Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

TL;DR

This paper introduces a generalized local Lipschitz number to characterize Sobolev functions and BV functions, establishing new links between differentiability, quasiconformal mappings, and metric space analysis.

Contribution

It proposes a new generalized Lipschitz number that unifies various concepts in analysis and geometry, extending classical theorems to broader contexts.

Findings

Characterizes Sobolev and BV functions using the generalized Lipschitz number.

Establishes connections between differentiability, quasiconformal mappings, and metric measure spaces.

Provides new tools for analyzing regularity and geometric properties of functions.

Abstract

We introduce a generalized version of the local Lipschitz number $\textrm{lip}\,u$, and show that it can be used to characterize Sobolev functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as well as functions of bounded variation. This concept turns out to be fruitful for studying, and for establishing new connections between, a wide range of topics including fine differentiability, Rademacher's theorem, Federer's characterization of sets of finite perimeter, regularity of maximal functions, quasiconformal mappings, Alberti's rank one theorem, as well as generalizations to metric measure spaces.