Intrinsic atomic characterization of 2-microlocal spaces with variable exponents on domains

Abstract

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.

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

“…Since then, several authors have devoted some attention to these spaces, expanding the knowledge about their properties. We mention [1,2,10,11,13,14,18,19,23].…”

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

“…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].…”

Intrinsic Characterization and the Extension Operator in Variable Exponent Function Spaces on Special Lipschitz Domains

“…For variable exponents there are not so many results on intrinsic characterizations and on the extension operator known. An intrinsic characterization for our spaces has been provided in [8] with the help of non smooth atomic characterizations. This approach also works for more general domains than special Lipschitz domains.…”