\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2002(2002), No. 86, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2002 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2002/86\hfil An embedding norm] {An embedding norm and the Lindqvist trigonometric functions} \author[Christer Bennewitz \& Yoshimi Sait\=o\hfil EJDE--2002/86\hfilneg] {Christer Bennewitz \& Yoshimi Sait\=o} \address{Christer Bennewitz \hfill\break Department of Mathematics, Lund University, \hfill\break Box 118, 221\,00 Sweden.} \email{christer.bennewitz@math.lu.se} \address{Yoshimi Sait\=o \hfill\break Dept. of Mathematics, University of Alabama at Birmingham, \hfill\break Birmingham, AL 35294, USA.} \email{saito@math.uab.edu} \date{} \thanks{Submitted September 12, 2002. Published October 9, 2002.} \subjclass[2000]{46E35, 33D05} \keywords{Sobolev embedding operator, Volterra operator} \begin{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 $10$ we can not have $F=0$, so that $\|f\|_p>0$. Since $\|T\|_p=\|F\|_p\le\|T\|_p\|f\|_p$ it actually follows that $\|f\|_p=1$. The lemma is proved. \end{proof} We are now ready to prove Theorem~\ref{t1}. We will do this by applying standard methods of the calculus of variations. \begin{proof}[Proof of Theorem \ref{t1}] We need only consider the case $10$ is a constant and we should note that \begin{equation*} (p-1)p^{-1}\frac{d}{dx}\Big[((F')^{p-1})^{p/(p-1)}\Big] = ((F')^{p-1})'F'. \end{equation*} Here, after multiplying $F$ by an appropriate constant, we may assume that $C=1$ in \eqref{6}. Although \eqref{6} is then satisfied in any interval where $F=1$, this will not satisfy \eqref{4}. In any point where $F\ne1$ we then obtain \begin{equation*} \frac{F'}{(1-F^p)^{1/p}}=a, \end{equation*} where $a=(\lambda(p-1))^{-1/p}$. Integrating again we obtain $F(x)=\sin_p(ax)$, since $F(0)=0$. Thus $f(x)=F'(x)=a\cos_p(ax)$. From $f(1)=0$ it follows that $a$ is a zero of $\cos_p$, and since we have $f\ge0$, $a$ is the first positive zero of $\cos_p$. Thus $a=\pi_p/2$. We also have $\lambda=a^{-p}/(p-1)$ so that $\|T\|_p=(p-1)^{-1/p}/a=2(p-1)^{-1/p}/\pi_p$. Now, if $q$ is the conjugate exponent to $p$ we have $p(p-1)^{-1/p}=p^{1/q}q^{1/p}$ and we are done, in view of the formula \eqref{1}. \end{proof} \section{Generalizations}\label{s4} One may of course also consider the operator \begin{equation*} T:L^p(0,1)\ni f\mapsto Tf(x)=\int_0^xf\in L^q(0,1), \end{equation*} where now $p$ and $q$ are unrelated exponents in $[1,\infty]$. Even in this case it is possible to calculate the norm $\|T\|_{p,q}$. We introduce the constant $\pi_{p,q}=2\int_0^1\frac{dt}{(1-t^q)^{1/p}}$, which is finite unless $p=1$, $q<\infty$, and then the function $\sipq$, first on the interval $[0,\pi_{p,q}/2)$ as the inverse of the strictly increasing function \begin{equation*} [0,1)\ni x\mapsto\int_0^x\frac{dt}{(1-t^q)^{1/p}}, \end{equation*} and then suitably extended to an odd function, which is $2\pi_{p,q}$-periodic and even around $\pi_{p,q}/2$ if $\pi_{p,q}$ is finite. We next define $\copq x=\frac{d}{dx}\sipq x$ and may the easily deduce that \begin{equation*} |\copq x|^p+|\sipq x|^q=1, \end{equation*} except if $p=\infty$, $q<\infty$. Drabek and Man\'asevich \cite{DM} also introduced $\sipq$ and $\copq$ functions. They are quite similar to ours but not the same. \begin{prp} For $p,q\in(1,\infty)$ we have $\pi_{p,q}=\frac2qB(1/p',1/q)$, where $p'$ is the dual exponent of $p$ and $B$ is the classical beta function. Furthermore, \begin{itemize} \item $\pi_{p,\infty}=2$, $1\le p\le\infty$, \item $\pi_{\infty,q}=2$, $1\le q\le\infty$, \item $\pi_{p,1}=2p'$, $1\le p\le\infty$, \item $\pi_{1,q}=\infty$, $1\le q<\infty$. \end{itemize} \end{prp} We may now carry out the analysis of the operator $T$ in much the same way as in Section~\ref{s3}, with the following conclusion. \begin{thm} For $p,q\in(1,\infty)$ we have \begin{equation*} \|T\|_{p,q}=(p'+q)^{1-\frac1{p'}-\frac1q}(p')^{1/q}q^{1/p'}/B(1/p',1/q), \end{equation*} where $p'$ is the dual exponent of $p$ and $B$ is the classical beta function. Extremals are all non-zero multiples of $\copq(\pi_{p,q}x/2)$. Furthermore, \begin{itemize} \item $\|T\|_{p,\infty}=1$, $1\le p\le\infty$. Extremals are all constants $\ne0$. In the case $p=1$ any non-zero multiple of a non-zero positive measure is an extremal. \item $\|T\|_{\infty,q}=(1+q)^{-1/q}$, $1\le q<\infty$. Extremals are all constants $\ne0$. \item $\|T\|_{p,1}=(1+p')^{-1/p'}$, \$1