The Vector Valued Quartile Operator

Details The Vector Valued Quartile Operator The Vector Valued Quartile Operator

2012-03-26

arxiv.org/abs/1203.5604v1

Mathematics

Classical Analysis and ODEs

32 pages, 5 figures

Scientific paper

Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach space X. Every known UMD Banach space has finite quartile type, and it was recently shown that the Walsh analog of Carleson's Theorem holds under a closely related assumption of finite tile type. For a Walsh model of the bilinear Hilbert transform however, the quartile type should be sufficiently close to that of a Hilbert space for our results to hold. A full set of inequalities is quantified in terms of quartile type.

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