Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications

TL;DR

This paper investigates atomic and molecular decompositions in variable exponent 2-microlocal Besov and Triebel--Lizorkin spaces, establishing convergence properties, denseness of Schwartz functions, and deriving Sobolev embeddings.

Contribution

It provides new results on convergence in these decompositions and proves the denseness of Schwartz class functions in variable exponent 2-microlocal spaces.

Findings

Convergence of decompositions in the spaces themselves

Denseness of Schwartz class functions

Sobolev embedding properties

Abstract

In this article we study atomic and molecular decompositions in $2$-microlocal Besov and Triebel--Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spaces. Some other properties, like Sobolev embeddings, are also obtained via atomic representations.