Wavelet decomposition and embeddings of generalised Besov-Morrey spaces

We study embeddings between generalised Besov-Morrey spaces. Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov-Morrey spaces into the Lebesgue spaces are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.

Also smoothness function spaces built upon Morrey spaces M u,p (R d ), in particular Besov-Morrey spaces N s u,p,q (R d ), 0 < p ≤ u < ∞, 0 < q ≤ ∞, s ∈ R, were investigated intensively in recent years. Yu.V. Netrusov was the first who combined the Besov and Morrey norms cf. [19]. He considered function spaces on domains and proved some embedding theorem, but the further attention paid to the spaces was motivated first of all by possible applications to PDEs. The Besov-Morrey spaces N s u,p,q (R d ) were introduced by H. Kozono and M. Yamazaki in [12] and used by them to study Navier-Stokes equations. Further applications of the spaces to PDEs can be found e.g. in the papers written by A.L. Mazzucato [15], by L.C.F. Ferreira, M. Postigo [4] or by M. Yang, Z. Fu, J. Sun, [31].
Here we study the Besov spaces N s ϕ,p,q (R d ) built upon generalised Morrey spaces. The spaces were introduced and studied by S. Nakamura, T. Noi and Y. Sawano [18], cf. also [1]. In particular they proved the atomic decomposition theorem for the spaces. In the recent paper [10] M. Izuki and T. Noi investigated the spaces on domains. The generalised Besov-Morrey spaces cover Besov-Morrey spaces and local Besov-Morrey spaces considered by H. Triebel [27] as special cases. Our main aim here is to find the sufficient and necessary conditions for the embeddings N s1 ϕ1,p1,q1 (R d ) ֒→ N s2 ϕ2,p2,q2 (R d ).
Our main tools are the atomic decomposition and the wavelet characterisation. This approach allows us to consider first embeddings on the level of sequence spaces, cf. Theorem 4.1, and afterwards to transfer the result to function spaces, cf. Theorem 5.1. In particular we regain the characterisation of embeddings of Besov-Morrey spaces N s u,p,q (R d ) proved in [7].
The paper is organised as follows. In Section 2 we present some preliminaries. We recall definitions and facts needed later on. In Section 3 we obtain the wavelet characterisation of the generalised Besov-Morrey spaces, cf. Theorem 3.1. Section 4 deals with the sequence spaces n s ϕ,p,q that correspond to N s ϕ,p,q (R d ) via the wavelet characterisation theorem. Theorem 4.1 contains the sufficient and necessary conditions for the embeddings. In the concluding Section 5 we transfer the results to the function spaces. We discuss several concrete examples.

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. Let N d 0 , where d ∈ N, be the set of all multi-indices, α = (α 1 , . . . , α d ) with α j ∈ N 0 and |α| := d j=1 α j . If x = (x 1 , . . . , x d ) ∈ R d and α = (α 1 , . . . , α d ) ∈ N d 0 , then we put x α := x α1 1 · · · x α d d . For a ∈ R, let ⌊a⌋ := max{k ∈ Z : k ≤ a}, ⌈a⌉ = min{k ∈ Z : k ≥ a}, and a + := max(a, 0). Given any u ∈ (0, ∞], it will be denoted by u ′ the number, possible ∞, defined by the expression 1 u ′ = (1 − 1 u ) + ; in particular when 1 ≤ u ≤ ∞, u ′ is the same as the conjugate exponent defined through 1 u + 1 u ′ = 1. All unimportant positive constants will be denoted by C, occasionally the same letter C is used to denote different constants in the same chain of inequalities. 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 . For each cube Q ⊂ R d we denote its side length by ℓ(Q), and, for a ∈ (0, ∞), we denote by aQ the cube concentric with Q having the side length aℓ(Q). For x ∈ R d and r ∈ (0, ∞) we denote by Q(x, r) the compact cube centred at x with side length r, whose sides are parallel to the axes of coordinates. We write simply Q(r) = Q(0, r) when x = 0. By Q we denote the collection of all dyadic cubes in R d , namely, In this paper we consider generalised Morrey spaces where the parameter u is replaced by a function ϕ according to the following definition.
Remark 2.2. The above definition goes back to [17]. When ϕ(t) = t d u for t > 0 and 0 < p ≤ u < ∞, then M ϕ,p (R d ) coincides with M u,p (R d ), which in turn recovers the Lebesgue space L p (R d ) when u = p. In the definition of · | M ϕ,p (R d ) balls or all cubes with sides parallel to the axes of coordinates can be taken. This change leads to equivalent quasi-norms. Note that for ϕ 0 ≡ 1 (which would correspond to For M ϕ,p (R d ) it is usually required that ϕ ∈ G p , where G p is the set of all nondecreasing functions ϕ : (0, ∞) → [0, ∞) such that ϕ(t)t −d/p is a nonincreasing function, i.e., A justification for the use of the class G p comes from the lemma below, cf. e.g. [18,Lemma 2.2]. One can easily check that G p2 ⊂ G p1 if 0 < p 1 ≤ p 2 < ∞. 18,24]). Let 0 < p < ∞ and ϕ : (0, ∞) → [0, ∞) be a function satisfying ϕ(t 0 ) = 0 for some t 0 > 0.
in the sense of equivalent (quasi-)norms.
if and only if there exists some C > 0 such that for all t > 0, ϕ 1 (t) ≤ Cϕ 2 (t). The argument can be immediately extended to 0 < p 2 ≤ p 1 < ∞.
We consider the following examples.
with 0 < u, v < ∞ belongs to G p with p = min(u, v). In particular, taking u = v, the function ϕ(t) = t d u belongs to G p whenever 0 < p ≤ u < ∞.
(iv) The function ϕ(t) = t d/u (log(L + t)) a , with L being a sufficiently large constant, belongs to G u if 0 < u < ∞ and a ≤ 0. Other examples can be found e.g. in [24,Ex. 3.15].
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 f ∈ S ′ (R d ), we use F f to denote its Fourier transform, and F −1 f for its inverse. Now let us define the generalised Besov-Morrey spaces introduced in [18].
Let η 0 , η ∈ S(R d ) be nonnegative compactly supported functions satisfying Definition 2.6. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and ϕ ∈ G p . The generalised Besov-Morrey space N s ϕ,p,q (R d ) is defined to be the set of all f ∈ S ′ (R d ) such that with the usual modification made in case of q = ∞.
Remark 2.7. The above spaces have been introduced in [18]. There the authors have proved that those spaces are independent of the choice of the functions η 0 and η considered in the definition, as different choices lead to equivalent quasi-norms, cf. [18,Thm 1.4]. When ϕ(t) = t d u for t > 0 and 0 < p ≤ u < ∞, then are the usual Besov-Morrey, which are studied in [32] or in the recent survey papers by W. Sickel [25,26]. Of course, we can recover the classical Besov spaces B s p,q (R d ) for any 0 < p < ∞, 0 < q ≤ ∞, and s ∈ R, When ϕ(t) = min(t d u , 1), then we recover the local Besov-Morrey spaces introduced by H. Triebel, when ϕ satisfies the additional condition for some constants ε > 0 and c > 0. This is a consequence of Corollary 6.17 of [18].

The atomic decomposition
An important tool in our later considerations is the characterisation of the generalised Besov-Morrey spaces by means of atomic decompositions. We follow [18] and start by defining the appropriate sequence spaces and atoms.
Definition 2.8. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and ϕ ∈ G p . The generalised Besov-Morrey sequence space n s ϕ,p,q (R d ) is the set of all double-indexed sequences λ := {λ j,m } j∈N0,m∈Z d ⊂ C for which the quasi-norm is finite (with the usual modification if q = ∞).
Remark 2.9. When ϕ(t) = t d u for t > 0 and 0 < p ≤ u < ∞, then n s ϕ,p,q (R d ) = n s u,p,q (R d ) are the usual Besov-Morrey sequence spaces. Moreover if u = p, then the space n s ϕ,p,q (R d ) coincides with a classical Besov sequence space b s p,q (R d ) since M ϕ,p (R d ) = L p (R d ) in that case.
for all x ∈ R d and for all α ∈ N d 0 with |α| ≤ K, and when for all β ∈ N d 0 with |β| ≤ L when L ≥ 0. In the sequel we write a j,m instead of a if the atom is located at Q j,m , i.e., supp a j,m ⊂ cQ j,m .
(i) Let f ∈ N s ϕ,p,q (R d ). Then there exists a family {a j,m } j∈N0,m∈Z d of (K, L, c)-atoms and a sequence λ = {λ j,m } j∈N0,m∈Z d ∈ n s ϕ,p,q (R d ) such that (ii) Let {a j,m } j∈N0,m∈Z d be a family of (K, L, c)-atoms and λ = {λ j,m } j∈N0,m∈Z d ∈ n s ϕ,p,q (R d ). Then The next lemma will be useful in the sequel and shows that the sequence spaces n s ϕ,p,q can be defined through a more convenient equivalent norm, extending the result for n s u,p,q from [7, Prop. 3.1]. Lemma 2.12. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and ϕ ∈ G p . Then Proof. For each j ∈ N 0 we calculate the quasi-norm If j < ν, there exists only one m 0 ∈ Z d such that Q = Q ν,k ⊂ Q j,m0 , and, moreover, since ϕ is nondecreasing, we obtain The reverse inequality is clear, from the definition of · | M ϕ,p (R d ) and (2.7). Therefore The result follows from (2.9) taking into account (2.4).

The wavelet characterisation
We assume that the reader is familiar with the basic notation and assertions of the wavelet theory.
There is a variety of excellent books that present general background material on wavelets, we can refer, in particular, to [2], [9] and [30]. We will follow the approach presented in [6] and consider here the compactly supported Daubechies wavelets. Let L ∈ N and let ψ F , ψ M ∈ C L (R) are real-valued compactly supported (L 2 -normalised) functions with The function ψ F is called scaling function (or father wavelet) and ψ M is called an associated function (mother wavelet).
where * indicates that at least one of the components of G must be an M . Then we set We will need the following modified versionñ s ϕ,p,q (R d ) of n s ϕ,p,q (R d ) spaces. The spaceñ s ϕ,p,q (R d ) collects all sequences Theorem 3.1. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and ϕ ∈ G p . For the wavelets defined in (3.2) we take The representation is unique with and Proof.
Step 1. We prove that the theorem follows from Theorem 5.1 in [6]. The space N s ϕ,p,q (R d ) is an (isotropic, inhomogeneous) quasi-Banach function space which satisfies and which can be characterised in terms of an L-atomic decomposition with L = K, cf. Theorem 2.11. Please note that the inequality L > d p − s implies L > σ p − s. So it is sufficient to prove that the sequence spaceñ s ϕ,p,q (R d ) is a κ-sequence space for some κ, 0 < κ < L, cf. Definition 4.1 in [6]. One can easily check that it is sufficient to prove that the space n s ϕ,p,q (R d ) is a κ-space, so to simplify the notation we restrict our attention to the space n s ϕ,p,q = n s ϕ,p,q (R d ). Let b > 1 and C 1 > 0. For j, J ∈ N 0 and m, M ∈ Z d we put Note that the cardinalities of #I j J (m) and #Î j J (M ) satisfy We prove that there exists κ, 0 < κ < L, such that (i) for any b > 1, C 1 > 0, and all µ ∈ n s ϕ,p,q , any sequence λ = {λ j,m } with belongs to n s ϕ,p,q and satisfies λ|n s ϕ,p,q ≤ C µ|n s ϕ,p,q ; Step 2. We prove the property (i) for 0 < p ≤ 1. We decompose the sum in (3.6) into two parts for J ≤ j and for J > j. Let ν ∈ Z and k ∈ Z d be fixed, with ν ≤ j. Then where the last inequality follows from (3.5) and the last but one follows from the definition of the set If q p > 1, then for any ε > 0 we get, using the Hölder inequality, So in both cases we have The first inequality follows from (3.10) and (3.11) and the fact that any k appoints one ℓ = ℓ(k), so the supremum over k can be dominated by the supremum over ℓ. The second inequality follows by rescaling.
In consequence, This is always possible if κ > max{σ p − s, s}, that is, we need L > max{σ p − s, s} here which is implied by (3.4). This finishes the proof of (i) for 0 < p ≤ 1.
Step 3. Now we prove the property (i) for p > 1. Applying the Hölder inequality twice yields for some in view of (3.5), see also [6]. Now using (3.14) and (3.15) and a similar method as in (3.9) we can prove that The rest of the proof goes similarly as in the case p ≤ 1. Now we should choose ε > 0 such that κ − 2ε − σ p + s = κ − 2ε + s > 0 and κ − 2ε − s > 0. In other words, we need κ to satisfy L > κ > |s|, but this is again possible in view of (3.4).
Step 4. The proof of the property (ii) is straightforward. Let Q be some cube, J ∈ N 0 and M ∈ Z d such that Q J,M ⊂ Q. Then we have So the estimate holds with the same constant for any cube Q and κ > ( d p − s) + . In view of (3.4) it is always possible to find κ such that L > κ > ( d p − s) + . This concludes the proof.
Remark 3.2. As in the paper [6] we do not claim the condition in (3.4) to be sharp, the assumption on L is just taken for convenience, following the argument in [6]. Moreover, for our purposes, that is, to transfer our sequence space results from Section 4 to the function space counterparts in Section 5, it is absolutely sufficient to find some number L satisfying (3.4). But we did not care for minimal assumptions.

Embeddings of generalised Besov-Morrey sequence spaces
First we deal with the embeddings of generalised Besov-Morrey sequence spaces n s ϕ,p,q , for the definitions we refer to Section 2. These sequence spaces appear naturally when applying the wavelet decomposition result Theorem 3.1 for generalised Besov-Morrey (function) spaces.
Step 1. First we consider the sufficiency of the conditions (4.2)-(4.3). Please note that it follows from (4.2) that the supremum defining α j is finite, so the sequence (α j ) j is well defined. We start by proving some inequalities for any fixed j ∈ N 0 . If p 2 ≤ p 1 , i.e., ̺ = 1, then we have the following inequality sup ν:ν≤j k∈Z d Indeed, for any ν ≤ j we have where the first inequality follows by Hölder's inequality. Taking the supremum over ν ≤ j and k ∈ Z d we get (4.4).
Step 3. It remains to prove the necessity of the conditions. First we prove that the embedding (4.1) implies (4.2).
Substep 3.1 We fix j 0 ≥ 0, ν 0 ≤ j 0 and consider the sequence λ (j0,ν0) defined as follows Then The function ϕ 1 belongs to the class G p1 therefore In a similar way we prove that So if the embedding (4.1) holds, then Moreover the constant C is independent of j 0 and ν 0 . So if p 1 ≥ p 2 , i.e., ̺ = 1, we can fix j 0 = 0. This proves (4.2).

(4.19)
We group the elements λ j0,m of the sequence in such a way that exactly N (1) ki elements related to the cube Q ν0+1,ki are equal to 1.
Next we repeat the procedure for any cube Q ν0+1,ki . We define M j0,m = 1 for at most one cube Q j0,m in any cube Q ν0+2,k ⊂ Q ν0+1,ki in such a way that we do not exceed the total number N  In the next steps we define M ki,kj ⌉ and so on. The procedure stops after at most 2 (j0−ν0) steps. One can easily see that for any ν, ν 0 < ν = ν 0 + η < j 0 we have (4.21) Now we take j 0 = 0 and since ν 0 < ν ≤ 0, ϕ 1 ∈ G p1 and ϕ 1 (1) = 1. The constant C is independent of ν 0 . In consequence λ (N ) |n s1 ϕ1,p1,q1 ≤ C. So if the embedding (4.1) holds, then proving (4.2) when p 1 < p 2 .
Moreover, since we recover exactly the conditions for classical Besov-Morrey sequence spaces in Theorem 3.2 of [7].
(b) Besides the 'classical' example given above, we consider the functions ϕ ui,vi defined by (2.3), 0 < u i , v i < ∞, i = 1, 2. Now one can easily calculate that condition (4.2) is equivalent to v1 v2 ≤ ̺ and the condition (4.3) is equivalent to (4.30) Please note that (4.30) coincides with the conditions formulated in [8] for embeddings of Besov-Morrey spaces defined on bounded domains.
(c) Finally we return to the setting in Example 2.5 (iii), where u i ≥ p i , i = 0, 1. Formally this can be seen as an extension of the previous example to v i = ∞, i = 1, 2. Please note that the sequence spaces correspond via Theorem 3.1 to the local Besov-Morrey spaces, cf. Remark 2.7. Since sup t>1 ϕ 2 (t)/ϕ 1 (t) ̺ = 1, (4.2) is satisfied, thus it remains to deal with the condition (4.3), which leads to (4.30) again.
Next we collect a number of interesting and useful implications of Theorem 4.1.
(i) If p 1 ≥ p 2 , then the embedding (4.1) is continuous if and only if s 1 > s 2 or s 1 = s 2 and q 1 ≤ q 2 .
(ii) If p 1 < p 2 , then the embedding (4.1) is continuous if and only if We recall that b s p,q , s ∈ R, 0 < p, q ≤ ∞, denote the classical Besov sequence spaces, cf. Remark 2.9. Now we extend the above definition to the case p = ∞.
We have the following observation from Lemma 2.12.
Remark 4.6. This can be seen as some sequence space counterpart of Remark 2.2. holds. Proof.
Example 4.10. We explicate Corollary 4.8 for some function ϕ. We restrict ourselves to the situation p 1 < p 2 , i.e., Corollary 4.8 (ii). Let where a ∈ R. Since sup t>0 ϕ(t) < ∞, we deal with the second condition in (4.39), which leads to and a ≥ 0, if q 1 ≤ q 2 , or, and a > 1 We used that in this case j . Now we focus on embeddings where either the target or the source space is a Besov sequence space, for what we recall that b s p,q = n s p,p,q , 0 < p < ∞, 0 < q ≤ ∞, s ∈ R.
Proof. This is the counterpart for function spaces of Corollary 4.8.
Proof. This corresponds to Corollary 4.11.
In combination with the well-known embedding B 0 r,1 (R d ) ֒→ L r (R d ), 1 ≤ r < ∞, we thus obtain from Corollary 5.6 the following result.
Corollary 5.8. Let s i ∈ R, 0 < p i < ∞, 0 < q i ≤ ∞ for i = 1, 2, and ϕ 2 ∈ G p2 . Denote again Remark 5.9. Obviously one can also explicate Theorem 5.1 for the example functions, similar to Examples 4.3, Example 4.10 etc. For instance, we can prove that the formula (4.30) gives sufficient and necessary conditions for the embedding of two local Besov-Morrey spaces.