Sobolev Spaces in Metric Spaces

Abstract

We study Sobolev type spaces (called Newtonian spaces) in metric measure spaces equipped with a doubling measure and supporting a p −Poincaré inequality. The Sobolev spaces are defined using the minimal upper gradient which is a substitute of the modulus of the usual gradient. We show that they are the right extension of the usual Sobolev spaces in Rⁿ . In particular Newtonian functions are quasicontinuous and that they are absolutely continues on almost every curve. Moreover, Newtonian functions are continuous on the complement of small sets.