Year: 2026
Type: preprint
Abstract: We establish noncommutative maximal inequalities for averages over submanifolds in the $L^p$-spaces associated with semifinite von Neumann algebras. For every $1n/(n-1)$. To our knowledge, these are the first such noncommutative inequalities for general lower-dimensional submanifolds and polynomial images. The key step is a weak type $(1,1)$ estimate for a regularized auxiliary family adapted to non-isotropic dilations. The geometric kernels need not satisfy the pointwise smoothness or $L^2$-Dini conditions required by existing noncommutative Calderón--Zygmund theories. We combine a Calderón--Zygmund decomposition based on Cuculescu projections with integral regularity estimates, a non-isotropic Littlewood--Paley decomposition and Fourier decay to prove this endpoint estimate. Marcinkiewicz interpolation yields maximal $L^p$ bounds for the regularized averages, and interpolation with Fourier-based $L^2$ bounds recovers the original averages. The variable-hypersurface result uses oscillatory $TT^*$ estimates and localization. As applications, we obtain maximal ergodic inequalities for trace-preserving actions of $\mathbb{R}^n$ and, under additional assumptions, bilaterally almost uniform convergence results for the associated normalized averages.
Source: arXiv (Cornell University)
Authors: Lai, Xudong, Liu, Siyu
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4232864832, G5214839347
OpenAlex ID: W7214747065