Schatten-Class Wick Multipliers for Gaussian Fields on Closed Riemannian Manifolds: Oriented Tensor Estimates and Covariance-Profile Transfer

We develop a covariance-profile method for constructing Schatten-class multiplication operators generated by Wick powers of Gaussian fields on smooth closed Riemannian manifolds. The main result converts dyadic bounds for mixed covariance kernels into moment estimates, convergence, and regularization independence for suitably sandwiched Wick multipliers. The admissible Schatten range is determined by the covariance growth exponent, the dimension of the manifold, and the amount of smoothing on the two sides of the multiplier. The framework applies to fractional Gaussian fields, massive Gaussian free fields, and Gaussian fields associated with positive elliptic covariance operators. In model settings, we also prove the sharpness of the resulting summability conditions. The probabilistic input is a dimension-free estimate for operator-valued Gaussian random tensors controlled by all oriented coefficient flattenings. A Gaussian regression argument transfers this estimate to same-field Wick chaos and yields the optimal universal factor in increasing convex order.