Tietze type extensions for absolutely continuous functions in the plane

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. We show that this can be done for a large family of initial domains whose components consist of polygons and convex curves. The connection between these extension theorems and the spectral theory of $AC(\sigma )$ operators is discussed.