The sharp $A_p$ constant for weights in a reverse-Hölder class

Abstract

Coifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Hölder inequality". In a recent paper V. Vasyunin [The exact constant in the inverse Hölder inequality for Muckenhoupt weights. St. Petersburg Math. J. 15 (2004), no. 1, 49-79] presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp $A_p$ constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [The $L_p$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277]. Additionally, we find sharp bounds for the $A_p$ constants of reverse-Hölder-class weights defined on rectangles in ${\mathbb{R}}^n$, as well as bounds on the $A_p$ constants for reverse-Hölder weights defined on cubes in ${\mathbb{R}}^n$, without claiming the sharpness.