On equicontinuity of homeomorphisms with finite distortion in the plane

TL;DR

This paper investigates conditions under which families of homeomorphisms with finite distortion in the plane are equicontinuous and normal, establishing both sufficiency and necessity of certain integral and oscillation constraints on their distortion functions.

Contribution

It provides a comprehensive characterization of equicontinuity and normality conditions for finite distortion homeomorphisms, including necessary and sufficient criteria involving the distortion function.

Findings

Conditions on $K_f(z)$ ensure equicontinuity and normality.

Necessary and sufficient conditions on $\

contribution

Abstract

It is stated equicontinuity and normality of families $\frak{F}^{\Phi}$ of the so--called homeomorphisms with finite distortion on conditions that $K_{f}(z)$ has finite mean oscillation, singularities of logarithmic type or integral constraints of the type $\int\Phi\left(K_{f}(z)\right)dx\,dy<\infty$ in a domain $D\subset{\C}.$ It is shown that the found conditions on the function $\Phi$ are not only sufficient but also necessary for equicontinuity and normality of such families of mappings.