Sobolev interpolation inequalities on generalized John domains

Seng-Kee Chua

Vol. 242 (2009), No. 2, 215–258

DOI: 10.2140/pjm.2009.242.215

Abstract

We obtain weighted Sobolev interpolation inequalities on generalized John domains that include John domains (bounded or unbounded) for δ-doubling measures satisfying a weighted Poincaré inequality. These measures include ones arising from power weights d(x,∂Ω)^α and need not be doubling. As an application, we extend the Sobolev interpolation inequalities obtained by Caffarelli, Kohn and Nirenberg. We extend these inequalities to product spaces and give some applications on products Ω₁ × Ω₂ of John domains for A_p(ℝⁿ × ℝ^m) weights and power weights of the type w(x,y) = dist(x,G₁)^α dist(y,G₂)^β, where G₁ ⊂ ∂Ω₁ and G₂ ⊂ ∂Ω₂. For certain cases, we obtain sharp conditions.

Keywords

δ-balls, δ-doubling, Boman domains, Poincaré inequalities

Mathematical Subject Classification 2000

Primary: 26D10, 46E35

Milestones

Received: 17 April 2008

Revised: 25 May 2009

Accepted: 31 July 2009

Published: 1 October 2009