On the extension of Muckenhoupt weights in metric spaces

Abstract

A theorem by Wolff states that weights defined on a measurable subset of ${\mathbb{R}}^n$ and satisfying a Muckenhoupt-type condition can be extended into the whole space as Muckenhoupt weights of the same class. We give a complete and self-contained proof of this theorem generalized into metric measure spaces supporting a doubling measure. Related to the extension problem, we also show estimates for Muckenhoupt weights on Whitney chains in the metric setting.

20 pages. We have rewritten part of the introduction, as well as the beginnings of Sections 3 and 4, to state our relation to an earlier result in more explicit terms