发表机构
Harbin Institute of Technology（哈尔滨工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立了非交换Lp空间中几何平均的极大不等式，涵盖有限型子流形、多项式参数化及可变超曲面情形，并应用于非交换极大遍历不等式与双边几乎一致收敛。
AI 中文摘要
我们建立了与半有限冯·诺依曼代数相关的非交换\\(L^p\\)空间中算子值函数的几何平均的极大不等式。对于在参数原点处有限型的固定光滑子流形上的平均，我们证明了对于每个\\(1<p\leq\infty\\)的局部极大界。对于多项式参数化，我们在没有有限型假设的情况下获得了所有正尺度上的界，常数仅取决于次数和维数。我们还证明了在\\(\mathbb R^n\\)（\\(n\geq3\\)）中满足一致旋转曲率条件的可变超曲面的局部极大不等式，范围是\\(p>n/(n-1)\\)。有限型和多项式估计依赖于一个适用于非各向同性膨胀的正则化辅助族的弱\\((1,1)\\)型不等式。其证明使用了基于Cuculescu投影的非交换Calderón-Zygmund分解。与基于Fourier的\\(L^2\\)界的插值恢复了原始平均的极大不等式。可变超曲面结果使用了基于振荡\\(L^2\\)估计和局部化的单独论证。作为应用，我们获得了\\(\mathbb R^n\\)的保迹作用的一些非交换极大遍历不等式（对应于上述考虑的几何平均）以及归一化遍历平均的双边几乎一致收敛。
英文摘要
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 \(1<p\leq\infty\). For polynomial parametrizations, we obtain bounds over all positive scales without a finite-type assumption, with constants only depending on the degree and dimensions. We also prove local maximal inequalities for variable hypersurfaces in \(\mathbb R^n\), \(n\geq3\), satisfying a uniform rotational curvature condition, in the range \(p>n/(n-1)\). 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.
Comments82 pages
