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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.