The quartile operator and pointwise convergence of Walsh series

The bilinear Hilbert transform is given by H(f; g)(x) :=p:v: integral f(x - t)g(x + t) dt/t. It satisfies estimates of the type parallel to H(f; g) parallel to r less than or equal to C(s; t) parallel to f parallel to s parallel to g parallel to 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 th e 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 exp osition is to clarify the connection between these two results and to prese nt an easy approach to recent methods of time-frequency analysis.