Noncommutative maximal weak type $(1,1)$ estimates for convolution operators under $L_1$-Hörmander conditions

Abstract

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 $1<r\leq\infty$, 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 $L\log L$ 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.