APPROXIMATION BY LIPSCHITZ FUNCTIONS IN ABSTRACT SOBOLEV SPACES ON METRIC SPACES

We prove that the density of locally Lipschitz functions in a global Sobolev space based on a Banach function space implies the density of Lipschitz functions, with compact support in a given open set, in the corresponding Sobolev space with zero boundary values. In the case, when the Banach function space is a Lebesgue space, we recover some density results of Bjorn, Bjorn and Shanmugalingam (2008). Our results require neither a doubling measure nor the validity of a Poincare inequality in the underlying metric measure space.