Entropy conditions in two weight inequalities for singular integral operators

TL;DR

This paper introduces a novel entropy-based bumping method for weights in two-weight inequalities for singular integral operators, leading to stronger results than previous bump conjecture solutions.

Contribution

It develops a new entropy bumping technique that surpasses Orlicz norm bumping, providing improved results for two-weight inequalities and one-sided bumping conjectures.

Findings

Established entropy bumping as a stronger alternative to Orlicz bumping.

Proved the bump conjecture under new entropy conditions.

Extended results to non-homogeneous settings.

Abstract

The new type of "bumping" of the Muckenhoupt $A_2$ condition on weights is introduced. It is based on bumping the entropy integral of the weights. In particular, one gets (assuming mild regularity conditions on the corresponding Young functions) the bump conjecture, proved earlier by A. Lerner and independently by Nazarov--Reznikov--Treil--Volberg, as a corollary of entropy bumping. But our entropy bumps cannot be reduced to the bumping with Orlicz norms in the solution of bump conjecture, they are effectively smaller. Henceforth we get somewhat stronger result than the one that solves the bump conjecture. New results concerning one sided bumping conjecture are obtained. All the results hold in the general non-homogeneous situation.