Quasiconformal, Lipschitz, and BV mappings in metric spaces

TL;DR

This paper investigates generalized Lipschitz and distortion measures for mappings between metric measure spaces, providing conditions under which such mappings belong to BV or Newton-Sobolev classes, extending quasiconformal theory.

Contribution

It introduces generalized local Lipschitz and distortion numbers in metric spaces and establishes new criteria for BV and Newton-Sobolev regularity of mappings.

Findings

Sufficient conditions for BV regularity of mappings.

Criteria for Newton-Sobolev regularity.

Extension of quasiconformal concepts to metric spaces.

Abstract

Consider a mapping $f\colon X\to Y$ between two metric measure spaces. We study generalized versions of the local Lipschitz number $\mathrm{Lip} f$, as well as of the distortion number $H_f$ that is used to define quasiconformal mappings. Using these, we give sufficient conditions for $f$ being a BV mapping $f\in BV_{\mathrm{loc}}(X;Y)$ or a Newton-Sobolev mapping $f\in N_{\mathrm{loc}}^{1,p}(X;Y)$, with $1\le p<\infty$.