﻿ 双曲空间中的Moser-Trudinger不等式

# 双曲空间中的Moser-Trudinger不等式Moser-Trudinger Inequalities in Hyperbolic Spaces

Abstract: In this paper, the Moser-Trudinger inequality is extended to hyperbolic space by using the technique of level set segmentation, non increasing rearrangement theory and O’Neil’s lemma. The results generalize and improve the recent results.

$g\left(x\right)=\frac{4}{1-{|x|}^{2}}|{}^{2}\underset{i=1}{\overset{n}{\sum }}d{x}_{i}^{2}$.

$\text{d}Vl{o}_{g}=\frac{{2}^{n}}{{\left(1-{|x|}^{2}\right)}^{2}}\text{d}x$.

$x\in {B}^{n}$，用 $\rho \left(x\right)=d\left(x,0\right)=\mathrm{ln}\frac{1+|x|}{1-|x|}$，表示从x到0的测地距离， $r>0$ 时，用 ${B}_{g}\left(0,r\right)$ 表示球心在原点 半径为 $r>0$ 的测地开球。仍然用 $\nabla$ 表示 ${R}^{n}$ 中Euclidean梯度，同时用 $〈.,.〉$ 表示 ${R}^{n}$ 中的标准内积，相对于距离g，在任何切向空间中，双曲梯度 ${\nabla }_{g}$ 和内积 ${〈.,.〉}_{g}$ 为如下形式：

${\nabla }_{g}=\frac{{\left(1-{|x|}^{2}\right)}^{2}}{4}\nabla ,\text{}{〈.,.〉}_{g}=\frac{4}{{\left(1-{|x|}^{2}\right)}^{2}}〈.,.〉$.

${\int }_{{B}^{n}}{|{\nabla }_{g}u|}_{g}^{n}\text{d}{V}_{0}{l}_{g}={\int }_{{B}^{n}}{|\nabla u|}^{n}\text{d}x$. (1.1)

${V}_{0}{l}_{g}\left(\left\{x\in {H}^{n}:|u\left(x\right)|>t\right\}\right)={\int }_{\left\{x\in {H}^{n}:|u\left(x\right)|>t\right\}}\text{d}{V}_{0}{l}_{g}<\infty ,\text{}\forall t>0$.

${\mu }_{u}={V}_{0}{l}_{g}\left(\left\{x\in {H}^{n}:|u\left(x\right)|>t\right\}\right),\text{}t>0$.

${u}^{\ast }\left(t\right)=\mathrm{sup}\left\{s>0:{\mu }_{u}\left(s\right)>t\right\}$.

${u}_{g}^{#}={u}^{\ast }\left({V}_{0}{l}_{g}\left({B}_{g}\left(0,\rho \left(x\right)\right)\right)\right),\text{}x\in {B}^{n}$. (1.2)

${u}_{g}^{#}={u}^{\ast }\left({\sigma }_{n}{|x|}^{n}\right),\text{}x\in {B}^{n}$. (1.3)

${\int }_{{B}^{n}}\Phi \left(|u|\right)\text{d}{V}_{0}{l}_{g}={\int }_{{B}^{n}}\Phi \left({u}_{g}^{#}\right)\text{d}{V}_{0}{l}_{g}={\int }_{{R}^{n}}\Phi \left({u}_{E}^{#}\right)\text{d}x={\int }_{0}^{\infty }\Phi \left({u}^{\ast }\left(t\right)\right)\text{d}t$. (1.4)

${\int }_{{B}^{n}}{|{\nabla }_{g}{u}_{g}^{#}|}_{g}^{n}\text{d}{V}_{0}{l}_{g}\le {\int }_{{B}^{n}}{|{\nabla }_{g}u|}_{g}^{n}\text{d}{V}_{0}{l}_{g}$.

${\int }_{{B}^{n}}\frac{\varphi \left({\alpha }_{n}\left(1-\frac{\beta }{\alpha }\right){|u|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }J\left(\theta ,\rho \right)}\text{d}Vo{l}_{g}<\infty$.

$\begin{array}{c}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{\alpha }\right){|u|}^{\frac{n}{n-1}}\right)}{\rho {\left(x\right)}^{\beta }J\left(\theta ,\rho \right)}\text{d}V\le {\int }_{\Omega \left(u\right)}\frac{\mathrm{exp}\left({\alpha }_{n}\left(1-\frac{\beta }{\alpha }\right){|u|}^{\frac{n}{n-1}}\right)}{\rho {\left(x\right)}^{\beta }J\left(\theta ,\rho \right)}\text{d}V+{\int }_{\beta \\Omega \left(u\right)}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{\alpha }\right){|u|}^{\frac{n}{n-1}}\right)}{\rho {\left(x\right)}^{\beta }J\left(\theta ,\rho \right)}\text{d}V\\ \le I+II\end{array}$ (1.5)

$\begin{array}{c}{\int }_{\beta \\Omega \left(u\right)}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{\rho {\left(x\right)}^{\beta }J\left(\theta ,\rho \right)}\text{d}V\underset{˜}{<}{\int }_{{B}^{n}\\Omega \left(u\right)}\frac{{|u|}^{n}}{\rho {\left(x\right)}^{\beta }}\text{d}V\underset{˜}{<}{\int }_{\left\{|u|<1\right\}\cap \left\{\rho \le {‖u‖}_{n}\right\}}\frac{{|u|}^{n}}{\rho {\left(x\right)}^{\beta }}\text{d}V+{\int }_{\left\{|u|<1\right\}\cap \left\{\rho \ge {‖u‖}_{n}\right\}}\frac{{|u|}^{n}}{\rho {\left(x\right)}^{\beta }}\text{d}V\\ \underset{˜}{<}{\int }_{\left\{|u|<1\right\}\cap \left\{\rho \le {‖u‖}_{n}\right\}}\frac{1}{\rho {\left(x\right)}^{\beta }}\text{d}V+{\int }_{\left\{|u|<1\right\}\cap \left\{\rho \ge {‖u‖}_{n}\right\}}\frac{{|u|}^{n}}{\rho {\left(x\right)}^{\beta }}\text{d}V\\ \underset{˜}{<}{\int }_{0}^{{‖u‖}_{n}}\frac{{\mathrm{sinh}}^{3}t}{{t}^{\beta }}\text{d}t+\frac{1}{{‖u‖}_{n}^{\beta }}{\int }_{\left\{|u|<1\right\}\cap \left\{\rho \ge {‖u‖}_{n}\right\}}{u}^{n}\text{d}V\le C\end{array}$

${\int }_{\Omega \left(u\right)}\frac{\mathrm{exp}\left({\alpha }_{n}\left(1-\frac{\beta }{\alpha }\right){|u|}^{\frac{n}{n-1}}}{\rho {\left(x\right)}^{\beta }J\left(\theta ,\rho \right)}$.

$u=u\ast {\left(|{\nabla }_{g}|-{\lambda }^{\frac{1}{n}}\right)}^{n}=v\ast {\phi }_{1}$

$\begin{array}{c}I={\int }_{0}^{|\Omega |}\text{exp}\left(\left(1-\frac{\beta }{n}\right){\alpha }_{n}{u}^{\ast }{\left(t\right)}^{\frac{n}{n-1}}\right){g}^{\ast }\left(t\right)\text{d}t\\ \le {\int }_{0}^{\Omega }\mathrm{exp}\left(\left(1-\frac{\beta }{n}\right){\alpha }_{n}|\frac{1}{t}{\int }_{0}^{t}{v}^{\ast }\left(s\right)\text{d}s{\int }_{0}^{t}{\phi }_{1}^{\ast }\left(s\right)\text{d}s+{\int }_{0}^{\infty }{v}^{\ast }\left(s\right){\phi }_{1}^{\ast }\left(s\right)\text{d}s|{}^{\frac{n}{n-1}}\right){g}^{\ast }\left(t\right)\text{d}t\\ \le {\int }_{0}^{\Omega }{\int }_{0}^{+\infty }\mathrm{exp}\left(\left(1-\frac{\beta }{n}\right){\alpha }_{n}|\frac{1}{{\Omega }_{0}{\text{e}}^{-t}}{\int }_{0}^{{\Omega }_{0}{\text{e}}^{-t}}{v}^{\ast }\left(s\right)\text{d}s{\int }_{0}^{{\Omega }_{0}{\text{e}}^{-t}}{\phi }_{1}^{\ast }\left(s\right)\text{d}s+{\int }_{{\Omega }_{0}{\text{e}}^{-t}}^{\infty }{v}^{\ast }\left(s\right){\phi }_{1}^{\ast }\left(s\right)\text{d}s|{}^{\frac{n}{n-1}}\right){g}^{\ast }\left(t\right)\text{d}t\\ ={\Omega }_{0}{\int }_{0}^{+\infty }{\text{e}}^{-F\left(t\right)}\text{d}t\end{array}$

$F\left(t\right)=t-\left(1-\frac{\beta }{\alpha }\right){\alpha }_{n}|\frac{1}{{\Omega }_{0}{\text{e}}^{-t}}{\int }_{0}^{{\Omega }_{0}{\text{e}}^{-t}}{v}^{\ast }\left(s\right)\text{d}s{\int }_{0}^{{\Omega }_{0}{\text{e}}^{-t}}{\phi }_{1}^{\ast }\left(s\right)\text{d}s+{\int }_{{\Omega }_{0}{\text{e}}^{-t}}^{\infty }{v}^{\ast }\left(s\right){\phi }_{1}^{\ast }\left(s\right)\text{d}s|{}^{\frac{n}{n-1}}-\mathrm{ln}{g}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)$

$\begin{array}{l}\psi \left(t\right)=\sqrt{{\Omega }_{0}{\text{e}}^{-t}}{v}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right),\\ \phi \left(t\right)=\sqrt{{\alpha }_{n}{\Omega }_{0}{\text{e}}^{-t}}{\phi }_{1}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)\end{array}$

$F\left(t\right)$ 可以表示成如下的形式：

$\begin{array}{c}F\left(t\right)=t-\left(1-\frac{\beta }{n}\right){\left({\text{e}}^{t}{\int }_{t}^{\infty }{\text{e}}^{-\frac{s}{2}}\psi \left(s\right)\text{d}s{\int }_{t}^{\infty }{\text{e}}^{-\frac{s}{2}}\phi \left(s\right)\text{d}s+{\int }_{-\infty }^{t}\psi \left(s\right)\phi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}-\mathrm{ln}{g}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)\\ =t-\left(1-\frac{\beta }{n}\right){\left({\int }_{-\infty }^{+\infty }a\left(s,t\right)\phi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}-\mathrm{ln}{g}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)\end{array}$

$a\left(s,t\right)$ 为如下形式

$a\left(s,t\right)=\left\{\begin{array}{l}\phi \left(s\right),\text{}st\hfill \end{array}$

1) 存在与 $\phi$ 有关的常数满足 ${\mathrm{inf}}_{t\ge 0}F\left(t\right)\ge -C$

2) 设 ${E}_{\lambda }=\left\{t\ge 0:F\left(t\right)\le \lambda \right\}$，那么存在依赖于 $\phi$ 的两个常数 ${C}_{1}$${C}_{2}$ 满足

$|{E}_{\lambda }|\le {C}_{1}|\lambda |+{C}_{2}$.

${\left({\int }_{-\infty }^{+\infty }\alpha \left(s,t\right)\psi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}\le t+C$.

$\begin{array}{c}F\left(t\right)=t-\left(1-\frac{\beta }{n}\right){\left({\int }_{-\infty }^{+\infty }a\left(s,t\right)\psi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}-\mathrm{ln}{g}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)\\ \ge t-\left(1-\frac{\beta }{n}\right)\left(t+C\right)+\frac{\beta }{n}\left(\mathrm{ln}\frac{n{\Omega }_{0}}{{\omega }_{n-1}}-t\right)\\ =\left(\frac{\beta }{n}-1\right)+\frac{\beta }{n}\mathrm{ln}\frac{n{\Omega }_{0}}{{\omega }_{n-1}}\\ =C\end{array}$

1) 得证

$\begin{array}{c}{t}_{2}-\lambda \le \left(1-\frac{\beta }{n}\right){\left({\int }_{-\infty }^{+\infty }a\left(s,t\right)\psi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}+\mathrm{ln}{g}^{\ast }\left({\Omega }_{0}{\text{e}}^{-t}\right)\\ \le \left(1-\frac{\beta }{n}\right){\left({\int }_{-\infty }^{+\infty }a\left(s,t\right)\psi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}-\frac{\beta }{n}\left(\mathrm{ln}\frac{n{\Omega }_{0}}{{\omega }_{3}}-{t}_{2}\right)\end{array}$

${t}_{2}-\lambda \le \left(1-\frac{\beta }{n}\right){\left({\int }_{-\infty }^{+\infty }a\left(s,t\right)\psi \left(s\right)\text{d}s\right)}^{\frac{n}{n-1}}-\frac{\beta }{n}\left(\mathrm{ln}\frac{n{\Omega }_{0}}{{\omega }_{3}}-{t}_{2}\right)$

$\underset{u\in {W}^{1,n}\left({H}^{n}\right),{‖{\nabla }_{g}u‖}_{n,g}^{n}-\lambda {‖u‖}_{n,g}^{n}\le 1}{\mathrm{sup}}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}<\infty$ (1.6)

$\frac{1}{{\rho }^{\beta }}\le 1+\frac{1}{{\rho }^{\beta }J\left(\theta ,\rho \right)}\cdot \underset{\theta \in {R}^{n-1},\rho \in \left[0,1\right]}{J}\left(\theta ,\rho \right)\le \left(1+\frac{1}{{\rho }^{\beta }J\left(\theta ,\rho \right)}\right)C$.

$\begin{array}{l}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}\\ \le {\int }_{{B}^{n}}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|{u}_{g}^{#}\left(x\right)|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}\\ \le c{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|{u}_{g}^{#}|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}+C{\int }_{{B}^{n}}{\varphi }_{n}\left({\alpha }_{n}\left(1-\frac{\beta }{n}\right){|{u}_{g}^{#}|}^{\frac{n}{n-1}}\right)\text{d}{V}_{0}{l}_{g}\\ =C\left(I+II\right)\end{array}$

$\underset{u\in {W}^{1,n}\left({H}^{n}\right),{‖{\nabla }_{g}u‖}_{n,g}^{n}-\frac{n-1}{n}{‖u‖}_{n,g}^{n}\le 1}{\mathrm{sup}}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}\le C\left({‖u‖}_{n,g}^{n-\beta }+{‖u‖}_{n,g}^{n}\right)$ (1.7)

$\begin{array}{l}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}\\ \le C{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}{\rho }^{\beta }J\left(\theta ,\rho \right)}\text{d}{V}_{0}{l}_{g}+C{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}}\text{d}{V}_{0}{l}_{g}\\ =C\left({I}_{1}+{I}_{2}\right)\end{array}$

$\begin{array}{c}{I}_{2}={\int }_{{B}^{n}}\frac{\varphi \left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}}\text{d}{V}_{0}{l}_{g}\\ \le {\int }_{{B}^{n}}\frac{\varphi \left(\alpha {|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}}\text{d}{V}_{0}{l}_{g}\\ \le {‖u‖}_{n,g}^{n}\end{array}$

$J\left(\theta ,\rho \right)={\left(\frac{\mathrm{sinh}\rho }{\rho }\right)}^{n-1}$，令 $g\left(\rho \right)=\frac{1}{{\rho }^{\beta }J\left(\theta ,\rho \right)}$ 那么 ${g}^{\ast }\left(t\right)={\left(\frac{nt}{{\omega }_{n-1}}\right)}^{-\frac{\beta }{n}}$，再令 $h\left(x\right)=\frac{1}{{|x|}^{\beta }}$，那么 ${h}^{\ast }\left(t\right)={\left(\frac{nt}{{\omega }_{n-1}}\right)}^{-\frac{\beta }{n}}$，由1.4和 [14] 可得当 $\alpha \le {\alpha }_{n}$

$\begin{array}{c}{I}_{1}\le {\int }_{{R}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){u}_{e}^{#\frac{n}{n-1}}\right)}{{\left(1+|{u}_{e}^{#}|\right)}^{\frac{n}{n-1}}{|x|}^{\beta }}\text{d}x\\ \le {\int }_{{R}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){u}_{e}{}^{#\frac{n}{n-1}}\right)}{{\left(1+|{u}_{e}^{#}|\right)}^{\frac{n}{n-1}\left(1-\frac{\beta }{n}\right)}{|x|}^{\beta }}\text{d}x\\ \le C{‖{u}_{e}^{#}‖}_{n}^{n-\beta }\\ \le C{‖u‖}_{n,g}^{n-\beta }\end{array}$

${\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}{\rho }^{\beta }}\text{d}{V}_{0}{l}_{g}\le C\left({‖u‖}_{n,g}^{n-\beta }+{‖u‖}_{n,g}^{n}\right)$

$\underset{u\in {W}^{1.n}\left({H}^{n}\right),{‖{\nabla }_{g}u‖}_{n,g}^{n}-{\left(\frac{n-1}{n}\right)}^{n}{‖u‖}_{n,g}^{n}\le 1}{\mathrm{sup}}\frac{1}{{‖u‖}_{n,g}^{n-\beta }+{‖u‖}_{n,g}^{n}}{\int }_{{B}^{n}}\frac{{\varphi }_{n}\left(\alpha \left(1-\frac{\beta }{n}\right){|u|}^{\frac{n}{n-1}}\right)}{{\left(1+|u|\right)}^{\frac{n}{n-1}}{\rho }^{\beta }}<\infty$

NOTES

*通讯作者。

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