On Muckenhoupt–Wheeden Conjecture

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 failssupt>0tw({x∈R:|Tf(x)|>t})⩽C∫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 L2L2 consequence of the inequality above does not hold.