The sharp Poincar\'e--Sobolev type inequalities in the hyperbolic spaces ${\mathbb{H}}^n$

In this note, we establish a $L^p-$version of the Poincar'e–Sobolev inequalities in the hyperbolic spaces ${\mathbb{H}}^n$. The interest of this result is that it relates both the Poincar'e (or Hardy) inequality and the Sobolev inequality with the sharp constant in ${\mathbb{H}}^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincar'e–Gagliardo–Nirenberg and Poincar'e–Morrey–Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. Finally, we discuss several other Sobolev inequalities in the hyperbolic spaces ${\mathbb{H}}^n$ which generalize the inequalities due to Mugelli and Talenti in ${\mathbb{H}}^2$.