Pointwise convergence of the ergodic bilinear Hilbert transform

Abstract

Let $\mathbf{X}=(X,\Sigma ,m,\tau )$ be a dynamical system. We prove that the bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(τ^nx)g(τ^{-n}x)}{n}$ converges almost everywhere for each $f,g\in L^{\infty }(X).$ We also give a proof along the same lines of Bourgain's analog result for averages.

28 pages, no figures