A.E. Convergence vs Boundedness
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910026112696320 |
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| author | Gao, Xinyu Grafakos, Loukas |
| author_facet | Gao, Xinyu Grafakos, Loukas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16654 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A.E. Convergence vs Boundedness Gao, Xinyu Grafakos, Loukas Classical Analysis and ODEs 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. |
| title | A.E. Convergence vs Boundedness |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2602.16654 |