Quasilinearization with regularizing tensor paraproducts

We extend Bony's celebrated work on paraproducts to continous and multiscale \emph{tensor} paraproducts. For $A \in \mathcal{C}^2(\mathbb{R})$ and $f \in \Lambda_{\alpha}([0,1]^2, d_d(x,y)^{\alpha} \times d'_d(x',y')^{\alpha})$, we construct an approximation, $\tilde{A}_{(N,N')}(f)$ to $A(f)$, replacing the operator $T: f \to A(f)$ with the continous tensor paraproduct, $\Pi^{(t,t')}_{(A',A'')}$, and the multiscale tensor paraproduct $\Pi^{(N,N')}_{(A',A'')}:f \to \tilde{A}_{(N,N')}(f) + \Delta_{ (N,N')}(A,f)$. In the multiscale case, we provide estimates on the residual, $\Delta_{(N,N')}(A,f)$, and show it has twice the regularity of $f$ such that $\Delta_{(N,N')}(A,f) \in \Lambda_{2 \alpha}([0,1]^2)$ and $\lVert \Delta_{(N,N')}(A,f) \rVert_{\Lambda_{2\alpha}([0,1]^2)} \leq C_A \lVert f \rVert_{\Lambda_{\alpha}([0,1]^2)}$. Our theoretical findings are supplemented with a computational example.