On Muckenhoupt-Wheeden Conjecture

Details On Muckenhoupt-Wheeden Conjecture On Muckenhoupt-Wheeden Conjecture

2010-08-24

arxiv.org/abs/1008.3943v2

Mathematics

Classical Analysis and ODEs

14 pages, 2 figures, corrected typos

Scientific paper

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.

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