A.E. Convergence vs Boundedness

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 $1\mathrm{/}2\le r\le 1$ satisfy $1\mathrm{/}p+1\mathrm{/}q=1\mathrm{/}r$. For a family of translation-invariant bilinear operators $$T_m:L^p(M)\times L^q(M)\to L^r(M),m\in \mathbb{N},$$ that converge almost everywhere, we prove that the associated maximal operator $$T^{\ast }(f,g)=\underset{m}{\sup }\mathrm{\mid }{T}_m(f,g)\mathrm{\mid }$$ is of weak type $L^p(M)\times L^q(M)\to L^{r,\mathrm{\infty }}(M)$. 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 $r=(1\mathrm{/}p+1\mathrm{/}q{)}^{-1}$ for $p,q>1$, as well as almost everywhere convergence results for bilinear Bochner–Riesz means and other bilinear ergodic averages on the torus.