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

In this paper we prove a reverse Hölder 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 Hölder 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.