Dimension-Free Square Function Estimates for Dunkl Operators

Abstract

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in ${\mathbb{R}}^d$ is introduced by employing the full "gradient" induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in (1,\infty )$. For $p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in [2,\infty )$, we restrict to a particular case when the corresponding Weyl group is isomorphic to ${\mathbb{Z}}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry--Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.

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References in corpus (8)

• Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
• A chaotic representation property of the multidimensional Dunkl processes
• Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting
• Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms
• Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces
• Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$
• Upper and lower bounds for Littlewood-Paley square functions in the Dunkl setting
• Littlewood--Paley--Stein inequalities on $\text{RCD}(K,\infty )$ spaces

Cited by in corpus (4)

• Sharp Li--Yau Inequalities for Dunkl Harmonic Oscillators
• Littlewood--Paley--Stein Square Functions for the Fractional Discrete Laplacian on $\mathbb{Z}$
• Li--Yau Inequalities for Dunkl Heat Equations
• Weak Type Estimates for Square Functions of Dunkl Heat Flows