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 $f\in {\mathrm{\Lambda }}_{\alpha }([0,1{]}^2,d_d(x,y{)}^{\alpha }\times d_d^{\mathrm{\prime }}(x^{\mathrm{\prime }},y^{\mathrm{\prime }}{)}^{\alpha })$, we construct an approximation, ${\tilde{A}}_{(N,N^{\mathrm{\prime }})}(f)$ to $A(f)$, replacing the operator $T:f\to A(f)$ with the continous tensor paraproduct, ${\mathrm{\Pi }}_{(A^{\mathrm{\prime }},A^{\mathrm{\prime }\mathrm{\prime }})}^{(t,t^{\mathrm{\prime }})}$, and the multiscale tensor paraproduct ${\mathrm{\Pi }}_{(A^{\mathrm{\prime }},A^{\mathrm{\prime }\mathrm{\prime }})}^{(N,N^{\mathrm{\prime }})}:f\to {\tilde{A}}_{(N,N^{\mathrm{\prime }})}(f)+{\mathrm{\Delta }}_{(N,N^{\mathrm{\prime }})}(A,f)$. In the multiscale case, we provide estimates on the residual, ${\mathrm{\Delta }}_{(N,N^{\mathrm{\prime }})}(A,f)$, and show it has twice the regularity of $f$ such that ${\mathrm{\Delta }}_{(N,N^{\mathrm{\prime }})}(A,f)\in {\mathrm{\Lambda }}_{2\alpha }([0,1{]}^2)$ and $\parallel {\mathrm{\Delta }}_{(N,N^{\mathrm{\prime }})}(A,f){\parallel }_{{\mathrm{\Lambda }}_{2\alpha }([0,1{]}^2)}\le C_A\parallel f{\parallel }_{{\mathrm{\Lambda }}_{\alpha }([0,1{]}^2)}$. Our theoretical findings are supplemented with a computational example.