A weighted weak-type bound for Haar multipliers. Issue 3 (22nd April 2018) - Osȩkowski, Adam
Abstract : We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [ − 1, 1], and is the r -maximal operator, then for any weight w and any function f on [0, 1) we have for some constant Cr depending only on r . We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md . The proof rests on the Bellman function method; using this technique we establish related weighted L p estimates for p close to 1, and then deduce the main result by extrapolation arguments.