Multilinear paraproducts on Sobolev spaces

TL;DR

This paper characterizes the boundedness of multilinear paraproducts on Sobolev spaces using Triebel-Lizorkin norms, leading to new $T(1)$-type theorems for multilinear Calderón-Zygmund operators.

Contribution

It introduces a novel characterization of multilinear paraproducts' Sobolev space boundedness via Triebel-Lizorkin norms, extending classical $T(1)$ theorems.

Findings

Sobolev space boundedness characterized by Triebel-Lizorkin norms

Wavelet representation theorem facilitates the analysis

New $T(1)$-type theorems for multilinear Calderón-Zygmund operators

Abstract

Paraproducts are a special subclass of the multilinear Calder\'on-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the $\mathrm{BMO}$ norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space $T(1)$-type theorems for multilinear Calder\'on-Zygmund operators.