Optimal Sobolev Inequalities of Arbitrary Order on Compact Riemannian Manifolds

✍ Scribed by Olivier Druet

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 341 KB
• Volume: 159
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3264

✦ Synopsis

Let (M, g) be a smooth compact Riemannian N-manifold, N 2, let p # (1, N) real, and let H p 1 (M) be the Sobolev space of order p involving first derivatives of the functions. By the Sobolev embedding theorem, H p 1 (M)/L p* (M) where p*=NpÂ(N& p). Classically, this leads to some Sobolev inequality (I 1 p ), and then to some Sobolev inequality (I p p ) where each term in (I 1 p ) is elevated to the power p. Long-standing questions were to know if the optimal versions with respect to the first constant of (I 1 p ) and (I p p ) do hold. Such questions received an affirmative answer by Hebey Vaugon for p=2. We prove here that for p>2, and p 2 <N, the optimal version of (I p p ) is false if the scalar curvature of g is positive somewhere. In particular, there exist manifolds for which the optimal versions of (I 1 p ) are true, while the optimal versions of (I p p ) are false. Among other results, we prove also that the assumption on the sign of the scalar curvature is sharp by showing that for any p # (1, N), the optimal version of (I p p ) holds on flat tori.