The embedding \(W^{n,1}\) into \(C_0\) on compact Riemannian manifolds
type: Item
instance of: scholarly article
title: The embedding \(W^{n,1}\) into \(C_0\) on compact Riemannian manifolds
zbMATH Open document ID: 1040.53049
DOI: 10.1023/A:1024176018875
author: Emmanuel Humbert
author: Bruno Nazaret
published in: Potential Analysis
publication date: 2003-08-21T00:00:00Z
review text: The authors are concerned with the Sobolev embedding of \(W^{n,1}(\mathbb{R}^n)\) (= the space of functions whose derivatives of order lower than \(n\in \mathbb{N}^{*}\) belong to \(L^1\)) into \(C_0(\mathbb{R}^n)\) (= the space of bounded continuous functions). The corresponding inequality asserts that there exists \(C>0\) such that for all \(u\in C_c^{\infty}(\mathbb{R}^n)\), \[ \| u\| _{\infty}\leq C\int_{\mathbb{R}^n} | \nabla^n u| \,d\xi, \tag{S(C)} \] where \(\xi\) stands for the standard Euclidean metric on \(\mathbb{R}^n\) and where \[ | \nabla^n u| :=\sqrt{\sum_{i_1, \dots,i_n\in \{1,\dots,n\}} \left(\frac{\partial^n u}{\partial x_{i_1}\cdots\partial x_{i_n}} \right)^2 }. \] They compute the values of the best constants in S(C) for \(n\in \{1,2\}\). The previous inequality is adapted for smooth compact Riemannian manifolds of dimension \(n\geq 1\), which involves two constants, and its optimality is discussed. Similar results concerning inequalities involving derivatives of order one have been obtained by \textit{T. Aubin} and \textit{Y. Y. Li} [J. Math. Pures Appl., IX. Sér. 78, 353--387 (1999; Zbl 0944.46027)], \textit{O. Druet} [Math. Ann. 314, 327--346 (1999; Zbl 0934.53028)], \textit{E. Hebey} and \textit{M. Vaugon} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, 57--93 (1996; Zbl 0849.53035)] and the first author [Proc. R. Soc. Edinb., Sect. A, Math. 131, 621--646 (2001; Zbl 1040.53048)].
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1968877
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: Sobolev embedding
zbMATH Keywords: optimal inequality
zbMATH Keywords: best constant
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
full work available at URL: a:1024176018875
OpenAlex ID: W2601304190
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description: scientific article; zbMATH DE number 1968877