Quasilinearization with regularizing tensor paraproducts

Abstract

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

21 pages, 1 figure