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 $\Phi$-modulus and $\Phi$-capacity. The resulting space $N_{\Phi }^1$ is a Banach space. The relationship between $N_{\Phi }^1$ and $M_{\Phi }^1$ (the first extension defined in Aïssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which $N_{\Phi }^1$ is strictly smaller than the Orlicz space ${\mathbf{L}}_{\Phi }$.