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The volume of the boundary of a Sobolev extension domain

Abstract

In [J. Funct. Anal, 2008], Hajlasz, Koskela and Tuominen proved a Sobolev (p, p)-extension domain must be Ahlfors regular. By the Lebesgue differentiation theorem, its boundary must be of volume zero. Then a natural question to ask is how about the boundary of a Sobolev (p, q)-extension domain with q<p. We show for q>n-1 (q>=1, when n=2), the boundary of an arbitrary Sobolev (p, q)-extension domain is of volume zero. For q<n-1, we will construct a Sobolev (p, q)-extension domain with the boundary is of positive volume. This talk bases on a joint project with Koskela and Ukhlov.