On Muckenhoupt-Wheeden Conjecture

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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.

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Reguera, M. C. (2011). On Muckenhoupt-Wheeden Conjecture. Advances in Mathematics, 227(4), 1436–1450. https://doi.org/10.1016/j.aim.2011.03.009

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