Functional inequalities on symmetric spaces of noncompact type and applications

TL;DR

This paper systematically studies functional inequalities on symmetric spaces of noncompact type, establishing key inequalities like Stein-Weiss and applying them to analyze nonlinear PDEs, including global existence results for wave equations.

Contribution

It introduces the first systematic derivation of Stein-Weiss and related inequalities on symmetric spaces of noncompact type, and demonstrates their applications to nonlinear PDEs.

Findings

Established Stein-Weiss inequality on symmetric spaces

Derived Hardy-Sobolev and Gagliardo-Nirenberg inequalities in this setting

Proved global existence results for semilinear wave equations with damping

Abstract

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein-Weiss inequality, also known as a weighted Hardy-Littlewood-Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein-Weiss inequality, we deduce Hardy-Sobolev, Hardy-Littlewood-Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace-Beltrami operator on symmetric spaces.