Sobolev Spaces on Metric Spaces (axiomatic approach)
by
V. Goldshtein
Ben Gurion University, Beer Sheva, Israel
Coauthors: M.Troyanov, EPFL, Lausanne, Switzerland

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples.We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic metric spaces are then discussed as a weak form of embedding theorems for the axiomatic Sobolev spaces.

We discuss also the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove results about quasi-continuity of functions that belongs to the axiomatic Sobolev spaces.

Date received: August 14, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caia-10.