Second order Sobolev type inequalities in the hyperbolic spaces

2018/05/05 by Nguyen, Van Hoang
#26D10 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1805.02055

Abstract

We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator Δ_g in the hyperbolic space \mathbb Hⁿ with n≥ 5. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in \mathbb Hⁿ. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for Δ_g in \mathbb Hⁿ. As a consequence, we obtain the Poincaré--Sobolev inequality for the second order GJMS operator P₂ in \mathbb Hⁿ. In dimension 4, we obtain an improvement of the sharp Adams inequality and an Adams inequality with exact growth for radial functions in \mathbb H⁴.

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