Logarithmic bump conditions and the two weight boundedness of Calder\'on-Zygmund operators

TL;DR

This paper establishes new two-weight boundedness conditions for Calderón-Zygmund operators using logarithmic bump conditions, partially resolving longstanding conjectures and providing counterexamples to previous conjectures.

Contribution

It introduces sharp log bump conditions for two-weight inequalities, advancing understanding of Calderón-Zygmund operator boundedness and disproving a conjecture by Muckenhoupt and Wheeden.

Findings

Haar shifts map $L^p(v)$ to $L^p(u)$ with quadratic dependence on complexity.

Two-weight boundedness for all Calderón-Zygmund operators under new conditions.

Disproof of a conjecture on weak-type inequalities for the Hilbert transform.

Abstract

We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calder\'on-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the $A_2$ conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele.