Lipschitz and Triebel--Lizorkin spaces, commutators in Dunkl setting

2023/07/02 by Han, Yongsheng, Lee, Ming-Yi, Li, Ji +1 · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2307.00502

Abstract

We first study the Lipschitz spaces Λ_d^β associated with the Dunkl metric, β∈(0,1), and prove that it is a proper subspace of the classical Lipschitz spaces Λ^β on \mathbb R^N, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces Λ^β connects to the Triebel--Lizorkin spaces F^α,q_p,\rm D associated with the Dunkl Laplacian \triangle_{\rm D} in \mathbb R^ N and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian \triangle_{\rm D}^-α/2, 0<α<N (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup e^-t\triangle_{\rm D}. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel--Lizorkin spaces F^α,q_p,\rm D.

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• Sparse domination for singular integral operators and their commutators in Dunkl setting with applications

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