Electron. J. Diff. Eqns., Vol. 2002(2002), No. 86, pp. 1-6.

An embedding norm and the Lindqvist trigonometric functions

Christer Bennewitz & Yoshimi Saito

Abstract:
We shall calculate the operator norm $\|T\|_p$ of the Hardy operator
$Tf = \int_0^x f $, where $1\le p\le \infty$.
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.

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Christer Bennewitz
Department of Mathematics, Lund University,
Box 118,22100 Sweden
e-mail: christer.bennewitz@math.lu.se

Yoshimi Saito
Dept. of Mathematics, University of Alabama at Birmingham,
Birmingham, AL 35294, USA.
e-mail: saito@math.uab.edu

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