A.E. Convergence vs Boundedness

Abstract

We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1\le p,q\le 2$ and $1/2\le r\le 1$ satisfy $1/p+1/q=1/r$. For a family of translation-invariant bilinear operators \[ T_m : L^p(M) \times L^q(M) \to L^r(M), \qquad m \in \mathbb{N}, \] that converge almost everywhere, we prove that the associated maximal operator \[ T^*(f,g) = \sup_m |T_m(f,g)| \] is of weak type $L^p(M)\times L^q(M)\to L^{r,\infty }(M)$. The proof relies on probabilistic methods and a bilinear extension of Stein's lemma for double Rademacher series. We also establish a bilinear analogue of Sawyer's extension of Stein's theorem for positive bilinear operators commuting with a mixing family of measure-preserving transformations. Applications include strong-type boundedness of maximal bilinear tail operators associated with ergodic transformations in the natural exponent range $r=(1/p+1/q{)}^{-1}$ for $p,q>1$, as well as almost everywhere convergence results for bilinear Bochner--Riesz means and other bilinear ergodic averages on the torus.