论文标题
非恰当类型和应用的对称空间的功能不平等
Functional inequalities on symmetric spaces of noncompact type and applications
论文作者
论文摘要
本文的目的是开始系统地研究对非级别类型较高等级类型的对称空间的功能不平等。这项研究的第一个主要目标是建立Stein-Weiss的不平等，也称为加权强硬的木材 - 索伯夫不平等，以实现非脉动类型的对称空间的Riesz潜力。这是通过在对称空间上使用多面体距离的精细估计来实现的，并通过将Ruzhansky和Verma与Anker和JI获得的非合理类型的对称空间的尖锐的Bessel-Green-Riesz kernel估计相结合。由于Stein-Weiss的不平等，我们推断出Hardy-Sobolev，Hardy-Littlewood-Sobolev，Gagliardo-Nirenberg和Caffarelli-Kohn-Nirenberg不平等的不平等现象。本文的第二个主要目的是显示上述不平等现象在对称空间上研究非线性PDE的应用。具体而言，我们表明，Gagliardo-Nirenberg的不等式可用于为在对称空间上的Laplace-Beltrami操作员进行阻尼和质量项的半线性波方程的小数据全球存在结果。
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.
