© 1999 by London Mathematical Society
© The London Mathematical Society
Best Constants in Sobolev Inequalities on the Sphere and in Euclidean Space
School of Mathematics, University of Wales College of Cardiff Senghennydd Road, Cardiff CF2 4YH
Received 1 July 1996. Revision received 4 October 1996.
In this paper we shall be dealing with best constants characterising the embedding of a Sobolev space of L2-type Hl=Wl2 into the space of bounded continuous functions when l>n/2. More specifically, we are interested in the value of the best constant cM(p, l) in the inequality
||f||c 
cM(p, l)||(–
)p/2f||
||(–)l/2f||1– (0.1)
where M stands for Euclidean space Rn or the n-sphere Sn (in the latter case f is assumed to have zero average (f, 1)=0). Accordingly, is either the classical Laplace operator or the Laplace–Beltrami operator acting on the surface of Sn:f(s)=f(x/|x|)|x=s, s
Sn, n
2 (on S1, of course, f=f'').
Throughout ||·|| is the L2-norm, p and l are real numbers satisfying
 |
and =(2l–n)/(2(l–p)), 1–=(n–2p)/(2(l–p)).
Before describing the contents of the paper we recall the well-known references [3, 10, 11, 16] and the survey [18] where best constants and corresponding extremal functions of the Sobolev embeddings in Rn were dealt with.
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