根系相关正交展开的Littlewood-Paley理论
Littlewood-Paley theory for orthogonal expansions associated with root systems
- Universität Paderborn（帕德博恩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入紧致Heckman-Opdam拉普拉斯算子的非对称热与Poisson半群，证明Littlewood-Paley $g$-函数及Riesz变换、位势的$L^p$有界性，并统一了Jacobi展开等特例。
AI中文摘要：
我们引入了紧致情形下与Heckman-Opdam拉普拉斯算子相关的非对称热半群和Poisson半群。基于Poisson半群，我们以Stein对紧致李群的研究精神，研究了若干Littlewood-Paley $g$-函数，并证明了它们在$1<p<\infty$时的$L^{p}$有界性。作为应用，我们定义了相关的Riesz变换和Riesz位势，并证明了它们在$1<p<\infty$时的$L^{p}$连续性。通过对相关反射群作用取平均，我们的框架特别涵盖了Jacobi多项式展开及其相应直积情形的Littlewood-Paley-Stein理论作为特例。
英文摘要：
We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein's work for compact Lie groups and prove their $L^{p}$-boundedness for $1<p\leq 2$ and for some of them also for $1<p<\infty$. As an application, we define associated Riesz transforms and imaginary powers and prove their $L^{p}$-continuity for $1<p<\infty$. Passing to the average with respect to the action of the associated reflection group, we obtain $L^{p}$-boundedness of $g$-functions for the symmetric Poisson semigroup for all $1<p<\infty$. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.
