{ "id": "1802.08777", "version": "v1", "published": "2018-02-24T01:23:48.000Z", "updated": "2018-02-24T01:23:48.000Z", "title": "The sharp Poincaré--Sobolev type inequalities in the hyperbolic spaces $\\mathbb H^n$", "authors": [ "Van Hoang Nguyen" ], "comment": "14 pages, to appear in Journal of Mathematical Analysis and Applications", "categories": [ "math.FA", "math.AP" ], "abstract": "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 $\\mathbb H^2$.", "revisions": [ { "version": "v1", "updated": "2018-02-24T01:23:48.000Z" } ], "analyses": { "subjects": [ "26D10", "46E35" ], "keywords": [ "sharp poincaré-sobolev type inequalities", "hyperbolic space", "sobolev inequality", "euclidean space", "sharp sobolev inequalities" ], "note": { "typesetting": "TeX", "pages": 14, "language": "en", "license": "arXiv", "status": "editable" } } }