Weighted Boundedness of a Composition of Paraproducts

TL;DR

This paper extends the boundedness results of a composition of paraproduct operators from the classical $L^2$ space to weighted $L^p(w)$ spaces, providing new bounds and a full characterization in the weighted $L^2(w)$ case.

Contribution

It introduces an alternative upper bound for the operator $ ext{ extbackslash Pi}_b^* ext{ extbackslash Pi}_d$ and extends boundedness results to weighted $L^p(w)$ spaces, including a complete characterization in $L^2(w)$.

Findings

New upper bounds for the operator in weighted $L^p(w)$ spaces.

Full characterization of boundedness in weighted $L^2(w)$.

Extension of known bounds from $L^2( eal)$ to $L^p(w)$.

Abstract

In this paper we offer alternate upper bound for the operator $\Pi_b^*\Pi_d$ to the ones present in literature, thus extending the known upper bounds from the $L^2(\mathbb{R})$ setting to $L^p(w)$, for $1<p<\infty,$ and a Muckenhoupt weight $w$. In the $L^2(w)$ setting, we fully characterize the boundedness of the operator.