The Vector Valued Quartile Operator
Classical Analysis and ODEs
2015-09-07 v2
Abstract
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.
Cite
@article{arxiv.1203.5604,
title = {The Vector Valued Quartile Operator},
author = {Tuomas P. Hytönen and Michael T. Lacey and Ioannis Parissis},
journal= {arXiv preprint arXiv:1203.5604},
year = {2015}
}
Comments
32 pages, 5 figures, incorporates referee's report, to appear in Collect. Math