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BEST CONSTANTS IN SOBOLEV INEQUALITIES ON THE SPHERE AND IN EUCLIDEAN SPACE

Published online by Cambridge University Press:  01 February 1999

A. A. ILYIN
A. A. ILYIN
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghennydd Road, Cardiff CF2 4YH Permanent address: Keldysh Institute of Applied Mathematics, 4 Miusskaya Square, Moscow 125047, Russia. E-mail: [email protected]
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Abstract

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

formula here

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, Δ is either the classical Laplace operator or the Laplace–Beltrami operator acting on the surface of Sn[ratio ]Δf(s)= Δf(x/[mid ]x[mid ])[mid ]x=s, s∈Sn, n[ges ]2 (on S1, of course, Δf=f″).

Throughout ∥·∥ is the L2-norm, p and l are real numbers satisfying

formula here

and θ=(2l−n)/(2(l−p)), 1−θ=(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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 1 , February 1999 , pp. 263 - 286
Copyright
The London Mathematical Society 1999

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