Variable Besov spaces: continuous version

TL;DR

This paper introduces variable Besov spaces using continuous Calderón reproducing formulas, establishing their properties, embeddings, and atomic decompositions, thus expanding the functional analysis framework for variable smoothness spaces.

Contribution

It defines variable Besov spaces via continuous Calderón formulas and proves their independence, embeddings, and atomic decompositions, advancing the theory of variable smoothness function spaces.

Findings

Spaces are well-defined and basis-independent.

Sobolev embeddings are established.

Atomic decomposition is proved.

Abstract

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calder\`on reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We characterize these function spaces by so-called Peetre maximal functions and we obtain the Sobolev embeddings for these function spaces. We use these results to prove the atomic decomposition for these spaces.