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An elementary proof of the A 2 bound

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Abstract

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 A p 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-L 1 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 A 2 bounds in that setting.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey

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  1. Michael T. Lacey
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Correspondence to Michael T. Lacey.

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Research supported in part by grant NSF-DMS 1265570.

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Lacey, M.T. An elementary proof of the A 2 bound. Isr. J. Math. 217, 181–195 (2017). https://doi.org/10.1007/s11856-017-1442-x

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  • DOI: https://doi.org/10.1007/s11856-017-1442-x

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