Fiber-wise Calder\'on-Zygmund decomposition and application to a bi-dimensional paraproduct

TL;DR

This paper introduces a fiber-wise Calderón-Zygmund decomposition tailored for a new class of bi-dimensional bilinear paraproducts that are not covered by traditional Calderón-Zygmund theory.

Contribution

The paper develops a novel fiber-wise Calderón-Zygmund decomposition specifically designed for bi-dimensional bilinear paraproducts outside the scope of classical Calderón-Zygmund operators.

Findings

Established a new decomposition method for bi-dimensional paraproducts

Extended Calderón-Zygmund theory to new bilinear operators

Provided tools for analyzing complex bi-dimensional harmonic analysis problems

Abstract

We are interested in a new kind of bi-dimensional bilinear paraproducts (appearing in [6]), which do not fit into the setting of bilinear Calder\'on-Zygmund operators. In this paper we propose a fiber-wise Calder\'on-Zygmund decomposition, which is specially adapted to this kind of bi-dimensional paraproduct.