An embedding norm and the Lindqvist trigonometric functions

Abstract

We shall calculate the operator norm $|T|_p$ of the Hardy operator $Tf = int_0^x f $, where $1le ple infty$. This operator is related to the Sobolev embedding operator from $W^{1,p}(0,1)/mathbb{C}$ into $W^p(0,1)/mathbb{C}$. For $1<p<infty$, 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]. Submitted September 12, 2002. Published October 9, 2002. Math Subject Classifications: 46E35, 33D05. Key Words: Sobolev embedding operator; Volterra operator.

Keywords

• Sobolev embedding operator
• Volterra operator.