Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications
Kassymov A. Kumar V. Ruzhansky M.
July 2024Springer
Journal of Geometric Analysis
2024#34Issue 7
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.
22E30 , 26D10 , 35A23 45J05 , 35L05 43A90 , 43A85 , Caffarelli–Kohn–Nirenberg inequality , Gagliardo–Nirenberg inequality , Global wellposedness , Hardy inequality , Hardy–Littlewood–Sobolev inequality , Laplace–Beltrami operator , Nonlinear wave equations , Riesz potential , Sobolev inequality , Stein–Weiss inequality , Symmetric spaces of noncompact type
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
Department of Mathematics: Analysis
Institute of Mathematics and Mathematical Modeling
School of Mathematical Sciences
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