Title: Functional inequalities on symmetric spaces of noncompact type and applications Show Chinese title

Title: 非紧类型对称空间上的函数不等式及其应用

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. Show Chinese abstract

Abstract: 本文的目的是开始对高阶非紧对称空间上的函数不等式进行系统研究。 本研究的第一个主要目标是建立Stein-Weiss不等式，也称为加权Hardy-Littlewood-Sobolev不等式，适用于非紧对称空间上的Riesz位势。 这是通过使用非紧对称空间上的多面体距离来执行基本球函数的精细估计，并结合Ruzhansky和Verma开发的积分Hardy不等式，以及Anker和Ji获得的非紧对称空间上的精确Bessel-Green-Riesz核估计来实现的。 作为Stein-Weiss不等式的推论，我们推导出非紧对称空间上的Hardy-Sobolev、Hardy-Littlewood-Sobolev、Gagliardo-Nirenberg和Caffarelli-Kohn-Nirenberg不等式。 本文的第二个主要目的是展示上述不等式在研究非紧对称空间上的非线性偏微分方程中的应用。 具体而言，我们表明Gagliardo-Nirenberg不等式可以用于建立带有阻尼和质量项的对称空间上Laplace-Beltrami算子的半线性波动方程的小数据全局存在结果。