论文标题
Lorentz-Sobolev空间中的Sobolev型不平等现象
The sharp Sobolev type inequalities in the Lorentz--Sobolev spaces in the hyperbolic spaces
论文作者
论文摘要
令$ w^1l^{p，q}（\ mathbb h^n）$，$ 1 \ leq q，p <\ infty $表示lorentz-sobolev spaces of Order One的lorentz-sobolev空间$ \ mathbb h^n $。我们在本文中的目标是三个方面。首先，我们用$ w^1l^{p，q}（\ Mathbb h^n）$建立了一个尖锐的庞加莱不平等，并用$ 1 \ leq q \ leq p $ p $，在\ cite {ngonguyenamv}中概括到lorentz-sobolev空间的设置中。其次，我们在$ W^1l^{p，q}中证明了几种尖锐的Poincaré-Sobolev类型不等式（\ Mathbb H^n）$，$ 1 \ leq Q \ leq Q \ leq P <n $将结果推广到\ cite {nguyenps2018}中，以{nguyenps2018}到Lorentz-sobolev Space的设置。最后，我们将改进的Moser-Trudinger类型不平等为$ W^1l^{n，q}（\ Mathbb {h}^n $中的$ p = n $，$ 1 \ leq q \ leq n $在\ cite \ cite {nguyenmt2018}中概括，并改善了结果{y \ C.为了证明主要结果，我们将在$ w^1 l^{p，q}中证明pólya-szegö类型原理（\ mathbb h^n）$，$ 1 \ leq q \ leq q \ leq p $，这可能是独立的。
Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p$ which generalizes the result in \cite{NgoNguyenAMV} to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincaré-Sobolev type inequalities in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p < n$ which generalize the results in \cite{NguyenPS2018} to the setting of Lorentz-Sobolev spaces. Finally, we provide the improved Moser-Trudinger type inequalities in $W^1L^{n,q}(\mathbb{H}^n)$ in the critical case $p= n$ with $1\leq q \leq n$ which generalize the results in \cite{NguyenMT2018} and improve the results in \cite{YangLi2019}. In the proof of the main results, we shall prove a Pólya--Szegö type principle in $W^1 L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p$ which maybe is of independent interest.
