Best Constants in Sobolev Inequalities on the Sphere and in Euclidean Space

In this paper we shall be dealing with best constants characterising the embedding of a Sobolev space of L 2 ‐type H l = W l 2 into the space of bounded continuous functions when l > n /2. More specifically, we are interested in the value of the best constant c M ( p , l ) in the inequality ∥ f ∥ c ⩽ c M ( p , l )∥(−Δ) p /2 f ∥ θ ∥(−Δ) l /2 f ∥ 1−θ (0.1) where M stands for Euclidean space R n or the n ‐sphere S n (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 S n :Δ f ( s )=Δ f ( x /∣ x ∣)∣ x = s , s ∈ S n , n ⩾2 (on S 1 , of course, Δ f = f ″). Throughout ∥·∥ is the L 2 ‐norm, p and l are real numbers satisfying 0.2 p < n 2 < 1and θ=(2 l − n )/(2( l − p )), 1−θ=( n −2 p )/(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 R n were dealt with.