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. sup tw({x∈R{double-struck}:|Tf(x)|>t})≥C∫R|f|Mw(x)dx. t≥0 (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 L2 consequence of the inequality above does not hold. © 2011 Elsevier Inc.