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Weighted Bilinear Multiplier Theorems in Dunkl Setting via Singular Integrals

Suman Mukherjee, Sanjay Parui

TL;DR

This work extends harmonic analysis in the Dunkl setting to multilinear and weighted contexts by developing Littlewood–Paley theory and a multilinear Calderón–Zygmund framework, then proving one- and two-weight inequalities for multilinear Dunkl–CZ operators. It also establishes a Dunkl analogue of the Coifman–Meyer theorem for bilinear multipliers, providing weighted weak- and strong-type bounds under $A^k_{\infty}$ and $A^k_{\overrightarrow P}$ weights. The results handle the complications arising from reflection groups, including orbit considerations, and rely on a robust combination of LP theory, Banach-valued singular integrals, and dyadic decomposition arguments. The findings pave the way for broader multilinear Dunkl analysis and potential applications to Dunkl-type Leibniz rules and related operator theory.

Abstract

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood-Paley type theorems and weighted inequalities for multilinear Calderón-Zygmund operators in Dunkl setting are also proved.

Weighted Bilinear Multiplier Theorems in Dunkl Setting via Singular Integrals

TL;DR

This work extends harmonic analysis in the Dunkl setting to multilinear and weighted contexts by developing Littlewood–Paley theory and a multilinear Calderón–Zygmund framework, then proving one- and two-weight inequalities for multilinear Dunkl–CZ operators. It also establishes a Dunkl analogue of the Coifman–Meyer theorem for bilinear multipliers, providing weighted weak- and strong-type bounds under and weights. The results handle the complications arising from reflection groups, including orbit considerations, and rely on a robust combination of LP theory, Banach-valued singular integrals, and dyadic decomposition arguments. The findings pave the way for broader multilinear Dunkl analysis and potential applications to Dunkl-type Leibniz rules and related operator theory.

Abstract

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood-Paley type theorems and weighted inequalities for multilinear Calderón-Zygmund operators in Dunkl setting are also proved.
Paper Structure (9 sections, 15 theorems, 173 equations)

This paper contains 9 sections, 15 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Dunkl Theory
  3. Littlewood-Paley Type Theorems in Dunkl Setting
  4. Spaces of Homogeneous Type and Muckenhoupt Weights
  5. Statements of Main Theorems
  6. Multilinear Calderón-Zygmund type Singular Integrals in Dunkl setting
  7. Bilinear Multipliers Operators in Dunkl setting
  8. Proofs of The Weighted Inequalities for Singular Integrals
  9. Proofs of The Weighted Inequalities for Multipliers

Key Result

Theorem 3.1

Let $u\in \mathbb{R}^d, 1<p<\infty$. Let $\psi$ be a smooth function on $\mathbb{R}^d$ such that $supp\, \psi\subset\{\xi\in \mathbb{R}^d:1/r\leq |\xi|\leq r\}$ for some $r>1$. For $j\in \mathbb{Z}$, define $\psi_j(\xi)=\psi (\xi/2^j)$ and for $f\in \mathcal{S}(\mathbb{R}^d)$ define Then where $n=\lfloor d_k \rfloor+2$ and $C$ is independent of $u$.

Theorems & Definitions (35)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • ...and 25 more