Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces

@misc{lai2026noncommutative,
  title = {{Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces}},
  author = {Xudong Lai and Siyu Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34906},
  eprint = {2609.34906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34906},
}

TY  - UNPB
TI  - Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces
AU  - Xudong Lai
AU  - Siyu Liu
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34906
UR  - https://arxiv.org/abs/2609.34906
AB  - We establish noncommutative maximal inequalities for averages over submanifolds in the L^p-spaces associated with semifinite von Neumann algebras. For every 1<p≤∞, we prove local maximal bounds for smooth submanifolds of finite type at the parameter origin, and global bounds over all positive scales for polynomial parametrizations, without a finite-type assumption and with constants depending only on p, the degree and the dimensions. For variable hypersurfaces in ℝ^n, n≥3, satisfying a uniform rotational curvature condition, we obtain local maximal bounds for p>n/(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 ℝ^n and, under additional assumptions, bilaterally almost uniform convergence results for the associated normalized averages.
ER  -