Sharp Adams-Moser-Trudinger type inequalities in the hyperbolic space

2016/06/22 by Ngô, Quôc-Anh, Nguyen, Van Hoang
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1606.07094

Abstract

The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \mathbb Hⁿ. First, we prove a sharp Adams inequality of order two with the exact growth condition in \mathbb Hⁿ. Then we use it to derive a sharp Adams-type inequality and an Adachi-Tanaka-type inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \mathbb Hⁿ, which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \mathbb Hⁿ. Our proofs rely on the symmetrization method extended to hyperbolic spaces.

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