The reverse H\"older inequality for $\mathcal{A}_{p(\cdot)}$ weights with applications to matrix weights

TL;DR

This paper establishes a reverse H"older inequality for variable exponent Muckenhoupt weights and applies it to demonstrate openness properties of matrix weights, advancing understanding in weighted harmonic analysis.

Contribution

It proves a quantitative reverse H"older inequality for $\, ext{A}_{p(\, ext{·})}\, ext{weights}$ and shows new openness properties for matrix weights, including scalar cases.

Findings

Quantitative reverse H"older inequality for $\, ext{A}_{p(\, ext{·})}\, ext{weights}$

Openness properties of matrix $ ext{A}_{p(\, ext{·})}$ weights

Results are new even for scalar weights

Abstract

In this paper we prove a reverse H\"{o}lder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse H\"{o}lder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.