The sharp Poincaré--Sobolev type inequalities in the hyperbolic spaces \mathbb Hⁿ

2018/02/24 by Van Hoang Nguyen, Nguyen, Van Hoang
Mathematics · #26D10 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1802.08777

openalex publication_date 2018/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we establish a L^p-version of the Poincaré--Sobolev inequalities in the hyperbolic spaces \mathbb Hⁿ. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in \mathbb Hⁿ. Our approach is based on the comparison of the L^p-norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré--Gagliardo--Nirenberg and Poincaré--Morrey--Sobolev inequalities in the hyperbolic spaces \mathbb Hⁿ. Finally, we discuss several other Sobolev inequalities in the hyperbolic spaces \mathbb Hⁿ which generalize the inequalities due to Mugelli and Talenti in \mathbb H².

Citations

• Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces

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