The sharp $A_p$ constant for weights in a reverse-H\"older class

TL;DR

This paper determines the exact $A_p$ constants for weights in a reverse-H"older class using Bellman functions, extending results to higher dimensions and related integrability properties.

Contribution

It introduces a Bellman function approach to find sharp $A_p$ constants for reverse-H"older weights, including multidimensional bounds and higher-integrability constants.

Findings

Sharp $A_p$ constants for weights in reverse-H"older class on intervals

Bounds for $A_p$ constants on rectangles and cubes in n dimensions

Precise constants for Gehring's higher-integrability result

Abstract

In a recent paper V. Vasyunin presented a proof of the reverse H\"older 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-H\"older class on an interval; we also find the sharp constants for the higher-integrability result of Gehring. Additionally, we find bounds for the $A_p$ constants of reverse-H\"older-class weights defined on rectangles and on cubes in n dimensions.