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arXiv · 1311.6025

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

Rodrigo Banuelos·Adam Osekowski
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Abstract

We study classical weighted $L^p\to L^q$ inequalities for the fractional maximal operators on $\R^d$, 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 ``$A_{p-\varepsilon}$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Pérez and Torres.

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Rodrigo Banuelos, Adam Osekowski. 2013-11-23. Weighted norm inequalities for fractional maximal operators--a Bellman function approach. https://arxiv.org/abs/1311.6025

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