D-tensor paraproducts and its caricatures

We generalize the $2$-tensor paraproduct decomposition result of arXiv:2503.12629 to d-tensors and expound on its caricatures. In particular, we show that for A ∈ Cᵈ, f ∈ Λ^α([0,1]ᵈ), $A(f)$ has an approximation Ã(f) = (∑α=1ᵈ A^α(Pj₁,j₂, …, jd(f)) ṽ^α(f) ) for a fixed sequence of scales, j = (j₁, j₂, …, jd), and the series ∑α=1ᵈ A^α(Pj₁,j₂, …, jd(f)) ṽ^α(f) is a taylor expansion of A. Additionally, we show the sequence of operators (ṽ¹(f),ṽ²(f), …, ṽᵈ(f)) form a basis for the subspace comprised of linear functionals of the form $A(f)$, and the residual, Δ(A,f) = Ã(f) - A(f) ∈ Λ2α([0,1]ᵈ). Consequentially, we show one can obtain a Calderon-Zygmund type decomposition, Δ(A,f)_λ+ Ã(f)_λ with the obtained paraproduct decomposition A(f) = Δ(A,f) + Ã(f). Our theoretical findings are supported by computational examples for d=2,3.