Muckenhoupt-Wheeden conjectures in higher dimensions

TL;DR

This paper extends recent disproofs of Muckenhoupt-Wheeden conjectures from one-dimensional to higher-dimensional settings, demonstrating that certain weighted inequalities for Calderón-Zygmund operators do not hold.

Contribution

It constructs weights in higher dimensions that show the conjectures are false for classical Calderón-Zygmund operators, generalizing previous one-dimensional results.

Findings

Construction of weights in higher dimensions with large Hilbert transform action

Disproof of Muckenhoupt-Wheeden conjectures in higher dimensions

Extension of one-dimensional counterexamples to multiple dimensions

Abstract

In recent work by Reguera and Thiele and by Reguera and Scurry, two conjectures about joint weighted estimates for Calder\'on-Zygmund operators and the Hardy-Littlewood maximal function have been refuted in the one-dimensional case. One of the key ingredients for these results is the construction of weights for which the action of the Hilbert transform is substantially bigger than that of the maximal function. In this work, we show that a similar construction is possible for classical Calder\'on-Zygmund operators in higher dimensions. This allows us to fully disprove the conjectures.