The Vector Valued Quartile Operator

TL;DR

This paper establishes vector-valued inequalities for a Walsh analog of the bilinear Hilbert transform, linking these inequalities to the quartile type of UMD Banach spaces and extending known results in harmonic analysis.

Contribution

It introduces a new framework connecting quartile type of UMD spaces with Walsh models of bilinear operators, expanding the understanding of vector-valued inequalities.

Findings

Vector-valued inequalities hold under finite quartile type.

Results depend on the quartile type being close to that of a Hilbert space.

Quantitative inequalities are established in terms of quartile type.

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.