AI 中文总结
研究算子值高斯混沌的代数转移演算，通过链接不等式等传播轮廓界，应用于奇异Wick乘子，在多项式增长群及\(\mathbb{Z}^D\)、\(\mathbb{T}^D\)上取得成果，如确定收敛阈值、相图及相关算子性质。
AI 中文摘要
我们为算子值高斯混沌基础核的定向Schatten轮廓开发了一种代数转移演算。一个无维链接不等式通过切割分解、张量积、系数映射和有序收缩来传播轮廓界。结合定向扁平化高斯估计，该演算在具有非交换系数的完备Wick混沌上产生连续乘法，一个阶乘加权解析Wick级数的结合代数，以及关于无环Peter - Weyl融合树的局部到全局定理。然后我们将该方法应用于多项式增长群上的奇异Wick乘子。对于二阶乘子，我们得到了尖锐的Schatten收敛阈值的充要条件；在\(\mathbb{Z}^D\)上，我们确定了每个阶的完整相图。傅里叶转移给出了\(\mathbb{T}^D\)上夹心Wick乘法算子的精确Sobolev、Schatten类、紧性和迹类阈值，以及尖锐的傅里叶 - Galerkin速率和近似数衰减。
英文摘要
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.
Comments54 pages
