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<p>We establish Triebel–Lizorkin spaces in the Dunkl setting which are associated with finite reflection groups on the Euclidean space. The group structures induce two non-equivalent metrics: the Euclidean metric and the Dunkl metric. In this paper, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> space and the Dunkl–Calderón–Zygmund singular integral operator in the Dunkl setting play a fundamental role. The main tools used in this paper are as follows: (i) the Dunkl–Calderón–Zygmund singular integral operator and a new Calderón reproducing formula in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with the Triebel–Lizorkin space norms; (ii) new test functions in terms of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> functions and distributions; (iii) the Triebel–Lizorkin spaces in the Dunkl setting which are defined by the wavelet-type decomposition and the analogous atomic decomposition of the Hardy spaces.</p>

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Triebel–Lizorkin spaces in Dunkl setting

  • 期刊论文

  • 发表日期：

    2026年3月16日

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  • Chuhan Sun,
  • Zhiming Wang

摘要

We establish Triebel–Lizorkin spaces in the Dunkl setting which are associated with finite reflection groups on the Euclidean space. The group structures induce two non-equivalent metrics: the Euclidean metric and the Dunkl metric. In this paper, the \(L^2\) L 2 space and the Dunkl–Calderón–Zygmund singular integral operator in the Dunkl setting play a fundamental role. The main tools used in this paper are as follows: (i) the Dunkl–Calderón–Zygmund singular integral operator and a new Calderón reproducing formula in \(L^2\) L 2 with the Triebel–Lizorkin space norms; (ii) new test functions in terms of the \(L^2\) L 2 functions and distributions; (iii) the Triebel–Lizorkin spaces in the Dunkl setting which are defined by the wavelet-type decomposition and the analogous atomic decomposition of the Hardy spaces.