Tietze type extensions for absolutely continuous functions in the plane

TL;DR

This paper investigates the extension problem for absolutely continuous functions in the plane, demonstrating that such extensions are possible for domains composed of polygons and convex curves, with applications to spectral theory.

Contribution

It introduces conditions under which absolutely continuous functions can be extended from certain planar domains, advancing understanding in function extension problems.

Findings

Extensions are possible for domains with polygonal and convex curve components.

The results have implications for the spectral theory of AC(σ) operators.

Provides new methods for function extension in the plane.

Abstract

It is an open problem whether one can always extend an absolutely continuous function (in the sense of Ashton and Doust) on a compact subset of the plane to a larger compact set. In this paper we show that this can be done for a large family of initial domains whose components consist of polygons and convex curves. An application is given to the spectral theory of $AC(\sigma)$ operators.