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

Abstract

In this note, we establish a $L^p-$version of the Poincaré--Sobolev inequalities in the hyperbolic spaces ${\mathbb{H}}^n$. The interest of this result is that it relates both the Poincaré (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 Poincaré--Gagliardo--Nirenberg and Poincaré--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 ${\mathbb{H}}^2$.

14 pages, to appear in Journal of Mathematical Analysis and Applications

References in corpus (4)

• Improved L${}^p$-Poincaré inequalities on the hyperbolic space
• Sharp Lp-entropy inequalities on manifolds
• Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces
• Sharp constant in Riemannian L^p-Gagliardo-Nirenberg inequalities