Two weight bump conditions for matrix weights

TL;DR

This paper extends two weight bump conditions to matrix weights and establishes new inequalities for various operators, providing quantitative estimates and applications to PDE-related inequalities.

Contribution

It introduces matrix weight inequalities for several operators and applies these to derive quantitative one weight estimates and a Poincaré inequality.

Findings

Proved inequalities for fractional maximal operators, fractional and singular integrals, sparse and averaging operators.

Established quantitative one weight estimates based on matrix $A_p$ constants.

Derived a Poincaré inequality relevant to degenerate elliptic PDEs.

Abstract

In this paper we extend the theory of two weight, $A_p$ bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix $A_p$ constant, for singular integrals, and prove a Poincar\'e inequality related to those that appear in the study of degenerate elliptic PDEs.