Riesz transforms and multipliers for the Bessel-Grushin operator

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• 作者：Victor Almeida ; Jorge J. Betancor ; Alejandro J. Castro…
• 刊名：Journal d'Analyse Math¨¦matique
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：128
• 期：1
• 页码：51-106
• 全文大小：489 KB
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• 作者单位：Victor Almeida (1)
  Jorge J. Betancor (1)
  Alejandro J. Castro (2)
  Kishin Sadarangani (3)
  
  1. Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, 38271, La Laguna (Sta. Cruz de Tenerife), Spain
  2. Department of Mathematics, Uppsala University, S-751 06, Uppsala, Sweden
  3. Departamento de Matemáticas, Universidad de las Palmas de Gran Canaria, Campus de Tafira Baja, 35017, Las Palmas de Gran Canaria, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Functional Analysis
  Dynamical Systems and Ergodic Theory
  Abstract Harmonic Analysis
  Partial Differential Equations
  
• 出版者：Hebrew University Magnes Press
• ISSN：1565-8538

文摘

We establish that the spectral multiplier \(m({G_\alpha })\) associated to the differential operator $$G_\alpha = - \Delta _x + \sum\limits_{j = 1}^m {\frac{{\alpha _j^2 - 1/4}} {{x_j^2 }} - \left| x \right|^2 \Delta _y on (0,\infty )^m \times \mathbb{R}^n ,} $$, which we call the Bessel-Grushin operator, is of weak type (1, 1) provided that M is in a suitable local Sobolev space. In order to do this, we prove a suitable weighted Plancherel estimate. Also, we study L p -boundedness properties of Riesz transforms associated to \({G_\alpha }\) in the case n = 1. The first, second, and the fourth authors are partially supported by MTM2013/44357-P.