Mixed $A_p$-$A_\infty$ estimates with one supremum

TL;DR

This paper derives new mixed $A_p$-$A_$ bounds for Calder3n-Zygmund operators involving only one supremum, addressing open questions and improving upon previous bounds with logarithmic factors.

Contribution

It introduces novel mixed bounds involving a single supremum for Calder3n-Zygmund operators, resolving longstanding questions and conjectures in harmonic analysis.

Findings

Established bounds with one supremum for $A_p$-$A_$ constants.

Provided answers to open questions and conjectures in the field.

Showed bounds with logarithmic factors can be significantly smaller than previous bounds.

Abstract

We establish several mixed $A_p$-$A_\infty$ bounds for Calder\'on-Zygmund operators that only involve one supremum. We address both cases when the $A_\infty$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, of a conjecture of Hyt\"onen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller than the previously known bounds (both one supremum and two suprema).