Calder\'on-Zygmund operators related to Jacobi expansions

TL;DR

This paper establishes that key harmonic analysis operators related to Jacobi expansions are Calderón-Zygmund operators, enabling their boundedness properties to be derived from general theory, using explicit kernel formulas.

Contribution

It proves that several fundamental Jacobi-related operators are Calderón-Zygmund, providing a unified framework for their analysis based on explicit kernel formulas.

Findings

Operators are Calderón-Zygmund in the Jacobi setting

Mapping properties follow from general Calderón-Zygmund theory

Explicit Jacobi-Poisson kernel formula derived

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) Calder\'on-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.