Another extension of Orlicz-Sobolev spaces to metric spaces

Abstract

We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of the Φ-modulus and Φ-capacity. The resulting space NΦ1 is a Banach space. The relationship between NΦ1 and MΦ1 (the first extension defined in Aïssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which NΦ1 is strictly smaller than the Orlicz space LΦ.