Random Tensor Estimates and Deterministic Diagonal Resolutions for Mixed Paracontrolled Operators

Abstract

We prove an abstract operator-level convergence criterion for mixed paracontrolled random operators \[ T^{i;j,k}_Λ(w) = I_i(w<Ψ_{j,Λ})\circ Ψ_{k,Λ} \] on \(\mathbb T^d\). These operators are the second-level mixed blocks generated by quadratic paracontrolled expansions, and are singular in the range where the deterministic resonant-product threshold fails. At finite cutoff we separate \[ T^τ_Λ= D^τ_Λ+B^τ_Λ, \] where \(D^τ_Λ\) is the Fourier-diagonal covariance contraction supported on the external diagonal \(n=q\), and \(B^τ_Λ\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered component is controlled by a finite second-order random tensor estimate, a Banach-valued time lift, and Sobolev--Besov dyadic summation. The deterministic component is treated through a diagonal-resolution interface: retained raw branches or model-determined orbit sums are required only to satisfy scalar Volterra dyadic bounds after the relevant finite cancellations have been formed. The result applies in arbitrary dimension, for finite color/label families, Fourier-diagonal weak covariance, random initial data, colored-in-time Gaussian profiles, and frequency-dependent amplitudes. It abstracts the reusable mixed-operator mechanism underlying concrete color-phase diagonal analyses, while leaving model-specific diagonal classification and scalar Volterra estimates as explicit inputs.