On various approaches to Besov-type spaces of variable smoothness

TL;DR

This paper explores different methods for defining Besov spaces with variable smoothness, establishing relationships between various formulations based on convolutions and other parameters.

Contribution

It introduces a connection between two types of Besov spaces of variable smoothness, expanding understanding of their structure and properties.

Findings

Established a relation between $B^{oldsymbol{ ho}}_{p,q}$ and $ ilde{B}^{l}_{p,q,r}$ spaces.

Provided new insights into the structure of Besov spaces with variable smoothness.

Enhanced the theoretical framework for analyzing function spaces with non-uniform smoothness.

Abstract

The paper is concerned with Besov spaces of variable smoothness $B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the spaces $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$, which were introduced earlier by the author.