A weighted weak-type bound for Haar multipliers. Issue 3 (22nd April 2018)

• Record Type: Journal Article
• Authors: Osȩkowski, Adam
• Is Part Of: Proceedings. Volume 148:Issue 3(2018)
• Page Start: 643
• Page End: 658
• Publication Date: 2018-04-22
• Subjects: maximal, -- dyadic, -- Bellman function, -- best constants; Primary 42B25, -- Secondary 46E30, -- 60G42; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210517000415
• Languages: English
• ISSNs: 0308-2105

Abstract

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.