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, \ldots, 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.