Matrix Weighted $L^p$ Estimates in the Nonhomogeneous Setting

TL;DR

This paper extends matrix weighted $L^p$ estimates and convex body domination techniques to the nonhomogeneous setting, overcoming the lack of reverse-H"older inequality by using a generalized weighted Carleson embedding theorem.

Contribution

It introduces a modified pointwise convex body domination for vector-valued Haar shifts and extends matrix weighted $L^p$ estimates to nonhomogeneous measures without regularity assumptions.

Findings

Established convex body domination for vector-valued Haar shifts in nonhomogeneous setting

Identified $L^1$-normalized shifts where standard domination holds without regularity

Extended matrix weighted $L^p$ estimates for sparse forms to nonhomogeneous measures

Abstract

We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv:2309.13943. Moreover, we identify a subclass of shifts, called $L^1$-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted $L^p$ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-H\"older inequality for scalar weights, which was used in arXiv:1710.03397 to establish $L^p$ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition.