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

Abstract

In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel-Lizorkin-type spaces with variable exponents B w,φ p(•),q(•) (R n ) and F w,φ p(•),q(•) (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 w p(•),q(•) (R n ) and F w p(•),q(•) (R n ), but also the more classical smoothness Morrey spaces B s,τ p,q (R n ) and F s,τ p,q (R n ). Afterwards, we state a pointwise multipliers assertion for this scale.

Variable Besov-type spaces

“…We mention that the variable Triebel-Lizorkin version of our spaces introduced on this paper is given in [13]. Variable Besov-Morrey spaces are given in [2], see [16] and [31] for the variable 2-microlocal Besov-Triebel-Lizorkin-type spaces.…”