Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The sharp Sobolev type inequalities in the Lorentz--Sobolev spaces in
  the hyperbolic spaces   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 Poincar\'e
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 Poincar\'e-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
P\'olya--Szeg\"o type principle in $W^1 L^{p,q}(\mathbb H^n)$ with $1\leq q
\leq p$ which maybe is of independent interest.

13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg