Improved Moser-Trudinger type inequalities in the hyperbolic space ${\mathbb{H}}^n$

Abstract

We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space ${\mathbb{H}}^n$, $n\ge 2$. Namely, we prove the following result: for any $0\le \lambda <{\left(\frac{n-1}{n}\right)}^n$, then we have $$\underset{\begin{array}{c}u\in C_0^{\infty }({\mathbb{H}}^n){\int }_{{\mathbb{H}}^n}|{\nabla }_{g}u{|}_g^{n}d{\text{Vol}}_g-\lambda {\int }_{{\mathbb{H}}^n}|u{|}^{n}d{\text{ Vol}}_g\le 1\end{array}}{\sup }{\int }_{{\mathbb{H}}^n}{\Phi }_n({\alpha }_n|u{|}^{\frac{n}{n-1}})d{\text{ Vol}}_g<\infty ,$$ where ${\alpha }_n=n{\omega }_{n-1}^{\frac{1}{n-1}}$, ${\omega }_{n-1}$ denotes the surface area of the unit sphere in ${\mathbb{R}}^n$ and ${\Phi }_n(t)=e^t-{\sum }_{j=0}^{n-2}\frac{t^j}{j!}$. This improves the Moser-Trudinger inequality in hyperbolic spaces obtained recently by Mancini and Sandeep, by Mancini, Sandeep and Tintarev and by Adimurthi and Tintarev. In the limiting case $\lambda =(\frac{n-1}{n}{)}^n$, we prove a Moser-Trudinger inequality with exact growth in ${\mathbb{H}}^n$, $$\underset{\begin{array}{c}u\in C_0^{\infty }({\mathbb{H}}^n){\int }_{{\mathbb{H}}^n}|{\nabla }_{g}u{|}_g^{n}d{\text{ Vol}}_g-(\frac{n-1}{n}{)}^n{\int }_{{\mathbb{H}}^n}|u{|}^{n}d{\text{ Vol}}_g\le 1\end{array}}{\sup }\frac{1}{{\int }_{{\mathbb{H}}^n}|u{|}^{n}d{\text{ Vol}}_g}{\int }_{{\mathbb{H}}^n}\frac{{\Phi }_n({\alpha }_n|u{|}^{\frac{n}{n-1}})}{(1+|u|{)}^{\frac{n}{n-1}}}d{\text{ Vol}}_g<\infty .$$ This improves the Moser-Trudinger inequality with exact growth in ${\mathbb{H}}^n$ established by Lu and Tang.

14 pages, comment are welcome, update references, to appear in Nonlinear Analysis

References in corpus (1)

• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space