An Embedding Norm and the Lindqvist Trigonometric Functions

Abstract

We shall calculate the operator norm ||T||p of the Hardy operator Tƒ = ∫x0 ƒ, where 1 ≤ p ≤ ∞. This operator is related to the Sobolev embedding operator from W1,p (0,1)/ℂ into Wp (0,1)/ℂ. For 1 < p < ∞, the extremal, whose norm gives the operator norm ||T||p, is expressed in terms of the function sin p which is a generalization of the usual sine function and was introduced by Lindqvist [6].