On 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: $$\underset{t>0}{\sup }tw\left\{x\in \mathbb{R}\mid |Tf(x)|>t\right\}\le C{\int }_{\mathbb{R}}|f|Mw(x)dx.$$ (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.

14 pages, 2 figures, corrected typos