The quartile operator and pointwise convergence of Walsh series

Thiele, Christoph

doi:10.1090/S0002-9947-00-02577-0

The quartile operator and pointwise convergence of Walsh series
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Trans. Amer. Math. Soc. 352 (2000), 5745-5766 Request permission

Abstract:

The bilinear Hilbert transform is given by \[ H(f,g)(x):= p.v.\ \int f(x-t)g(x+t)\frac {dt}{t}. \] It satisfies estimates of the type \[ \|H(f,g)\|_r\le C(s,t)\|f\|_s \|g\|_t.\] In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in $L^p[0,1]$, with $1<p$ converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

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