Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

2025/07/14 by Bui, The Anh, Mukherjee, Suman · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.10042

Abstract

We establish fractional Leibniz rules for the Dunkl Laplacian Δ_k of the form ‖(-Δ_k)^s(fg)‖_L^p(dμk) \lesssim ‖(-Δ_k)^s f‖_L^p1(dμk) ‖g‖_L^p2(dμk) + ‖f‖_L^p1(dμk) ‖(-Δ_k)^s g‖_L^p2(dμk). 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.

Citations

• Remarks on Dunkl translations of non-radial kernels
• The Kato-Ponce Inequality with Polynomial Weights
• Hörmander's multiplier theorem for the Dunkl transform
• Hörmander's multiplier theorem for the Dunkl transform
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• The Kato-Ponce Inequality
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• Dunkl operators: Theory and applications
• Differential-difference operators associated to reflection groups

Cited by

• Strichartz estimates for higher order Schrödinger equations with Partial regular initial data

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