Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces

In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel-Lizorkin-type spaces with variable exponents $B^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu at al. and cover not only variable 2-microlocal Besov and Triebel-Lizorkin spaces $B^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$, but also the more classical smoothness Morrey spaces $B^{s, \tau}_{p,q}(\mathbb{R}^n)$ and $F^{s,\tau}_{p,q}(\mathbb{R}^n)$. Afterwards, we state a pointwise multipliers assertion for this scale.


Introduction
The introduction of function spaces with variable integrability, also known as variable exponent function spaces L p(·) (R n ), goes back to Orlicz [27] in 1931. However, only several decades later they were substantially studied, in the papers [21] of Kováčik and Rákosník, as well as [9] of Edmunds and Rákosník and [5] of Diening. The spaces L p(·) (R n ) have several applications, such as in fluid dynamics, image processing, PDEs and variational calculus. For an overview we refer to [6].
The merger of the concepts of variable integrability and variable smoothness was done by Diening, Hästö and Roudenko in [7], where the authors defined Triebel-Lizorkin spaces with variable exponents F s(·) p(·),q(·) (R n ). The interplay between the three parameters s, q and p can, easily and interestingly, be verified on the trace theorem on R n−1 proved by these authors [7,Theorem 3.13]. This interaction is also clear on the Sobolev embedding results obtained for these spaces by Vybíral in [36].
For the Besov spaces it is not so easy to have also the parameter q as a variable one. Almeida and Hästö introduced in [3] Besov spaces B s(·) p(·),q(·) (R n ) with all three indices variable, using for that a different modular which already uses the variable structure on q(·). They proved the Sobolev and other usual embeddings in this scale.
A more general approach to spaces of variable smoothness are the so-called 2-microlocal function spaces, where the smoothness gets measured by a weight sequence w = (w j ) j∈N 0 . Besov spaces with such weight sequences appeared first in the works of Peetre [28] and Bony [4]. The variable 2-microlocal Besov and Triebel-Lizorkin spaces B w p(·),q (R n ) and F w p(·),q(·) (R n ) were introduced by Kempka in [16,17]. Since then, several authors have devoted some attention to these spaces, expanding the knowledge about their properties. We mention [1,2,[10][11][12][13]18,19,23].
Function spaces with variable exponents represent a kind of approach that generalizes classical function spaces. However, there are different approaches, which also lead us to generalized Besov and Triebel-Lizorkin spaces. The Besov-type spaces B s,τ p,q (R n ) and the Triebel-Lizorkintype spaces F s,τ p,q (R n ) are an example of that. They were introduced in [37] and, besides the classical Besov and Triebel-Lizorkin spaces, they also cover Triebel-Lizorkin-Morrey spaces introduced by Tang and Xu in [31] and the hybrid functions spaces introduced and studied by Triebel in [33,34], together with their use in heat and Navier-Stokes equations.
In this paper we aim to derive a non-smooth atomic characterization for the spaces B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ). An essential tool here is their characterization via local means, which follows immediately from the characterization by maximal functions. Although this characterization was already considered in [41], now we prove a more general version, which is more in line with the results of this type present in the literature.
As for the characterization via non-smooth atoms, recently in [14] the authors proved a result for the spaces F s(·),φ p(·),q(·) (R n ), which was the first result on this subject for this type of function spaces, even for the case of constant exponents. Now we extend it to the 2-microlocal spaces F w,φ p(·),q(·) (R n ) and also complete this study by obtaining the counterpart for the Besov scale B w,φ p(·),q(·) (R n ). Covered by these results will be also the results obtained in [11] for 2-microlocal variable Besov and Triebel-Lizorkin spaces.
Implicit in the name of this characterizationnon-smooth -is the fact that we replace the usual (smooth) atoms by more general ones, in the sense that they have weaker assumptions on the smoothness. We then show that, also in this case, all the crucial information comparing to smooth atomic decompositions is kept. This modification appeared first in [35], where Triebel and Winkelvoß suggested the use of these more relaxed conditions to define classical Besov and Triebel-Lizorkin spaces intrinsically on domains. More recent is the work [30] of Scharf, where a non-smooth atomic characterization for B s p,q (R n ) and F s p,q (R n ) was derived, using even more general atoms. Here we follow this approach to prove our main result. Moreover, as an application, we provide an assertion on pointwise multipliers for the spaces B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ).

Notation and definitions
We start by collecting some general notation used throughout the paper. As usual, we denote by N the set of all natural numbers, N 0 = N∪{0}, and R n , n ∈ N, the ndimensional real Euclidean space with |x|, for x ∈ R n , denoting the Euclidean norm of x. By Z n we denote the lattice of all points in R n with integer components. For β := (β 1 , · · · , β n ) ∈ Z n , let |β| := |β 1 | + · · · + |β n |. If a, b ∈ R, then a ∨ b := max{a, b}. We denote by c a generic positive constant which is independent of the main parameters, but its value may change from line to line. The expression A B means that A ≤ c B. If A B and B A, then we write A ∼ B.
Given two quasi-Banach spaces X and Y , we write X ֒→ Y if X ⊂ Y and the natural embedding is bounded.
If E is a measurable subset of R n , we denote by χ E its characteristic function and by |E| its Lebesgue measure. By supp f we denote the support of the function f . For each cube Q ⊂ R n we denote its center by c Q and 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 n and r ∈ (0, ∞), we denote by Q(x, r) the cube centered at x with side lenght r, whose sides are parallel to the axes of coordinates.
Given k ∈ N 0 , C k (R n ) is the space of all functions f : R n → C which are k-times continuously differentiable (continuous in k = 0) such that The Hölder space C s (R n ) with index s > 0 is defined as the set of all functions f ∈ C ⌊s⌋ − (R n ) with where ⌊s⌋ − ∈ N 0 and {s} + ∈ (0, 1] are uniquely determined numbers so that s = ⌊s⌋ − + {s} + . If s = 0 we set C 0 (R n ) := L ∞ (R n ). By S (R n ) we denote the usual Schwartz class of all infinitely differentiable rapidly decreasing complex-valued functions on R n and S ′ (R n ) stands for the dual space of tempered distributions. The Fourier transform of f ∈ S (R n ) or f ∈ S ′ (R n ) is denoted by f , while its inverse transform is denoted by f ∨ . Now we give a short survey on variable exponents. For a measurable function p : R n → (0, ∞], let p − := ess inf x∈R n p(x) and p + := ess sup x∈R n p(x).
In this paper we denote by P(R n ) the set of all measurable functions p : R n → (0, ∞] (called variable exponents) which are essentially bounded away from zero. For p ∈ P(R n ) and a measurable set E ⊂ R n , the space L p(·) (E) is defined to be the set of all (complex or realvalued) measurable functions f such that It is known that L p(·) (E) is a quasi-Banach space, a Banach space when p − ≥ 1. If p(·) ≡ p is constant, then L p(·) (E) = L p (E) is the classical Lebesgue space. For later use we recall that L p(·) (E) has the lattice property. Moreover, we have In the setting of variable exponent function spaces it is needed to require some regularity conditions to the exponents. We recall now the standard conditions used.
Definition 2.1. Let g ∈ C(R n ). We say that g is locally log-Hölder continuous, abbreviated g ∈ C log loc (R n ), if there exists c log (g) > 0 such that for all x, y ∈ R n .
We say that g is globally log-Hölder continuous, abbreviated g ∈ C log (R n ), if g is locally log-Hölder continuous and there exists g ∞ ∈ R such that for all x ∈ R n .
Note that all functions in C log loc (R n ) are bounded and if g ∈ C log (R n ) then g ∞ = lim |x|→∞ g(x). Moreover, for g ∈ P(R n ) with g + < ∞, we have that g ∈ C log (R n ) if, and only if, 1/g ∈ C log (R n ). The notation P log (R n ) is used for those variable exponents p ∈ P(R n ) with p ∈ C log (R n ).

Mixed sequence-Lebesgue spaces
We introduce now mixed sequence-Lebesgue spaces and, in the next subsection, we present some properties about them, which will be very useful throughout this work.
Definition 2.2. Let p, q ∈ P(R n ) and E be a measurable subset of R n .
(ii) The space ℓ q(·) (L p(·) (E)) is defined to be the set of all sequences of measurable functions where, for all sequences (g j ) j∈N 0 of measurable functions, with the convention λ 1/∞ = 1 for all λ ∈ (0, ∞).
The following is our convention for dyadic cubes: For j ∈ Z and k ∈ Z n , denote by Q jk the dyadic cube 2 −j ([0, 1) n + k) and x Q jk its lower left corner.
When the dyadic cube Q appears as an index, such as Q∈Q and (· · · ) Q∈Q , it is understood that Q runs over all dyadic cubes in R n .
For the function spaces under consideration in this paper, the following modified mixed-Lebesgue sequence spaces are of special importance.
(i) We denote by ℓ φ q(·) (L p(·) ) the set of all sequences (g j ) j∈N 0 of measurable functions on R n such that where the supremum is taken over all dyadic cubes P in R n .
(ii) We denote by L φ p(·) (ℓ q(·) ) the set of all sequences (g j ) j∈N 0 of measurable functions on R n such that where the supremum is taken over all dyadic cubes P in R n .

Auxiliary results
Although the Hardy-Littlewood maximal operator M t constitutes a great tool in the theory of classical function spaces and also in the scale of variable Lebesgue spaces, it is not, in general, a good instrument in the mixed spaces L p(·) (ℓ q(·) ) and ℓ q(·) (L p(·) ). It was actually proved in [3] and in [7] that this operator is not bounded in these spaces if one considers q non-constant. However, this adversity can be overcome by the use of convolution inequalities involving radially decreasing kernels, namely the so-called η-functions, defined by for ν ∈ N 0 and R > 0.
The next result was proved in [6, Lemma 4.6.3] and shows that the convolution operator is well-behaved in L p(·) (R n ) for p ∈ P log (R n ), when considering radially decreasing integrable functions.
where the implicit constant depends only on n and p.
Remark 2.7. Note that we can use the previous lemma for the η-functions defined above. Namely, taking ψ = η 0,m with m > n, then we have Ψ = η 0,m ∈ L 1 (R n ). Thus, setting The following two results show that the η-functions are well suited for the mixed Lebesguesequence spaces. The first one was proved in [7, Theorem 3.2] and the second goes back to [19,Lemma 10].
holds for every sequence (f ν ) ν∈N 0 of L loc 1 (R n ) functions and constant R > n.
In the next result we state the corresponding counterparts for the modified mixed Lebesgue sequence spaces from Definition 2.4.
then there exists c > 0 such that for all sequences (f ν ) ν∈N 0 ∈ ℓ φ q(·) (L p(·) ) it holds Proof. We will prove part (i), as the second follows similarly. For any given dyadic cube P ∈ Q and any ν ∈ N 0 , we decompose each f ν into the sum for i ∈ N and c P being the center of the cube P . Then we have We claim that, for j = 1, 2, which, together with the arbitrariness of P ∈ Q, allows us to conclude the proof.
Step 1. Let us prove (9) for j = 1. Here we apply Lemma 2.9 with R > n + c log (1/q) and use (7) to obtain as we desired.
Step 2. To estimate I 2 , we start by estimating the convolution appearing here. Note that, for some constant ε satisfying ε > max{0, log 2c1 (φ)} and R − ε > n + c log (1/q). Therefore, using this, Lemma 2.9 and (7), we get which allows us to conclude the proof of (9). Consequently the proof of Lemma 2.10 is complete. Lastly, we present a discrete convolution inequality, which extends [41, Lemma 5.6] slightly.
Then there exist constants C 1 , C 2 > 0, depending on p(·), q(·) and δ, such that and Proof. The inequality (11) was stated and proved in [14,Lemma 3.5]. Therefore, we are left to prove (10). Let us assume that p, q ≥ 1. The extension for all p, q ∈ P(R n ) can be done similarly as in Step 2 of the proof of [14, Lemma 3.5] and we omit it here.
Step 1. Firstly we show that there exists some constant c 1 > 0 such that I 1 (P ) ≤ c 1 . Due to the unit ball property, we turn to the modular and note that Let us define for j ∈ N 0 , ν ≥ j + 1 and some 0 < ε < D 1 . We claim that, for each ν ≥ j + 1, the sum ∞ k=j+1 I j,k 1 is not smaller than the infimum in (13). We may assume that this sum is finite. For any δ > 0 we have The claim follows then by the convergence of the second part of the series on the right-hand side and the arbitrariness of δ > 0. Now using this in (13), we have doing a convenient change of variables and with the choice of c 1 = c(ε) c(D 1 − ε). The first part is then proved.
Step 2. We prove now that I 2 (P ) ≤ c 2 , for some c 2 > 0. We proceed similarly as before, and get, for 0 < ε < min{D 2 , D 2 − log 2c1 (φ)}, for the same c(ε) as before. For H 2 (P ), after a proper change of variables, and choosing c 2 ≥c := 2 c(ε) c(D 2 − ε), we have As for H 1 (P ), we proceed similarly. We then get . In the fourth step we have used (7) and the fact that inf λ ∈ (0, ∞) : We then obtain I 2 (P ) ≤ c 2 by considering c 2 ≥ max{c,c}, as we wanted to prove.
Remark 2.13. Naturally, this statement holds also true if the indices k and ν run only over natural numbers.

Variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces
We will present now the definition of the spaces under consideration in this paper. To do this, we start by introducing the notions of admissible weight sequence and admissible pair of functions.
Definition 2.14. Let α ≥ 0 and α 1 , α 2 ∈ R with α 1 ≤ α 2 . A sequence of non-negative measurable functions in R n w = (w j ) j∈N 0 belongs to the class W α α 1 ,α 2 (R n ) if the following conditions are satisfied: (i) There exists a constant c > 0 such that 0 < w j (x) ≤ c w j (y) (1 + 2 j |x − y|) α for all j ∈ N 0 and all x, y ∈ R n .
(ii) For all j ∈ N 0 it holds Such a system (w j ) j∈N 0 ∈ W α α 1 ,α 2 (R n ) is called admissible weight sequence.
Properties of admissible weights may be found in [ and Further, we set ϕ j (x) := 2 jn ϕ(2 j x) for j ∈ N. Then (ϕ j ) j∈N 0 ⊂ S (R n ) and We are finally in a position to introduce variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces.
Remarks 2.17. (i) These spaces were introduced by Wu et al. in [41], where the authors have proved the independence of the spaces on the admissible pair.

Maximal functions and local means characterization
Let (ψ j ) j∈N 0 be a sequence in S (R n ). For each f ∈ S ′ (R n ) and a > 0, the Peetre's maximal functions were defined by Peetre in [28] by In [41,Theorems 4.5 and 4.7], the authors proved a characterization of B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ) using the Peetre's maximal functions, but where the sequence (ψ j ) j∈N 0 is the same as in Definition 2.16, which is built upon an admissible pair. Here we intend to extend those results, by showing that they still hold if one considers more general pairs of functions. Additionally, we also prove that one can replace the admissible pairs in the definition of B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ) by more general ones (in terms of equivalent quasi-norms). The main result of this section reads then as follows. the constant in (7), and let ψ 0 , ψ ∈ S (R n ) be such that and for some ε > 0 and k ∈]1, 2].
for all f ∈ S ′ (R n ).
for all f ∈ S ′ (R n ).  (17) and (18) are the so-called Tauberian conditions. If R = 0 then no moment conditions (16) on ψ are required.
(ii) The case φ ≡ 1 is covered by [1, Theorem 3.1(ii)], where we can find also a discussion on the importance of having a k in the conditions (17) and (18) in contrast with the case k = 2 usually found in the literature in such type of result, cf. e.g. [16,19,29].
(iv) In [41] the authors proved the independence of the spaces from the admissible pair as a consequence of the ϕ-transform characterization. The above theorem provides an alternative proof, since an admissible pair satisfies conditions (17) and (18) with ε = 6 5 and k = 25 18 . Moreover, it becomes clear that B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ) can be defined using more general pairs than the admissible ones (in the sense of equivalent quasi-norms). We refer in particular to the case stated in Corollary 3.4 below. Remark 3.3. The proof of Theorem 3.1 can be carried out following the proof done by Rychkov [29] in the classical case. Part (ii) is an extension of [14,Theorem 3.2], where the spaces F s(·),φ p(·),q(·) (R n ) were considered. As noticed in [14,Remark 3.9], the proof can easily be adapted for the more general scale F w,φ p(·),q(·) (R n ). In a similar way, one can prove the result for the variable 2-microlocal Besov-type spaces B w,φ p(·),q(·) (R n ). In this case, the discrete convolution inequality stated in Lemma 2.10(i) is of great importance.
Lastly in this section we present an important application of Theorem 3.1, that is when ψ 0 and ψ, satisfying (16)- (18), are local means. The name comes from the compact support of ψ 0 := k 0 and ψ := k, which is admitted in the following statement.

Non-smooth atomic decomposition
In [41] the authors obtained a characterization of the spaces B w,φ p(·),q(·) (R n ) and F w,φ p(·),q(·) (R n ) by smooth atomic decompositions, generalizing previous results obtained in [8,17,37,39,40] for the particular cases described in Remark 2.17. We recall the result from [41] and start by defining smooth atoms and appropriate sequence spaces, where we opted here for a different normalization.
Remark 4.2. As usual when L = 0 no moment conditions are required by (21).
(i) The sequence space b w,φ p(·),q(·) (R n ) is defined as the set of all sequences t := {t Q } Q∈Q * ⊂ C such that (ii) Assume p + , q + < ∞. The sequence space f w,φ p(·),q(·) (R n ) is defined as the set of all sequences Next we present the notion of non-smooth atoms already used in [11] in the context of 2-microlocal spaces with variable exponents and which were slightly adapted from [30]. Note that the usual parameters K and L are now non-negative real numbers instead of non-negative integer numbers.
and for every ψ ∈ C L (R n ) it holds Remark 4.5. Since C k (R n ) ֒→ C k (R n ) for k ∈ N 0 , it is clear that condition (23) follows from (20). Moreover, using a Taylor expansion, (24) can be derived from (21)  For the next two auxiliary results we refer to [11,Lemmas 3.6,3.7].
We are now ready to state the main theorems of this section. We start by presenting the result for the spaces F w,φ p(·),q(·) (R n ). However, we won't present the proof here, as it can be carried out in the same way as the proof of [14,Theorem 4.10]. Due to the use of an admissible weight sequence w ∈ W α α 1 ,α 2 (R n ) in our case, we only have to do some minor adjustments.
Step 3. We deal now with part (i). Firstly, as in the proof of the characterization of F s(·),φ p(·),q(·) (R n ) with non-smooth atoms stated in [14,Theorem 4.10], we will apply the local means characterization proved in the previous section. Let us assume that {t Q } Q∈Q * ∈ b w,φ p(·),q(·) (R n ) and that {a Q } Q∈Q * is a family of [K, L]-nonsmooth atoms with K, L ≥ 0 such that (26) holds. Since the convergence of f = Q∈Q * t Q a Q in S ′ (R n ) was already shown in the previous step, we are left to the proof of Without loss of generality, we may assume that t | b w,φ p(·),q(·) (R n ) = 1 and show that f | B w,φ p(·),q(·) (R n ) 1.
Let k j , j ∈ N 0 , be local means as in Corollary 3.4. Then, In what follows, let r ∈ (0, min{1, p − , q − }) such that L > n r − n − α 1 . Substep 3.1. Firstly we show that I 1. For any given P ∈ Q, we split the sum in ν in two parts as follows and prove that I i 1, for i = 1, 2. Observe that I 1 = 0 if j P ≤ 0. Therefore, it suffices to consider the case when j P > 0.
Moreover, note that, by the properties of the admissible weight sequence, the following holds true for all x ∈ R n and j ≥ ν.
We first estimate I 1 . By (30) and Lemma 4.7, we can follow the same procedure as in Step 2 of the proof of [39, Theorem 3.8] and obtain where a > n/r, c P is the center of P and c 0 ∈ N independent of x, P, i, ν and k.
Then there exists a positive number c such that for all ϕ ∈ C ρ (R n ) and all f ∈ F w,φ p(·),q(·) (R n ).