Matrix weighted estimates on spaces of homogeneous type

TL;DR

This paper extends matrix weighted estimates to spaces of homogeneous type, including endpoint and strong type estimates, by generalizing convex body domination results and applying them to vector-valued operators.

Contribution

It introduces new matrix weighted estimates on spaces of homogeneous type, extending convex body domination techniques and providing a $T(1)$-type result for this setting.

Findings

Extended convex body domination to homogeneous spaces

Established endpoint and strong type matrix estimates

Applied results to vector-valued Petermichl operators

Abstract

In this paper matrix quantitative weighted estimates on spaces of homogeneous type, such as endpoint estimates, strong type estimates are provided. To that end we extend some earlier results on convex body domination due to Nazarov, Petermichl, Treil and Volberg to this setting. We also provide a $T(1)$ alike convex body domination result analogous to the one provided by Lerner and Ombrosi, and an application to vector valued extensions of Petermichl operators.