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<p>In this work, we establish Plancherel-Pólya inequalities within the Dunkl setting associated with finite reflection groups on Euclidean space. The underlying geometric framework is governed by the standard Euclidean metric and the so-called Dunkl “metric” induced by the group action. We give Littlewood-Paley characterizations—via the Lusin area integrals and the Littlewood-Paley <i>g</i>-functions and, as applications, we obtain a new Littlewood-Paley theory tailored to Dunkl-Hardy spaces. The key tools developed include novel weak-type semi-discrete Calderón reproducing formulas in the Dunkl setting.</p>

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Plancherel-Pólya Inequality and its Applications in the Dunkl Setting

  • 期刊论文

  • 发表日期：

    2026年2月23日

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  • Yuying Chen,
  • Yanchang Han,
  • Yongsheng Han,
  • Chaoqiang Tan

摘要

In this work, we establish Plancherel-Pólya inequalities within the Dunkl setting associated with finite reflection groups on Euclidean space. The underlying geometric framework is governed by the standard Euclidean metric and the so-called Dunkl “metric” induced by the group action. We give Littlewood-Paley characterizations—via the Lusin area integrals and the Littlewood-Paley g-functions and, as applications, we obtain a new Littlewood-Paley theory tailored to Dunkl-Hardy spaces. The key tools developed include novel weak-type semi-discrete Calderón reproducing formulas in the Dunkl setting.