Pointwise behaviour of Orlicz–Sobolev functions

Abstract

We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space \(L^{\Psi}(X)\) , then each Orlicz–Sobolev function can be approximated by a Hölder continuous function both in the Lusin sense and in norm.