THE SHARP Ap CONSTANT FOR WEIGHTS IN A REVERSE-HÖLDER CLASS

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 [17] 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 Ap constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [7]. Additionally, we find sharp bounds for the Ap constants of reverse-Hölderclass weights defined on rectangles in R, as well as bounds on the Ap constants for reverse-Hölder weights defined on cubes in R, without claiming the sharpness.