Localized Centered Second-Chaos Operator

We prove a localized continuous-frequency operator estimate for centered Gaussian chaoses of order two. The result applies to operator-valued centered second chaoses, including Wick-centered same-family variants, between Hilbert spaces. In the model, two Gaussian frequency legs at scale $N$, an input leg at scale $Q$, and an output leg at scale $M$ are coupled through a soft incidence kernel; non-orthogonal Gaussian profiles are represented by covariance synthesis maps. The proof combines four oriented flattenings, rectangular non-commutative Khintchine inequalities, soft-incidence Schatten bounds, and Sobolev–Besov dyadic summation. The time lift gives $L^p$ operator convergence, while a Galerkin stabilization hypothesis gives pathwise full-cutoff convergence by the first Borel–Cantelli lemma. Under $\mathcal{G}(N)\lesssim N^{-\mathrm{\Gamma }}$ one obtains the window $$\mathrm{\Gamma }>\frac{d}{2},s<\lambda +\mathrm{\Gamma }-d,\max \{0,d-\mathrm{\Gamma }\}<\sigma <\lambda +\mathrm{\Gamma }-d\mathrm{.}$$ The theorem applies to the near-output Wick-centered branch of localized paracontrolled resonant products on ${\mathbb{R}}^d$.