Self-improving properties of inequalities of poincaré type on s-John Domains

Abstract

We derive weak- and strong-type global Poincaré estimates over s-John domains in spaces of homogeneous type. The results show that Poincaré inequalities over quasimetric balls with given exponents and weights are self-improving in the sense that they imply global inequalities of a similar kind, but with improved exponents and larger classes of weights. The main theorems are applications of a geometric construction for s-John domains together with self-improving results in more general settings, both derived in our companion paper J. Funct. Anal. 255 (2008), 2977-3007. We have reduced our assumption on the principal measure μ to be just reverse doubling on the domain instead of the usual assumption of doubling. While the primary case considered in the literature is p ≤ q, we will also study the case 1 ≤ q < p. © 2011 by Pacific Journal of Mathematics.