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Calderón-Zygmund Operators Related to Jacobi Expansions

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Abstract

We study several fundamental operators in harmonic analysis related to Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein square functions. We show that these 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. Our proofs rely on an explicit formula for the Jacobi-Poisson kernel, which we derive from a product formula for Jacobi polynomials.

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Acknowledgements

The first-named author was supported in part by MNiSW Grant N N201 417839.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warszawa, Poland

    Adam Nowak

  2. Institute of Mathematics and Computer Science, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370, Wrocław, Poland

    Adam Nowak

  3. Mathematical Sciences, University of Gothenburg and Mathematical Sciences, Chalmers University of Technology, 412 96, Göteborg, Sweden

    Peter Sjögren

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  1. Adam Nowak
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  2. Peter Sjögren
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Correspondence to Adam Nowak.

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Communicated by Fulvio Ricci.

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Nowak, A., Sjögren, P. Calderón-Zygmund Operators Related to Jacobi Expansions. J Fourier Anal Appl 18, 717–749 (2012). https://doi.org/10.1007/s00041-012-9217-6

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  • DOI: https://doi.org/10.1007/s00041-012-9217-6

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