Sharp Morrey-Sobolev inequalities on complete Riemannian manifolds
type: Item
instance of: scholarly article
title: Sharp Morrey-Sobolev inequalities on complete Riemannian manifolds
zbMATH Open document ID: 1321.58034
author: Alexandru Kristály
published in: Potential Analysis
publication date: 2015-02-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.1308
review text: The author recalls the well known result that \(W^{1,p}(\mathbb R^n)\), the Sobolev space of order one (such that \(p>n\)), is embedded into the space of bounded measurable functions, this embedding takes the name of Morrey Sobolev Inequality (MSI for short). In the present article, the author enhances the previous result, by giving a new embedding on a complete Riemannian manifold. Precisely, the author considers \((M,g)\), a smooth manifold, simply connected, complete, and endowed with a negative sectional curvature manifold, besides \((M,g)\) satisfies the Cartan-Hadamard conjecture, that is, roughly speaking, any compact domain with smooth boundary satisfies the isoperimetric inequality. Therefore, in the space of Lipschitz functions with compact support in \(M\), the author gives the MSI, besides, up to a multiplicative constant. The constant bound, \(C(p,n)\), is given in terms of the Hausdorff measure of a defined compact set. Also he provides necessary and sufficient conditions for \((M,g)\) to be isometric to the Euclidean space (Theorem 1.1). Concerning the second result, the author considers a smooth complete manifold equipped with a positive Ricci curvature and shows that the volume of a geodesic ball of radius \(\rho\) is lower bounded by a function comprised by a constant \(C\geq C(p,n)\), \(\rho\), and \(\omega_n\) (the volume of the Euclidean unit ball) (Theorem 1.2). Also through an interpolation, he gives versions of the previous results such that the constant bound is \(L_1\)-bound (Theorem 1.3).
reviewed by: Mohammed El Aïdi
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 6393741
zbMATH Keywords: Morrey-Sobolev inequality
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: sharp constant
zbMATH Keywords: curvature
zbMATH Keywords: rigidity
Wikidata QID: Q115381412
MaRDI profile type: Publication
OpenAlex ID: W1999359670
arXiv ID: 1408.1308
cites work: Problèmes isoperimetriques et espaces de Sobolev
cites work: Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature
cites work: Q4748596
cites work: Q5473394
cites work: Lipschitz spaces and Poincaré inequalities
cites work: A sharp four dimensional isoperimetric inequality
cites work: Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities
cites work: The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems
cites work: Q4719526
cites work: Normal and integral currents
cites work: Functional Inequalities: New Perspectives and New Applications
cites work: Q4953998
cites work: An isoperimetric comparison theorem
cites work: Q3124805
cites work: Sobolev Spaces
cites work: Comparison theorems in Finsler geometry and their applications
cites work: Best constant in Sobolev inequality
cites work: Q4839442
DOI: 10.1007/S11118-014-9427-4
description: scientific article