The quartile operator and pointwise convergence of Walsh series

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.

Locations

• Transactions of the American Mathematical Society - View - PDF