Adams inequality on the hyperbolic space

Abstract

In this article we establish the following Adams type inequality in the Hyperbolic space H N :As an application we prove the asymptotic behaviour of the best constants in Sobolev inequalities when 2k = N and also prove some existence results for the Q k curvature type equation in H N .MSC2010 Classification: 46E35, 26D10 Keywords: Adams inequality, Hyperbolic spacewhere Ω is a bounded domain in R N , and |Ω| denotes the volume of Ω and ω N −1 is the N − 1 dimensional measure of S N −1 . Moreover when †

Second order Sobolev type inequalities in the hyperbolic spaces

“…The Pólya-Szegö principle in H n asserts that if u ∈ W 1,p (H n ) then u ♯ ∈ W 1,p (H n ) and u ♯ W 1,p (H n ) ≤ u W 1,p (H n ) . Furthermore, the following Poincaré-Sobolev inequality holds (see [13] for p = 2 and [30] for general p ∈ (1, ∞))…”

“…where ∆ and ·, · denote the usual Laplace operator and the usual scalar product in R n respectively. The spectral gap of −∆ g on L 2 (H n ) is (n−1) 2 4 (see, e.g., [13,22]), i.e.,…”

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

“…See also [37,41,43] for the versions of the Moser-Trudinger inequality with a remainder term related to the metric of the Poincaré ball. We refer the reader to [14,26,36] for the higher order extensions of the Moser-Trudinger inequality (i.e., Adams inequality) in hyperbolic space.…”

The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces Hn

“…In [48], the author improves the inequality (1.5) by proving the following inequality sup u∈W 1,n (H n ), H n |∇gu| n g dVg −λ H n |u| n dVg≤1 H n Φ(α n |u| n n−1 )dV g < ∞, (1.6) for any λ < ( n−1 n ) n . The Adams inequality in the hyperbolic spaces were proved by Karmakar and Sandeep [26] in the following form…”