Pointwise convergence of bilinear ergodic averages for commuting transformations

Abstract

Let $(X,\mathcal B,μ)$ be a probability space and let $S,Q:X\to X$ be commuting probability-preserving transformations. Let $1\leq p,q\leq\infty$ satisfy $1/p+1/q\leq1$. We prove that, for every fixed $F\in L^p(μ)$ and $G\in L^q(μ)$, the averages $$ \frac1N\sum_{n=1}^N F(S^nx)G(Q^nx) $$ converge almost everywhere as $N\to\infty$. Our proof introduces a new analytic approach that relies on neither the circle method nor characteristic-factor theory. Exploiting the triangular structure arising from commuting transformations, we combine matrix-energy estimates with Gaussian heat-flow arguments to establish pointwise variation inequalities for suitable smooth averages. We then use even and odd Gaussian kernels to recover the convergence of the original averages.