In this article we study variable exponent Sobolev spaces on metric measure spaces. We employ two definitions: a Hajłasz type definition, which uses a pointwise maximal inequality, and a Newtonian type definition, which uses an upper gradient. We prove that these spaces are Banach, that Lipschitz functions are dense as well as other basic properties. We also study when these spaces coincide.
"Variable exponent Sobolev spaces on metric measure spaces." Funct. Approx. Comment. Math. 36 79 - 94, January 2006. https://doi.org/10.7169/facm/1229616443