A.e. convergence versus boundedness

In this paper, we extend Stein’s maximal theorem [26] to the bilinear setting; precisely, let M be a homogeneous space on which a compact abelian group acts transitively and let 1 ≤ p, q ≤ 2 , 1/2 ≤ r ≤ 1 be related by $$ 1/p + 1/q = 1/r $$ . Given a family of bounded bilinear operators Tₘ: Lᵖ(M) × Lq(M) → Lʳ(M) , m=1,2, , that commute with translations and converge a.e. as m→ ∞ , we show that the associated maximal operator T^*(f, g) = ₘ |Tₘ(f, g)| satisfies a weak-type Lᵖ(M)× Lq(M) → Lr,∞(M) estimate. Our proof is based on probabilistic arguments, properties of the Rademacher functions, and measure-theoretic constructions. The main obstacle we overcome is an extension of Stein’s lemma that provides an estimate for the L² norm of a tail of a double Rademacher series in terms of the L^∞ norm of the series over a measurable set. We also obtain a bilinear analogue of Sawyer’s [25] extension of Stein’s theorem to the wider range 1≤ p,q<∞ . This result is valid for positive bilinear operators that commute with family of measure-preserving transformations. Our main application concerns the boundedness of a maximal bilinear tail operator associated with an ergodic measure–preserving transformation on a finite measure space, which was shown by Assani and Buczolich [2] to be finite a.e.. Our results imply that this maximal operator is bounded from Lᵖ× Lq to Lʳ for the natural exponent r=(1/p+1/q)⁻¹ when $$p,q>1$$ ; this extends the integrability of the bilinear maximal tail operator from the subcritical region to the “natural” boundedness region. We discuss an additional application related to the a.e. convergence of bilinear Bochner-Riesz means.