Riesz transforms on compact Riemannian symmetric spaces of rank one

TL;DR

This paper establishes sharp mixed norm estimates for Riesz transforms on rank-one compact Riemannian symmetric spaces, leveraging kernel bounds and extrapolation techniques, advancing harmonic analysis on these geometric structures.

Contribution

It provides the first uniform kernel estimates for Jacobi--Riesz transforms and adapts Rubio de Francia's extrapolation theorem to this setting.

Findings

Sharp mixed norm estimates for Riesz transforms on symmetric spaces

Uniform kernel bounds for Jacobi--Riesz transforms

Extension of extrapolation techniques to geometric analysis

Abstract

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi--Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia's extrapolation theorem. The latter results are of independent interest.