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arXiv:1802.08777 [math.FA]AbstractReferencesReviewsResources

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

Van Hoang Nguyen

Published 2018-02-24Version 1

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 $\mathbb H^2$.

Comments: 14 pages, to appear in Journal of Mathematical Analysis and Applications
Categories: math.FA, math.AP
Subjects: 26D10, 46E35
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