Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces

Xudong Lai; Siyu Liu

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arXiv · 2609.34906

Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces

Xudong Lai·Siyu Liu

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Abstract

We establish maximal inequalities for geometric averages of operator-valued functions in noncommutative $L^p$-spaces associated with semifinite von Neumann algebras. For averages over a fixed smooth submanifold of finite type at the parameter origin, we prove local maximal bounds for every $1 n/(n-1)$. The finite-type and polynomial estimates rely on a weak type $(1,1)$ inequality for a regularized auxiliary family adapted to non-isotropic dilations. Its proof uses a noncommutative Calderón-Zygmund decomposition based on Cuculescu projections. Interpolation with Fourier-based $L^2$ bounds recovers the maximal inequalities for the original averages. The variable-hypersurface result uses a separate argument based on oscillatory $L^2$ estimates and localization. As applications, we obtain some noncommutative maximal ergodic inequalities for trace-preserving actions of $\mathbb R^n$ (corresponding to the geometric averages considered above) and bilateral almost uniform convergence for normalized ergodic averages.

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BibTeX RIS

Xudong Lai, Siyu Liu. 2026-09-28. Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces. https://arxiv.org/abs/2609.34906

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