D-tensor paraproducts and its caricatures

We generalize the $2$-tensor paraproduct decomposition result of [arXiv:2503.12629] to $d$-tensors. In particular, we show that for $A\in C^d(\mathbb{R}),f\in {\mathrm{\Lambda }}_{\alpha }([0,1{]}^d)$, $A(f)$ 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 ${\mathrm{\Delta }}_{(N_i{)}_{i=1}^d}(A,f)={\tilde{A}}_{(N_i{)}_{i=1}^d}(f)-A(f)\in {\mathrm{\Lambda }}_{2\alpha }([0,1{]}^d)$. Our theoretical findings are supported by a computational example for d=3.