On the extension of Muckenhoupt weights in metric spaces

TL;DR

This paper generalizes Wolff's theorem on extending Muckenhoupt weights from subsets to entire metric measure spaces with doubling measures, providing a complete proof and related estimates.

Contribution

It offers a complete, self-contained proof of the extension theorem for Muckenhoupt weights in metric spaces, extending prior Euclidean results.

Findings

Extension of Muckenhoupt weights in metric measure spaces

Estimates for weights on Whitney chains in metric spaces

Generalization of Wolff's theorem to broader settings

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.