IUMJ

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

Authors: Rodrigo Banuelos and Adam Osekowski

Issue: Volume 64 (2015), Issue 3, 957-972

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 1970s. 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 that enjoys appropriate majorization and concavity. From this result and an explicit version of the $A_{p-\varepsilon}$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, P\'erez, and Torres.