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A.E. Convergence vs Boundedness

Xinyu Gao, Loukas Grafakos

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.

A.E. Convergence vs Boundedness

Abstract

We extend Stein's maximal theorem to the bilinear setting. Let be a homogeneous space with a transitive action of a compact abelian group, and let and satisfy . For a family of translation-invariant bilinear operators that converge almost everywhere, we prove that the associated maximal operator is of weak type . 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 for , as well as almost everywhere convergence results for bilinear Bochner--Riesz means and other bilinear ergodic averages on the torus.
Paper Structure (8 sections, 21 theorems, 274 equations)

This paper contains 8 sections, 21 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Some preliminary results
  3. The proof of Theorem \ref{['thm1']}
  4. The Extension to Borel measures
  5. The Extension to Positive Operators
  6. An application concerning maximal bilinear averages
  7. An application concerning maximal bilinear Bochner--Riesz operators
  8. An application in differentiation theory

Key Result

Theorem 1

Let $1 \le p,q \leq 2$ and $\frac{1}{2} \leq r \leq 1$, such that $\frac{1}{r} = \frac{1}{p} + \frac{1}{q}$. Let $T_m$ be a sequence of bounded operators from $L^p(M) \times L^q(M)$ to $L^r(M)$ that commute with simultaneous translations. Suppose that for every $f \in L^p(M)$, $h \in L^q(M)$, the po exists for almost every $x \in M$. Then, there exists a constant $C > 0$ such that for all $f \in L

Theorems & Definitions (44)

  • Definition 1
  • Theorem 1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • ...and 34 more