedding is compact for is compact, where we used for the first embedding [44, Corollary 5.2] (with (2.10) for A = F) again, and (2.11) for the last one. But this implies s 1 > s 2 .

Remark 3.3
Usually one needs the condition s 1 − s 2 > 0 to prove compactness of this kind of embeddings. Curiously this is not the case when considering spaces of type B s,τ p,q and F s,τ p,q . This can easily be seen by condition (3.11), for instance when τ 1 ≥ 1 p 1 and τ 2 < 1 p 2 . In parallel to (2.11) and Proposition 2.7, this evidences the fact that the parameter τ modifies indeed the smoothness of these spaces. Remark 3. 4 We briefly return to our Remark 2.9 which referred to the coincidence of Triebel's hybrid spaces L r A s p,q with the spaces A s,τ p,q if τ = 1 p + r d . Obviously, defining both by restriction to Ω, this is transferred to spaces on domains. In that sense Theorem 3.2 can be formulated as follows: let Then the embedding is compact if, and only if, the following condition holds: i.e., We now collect some immediate consequences of the above compactness result. We begin with the case τ 1 = τ 2 = τ ≥ 0.
is compact if, and only if, Proof We apply Theorem 3.2 with τ 1 = τ 2 .

Remark 3.6
The result is well-known for τ = 0, where (3.33) reads as We find it interesting that for τ > 0 there is not a simple 'τ -shift', but an interplay between τ and the p i -parameters-however, only when p 1 ≥ p 2 . This again refers to the hybrid role played by the additional τ -parameters and makes it even more obvious that it influences both the smoothness and the integrability parameters s i and p i , respectively. Now we deal with special target spaces. In the case of L ∞ (Ω) and bmo(Ω) we have the following result.
Corollary 3.7 Let s ∈ R, 0 < p ≤ u < ∞ and q ∈ (0, ∞]. Then the following conditions are equivalent: The equivalence of (i) and (iii) was proved in [14,15], whereas the equivalence of (ii) and (iii) follows from Theorem 3.2 since bmo(Ω) This covers the case A s u, p,q = E s u, p,q . The extension to the case A s u, p,q = N s u, p,q is done via (2.14).
The counterpart of Corollary 3.7 for spaces of type A s,τ p,q reads as follows.
Then the following conditions are equivalent:

Proof
Step 1 We prove the equivalence of (i) and (iii). The case τ = 0 is well-known, so we assume τ > 0. Note that the continuity of that embedding was studied in [11,Proposition 2.18]  Step 2 We prove the equivalence of (ii) and (iii). However, in case of A = F this coincides with Theorem 3.2 for s 1 = s, s 2 = 0, p 1 = p 2 = p, τ 1 = τ , τ 2 = 1 p , q 1 = q and q 2 = 2 since bmo(Ω) = F is compact if, and only if, (3.34) Proof Case (i) was already shown in [14, Proposition 5.3] (for A = N ). In view of (2.14) and the independence of the condition with respect to q the counterpart for A = E follows (and slightly extends our recent result in [15,Corollary 5.4] to r = 1). We come to (ii) and start with the case r ,∞ (Ω), 1 ≤ r ≤ ∞, so we apply Theorem 3.2 for s 1 = s, s 2 = 0, p 1 = p, p 2 = r , τ 1 = τ , τ 2 = 0, q 1 = q, and q 2 = 1 or q 2 = ∞ to obtain the necessary and sufficient conditions. Again we benefit from the independence of (3.10) with respect to the q-parameters. Finally, the case A = F follows by (2.10) again.
There is no continuous embedding Proof Here we directly follow our proof of Theorem 3.2 and apply our continuity results [14, Theorem 3.1] (for N -spaces) and [15,Theorem 5.2] (for E-spaces). We again follow the splitting suggested by (3.11). So let us assume in the sequel that there is a continuous embedding id τ .
Here some influence of the fine parameters q i can also be expected. But this question is postponed to a separate study in the future.

Entropy numbers
First we return to the compact embedding id A given by ( with the same choice of q 0 as before. Then we get Remark 4.12 Due to their similarities, we do not present the special cases when the source or the target space matches the classical spaces B s p,q and F s p,q , that is when τ 1 = 0 or τ 2 = 0. However, we would like to remark that the result is not symmetric in the sense that we have different results for both cases. Namely, when τ 1 = 0, part (i) of Theorem 4.2 is excluded, while for the case where τ 2 = 0 both parts of the theorem are relevant, naturally with the proper adaptations for this particular case.

Approximation numbers
Finally we briefly collect some partial results about approximation numbers of the embedding id τ , recall their definition (2.24). Now we assume that Ω is a C ∞ domain. In [43] we obtained some first result for approximation numbers a k (id τ ) when the target space was L ∞ (Ω): let p ∈ [2, ∞] (with p < ∞ in the F-case), q ∈ (0, ∞], 0 ≤ τ < 1 p and d( 1 In [16] we studied the situation for the embedding id A with the following result. The above proposition coincides with [16, Corollary 3.4(i)] apart from the case when p i = u i = ∞ for i = 1 or i = 2. But this extension can easily be verified following the short proof in [16]. We also refer to [1,Section 6] where also the periodic case and more general Morrey type spaces were studied. Now we give some partial counterpart of Theorem 4.2 in terms of approximation numbers.