Calderón-Zygmund operators related to Jacobi expansions
Journal article, 2012

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.

Jacobi operator

Riesz transform

Jacobi polynomial

Jacobi expansion

Square function

Calderon-Zygmund operator

Maximal operator

Imaginary power

Jacobi-Poisson semigroup


Adam Nowak

Polish Academy of Sciences

Wrocław University of Science and Technology

Peter Sjögren

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Journal of Fourier Analysis and Applications

1069-5869 (ISSN) 15315851 (eISSN)

Vol. 18 4 717-749

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