Noncommutative maximal weak type (1,1) estimates for convolution operators under L1-Hörmander conditions
Abstract
We prove weak type (1,1) estimates for families of scalar convolution operators acting on operator-valued function spaces. A joint L1-Hörmander condition, together with a maximal strong type estimate at one exponent 1<r≤∞, suffices. The kernels may be complex-valued and need not have uniformly bounded L1 norms. Under a size condition, the result also yields a weak endpoint estimate for all sharp truncation radii of singular kernels. This resolves, for convolution kernels, the problem mentioned in \cite[Remark~5.5]{parcet09pseudo} and recalled in Parcet's recent survey \cite[Remark~1.3]{Parcet2026ICM}. The proof involves not only a simultaneous dyadic representation of the operators but also a noncommutative maximal weak type (1,1) estimates of the resulting dyadic operators. This approach seems entirely new even in the commutative setting. Numerous applications are provided. On the one hand, it yields various maximal inequalities and pointwise convergence results for integrable kernels and dyadic approximations of the identity, monotone Llog L majorants, and Poisson integrals on nilpotent groups and symmetric spaces. On the other hand, via various transference techniques, it implies maximal inequalities for Fourier multipliers, Schur multipliers and double operator integrals, together with bilateral almost uniform convergence of dyadic Fourier summation methods on L1 of group von Neumann algebras.
Related
- Sharp lifespan estimates for heat equations with Hörmander operators
- A Hörmander spectral multiplier theorem for fractional Hardy operators
- Strict weak Hörmander conditions for hypoellipticity and structure equations
- Kernel estimates for the Weyl-Hörmander calculus, Weak (1,1) boundedness and other regularity properties
- Reverse Translation and Convolution Estimates in the Framework of the Mehler-Fock Transform
BibTeX
@misc{hong2026noncommutative,
title = {{Noncommutative maximal weak type $(1,1)$ estimates for convolution operators under $L_1$-Hörmander conditions}},
author = {Guixiang Hong and Liang Wang and Xumin Wang and Bang Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25761},
eprint = {2610.25761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25761},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncommutative maximal weak type (1,1) estimates for convolution operators under L_1-Hörmander conditions
AU - Guixiang Hong
AU - Liang Wang
AU - Xumin Wang
AU - Bang Xu
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25761
UR - https://arxiv.org/abs/2610.25761
AB - We prove weak type (1,1) estimates for families of scalar convolution operators acting on operator-valued function spaces. A joint L_1-Hörmander condition, together with a maximal strong type estimate at one exponent 1<r≤∞, suffices. The kernels may be complex-valued and need not have uniformly bounded L_1 norms. Under a size condition, the result also yields a weak endpoint estimate for all sharp truncation radii of singular kernels. This resolves, for convolution kernels, the problem mentioned in \cite[Remark~5.5]{parcet09pseudo} and recalled in Parcet's recent survey \cite[Remark~1.3]{Parcet2026ICM}. The proof involves not only a simultaneous dyadic representation of the operators but also a noncommutative maximal weak type (1,1) estimates of the resulting dyadic operators. This approach seems entirely new even in the commutative setting. Numerous applications are provided. On the one hand, it yields various maximal inequalities and pointwise convergence results for integrable kernels and dyadic approximations of the identity, monotone Llog L majorants, and Poisson integrals on nilpotent groups and symmetric spaces. On the other hand, via various transference techniques, it implies maximal inequalities for Fourier multipliers, Schur multipliers and double operator integrals, together with bilateral almost uniform convergence of dyadic Fourier summation methods on L_1 of group von Neumann algebras.
ER -