Hierarchical paraproducts

Abstract

We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A \in C^{d}(\mathbb{R}), f \in \Lambda_{\alpha}([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A \in C^1(\mathbb{R}),f \in \Lambda_{\alpha}(X))$ and $ (A \in C^2(\mathbb{R}),f \in \Lambda_{\alpha}(X \times Y))$. To do so, we construct partition trees on $X$ and $X \times Y$ such that analysis with respect to scale is sensible. We obtain results resembling those of [arXiv:2503.12629] and [arXiv:2508.13322], but with the finite sets $X$ and $X \times Y $ as support. In particular we construct the paraproduct $\Pi_{A',A''}^{L,S}: f \to \tilde{A}_{L,S}(f) + \Delta_{L,S}(A,f)$ such that $\Delta_{L,S}(A,f) \in \Lambda_{2\alpha}(X \times Y)$ and $\lVert \Delta_{L,S}(A,f) \rVert_{\Lambda_{2\alpha}(X \times Y)} \leq C_A \lVert f \rVert_{\Lambda_{\alpha}(X \times Y)}$. Analogous results are obtained when the support is just one finite set, $X$. This extension is motivated by situations where one wishes to separate the singular and smooth components of such compositions in graph signal processing environments.