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.