Fractional integrals on compact Riemannian symmetric spaces of rank one

TL;DR

This paper investigates the boundedness of fractional integral operators on compact rank-one symmetric spaces, providing sharp estimates for their kernels and analyzing weighted inequalities related to Jacobi polynomial expansions.

Contribution

It introduces a new sharp estimate for Jacobi fractional integral kernels with explicit parameter dependence, advancing understanding of fractional integrals on symmetric spaces.

Findings

Established boundedness results for fractional integrals on symmetric spaces.

Derived explicit sharp estimates for Jacobi fractional integral kernels.

Analyzed weighted inequalities for operators associated with Jacobi polynomial expansions.

Abstract

In this paper we study mixed norm boundedness for fractional integrals related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.