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Homogeneous Besov Spaces in Dunkl setting

Mengmeng Dou, Jiashu Zhang

TL;DR

This work extends Besov space theory to the Dunkl setting, where the Fourier transform is unavailable, by constructing a discrete Calderón reproducing formula using the Dunkl-Poisson kernel that blends the Euclidean and Dunkl metrics. It introduces a new test/distribution framework and a Dunkl-Besov space $igdot{{f B}}_{p,d}^{oldsymbol{ mu},q}(doldsymbol{ u})$, proves norm equivalences with existing Besov models, and establishes duality via explicit index relations. A weak-type reproducing formula in distributions and completeness results are shown, enabling a robust Besov calculus in Dunkl harmonic analysis. The paper also proves boundedness of Dunkl-Calderón-Zygmund operators on these spaces under standard cancellation hypotheses, linking the new Besov theory to operator bounds in this noncommutative setting.

Abstract

The purpose of this paper is to characterize the homogeneous Besov space in the Dunkl setting. We utilize a new discrete reproducing formula, that is, the building blocks are differences of the Dunkl-Poisson kernel which involves both the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the Dunkl setting, new test functions and distributions are introduced, and a new decomposition is established.

Homogeneous Besov Spaces in Dunkl setting

TL;DR

This work extends Besov space theory to the Dunkl setting, where the Fourier transform is unavailable, by constructing a discrete Calderón reproducing formula using the Dunkl-Poisson kernel that blends the Euclidean and Dunkl metrics. It introduces a new test/distribution framework and a Dunkl-Besov space , proves norm equivalences with existing Besov models, and establishes duality via explicit index relations. A weak-type reproducing formula in distributions and completeness results are shown, enabling a robust Besov calculus in Dunkl harmonic analysis. The paper also proves boundedness of Dunkl-Calderón-Zygmund operators on these spaces under standard cancellation hypotheses, linking the new Besov theory to operator bounds in this noncommutative setting.

Abstract

The purpose of this paper is to characterize the homogeneous Besov space in the Dunkl setting. We utilize a new discrete reproducing formula, that is, the building blocks are differences of the Dunkl-Poisson kernel which involves both the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the Dunkl setting, new test functions and distributions are introduced, and a new decomposition is established.
Paper Structure (7 sections, 10 theorems, 164 equations)

This paper contains 7 sections, 10 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete reproducing formula in $L^2\cap \dot{B}_{p,d}^{\alphaup,q}$
  4. Duality estimate
  5. Dunkl-Besov space
  6. Boundedness of Dunkl-Calderón-Zygmund operators
  7. Declarations

Key Result

Theorem 1.1

Suppose that $|\alphaup|<1$, $\max\left\{\frac{N}{N+1},\frac{N}{N+1+\alphaup}\right\}< p<\infty$ and $0<q<\infty$. If $f\in L^2(d\omegaup)$ with ${\|f\|_{\dot{B}_{p,d}^{\alphaup,q}}~<\infty}$, then there exists a function $h\in L^2(d\omegaup)$, such that $\|f\|_{2}\sim\|h\|_{2}$, $\|f\|_{\dot{B}_{p, where the series converges in the $L^2(d\omegaup)$ norm and the Dunkl-Besov space norm.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.2
  • Theorem 1.4
  • Definition 1.3
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • ...and 11 more