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Riesz transforms and multipliers for the Bessel-Grushin operator

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Abstract

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.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, 38271, La Laguna (Sta. Cruz de Tenerife), Spain

    Victor Almeida & Jorge J. Betancor

  2. Department of Mathematics, Uppsala University, S-751 06, Uppsala, Sweden

    Alejandro J. Castro

  3. Departamento de Matemáticas, Universidad de las Palmas de Gran Canaria, Campus de Tafira Baja, 35017, Las Palmas de Gran Canaria, Spain

    Kishin Sadarangani

Authors
  1. Victor Almeida
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  2. Jorge J. Betancor
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  3. Alejandro J. Castro
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  4. Kishin Sadarangani
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Corresponding author

Correspondence to Victor Almeida.

Additional information

The first, second, and the fourth authors are partially supported by MTM2013/44357-P.

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Almeida, V., Betancor, J.J., Castro, A.J. et al. Riesz transforms and multipliers for the Bessel-Grushin operator. JAMA 128, 51–106 (2016). https://doi.org/10.1007/s11854-016-0002-3

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  • DOI: https://doi.org/10.1007/s11854-016-0002-3

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