The Vector Valued Quartile Operator

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.

32 pages, 5 figures, incorporates referee's report, to appear in Collect. Math

References in corpus (4)

• Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems
• Pointwise convergence of vector-valued Fourier series
• Pointwise convergence of Walsh--Fourier series of vector-valued functions
• On the growth of vector-valued Fourier series

Cited by in corpus (6)

• Multiple Vector Valued Inequalities via the Helicoidal Method
• Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems
• A variation norm Carleson theorem for vector-valued Walsh-Fourier series
• Banach-valued multilinear singular integrals
• Vector-valued extensions of operators through multilinear limited range extrapolation
• Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type