The estimation of the area of a disk image for Sobolev classes

TL;DR

This paper establishes area estimates for disk images under Sobolev class homeomorphisms with the Luzin N-property, extending classical results like the Ikoma-Schwartz lemma to this setting.

Contribution

It provides new area bounds for Sobolev homeomorphisms with the Luzin N-property, generalizing classical geometric function theory results.

Findings

Derived area estimates for Sobolev homeomorphisms

Extended the Ikoma-Schwartz lemma to Sobolev class mappings

Connected angular dilatation with disk image area

Abstract

For regular homeomorphisms of Sobolev class $W^{1,1}_{\textrm{loc}}$ having the Luzin $N$-property, it is established the estimation of the area of a disk image in terms of an angular dilatation. As a corollary, the analog of the well-known Ikoma-Schwartz lemma for such mappings is obtained.