Weighted norm inequalities for fractional maximal operators--a Bellman function approach

Abstract

We study classical weighted $L^p\to L^q$ inequalities for the fractional maximal operators on ${\mathbb{R}}^d$, proved originally by Muckenhoupt and Wheeden in the 70's. We establish a slightly stronger version of this inequality with the use of a novel extension of Bellman function method. More precisely, the estimate is deduced from the existence of a certain special function which enjoys appropriate majorization and concavity. From this result and an explicit version of the ``$A_{p-\epsilon }$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Pérez and Torres.