Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$

TL;DR

This paper explores the properties of trigonometric Heckman-Opdam polynomials of type A1, establishing connections with ultraspherical polynomials, deriving explicit kernels, and developing harmonic analysis tools like convolutions and Hilbert transforms.

Contribution

It introduces a new convolution structure and fractional integrals for these polynomials, and defines a bounded Hilbert transform within this framework.

Findings

Derived explicit Poisson kernel expression.

Established a convolution structure via product formula.

Proved boundedness of the generalized Hilbert transform on L^p.

Abstract

In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product formula, we introduce a natural convolution structure and develop a theory of fractional integrals associated with these polynomials. We also define a generalized Hilbert transform in the framework of the Cherednik operator and prove its boundedness on $L^p$-spaces. This work provides an alternative perspective on the approach of B. Muckenhoupt and E.M. Stein \cite{MS}.