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

Abstract

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.

Minor typos corrected. We corrected the definition of the reverse Holder exponent r, as it was missing a $p_{+}$ power. The constant $C_{\ast }$ was missing a dependence on the Diening constant $C_D$. We removed the incorrect dependence on $d$ in Lemma 5.10. We clarified where the correct constant in Lemma 5.10 comes from by more clearly stating the constant in Proposition 5.8