Boundary Values of Functions of Dirichlet Spaces $L^1_2$ on Capacitary Boundaries

TL;DR

This paper establishes that functions with square integrable gradients can be extended to capacitary boundaries of simply connected plane domains, clarifying boundary behavior and properties of these boundaries in complex analysis.

Contribution

It proves the extension of Sobolev functions to capacitary boundaries and details their properties, enhancing understanding of boundary behavior in conformal and quasi-conformal mappings.

Findings

Functions with square integrable gradients extend to capacitary boundaries.

Capacitary boundary coincides with Euclidean boundary for locally connected domains.

Main properties of the capacitary boundary are thoroughly analyzed.

Abstract

We prove that any weakly differentiable function with square integrable gradient can be extended to a capacitary boundary of any simply connected plane domain $\Omega\ne\mathbb R^2$ except a set of a conformal capacity zero. For locally connected at boundary points domains the capacitary boundary coincides with the Euclidean one. A concept of a capacitary boundary was proposed by V.~Gol'dshtein and S.~K.~Vodop'yanov in 1978 for a study of boundary behavior of quasi-conformal homeomorphisms. We prove in details the main properties of the capacitary boundary. An abstract version of the extension property for more general classes of plane domains is discussed also.