A weighted weak-type bound for Haar multipliers

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 fu...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 148; no. 3; pp. 643 - 658
Main Author Osȩkowski, Adam
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.06.2018
Cambridge University Press
Subjects
60G42
Secondary 46E30
dyadic
maximal
Bellman function
best constants
Primary 42B25
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ISSN0308-2105
1473-7124
DOI10.1017/S0308210517000415

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Summary: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 Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.
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ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210517000415