Random Tensor Estimates and Deterministic Diagonal Resolutions for Mixed Paracontrolled Operators

Abstract

We prove an operator-level convergence theorem for the mixed paracontrolled blocks \[ T^{i;j,k}_Λ(w) =I_i\bigl(w<Ψ_{j,Λ}\bigr)\circΨ_{k,Λ} \] on \(\mathbb T^d\), under Fourier-diagonal Gaussian covariance. At finite Galerkin cutoff, the Wick decomposition is \[ T^τ_Λ=D^τ_Λ+B^τ_Λ, \qquad τ=(i;j,k), \] where \(D^τ_Λ\) is a deterministic Volterra multiplier on the external diagonal \(n=q\), while \(B^τ_Λ\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered coefficient has two Gaussian frequency legs and one input-output pair. Write \(Δ_T=\{(t,s):0\le s\le t\le T\}\). An iterated rectangular non-commutative Khintchine argument, followed by the four associated oriented flattenings, yields \[ \bigl\|B^τ_{Λ,N,Q,M}\bigr\|_ {L^p\bigl(Ω;C(Δ_T;\mathcal L(\ell_q^2,\ell_n^2))\bigr)} \lesssim_{p,\varepsilon} N^{d/2-Γ_τ+\varepsilon} \bigl(M^{d/2}+Q^{d/2}\bigr), \qquad Γ_τ=λ_i+α_j+α_k. \] Under the resulting strict Sobolev--Besov summability conditions, the centered cutoffs converge in \(L^p(Ω)\) and almost surely in operator norm. The covariance contraction is kept separate from the tensor estimate: retained raw branches, or prescribed finite combinations of branches, enter through a scalar Volterra criterion imposed after the relevant cancellation has been formed. We give a frequency-envelope variant and verify the diagonal criterion for distinct-speed wave and Klein--Gordon contractions.