Besov-Type Spaces with Variable Smoothness and Integrability

TL;DR

This paper introduces and characterizes Besov-type spaces with variable smoothness and integrability, providing atomic, transform, and embedding descriptions, and establishing a trace theorem for these spaces.

Contribution

The paper develops a comprehensive framework for Besov-type spaces with variable parameters, including their characterizations and a trace theorem, advancing the understanding of variable smoothness function spaces.

Findings

Characterization via $ ext{phi}$-transforms, atoms, and maximal functions

Sobolev-type embedding for variable Besov spaces

Trace theorem for these variable spaces

Abstract

In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $\varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.