The sharp Poincaré-Sobolev type inequalities in the hyperbolic spaces Hn

In this note, we establish a Lp-version of the Poincaré-Sobolev inequalities in the hyperbolic spaces Hn. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in Hn. Our approach is based on the comparison of the Lp-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 Poincaré-Gagliardo-Nirenberg and Poincaré-Morrey-Sobolev inequalities in the hyperbolic spaces Hn. Finally, we discuss several other Sobolev inequalities in the hyperbolic spaces Hn which generalize the inequalities due to Mugelli and Talenti in H2.