No 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.

No abstract

We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, id τ : B s 1 ,τ 1 p 1 ,q 1 (Ω) → B s 2 ,τ 2 p 2 ,q 2 (Ω) and id τ : F s 1 ,τ 1 p 1 ,q 1 (Ω) → F s 2 ,τ 2 p 2 ,q 2 (Ω), where Ω ⊂ R d is a bounded domain, and obtain necessary and sufficient conditions for the compactness of id τ. 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.

We provide an intrinsic atomic characterization for 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability on domains, B w p(·),q(·) (Ω) and F w p(·),q(·) (Ω), where Ω is a regular domain. We make use of the non-smooth atomic decomposition result obtained in [12] for these spaces to get the main result. * The authors were supported by the German science foundation (DFG) within the project KE 1847/1-2. F w p(·),q(·) (R n ) as linear combinations of smooth atoms, which are the building blocks for atomic decompositions. More recently, a more general decomposition for these spaces was obtained in [12], where the authors show that one can replace the usual atoms used in smooth atomic decompositions by more general ones, called non-smooth atoms. Those atoms are characterized by a relaxation on the smoothness assumptions and, nevertheless, one keeps all the crucial information compared to smooth atomic decompositions. We devote Section 3 to this topic.Regarding intrinsic characterizations of function spaces on domains, in [32] Triebel and Winkelvoß suggested the use of these non-smooth atoms as a tool to define classical Besov and Triebel-Lizorkin spaces B s p,q (Ω) and F s p,q (Ω) on a class of (non-smooth) domains. Also Rychkov in [27] gave an intrinsic characterization for the same scale of spaces, but considering smooth domains. More recently, Tyulenev in [33] studied Besov-type spaces of variable smoothness on rough domains, namely bounded Lipschitz domains in R n , epigraph of Lipschitz functions or (ǫ, δ)-domains. Concerning 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents, Kempka presented recently in [19] two different intrinsic characterizations of these spaces using local means and the Peetre maximal operator, on special Lipschitz domains.Since a non-smooth atomic characterization for the scale of 2-microlocal Besov and Triebel-Lizorkin spaces B w p(·),q(·) (R n ) and F w p(·),q(·) (R n ) was already obtained, our aim is to get an intrinsic characterization of these spaces for more general domains, as considered in [32]. We deal with this problem in Section 4, where we study spaces on the scale of regular domains. We wish to emphasize that this class of domains includes not only bounded connected Lipschitz domains but also special Lipschitz domains and (ǫ, δ)-domains.