A two‐weight boundedness criterion on spaces of homogeneous type with its application to some elliptic boundary value problems

Abstract

Let be a space of homogeneous type in the sense of Coifman and Weiss. Namely, is a non‐empty set, is a quasi‐metric on , and is a positive measure satisfying the doubling condition. In this article, the authors establish a general weighted good‐ inequality for a pair of functions on . Using this weighted good‐ inequality, the authors further obtain a two‐weight boundedness criterion for a pair of functions on in the scale of weighted Lebesgue spaces. As an application, the authors prove the two‐weight global gradient estimates for some quasi‐linear elliptic equations on bounded Reifenberg flat domains of .