Self-improving properties of inequalities of Poincaré type on s-John domains

Seng-kee Chua and Richard L. Wheeden

Vol. 250 (2011), No. 1, 67–108

DOI: 10.2140/pjm.2011.250.67

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.

Keywords

global Poincaré estimates, domains with cusps, δ-doubling, reverse doubling, power-type weights, quasimetric spaces

Mathematical Subject Classification 2000

Primary: 26D10, 46E35

Milestones

Received: 12 November 2009

Revised: 4 October 2010

Accepted: 16 November 2010

Published: 1 March 2011

Proposed: Jie Qing