Localized Centered Second-Chaos Operator

Abstract

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 $\mathcal G(N)\lesssim N^{-Γ}$ one obtains the window \[ Γ>\frac d2, \qquad s<λ+Γ-d, \qquad \max\{0,d-Γ\}<σ<λ+Γ-d. \] The theorem applies to the near-output Wick-centered branch of localized paracontrolled resonant products on $\mathbb R^d$.