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 Ap(.), introduced in [8]. All of our estimates are quantitative, showing the dependence of the exponent function on the Ap(.) characteristic. As an application, we use the reverse Hölder inequality to prove that the matrix Ap(.) weights, introduced in [10], have both a right and leftopenness property. This result is new even in the scalar case.