On Muckenhoupt-Wheeden Conjecture

TL;DR

This paper demonstrates that a dyadic version of the Muckenhoupt-Wheeden Conjecture is false by constructing a counterexample involving a Haar multiplier and a specific weight, challenging previous assumptions in weighted inequalities.

Contribution

It provides a counterexample showing the failure of a dyadic Muckenhoupt-Wheeden type inequality, clarifying limitations of current weighted inequality techniques.

Findings

The weak-type inequality fails for certain weights and Haar multipliers.

A specific $L^2$ consequence of the inequality does not hold.

The result disproves the dyadic Muckenhoupt-Wheeden Conjecture.

Abstract

Let M denote the dyadic Maximal Function. We show that there is a weight w, and Haar multiplier T for which the following weak-type inequality fails: $$ \sup_{t>0}t w\left\{x\in\mathbb R \mid |Tf(x)|>t\right\}\le C \int_{\mathbb R}|f|Mw(x)dx. $$ (With T replaced by M, this is a well-known fact.) This shows that a dyadic version of the so-called Muckenhoupt-Wheeden Conjecture is false. This accomplished by using current techniques in weighted inequalities to show that a particular $L^2$ consequence of the inequality above does not hold.