Analysis Related to All Admissible Type Parameters in the Jacobi Setting

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• 作者：Adam Nowak ; Peter Sj?gren ; Tomasz Z. Szarek
• 关键词：Jacobi expansion ; Jacobi–Poisson kernel ; Maximal operator ; Riesz transform ; Square function ; Spectral Multiplier ; Calderón–Zygmund operator ; Primary 42C05 ; Secondary 42C10
• 刊名：Constructive Approximation
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：41
• 期：2
• 页码：185-218
• 全文大小：408 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

We derive an integral representation for the Jacobi–Poisson kernel valid for all admissible type parameters \(\alpha ,\beta \) in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting that works for all possible \(\alpha \) and \(\beta \) . As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. The new Jacobi–Poisson kernel representation also leads to sharp estimates of this kernel. The paper generalizes methods and results existing in the literature but valid or justified only for a restricted range of \(\alpha \) and \(\beta \) .