An elementary proof of the A 2 bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 217; no. 1; pp. 181 - 195
Main Author Lacey, Michael T.
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.03.2017
Subjects
Inequalities
Mathematics
Operators (mathematics)
Transformations (mathematics)
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Summary:A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-017-1442-x