D-tensor paraproducts and its caricatures

Abstract

We generalize the $2$-tensor paraproduct decomposition result of [arXiv:2503.12629] to $d$-tensors. In particular, we show that for $A\in C^d(\mathbb{R}),f\in {\Lambda }_{\alpha }([0,1{]}^d)$, $A(f)$ can be approximated by ${\tilde{A}}_{(N_i{)}_{i=0}^d}(f)=({\sum }_{\beta =1}^d{A}^{\beta }(P^{j_1,j_2,\dots ,j_d}(f)){\tilde{\mathbf{v}}}^{\beta }(f))$ with the residual ${\Delta }_{(N_i{)}_{i=1}^d}(A,f)={\tilde{A}}_{(N_i{)}_{i=1}^d}(f)-A(f)\in {\Lambda }_{2\alpha }([0,1{]}^d)$. Our theoretical findings are supported by a computational example for d=3.

14 pages and 1 figure