Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

Abstract

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents Fp(·),q(·)sfalse(·false),ϕfalse(Rnfalse). These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents Fp(·),q(·)s(·)false(Rnfalse) as well as the classical Triebel–Lizorkin spaces Fp,qsfalse(Rnfalse), even the case when p=∞. Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces Fp,qs,τfalse(Rnfalse) with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.

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

“…(i) These spaces were introduced by Wu et al in [37], where the authors have proved the independence of the spaces on the admissible pair. (ii) In the particular case of w j (⋅) = 2 js(⋅) , with s ∈ C log loc (ℝ n ) , we recover A s(⋅), p(⋅),q(⋅) (ℝ n ) , A ∈ {B, F} , introduced in [38,39] and also investigated in [12]. (iii) When ≡ 1 , then A w, p(⋅),q(⋅) (ℝ n ) = A w p(⋅),q(⋅) (ℝ n ) , A ∈ {B, F} , are the 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents.…”

“…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 [12,Theorem 3.2], where the spaces F s(⋅), p(⋅),q(⋅) (ℝ n ) were considered. As noticed in [12,Remark 3.9], the proof can easily be adapted for the more general scale F w, p(⋅),q(⋅) (ℝ n ) .…”

“…Part (ii) is an extension of [12,Theorem 3.2], where the spaces F s(⋅), p(⋅),q(⋅) (ℝ n ) were considered. As noticed in [12,Remark 3.9], the proof can easily be adapted for the more general scale F w, p(⋅),q(⋅) (ℝ n ) . In a similar way, one can prove the result for the variable 2-microlocal Besov-type spaces B w, p(⋅),q(⋅) (ℝ n ) .…”

“…We start by presenting the result for the spaces F w, 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 [12,Theorem 4.10]. Due to the use of an admissible weight sequence w ∈ W 1 , 2 (ℝ n ) in our case, we only have to do some minor adjustments.…”

“…As for the characterization via non-smooth atoms, recently in [12] the authors proved a result for the spaces F s(⋅), 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, p(⋅),q(⋅) (ℝ n ) and also complete this study by obtaining the counterpart for the Besov scale B w, p(⋅),q(⋅) (ℝ n ) .…”