Hardy--Littlewood inequalities for the Heckman--Opdam transform

TL;DR

This paper extends Hardy--Littlewood inequalities to the Heckman--Opdam transform, providing a more precise Hausdorff--Young inequality that generalizes previous results for symmetric spaces and spherical Fourier transforms.

Contribution

It introduces Hardy--Littlewood inequalities for the Heckman--Opdam transform, broadening the scope of harmonic analysis on root systems and symmetric spaces.

Findings

Established Hardy--Littlewood inequalities for the Heckman--Opdam transform.

Derived a more precise Hausdorff--Young inequality for the transform.

Generalized results from symmetric spaces to root datum settings.

Abstract

We establish Hardy--Littlewood inequalities for the Heckman--Opdam transform associated to a general root datum $(\mathfrak{a},\Sigma,m)$ that generalizes an analogous result for the spherical Fourier transform on a Riemannian symmetric space of the non-compact type due to Eguchi and Kumahara. In particular we obtain a more precise Hausdorff--Young inequality that generalizes a recent result due to Narayanan, Pasquale, and Pusti.