Off-Diagonal Two Weight Bumps for Fractional Sparse Operators

TL;DR

This paper extends two weight boundedness results for fractional sparse operators to off-diagonal cases using entropy and comparison bumps, achieving sharp results with simplified proofs.

Contribution

It introduces new off-diagonal two weight bounds for fractional sparse operators utilizing entropy and comparison bump techniques, with nearly trivial proofs.

Findings

Established sharp off-diagonal two weight bounds

Applied entropy bump techniques for fractional operators

Simplified proofs using current machinery

Abstract

In this paper, we continue some recent work on two weight boundedness of sparse operators to the "off-diagonal" setting. We use the new "entropy bumps" introduced in by Treil-Volberg ([21]) and improved by Lacey-Spencer ([9]) and the "direct comparison bumps" introduced by Rahm-Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" in the sense that they are sharp in various particular cases. A feature is that given the current machinery, the proofs are almost trivial.