Conformally Invariant Besov Spaces on Chord-Arc Domains

TL;DR

This paper introduces Besov-type spaces on simply connected domains, explores their relation to domain geometry, characterizes chord-arc domains via these spaces, and shows they inherit conformal invariance.

Contribution

It defines new Besov spaces on domains, characterizes chord-arc domains through space isomorphisms, and demonstrates conformal invariance of these spaces.

Findings

Chord-arc domains are characterized by isomorphisms among the new Besov spaces.

The introduced Besov spaces inherit conformal invariance from classical settings.

The relation between domain geometry and Besov spaces is explicitly studied.

Abstract

Inspired by the classical Besov $p$-space ($1<p<\infty$) defined by means of higher-order derivatives on the upper half-plane, we introduce Besov-type spaces on simply connected domains. We study the relation between the geometric properties of the domain and these spaces, and characterize chord-arc domains in terms of the isomorphisms among these Besov spaces. Furthermore, we obtain that these spaces on chord-arc domains inherit the conformal invariance from the classical setting.