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

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.