Ein Kriterium für die Approximierbarkeit von Funktionen aus sobolewschen Räumen durch glatte Funktionen

Abstract

The present paper provides a necessary and sufficient criterion for an element of a Sobolev space W ^p_k (Ω) to be approximated in the Sobolev norm by C^k(Eⁿ)-smooth functions. Here Ω is a bounded open set of n-dimensional Euclidean space Eⁿ with convex closure$\bar \Omega$ and boundary ∂Ω having n-dimensional Lebesgue measure zero. No further boundary regularity (such as e.g. the segment property) is required.Our main tools are the Hardy-Littlewood maximal functions and a slightly strengthened version of a well-known extension theorem of Whitney.This work was inspired by and is very close in spirit to the pertinent parts of Calderon-Zygmund [6].