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 ${\Pi }_{A^{\prime },A^{\prime \prime }}^{L,S}:f\to {\tilde{A}}_{L,S}(f)+{\Delta }_{L,S}(A,f)$ such that ${\Delta }_{L,S}(A,f)\in {\Lambda }_{2\alpha }(X\times Y)$ and $\Vert {\Delta }_{L,S}(A,f){\Vert }_{{\Lambda }_{2\alpha }(X\times Y)}\le C_A\Vert f{\Vert }_{{\Lambda }_{\alpha }(X\times Y)}$. 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.

12 pages