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Weighted Boundedness of a Composition of Paraproducts

Ana Čolović

TL;DR

This work addresses weighted boundedness for the composition of dyadic paraproducts $\Pi_b^*\Pi_d$ on $L^p(w)$ with $1<p<\infty$ and $w\in A_p$, building on the foundational $L^2$-theory. The authors derive a sparse domination framework for $\Pi_b^*\Pi_d$ that preserves the $L^2$-level bounds, enabling sharp weighted bounds in $L^p(w)$ and a full $L^2(w)$ characterization. Central to the approach is the Pott–Smith identity, which decomposes $\Pi_b^*\Pi_d$ into tractable components whose sparse bounds are controlled by $\|\widehat{S}(b\circ d)\|_{CM}$ and $\|E(b\circ d)\|_{\ell^{\infty}}$, combined with multilinear sparse domination results from Culicu, Di Plinio and Ou. The paper also establishes a weighted $L^2(w)$ lower bound, with explicit dependence on weight characteristics, thereby aligning the weighted theory with the unweighted $L^2$-bounds and enhancing the understanding of commutator-type operators in Calderón–Zygmund theory.

Abstract

In this paper we offer alternate upper bound for the operator $Π_b^*Π_d$ 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.

Weighted Boundedness of a Composition of Paraproducts

TL;DR

This work addresses weighted boundedness for the composition of dyadic paraproducts on with and , building on the foundational -theory. The authors derive a sparse domination framework for that preserves the -level bounds, enabling sharp weighted bounds in and a full characterization. Central to the approach is the Pott–Smith identity, which decomposes into tractable components whose sparse bounds are controlled by and , combined with multilinear sparse domination results from Culicu, Di Plinio and Ou. The paper also establishes a weighted lower bound, with explicit dependence on weight characteristics, thereby aligning the weighted theory with the unweighted -bounds and enhancing the understanding of commutator-type operators in Calderón–Zygmund theory.

Abstract

In this paper we offer alternate upper bound for the operator to the ones present in literature, thus extending the known upper bounds from the setting to , for and a Muckenhoupt weight . In the setting, we fully characterize the boundedness of the operator.

Paper Structure

This paper contains 8 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposition of the (1,0,0,1) operator and the sparse bounds of its pieces
  4. Multivariable setting
  5. Proof of Theorem \ref{['T:Sparse-improved']}, Comparison of Bounds and the Proof of Theorem \ref{['T:Lower-bound-11']}
  6. Proof of Theorem \ref{['T:Sparse-improved']}
  7. Comparison of Bounds
  8. Proof of Theorem \ref{['T:Lower-bound-11']}

Key Result

Theorem 1.1

Let $b=\{b_I\}_{I\in \mathcal{D}},$ and $d=\{d_I\}_{I\in \mathcal{D}}$ be sequences of complex numbers. The operator $\Pi_b^*\Pi_d$ is bounded on $L^2(\mathbb{R})$ if and only if $\|\widehat{S}(b\circ d)\|_{CM}<\infty$ and $\|E(b\circ d)\|_{l^{\infty}}<\infty.$ Moreover,

Theorems & Definitions (20)

  • Theorem 1.1: Pott, Reguera, Sawyer and Wick - 2016
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 2.1: $BMO^{\mathcal{D}}$
  • Definition 2.2: Dyadic Square Function
  • Theorem 2.3
  • Proposition 2.4: Pott-Smith Identity
  • Lemma 2.5: Sparse bound of a paraproduct
  • ...and 10 more