Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity

TL;DR

This paper investigates how Triebel-Lizorkin regularity of functions is preserved under bi-Lipschitz transformations and examines the regularity of inverse functions in Lipschitz domains, providing new norm equivalences.

Contribution

It introduces an equivalent norm for Triebel-Lizorkin spaces with fractional smoothness in uniform domains, enhancing understanding of regularity under bi-Lipschitz maps.

Findings

Triebel-Lizorkin regularity is stable under bi-Lipschitz transformations.

Provided an equivalent norm for Triebel-Lizorkin spaces in uniform domains.

Established regularity results for inverse functions of bi-Lipschitz maps.

Abstract

We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.