Continuous characterization of Besov spaces of variable smoothness and integrability

TL;DR

This paper introduces new equivalent norms for Besov spaces with variable smoothness and integrability, utilizing advanced tools like Calderon reproducing formulas and variable exponent techniques, addressing challenges from parameter variability.

Contribution

It develops novel equivalent norms for Besov spaces with variable parameters, overcoming difficulties through regularity assumptions and advanced analytical tools.

Findings

New equivalent norms for variable Besov spaces

Effective handling of parameter variability with regularity conditions

Enhanced understanding of Besov space structure

Abstract

In this paper the authors obtain a new equivalent norms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of Calderon reproducing formula, maximal inequalities and variable exponent technique, but allowing the parameters to vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on these exponents.