The sharp Sobolev type inequalities in the Lorentz--Sobolev spaces in the hyperbolic spaces (2001.04018v1)
Abstract: Let $W1L{p,q}(\mathbb Hn)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb Hn$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincar\'e inequality in $W1L{p,q}(\mathbb Hn)$ 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 Poincar\'e-Sobolev type inequalities in $W1L{p,q}(\mathbb Hn)$ 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 $W1L{n,q}(\mathbb{H}n)$ 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 P\'olya--Szeg\"o type principle in $W1 L{p,q}(\mathbb Hn)$ with $1\leq q \leq p$ which maybe is of independent interest.