A new scale of function spaces characterizing homogeneous Besov spaces

TL;DR

This paper introduces a new scale of function spaces that precisely characterize homogeneous Besov spaces, extending previous work and unifying various related spaces with applications to boundary value problems.

Contribution

It defines and analyzes a new class of function spaces that complete and unify earlier characterizations of homogeneous Besov spaces, including their duality and interpolation properties.

Findings

New function spaces characterize homogeneous Besov spaces.

The spaces include previously studied weighted Z-spaces and relate to weighted tent spaces.

The paper establishes their completeness, duality, embeddings, and interpolation properties.

Abstract

We introduce and study a new scale of function spaces that characterize the homogeneous Besov spaces $\mathrm{\dot B}^{\beta}_{p,q}$, hence completing earlier work by Ullrich. These new spaces include the ones introduced by Barton and Mayboroda, and systematically studied by Amenta under the name of weighted $\mathrm{Z}$-spaces, for the purpose of boundary value problems with $\mathrm{\dot B}^{\beta}_{p,p}$ data. They are the counterparts to the weighted tent spaces with Whitney averages, developed by Huang, and arise as their real interpolants. We describe their functional analytic properties: completeness, duality, embeddings, as well as their real and complex interpolants.