Some Entropy Bump Conditions for Fractional Maximal and Integral Operators

TL;DR

This paper introduces entropy bump conditions within a new framework to establish weighted inequalities for fractional maximal and integral operators, advancing the theoretical understanding of these operators in harmonic analysis.

Contribution

It develops a novel entropy bump framework for fractional operators, leveraging recent techniques to prove weighted inequalities more efficiently.

Findings

Established new entropy bump conditions for fractional operators

Proved weighted inequalities using the entropy bounds framework

Enhanced understanding of fractional maximal and integral operators

Abstract

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil--Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.