Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces

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<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.