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Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

The Anh Bui, Suman Mukherjee

Abstract

We establish fractional Leibniz rules for the Dunkl Laplacian $Δ_k$ of the form $$\|(-Δ_k)^s(fg)\|_{L^p(dμ_k)} \lesssim \|(-Δ_k)^s f\|_{L^{p_1}(dμ_k)} \|g\|_{L^{p_2}(dμ_k)} + \|f\|_{L^{p_1}(dμ_k)} \|(-Δ_k)^s g\|_{L^{p_2}(dμ_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-Δ_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces.

Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

Abstract

We establish fractional Leibniz rules for the Dunkl Laplacian of the form Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function , the function satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces.

Paper Structure

This paper contains 15 sections, 11 theorems, 116 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Preliminaries of Dunkl theory
  3. Orbit distances and orbit of balls
  4. Dunkl operators
  5. Dunkl kernel and Dunkl transform
  6. Dunkl translations Dunkl convolution
  7. Dunkl heat kernel
  8. Ingredients for the proof of the Theorem \ref{['MAIN bddnees of 3paraproducts']}
  9. Estimates for the Dunkl translations of Schwartz functions
  10. Almost orthogonality type estimates
  11. Bilinear Dunkl setup and Bilinear Dunkl-Calderón-Zygmund theory
  12. Boundedness of Dunkl Paraproducts
  13. Proofs of the fractional Leibniz rules
  14. Paraproduct decomposition
  15. Proofs of the main theorem

Key Result

Theorem 1.1

WrobelABMVAFC Let $1 < p, p_1, p_2 < \infty$ be such that $1/p = 1/p_1 + 1/p_2$. Then for any $s > 0$ and $f,\, g \in \mathcal{S}(\mathbb{R}^d)$, with at least one of $f$ or $g$ being $\mathbb{Z}_2^d$-invariant, we have

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more