Morrey Sequence Spaces: Pitt’s Theorem and Compact Embeddings

Morrey (function) spaces and, in particular, smoothness spaces of Besov–Morrey or Triebel–Lizorkin–Morrey type have enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces mu,p=mu,p(Zd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{u,p}=m_{u,p}(\mathbb {Z}^d)$$\end{document}, 0<p≤u<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\le u<\infty $$\end{document}, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} spaces. We consider some basic features, embedding properties, a pre-dual, a corresponding version of Pitt’s compactness theorem, and further characterize the compactness of embeddings of related finite-dimensional spaces.


Introduction
Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type were studied in recent years quite intensively and systematically. Decomposition methods like atomic or wavelet characterizations require suitably adapted sequence spaces. This has been done to some extent already. We are Communicated by Pencho Petrushev.  Originally, Morrey spaces were introduced in [32], when studying solutions of second order quasi-linear elliptic equations in the framework of Lebesgue spaces. They can be understood as a complement (generalization) of the Lebesgue spaces L p (R d ). In particular, the Morrey space M u, p , 0 < p ≤ u < ∞, is defined as the collection of all complex-valued Lebesgue measurable functions on R d such that where B(x, R) = {y ∈ R d : |x − y| < R} are the usual balls centered at x ∈ R d with radius R > 0. Obviously, such that the usual assumption is p ≤ u < ∞. As can be seen from the definition, Morrey spaces describe the local behavior of the L p norm, which makes them useful when describing the local behavior of solutions of nonlinear partial differential equations, cf. [22, 25-26, 27, 30, 31, 50]. Furthermore, applications in harmonic analysis and potential analysis can be found in the papers [2][3][4]. For more information, we refer to the books [1] and [48] and, in particular, to the fine surveys [46,47] by Sickel. As for the smoothness spaces of Morrey type, aside from Besov-Morrey spaces N s u, p,q (R d ) in [22,30,31], and their counterparts Triebel-Lizorkin-Morrey spaces E s u, p,q (R d ), cf. [49], their atomic and wavelet characterizations were already described in the papers [39,[41][42][43][44], and we simplified the appearing sequence spaces n s u, p,q in [18] to some extent. There are further related approaches to Besov-type spaces B s,τ p,q (R d ) and Triebel-Lizorkin-type spaces F s,τ p,q (R d ), cf. [48] with forerunners in [12-14, 55, 56]. Triebel provided a third approach, so-called local and hybrid spaces, in [53,54], but they coincide with appropriately chosen spaces of type B s,τ p,q (R d ) or F s,τ p,q (R d ), cf. [57].
Recently, based on some discussion at the conference "Banach Spaces and Operator Theory with Applications" in Poznań in July 2017, we found that Morrey sequence spaces m u, p = m u, p (Z d ), 0 < p ≤ u < ∞, have been considered almost nowhere. The paper [6] concerns another type of discretization than we have in mind. To the best of our knowledge, there is only the paper [16] (and an interesting application in [17]) so far that is devoted to this subject. They are defined as natural generalizations Clearly, m p, p = p . We consider some basic features in Sect. 2 and present our main embedding result, Theorem 3.1, in Sect. 3, which reads as follows: Let 0 < p 1 ≤ u 1 < ∞ and 0 < p 2 ≤ u 2 < ∞. Then the embedding is continuous if, and only if, the following conditions hold: The embedding is never compact. In Sect. 4, we describe a pre-dual X u, p of m u, p , 1 ≤ p < u < ∞, which is a separable Banach space, unlike m u, p .
Dealing with the closure m 00 u, p of finite sequences in m u, p , we obtain a counterpart to Pitt's theorem [37] in our setting as follows, see Theorem 5.3 below: Let 1 < p < u < ∞ and 1 ≤ q < ∞. Then any bounded linear operator T : m 00 u, p → q is compact. The above sequence spaces are not rearrangement invariant. Further information about the Pitt theorem in rearrangement invariant setting can be found in [8] and [28]. Finally, we further characterize the compactness of embeddings of related finitedimensional spaces and obtain for the asymptotic behavior of the dyadic entropy numbers of such a finite-dimensional embedding that where j ∈ N, 0 < p i ≤ u i < ∞, i = 1, 2, and k ∈ N 0 with k 2 jd .

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{k ∈ Z : k ≤ a} and 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 | · | 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.
For 0 < p < ∞, we denote by p = p (Z d ), denote the Lorentz sequence spaces, as usual. Finally, we adopt the custom to denote by c = c(Z d ), c 0 = c 0 (Z d ), and c 00 = c 00 (Z d ) the corresponding subspaces of ∞ (Z d ) of convergent, null, and finite sequences, respectively; that is, As we mostly deal with sequence spaces on Z d , we shall often omit it from their notation, for convenience.

Remark 2.2
In [16] the corresponding one-dimensional counterpart was introduced and studied. Proof Part (i) is standard; the completeness can be shown similarly to the (onedimensional) counterpart in [16]. The first two assertions in (ii) are obvious; the monotonicity in p is a matter of Hölder's inequality. Concerning (iii), clearly for any m ∈ Z d , such that finally, taking the supremum over all m ∈ Z d , We prove (iv). Let {λ * ν } ν∈N be a nonincreasing rearrangement of a sequence λ = {λ k } k∈Z d . Then for any cube Q − j,m , we have It remains to deal with (v). Consider first the constant sequence λ 1 = {1} k∈Z d ∈ c. Then which disproves c ⊂ m u, p (and simultaneously strengthens (iii) by m u, p ∞ ). A slight modification disproves c 0 ⊂ m u, p : choose ε such that 0 < ε < d u , and consider λ = {λ k } k∈Z d given byλ k = |k| −ε . Thenλ ∈ c 0 . On the other hand, Now consider a special sequence λ = {λ k } k∈Z d that looks as follows: Obviously λ / ∈ c; in particular, λ / ∈ c 0 . Now, by construction, So the subspaces c 0 , c, and m u, p of ∞ are incomparable in the above sense.

Remark 2.4
Obviously, Definition 2.1 gives the discrete counterpart of M u, p (R d ) in view of (1.1). More precisely, given some sequence where χ A denotes the characteristic function of a set A ⊂ R d , as usual.
Conversely, let f 0 ∈ M u, p , and define Hence there is some obvious correspondence between the Morrey sequence and function spaces, respectively.

Remark 2.5 As in the case of the function spaces
where in the case of p = u = ∞, the latter sum has to be replaced by the supremum, as usual. Using this definition, we can show that m ∞, p = ∞ . This is obvious for p = u = ∞, so let us assume p < u = ∞. In view of Proposition 2.3(iii), it remains to show that ∞ → m ∞, p if p < u = ∞. Let λ ∈ ∞ . Thus for any j ∈ N 0 and m ∈ Z d ,

Remark 2.6
Let Q denote an arbitrary closed cube in R d with |Q| ≥ 1. We put Then λ|m u, p (1) and λ|m u, p (2) are equivalent norms in m u, p . It is obvious that λ|m u, p ≤ λ|m u, p (1) ≤ λ|m u, p (2) . On the other hand, let Q be a cube centered at x 0 with size r ≥ 1. We take a cube Q centered at x 0 with size r + 2. If we choose j in such a way that 2 j−1 < r + 2 ≤ 2 j , then there are at most 2 d dyadic cubes The rest is iterated application of Hölder's inequality.
Proof This follows by the definition immediately.
Proof If u = ∞, then m ∞, p = ∞ in view of Remark 2.6, and the result is well known. So assume 0 < p < u < ∞ now. Let E be a subset of N. We consider the following sequences λ (E) defined by It should be clear that So all the sequences belong to m u, For any cube Q − j,m , we have by construction, Thus m u, p (Z d ) contains a noncountable set of sequences such that the distance between two different elements of it is at least 1.

Remark 2.10
Let us mention briefly that in [16], further (one-dimensional) approaches to weak and generalized Morrey sequence spaces were considered.

Embeddings
We prove our main result about embeddings of different Morrey sequence spaces.
Here we also use and adapt some ideas of our paper [18], cf. the proof of Theorem 3.2 there. A similar construction was used by P. Olsen in [33], cf. the proof of Theorem 10 in [33].
is continuous if and only if the following conditions hold: The embedding (3.1) is never compact.

Proof
Step 1. First we prove the sufficiency of the conditions. If u 1 = u 2 , then p 2 ≤ p 1 , and by the Hölder inequality we get for any j ∈ N 0 and m ∈ Z d , The last inequality implies that So the statement follows from (3.3) and the monotonicity, see Proposition 2.3(ii).
Then straightforward calculation shows that λ|m u 1 , p 1 = 1. On the other hand, So {λ k } k does not belong to m u 2 , p 2 . Substep 2.2. Now we assume that u 1 ≤ u 2 and p 1 u 1 < p 2 u 2 , in particular, p 1 u 1 < 1. For any j ∈ N, we put For convenience let us assume that c p 1 q 1 = 1 (otherwise the argument below has to be modified in an obvious way). For any j ∈ N, we define a sequence λ ( j) = λ ( j) k k in the following way. We assume that k j elements of the sequence equal 1 and the rest is equal to 0. If Q 0,k Q − j,0 , then we put λ ( j) k = 0. Moreover, because of the inequality (3.4), we can choose the elements that equal 1 in such a way that the following property holds: (3.5) and the last sum is equal to k ν if ν = j. Thus Furthermore, the assumption 0 So for any N ∈ N, there exists a number j N ∈ N such that But this immediately implies that However, since we assume that the embedding (3.1) holds, there is a positive constant c > 0 such that In view of the last inequalities (3.6) and (3.7), we get and this leads to a contradiction for large N .
Step 3. The noncompactness of (3.1) immediately follows from Proposition 2.3, and the noncompactness of u 1 → ∞ . Proof This follows immediately from Theorem 3.1. Proof This follows immediately from Theorem 3.1.

Remark 3.4
Let us mention the following essential feature: if 0 < p < u < ∞, that is, we are in the proper Morrey situation, then there is never an embedding into any space r whenever 0 < r < ∞, but we always have m u, p → ∞ , in view of Proposition 2.3(iii) and Corollary 3.3(i).

Remark 3.5
We briefly want to compare Theorem 3.1 with forerunners in [16] and in the parallel setting of Morrey function spaces.
In [16,Prop. 2.4] the one-dimensional counterpart of Theorem 3.1 can be found in the case when (in our notation) 1 ≤ p 2 ≤ p 1 ≤ u 1 = u 2 < ∞, with some discussion about the sharpness of that result. Obviously condition (3.2) is automatically satisfied in this case. The method of their proof in [16] is different from ours.
We turn to function spaces and first consider spaces M u, p (Q) defined on a cube Q, where (1.1) has to be adapted appropriately. Then by a result of Piccinini in [34], see also [35], for 0 < p i ≤ u i < ∞, i = 1, 2, if and only if p 2 ≤ p 1 and u 2 ≤ u 1 .
This result was extended to R d by Rosenthal in [38], reading as So a similar diversity as in the classical L p -setting (spaces on bounded domains versus R d versus sequence spaces p ) is obvious. What is, however, more surprising is the similarity with our result [18,Thm. 3.2] in the context of sequence spaces n s u, p,q appropriate for smoothness Morrey spaces.
In the limiting situation s 1 − d u 1 = s 2 − d u 2 (and in adapted notation), we have shown that if and only if (3.2) and q 1 ≤ q 2 .

A Pre-dual of m u,p
Results concerning (pre-)dual spaces in the setting of Morrey function spaces have some history; we refer to [5,20,58] and, more recently, to [2] in this respect. We rely on the paper [40,Sect. 4], where further discussion can also be found.
(in the sense of equivalent norms). Moreover, the embedding Proof We begin with (4.2). Note that Hence, by definition, which results in 1 → X ( j) u, p and thus finishes the proof of (4.2). Similarly to (4.4), we obtain for the embedding id j , since p > u > 1. Thus id j ≤ 1. Now let m 0 ∈ Z d be fixed, and consider λ 0 = {λ 0 k } k∈Z d given by such that finally id j ≥ 1. This completes the proof of (4.3).
Now we combine the above sequence spaces X ( j) u, p (Z d ) at level j ∈ N 0 as follows.
equipped with the norm where the infimum is taken over all admitted decompositions of λ according to (4.6).

Proposition 4.4 Let
The system {e (n) } n∈Z d forms a normalized unconditional basis in X u, p (Z d ). (iii) X u, p (Z d ) is a separable Banach space.

Proof
Step 1. Let λ ∈ 1 . Then Lemma 4.2 applied with j = 0, in particular (4.2), imply that we obtain an admitted representation of λ in (4.6) choosing λ (0) = λ ∈ X (0) u, p and λ ( j) = 0, j ∈ N. This ensures λ ∈ X u, p and, in view of (4.7) and (4.5), If λ ∈ X u, p , then there exists a decomposition according to (4.6), and we can conclude where we applied (4.3). Taking the infimum over all possible representations according to (4.6), we get by (4.7) that λ| u ≤ λ|X u, p , and thus id : To complete the proof of (i), we have to show the converse inequalities in (4.8).
However, X confirming the latter equality in (4.8). Now we are done, since Step 2. Concerning (ii), one can easily calculate that e (n) |X u, p = 1 and that the system is complete in X u, p . So the statement follows from the trivial inequality |n|≤ ε n λ n e (n) |X u, p ≤ |n|≤ λ n e (n) |X u, p , ε n = ±1, λ n ∈ C, cf., e.g., [19,Theorem 6.7].
Step 3. The proof of (iii) is standard.
Taking the infimum over all representations of λ, we get On the other hand, if f ∈ (X u, p (Z d )) , then where f = sup λ|X u, p =1 | f (λ)|, as usual. For any dyadic cube Q −ν,m , ν ∈ N 0 , we take .
By duality for the p spaces, we get {2 Next we define a closed proper subspace of m u, p as follows. Let c 00 denote the finite sequences in C, that is, sequences that possess only finitely many nonvanishing elements. We define m 00 u, p = m 00 u, p (Z d ) to be the closure of c 00 in m u, p , Obviously m 00 u, p is separable. We shall prove below that X u, p is the dual space of m 00 u, p . We begin with some general properties. For that reason, let us denote by m 0 u, p = m 0 u, p (Z d ) the subspace of null sequences which belong to m u, p , Then we have the following basic properties.
Now μ ∈ m 0 u, p ⊂ m u, p and m u, p is complete, thus λ ∈ m u, p . Moreover, applying (4.9) with j = 0 implies that However, μ ∈ m 0 u, p ⊂ c 0 thus leads to λ ∈ c 0 . So finally λ ∈ m u, p ∩ c 0 = m 0 u, p . It remains to verify that m 00 u, p m 0 u, p . First note that, by definition, m 00 u, p ⊆ m 0 u, p . Now consider special lattice points m j = (2 2 j , 0, . . . , 0) ∈ Z d , j ∈ N 0 , and put Then λ ∈ c 0 ∩ m u, p = m 0 u, p , but obviously λ / ∈ m 00 u, p . Remark 4.7 The above result sheds some further light on the difference of the two norms · | ∞ and · |m u, p , since in the classical setting, is well known, in contrast to m 00 u, p m 0 u, p . We need the following lemma.
Proof For any sequence λ ∈ m u, p and any dyadic cube Q, we shall denote by λ| Q the restriction of λ to Q, i.e., (λ| Q ) k = λ k if Q 0,k ⊂ Q and (λ| Q ) k = 0 otherwise. Without loss of generality, we may assume λ ≡ 0.
We choose a positive number ε < 1 − 2 . If λ ∈ m 00 u, p , then there exists a dyadic cube Q such that by the choice of ε. Therefore, and the lemma is proved.
Proof By Proposition 4.5, the space X u, p is a pre-dual space of m u, p . So it is sufficient to show that any functional on m 00 u, p can be represented by some element of X u, p with the equality of norms.
First we prove that the space m 00 u, p can be isometrically embedded into a closed subspace of an appropriate vector valued c 0 space. Let for j ∈ N 0 and m ∈ Z d , A j,m = p (Q − j,m , w j,m ) be a weighted finite-dimensional p space, equipped with the norm We equip c 0 (A j,m ) with the usual norm, i.e., u, p by (4.1). Hence definition (4.6) yields that the sequence μ = ∞ j=0 μ ( j) ∈ X u, p and Remark 4.10 Similar calculations for the Morrey function spaces can be found in [40] and [29]. In the last paper, Köthe dual spaces to Morrey-type spaces generated by a basis of measurable functions are studied. In particular, Theorems 2.1 and 2.2 ibidem are related to our Propositions 3.5 and 3.9.
Moreover, arguments similar to those used in the proof of Proposition 4.9 show that We recall that m u, p (Z d ) → ∞ (Z d ), cf. Proposition 2.3.

Pitt's Compactness Theorem
Now we prove Pitt's theorem for the Morrey sequence spaces. We follow the approach presented in [9] and [15]. The original result reads as follows.
Theorem 5.1 ([37]) Let 1 ≤ q < p < ∞. Every bounded linear operator from p into q or from c 0 into q is compact.
We start with the following lemma that shows the similarity of m 00 u, p (Z d ) to c 0 if p < u. Lemma 5.2 Let 0 < p < u < ∞ and w (n) be a sequence in m 00 u, p (Z d ), which is weakly convergent to zero, w n 0. Then for any λ ∈ m 00 u, p (Z d ), Proof Step 1. First we assume that the sequence λ ∈ m 00 u, p is finite. The sequences λ + w (n) and w (n) belong to m 00 u, p , therefore, according to Lemma 4.8, there exist dyadic cubes Q n and Q n such that By the definition of lim sup there is always a subsequence of cubes {Q n i } i such that lim sup The cubes Q n are dyadic cubes of size at least one; therefore we may assume that the subsequence satisfies one of the following alternative conditions: (1) there exists a dyadic cube Q such that Q n i ⊂ Q for any i, A similar statement holds for the cubes Q n . Please note that the weak convergence of the sequence w (n) to zero implies the uniform convergence to zero of the coordinates of w (n) on any dyadic cube Q. In the next steps we shall denote by Q(λ) the dyadic cube that contains the support of λ. We have that If the sequence of cubes Q n i satisfies the condition (5.4), then once more the sequences λ + w (n i ) k and w (n i ) k coincide for sufficiently large i.
Step 2. The general case is true by the density of finitely supported sequences in m 00 u, p since the norm is a Lipschitzian function. Now we can finally establish Pitt's theorem in our context.
Then any bounded linear operator T from m 00 u, p (Z d ) into q (Z d ) is compact. Proof Due to Proposition 4.4, Lemma 4.8, Proposition 4.9, and Lemma 5.2, we can follow the arguments presented in [9]. We only sketch the proof for the convenience of the reader.
The dual space to m 00 u, p is separable, cf. Proposition 4.4(ii) and Proposition 4.9, so every bounded sequence in m 00 u, p has a weak Cauchy subsequence and T is compact if it is weak-to-norm continuous.
We may assume that T = 1. Let 0 < ε < 1. We choose x ε ∈ m 00 u, p such that x ε |m u, p = 1 and 1 − ε ≤ T (x ε )| q ≤ 1. Let w n 0 in m 00 u, p , and let w (n) |m u, p ≤ M. Lemma 5.2 and the analogous statement for q , cf. [9], imply for t > 0 that This leads to Taking the limit with ε → 0, we have shown that T (w (n) )| q → 0.

Remark 5.4
One can find in the literature results that can be considered as a quantitative version of Pitt's theorem. For example, one can characterize the compactness of the diagonal operator D σ : p → q , 1 ≤ q < p < ∞, in terms of entropy numbers or some s-numbers, cf. [7] or [24] and the references given there. Such estimates are not the subject of our paper. In the next section, we comment a bit on the behavior of the entropy numbers of embeddings in the finite dimensional case.

Finite Dimensional Morrey Sequence Spaces
Finally we shall briefly deal with finite dimensional sequence spaces related to m u, p . We have at least two reasons for doing so. First, in view of Theorem 3.1, there is never a compact embedding between two sequence spaces of Morrey type-whereas any continuous embedding between finite-dimensional spaces is compact. Secondly, when dealing with smoothness Morrey spaces like N s u, p,q or E s u, p,q , for instance, wavelet decompositions usually lead to appropriate sequence spaces that should be studied in further detail. In this spirit it is quite natural and helpful to understand finite sequence spaces of Morrey type better than so far.
For the latter reason we do not consider finite Morrey sequence spaces as general as possible, but only a special 'level' version of it.
where the supremum is taken over all ν ∈ N 0 and m ∈ Z d such that Q −ν,m ⊂ Q − j,0 .

Remark 6.2
Similarly one can define spaces related to any cube Q − j,m , m ∈ Z d , but they are isometrically isomorphic to m 2 jd u, p , so we restrict our attention to the last space. Clearly, for u = p, this space coincides with the usual 2 jd -dimensional space 2 jd p , that is, m 2 jd p, p = 2 jd p . Lemma 6.3 Let 0 < p 1 ≤ u 1 < ∞, 0 < p 2 ≤ u 2 < ∞ and j ∈ N 0 be given. Then the norm of the compact identity operator if p 1 < p 2 and p 2 u 2 > p 1 u 1 . (6.4) Definition 6.5 Let A 1 and A 2 be two complex (quasi-) Banach spaces, k ∈ N, and let T : A 1 → A 2 be a linear and continuous operator from A 1 into A 2 . The k th (dyadic) entropy number e k of T is the infimum of all numbers ε > 0 such that there exist 2 k−1 balls in A 2 of radius ε that cover the image T U 1 of the unit ball U 1 = {a ∈ A 1 : a|A 1 ≤ 1}.
For details and properties of entropy numbers, we refer to [7,10,21,36] (restricted to the case of Banach spaces) and [11] for some extensions to quasi-Banach spaces. Among other features we only want to mention the multiplicativity of entropy numbers: let A 1 , A 2 , and A 3 be complex (quasi-) Banach spaces and T 1 : A 1 −→ A 2 , T 2 : A 2 −→ A 3 two operators in the sense of Definition 6.5. Then e k 1 +k 2 −1 (T 2 • T 1 ) ≤ e k 1 (T 1 ) e k 2 (T 2 ), k 1 , k 2 ∈ N.
Note that lim k→∞ e k (T ) = 0 if and only if T is compact, which explains the saying that entropy numbers measure "how compact" an operator acts.
One of the main tools in our arguments will be the characterization of the asymptotic behavior of the entropy numbers of the embedding N p 1 → N p 2 . We recall it for convenience. For all n ∈ N, we have in the case of 0 < p 1 ≤ p 2 ≤ ∞ that and in case 0 < p 2 < p 1 ≤ ∞, it holds that for all k ∈ N. (6.6) In the case 1 ≤ p 1 , p 2 ≤ ∞, this has been proved by Schütt [45]. For p 1 < 1 and/or p 2 < 1, we refer to Edmunds and Triebel [11] and Triebel [52, 7.2, 7.3] (with a little supplement in [23]). Corollary 6.6 Let j ∈ N, 0 < p i ≤ u i < ∞, i = 1, 2, and k ∈ N 0 with k 2 jd . Then e k (id j : m 2 jd u 1 , p 1 → m 2 jd u 2 , p 2 ) ∼ 2 −k2 − jd 2 Remark 6.7 It will be obvious from the proof below that the assumption for k to be sufficiently large, k 2 jd , is not needed in all cases. But for simplicity we have stated the result above in that setting only.
of Morrey sequence spaces as introduced in this paper (with some forerunner in [16]) and found in this last section that for sufficiently large k ∈ N, k 2 jd , e k id j : m 2 jd u 1 , p 1 → m 2 jd u 2 , p 2 ∼ e k id : 2 jd u 1 → 2 jd u 2 , though the corresponding sequence spaces are quite different.