The sharp Sobolev type inequalities in the Lorentz--Sobolev spaces in the hyperbolic spaces

Abstract

Let $W^1{L}^{p,q}({\mathbb{H}}^n)$, $1\le q,p<\infty$ 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 $W^1{L}^{p,q}({\mathbb{H}}^n)$ with $1\le q\le p$ which generalizes the result in \cite{NgoNguyenAMV} to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincaré-Sobolev type inequalities in $W^1{L}^{p,q}({\mathbb{H}}^n)$ with $1\le q\le p<n$ which generalize the results in \cite{NguyenPS2018} to the setting of Lorentz-Sobolev spaces. Finally, we provide the improved Moser-Trudinger type inequalities in $W^1{L}^{n,q}({\mathbb{H}}^n)$ in the critical case $p=n$ with $1\le q\le n$ 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 $1\le q\le p$ which maybe is of independent interest.

24 pages, comment are welcome. arXiv admin note: text overlap with arXiv:2001.04017, arXiv:2001.03950