Algebraic Transfer for Operator-Valued Gaussian Chaoses:Oriented Schatten Profiles and Singular Wick Multipliers

Abstract

We develop an algebraic transfer calculus for the oriented Schatten profiles of kernels underlying operator-valued Gaussian chaoses. A dimension-free link inequality propagates profile bounds through cut factorizations, tensor products, coefficient maps, and ordered contractions. Besides the constant-one strong Schatten theorem, we prove the sharp weak endpoint \[ \mathfrak{S}_{r,\infty}\times\mathfrak{S}_{r,\infty} \longrightarrow \mathfrak{S}_{r,\infty}(\log\mathfrak{S})^{-1/r}, \qquad 1<r<\infty. \] The estimate holds uniformly over all contracted Hilbert spaces, and the exponent $1/r$ cannot be decreased within the displayed $q=\infty$ Lorentz--Zygmund scale. A separate finite-complexity argument gives sharp effective-rank and finite-cut-rank logarithmic bridges from all-cut operator profiles to Gaussian operator norms. Combined with oriented-flattening Gaussian estimates, the calculus yields continuous multiplication on completed Wick chaoses with noncommuting coefficients, an associative algebra of factorially weighted analytic Wick series, and a local-to-global theorem for loop-free Peter--Weyl fusion trees. We then apply the method to singular Wick multipliers on groups of polynomial growth. For second-order multipliers we obtain sharp necessary and sufficient Schatten convergence thresholds; on ${\mathbb{Z}}^D$ we determine the full singular phase diagram at every order. Fourier transfer gives exact Sobolev, Schatten-class, compactness, and trace-class thresholds for sandwiched Wick multiplication operators on ${\mathbb{T}}^D$, together with sharp Fourier--Galerkin rates and approximation-number decay.

54 pages