Adams inequality with exact growth in the hyperbolic space ℍ4 and Lions lemma

Abstract

In this article we prove Adams inequality with exact growth condition in the four dimensional hyperbolic space H 4 ,We will also establish an Adachi-Tanaka type inequality in this settings. Another aspect of this article is the P.L.Lions lemma in the hyperbolic space. We prove P.L.Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

Moser–Trudinger–Adams inequalities and related developments

“…There has been many more works on Moser-Trudinger and Adams in the Hyperbolic space, significantly improving the above results, for example [50,64,65,74].…”

Second order Sobolev type inequalities in the hyperbolic spaces

“…However, it is stated only for radial functions. In [14], an Adams inequality with exact growth in H 4 was proved for any function u on H 4 under the condition H 4 (P 2 u)udV g ≤ 1. Notice that H 4 (P 2 u)udV g is equivalent with a full Sobolev norm in H 4 which is different with our functional H 4 (∆ g u) 2 dV g − 81 16 H 4 u 2 dV g .…”

“…The choice of the sequence {ξ m } m is inspired by the sequence of Masmoudi and Sani [23]. Following the idea of Karmakar [14], we set…”

“…The choice of the sequence {ξ m } m is inspired by the sequence of Masmoudi and Sani [23]. Following the idea of Karmakar [14], we set ūm (x) = u m (3x) then the support of ūm is contained in {|x| ≤ 2 3 }. We can easily check that…”

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

“…Comparing with the inequality (1.10), our inequality (1.8) is stronger than the one of Lu and Tang. We refer the reader to [15,25,31,33,36] for the Adams inequality with exact growth both in the Euclidean and hyperbolic spaces.…”