\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 $1<p<\infty$, 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 \cite{Lin}.
\end{abstract}


\maketitle


\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{prp}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\sipq}{\operatorname{sin_{p,q}}}
\newcommand{\copq}{\operatorname{cos_{p,q}}}


\section{Introduction}

Evans, Harris and Sait\=o \cite{EHS} give the following result:
Let $W^{1,p}(0,1)$ be the complex first order Sobolev space given
by
\begin{equation*}
  W^{1,p}(0,1)=\big\{f\mid\int_0^1(|f|^p+|f'|^p)<\infty\big\},
\end{equation*}
where $1<p<\infty$. Then
\begin{equation}\label{0}
  \sup\frac{\|f\|_{p,s}}{\|f'\|_p}=\|T\|_p/2,
\end{equation}
where the supremum is over all non-zero functions in
$W^{1,p}(0,1)$, $\|\cdot\|_p$ is the norm of $L^p(0,1)$,
$\|\cdot\|_{p,s}$ is given by
\begin{equation*}
  \|f\|_{p,s}=\inf\limits_{C\in\mathbb{C}}\|f-C\|_p,
\end{equation*}
and $\|T\|_p$ is the operator norm of the Hardy operator
\begin{equation*}
  T:L^p(0,1)\ni f\mapsto Tf(x)=\int_0^xf\in L^p(0,1).
\end{equation*}
The left-hand side of \eqref{0} is the norm of the embedding from
the factor space $W^{1,p}(0,1)/\mathbb{C}$ into the factor space
$L^p(0,1)/\mathbb{C}$. Since the Poincar\'e inequality holds on
$(0,1)$, the norm $\|f'\|_p$, $f\in W^{1,p}(0,1)$, is one of the
equivalent norms of the space $W^{1,p}(0,1)/\mathbb{C}$. For more
details on the embedding operator we refer to Evans and Harris
\cite{EHa}, \cite{EHb}, in addition to \cite{EHS}.


In this note we calculate the norm
$\|T\|_p=p^{1/q}q^{1/p}\sin(\pi/p)/\pi$, $1<p<\infty$, where
$1/p+1/q=1$, and $\|T\|_1=\|T\|_\infty=1$. Our main arguments
depend on classical and elementary calculus of variations. We will
also calculate all extremals, {\em i.e.}, functions $f$ for which
$\|Tf\|_p=\|T\|_p\|f\|_p$. In particular, if $1<p<\infty$ these
are expressed in terms of `$\sin_p$' and `$\cos_p$' functions
which has been introduced in Lindqvist \cite{Lin}. These are
natural generalizations of the usual trigonometric functions when
the Euclidean norm in $\mathbb{R}^2$ is replaced by a $p$-norm.


After finishing our work, we found two recent works, Edmunds and
Lang \cite{EL} and Drabek and Man\'asevich \cite{DM}, which, among
others, produced quite a similar results in real $L^p$ spaces on
intervals using the results on the p-Laplacian.


In Section~\ref{s2} we shall give a short account on $\sin_p$ and
$\cos_p$ functions. In Section~\ref{s3} we shall first deal with
the cases that $p=1$ and $p=\infty$. Then we shall study the case
that $1<p<\infty$. After showing that we have only to find the
nonnegative extremals (Lemmas 3 and 4), we shall show that the
extremal is expressed in terms of the $\sin_p$ function and the
norm $\|T\|_p$ is computed using some properties of the $\sin_p$
and $\cos_p$ functions (Theorem \ref{t1}). We shall discuss the
operator
\begin{equation*}
  T:L^p(0,1)\ni f\mapsto Tf(x)=\int_0^xf\in L^q(0,1).
\end{equation*}
in Section~\ref{s4}.


\section{$\sin_p$, $\cos_p$ and $\tan_p$ functions}\label{s2}


Suppose $1<p<\infty$ and consider the function
$x\mapsto\int_0^x\frac{dt}{(1-t^p)^{1/p}}$, $0\le x\le1$. This is
strictly increasing, and for $p=2$ we obtain $\arcsin x$. By
analogy we denote the inverse for general $p$ by $\sin_p x$. It is
defined on the interval $[0,\pi_p/2]$ where
$\pi_p=2\int_0^1\frac{dt}{(1-t^p)^{1/p}}$, it is strictly
increasing on this interval, $\sin_p0=0$ and $\sin_p(\pi_p/2)=1$.
We may extend the definition to the interval $[0,\pi_p]$ by
setting $\sin_p x=\sin_p(\pi_p-x)$ for $\pi_p/2\le x\le\pi_p$, and
to $[-\pi_p,\pi_p]$ by extending it as an odd function. Finally,
we may define it on all of $\mathbb{R}$ by extending it as a
periodic function of period $2\pi_p$.


Next we define $\cos_p x=\frac{d}{dx}\sin_p x$, which is an even,
$2\pi_p$-periodic function, odd around $\pi_p/2$. In
$[0,\pi_p/2]$, setting $y=\sin_p x$, we have
\begin{equation*}
  \cos_p x=\tfrac{d}{dx}\sin_p x=(1-y^p)^{1/p}=(1-(\sin_p x)^p)^{1/p},
\end{equation*}
so that $\cos_p$ is strictly decreasing on this interval,
$\cos_p0=1$ and $\cos_p(\pi_p/2)=0$. Furthermore
\begin{equation*}
  |\cos_p x|^p+|\sin_p x|^p=1,
\end{equation*}
first in $[0,\pi_p/2]$, but then by symmetry and periodicity in
all of $\mathbb{R}$. One may also introduce an analogue of the
tangent function as $\tan_p x=\sin_p x/\cos_p x$. We then obtain
$\frac{d}{dx}\cos_p x= -|\tan_p x|^{p-2}\sin_p x$. We also get
$\frac{d}{dx}\tan_p x=1/|\cos_p x|^p=1+|\tan_p x|^p$, so that the
inverse of the restriction of $\tan_p$ to the interval
$(-\pi_p/2,\pi_p/2)$ has derivative $1/(1+|y|^p)$,
$y\in\mathbb{R}$. The constant $\pi_p$ is easily calculated. In
fact, by a change of variable $t=s^{1/p}$ we obtain
\begin{equation}\label{1}
  \pi_p=\tfrac2p\int_0^1(1-s)^{-1/p}s^{1/p-1}\,ds=\tfrac2pB(1-1/p,1/p)
   =2\frac{\pi/p}{\sin(\pi/p)},
\end{equation}
where $B$ is the classical beta function. If $q$ is the conjugate
exponent to $p$, so that $1/p+1/q=1$, this means in particular
that $p\pi_p=q\pi_q$.


The cases $p=1$ and $p=\infty$ are somewhat degenerate, especially
for $p=1$. For $p=\infty$ one gets $\pi_\infty=2$, and $\sin_p$
becomes a triangular wave, with $\sin_px=x$ on $[-1,1]$, and
$\cos_p$ the corresponding square wave. For $p=1$ one gets
$\pi_1=\infty$, and $\sin_p x$ is the odd extension of $1-e^{-x}$,
$x\ge0$, whereas $\cos_p x=e^{-|x|}$.


\section{An operator norm}\label{s3}


Consider the linear operator $Tf(x)=\int_0^xf$ on $L^p(0,1)$,
$1\le p\le\infty$. We shall determine the norm $\|T\|_p$ and all
{\em extremals}, {\em i.e.}, all functions $f\in L^p(0,1)$ for
which $\|Tf\|_p=\|T\|_p\|f\|_p$. In particular, we shall show that
if $q$ is the conjugate exponent to $p$, then $\|T\|_p=\|T\|_q$.
This may be proved by duality if $1<p<\infty$, but will follow
from our explicit calculations below.


\begin{thm}\label{t1}
For $1<p<\infty$,
$\|T\|_p=2p^{1/q}q^{1/p}/(p\pi_p)=p^{1/q}q^{1/p}\sin(\pi/p)/\pi$,
where $q$ is the dual exponent to $p$. The corresponding extremals
are all multiples of $\cos_p(\pi_px/2)$, where $\pi_p$ is given by
\eqref{1}. Furthermore, $\|T\|_1=\|T\|_\infty=1$, and the
extremals for $p=\infty$ are all constants, whereas no extremals
exist in $L^1(0,1)$ for the case $p=1$. If one extends $T$ to the
space of finite Borel measures on $[0,1]$, however, the extremals
are all multiples of the Dirac measure at 0.
\end{thm}
In particular $\|T\|_p=\|T\|_q$ since $p\pi_p=q\pi_q$. The
statement of the theorem indicates that it will be advantageous,
for $p=1$, to extend $T$ to operate on finite Borel measures on
$[0,1]$, normed by total variation, thus considering $L^1(0,1)$ as
the subset of absolutely continuous Borel measures on $[0,1]$. We
will do this in the sequel without further comment. The proof of
the theorem is almost immediate in the cases $p=1$ and $p=\infty$.


\begin{lem}
For $p=1$ extremals exist, they are precisely the non-zero
multiples of the Dirac measure at 0, and $\|T\|_1=1$.
\end{lem}


\begin{proof}
If $\mu$ is a finite Borel measure on $[0,1]$ and we define
$T\mu(x)=\int_{[0,x]}d\mu$, we obtain
\begin{multline}\label{2}
  \|T\mu\|_1=\int_0^1\big|\int_{[0,x]}d\mu\big|\,dx
  \le\int_0^1\int_{[0,x]}|d\mu|\,dx\\
  =\int_{[0,1]}(1-t)|d\mu(t)|\le\int_{[0,1]}|d\mu|=\|\mu\|_1,
\end{multline}
where the middle equality follows from Fubini's theorem. Thus
$\|T\|_1\le1$. However, we obtain equality throughout in \eqref{2}
precisely if $\int_{[0,1]}t|d\mu(t)|=0$, {\em i.e.}, the support
of $\mu$ consists of the point 0. Thus, if $\mu$ is a multiple of
the Dirac measure at 0.
\end{proof}


The proof in the case $p=\infty$ is even simpler.


\begin{lem}
For $p=\infty$ extremals exist, they are precisely the functions
which are a.e. constant, and $\|T\|_\infty=1$.
\end{lem}


\begin{proof}
If $f\in L^\infty(0,1)$ we obtain
\begin{equation*}
  \|Tf\|_\infty=\sup\limits_{0\le x\le1}\big|\int_0^xf\big|
  \le\int_0^1|f|\le\|f\|_\infty,
\end{equation*}
and we have equality throughout if and only if $f$ is a.e. a
multiple of $\|f\|_\infty$ (see also the proof of Lemma~\ref{l1}).
\end{proof}


The case when $1<p<\infty$ is less trivial, and the first step in
the proof is to show that we may restrict ourselves to positive
functions when looking for extremals.


\begin{lem}\label{l1}
Any extremal for $1\le p\le\infty$ is a multiple of a non-negative
extremal. Therefore, the norm $\|T\|_p$ when $T$ is defined on the
complex $L^p$ space is the same as the one when $T$ is defined on
the real $L^p$ space.
\end{lem}


\begin{proof}
We already know this if $p=1$ or $p=\infty$, so we now assume that
$1<p<\infty$. Suppose $f$ is an extremal and put $g=|f|$. Since
$\|f\|_p=\|g\|_p$ and $\big|\int_0^xf\big|\le\int_0^xg$ it follows
that if $f$ is an extremal, then so is $g$. Furthermore, it
follows that $\|Tf\|_p=\|Tg\|_p$. Now
$\big|\int_0^xf\big|\le\int_0^xg$ for all $x\in[0,1]$, so in order
that $\|Tf\|_p=\|Tg\|_p$ we must have equality throughout $[0,1]$.
In particular, we must have
\begin{equation*}
  \big|\int_0^1f\big|=\int_0^1g,
\end{equation*}
{\em i.e.}, we must have equality in the triangle inequality. This
proves the lemma, since it requires that $f$ and $g$ are a.e.
proportional. We remind the reader why this is so.


If we choose $\theta\in\mathbb{R}$ so that $e^{i\theta}\int_0^1f$
is positive, and we have equality in the triangle inequality
$\big|\int_0^1f\big|\le\int_0^1|f|$, then we have
$\int_0^1\Re(e^{i\theta}f)=\int_0^1|f|$ while at the same time
$\Re(e^{i\theta}f)\le|e^{i\theta}f|=|f|$. Thus $|f|=e^{i\theta}f$
a.e.
\end{proof}


The next step in the proof of Theorem~\ref{t1} is to show the
existence of extremals.


\begin{lem}
If $1<p<\infty$ there exists a function $f\in L^p(0,1)$ with
non-zero norm such that $\|Tf\|_p=\|T\|_p\|f\|_p$.
\end{lem}


\begin{proof}
Let $f_1,f_2,\dots$ be a sequence of unit vectors in $L^p(0,1)$
such that
\begin{equation*}
  \|Tf_n\|_p\to\|T\|_p \ \text{ as }n\to\infty.
\end{equation*}
We may assume that $f_n\rightharpoonup f\in L^p(0,1)$ weakly in
$L^p(0,1)$. For each fixed $x\in[0,1]$ the characteristic function
of $[0,x]$ is in $L^q(0,1)$, $q=p/(p-1)$, so we obtain
\begin{equation*}
  F_n(x)=\int_0^xf_n\to\int_0^xf=F(x) \ \text{ as }n\to\infty.
\end{equation*}
Since $|F_n(x)|\le\int_0^1|f_n|\le\|f_n\|_p=1$ by H\"{o}lder's
inequality, we obtain by dominated convergence that
$\|T\|_p=\lim\|F_n\|_p=\|F\|_p$, and since clearly $\|T\|_p>0$ 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 $1<p<\infty$. Suppose $f\ge0$ is an extremal with non-zero
norm and put $F(x)=Tf(x)=\int_0^xf\ge0$. Then setting
$\lambda=\|F\|_p^p/\|F'\|_p^p$ we have $\|T\|_p=\lambda^{1/p}$. If
$\varphi\in C^1(0,1)$ with $\varphi(0)=0$, then
$\|F+\varepsilon\varphi\|_p^p/\|F'+\varepsilon\varphi'\|_p^p$ has
a maximum for $\varepsilon=0$. Differentiating we get, for
$\varepsilon=0$, that
\begin{equation*}
  \int_0^1F^{p-1}\varphi-\lambda\int_0^1(F')^{p-1}\varphi'=0,
\end{equation*}
so integrating by parts we obtain
\begin{equation}\label{4}
  \int_0^1(\lambda(F')^{p-1}-\int_x^1F^{p-1})\varphi'=0
\end{equation}
for all $\varphi\in C^1(0,1)$ with $\varphi(0)=0$, so that, by du
Bois Reymond's lemma, $\lambda(F')^{p-1}-\int_x^1F^{p-1}$ is
constant. It follows that $(F')^{p-1}$ is continuously
differentiable, and differentiating we obtain
\begin{equation}\label{5}
  F^{p-1}+\lambda((F')^{p-1})'=0.
\end{equation}
Thus, integrating by parts in \eqref{4} we obtain
$(F'(1))^{p-1}\varphi(1)=0$, so that $F'(1)=f(1)=0$. Multiplying
\eqref{5} by $F'$ and integrating we obtain
\begin{equation}\label{6}
  F^p+\lambda(p-1)(F')^p=C,
\end{equation}
where $C>0$ 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 $\|T\|_{p,1}=(1+p')^{-1/p'}$, $1<p\le\infty$. Extremals
  are all non-zero multiples of $(1-x)^{1/(p-1)}$.
  \item $\|T\|_{1,q}=1$, $1\le q<\infty$. Extremals are all
  non-zero multiples of the Dirac measure at the origin.
\end{itemize}
\end{thm}


It will be seen that as a function of $(1/p,1/q)$ the norm
$\|T\|_{p,q}$ is continuous in the unit square $0\le1/p\le1$,
$0\le1/q\le1$. Furthermore, it is clear that
$\|T\|_{p,q}=\|T\|_{q',p'}$ for all $p,q\in[1,\infty]$, a fact
that could presumably also be proved by duality.


\begin{thebibliography}{0}


 \bibitem{DM} P.~Drabek and R.~Man\'asevich, {\em On the solutionto some
 p-Laplacian nonhmogeneous eigenvalue problems in closed form}, Preprint.


 \bibitem{EL} D.~E.~Edmunds and Jan Lang, {\em Behaviour of the approximate
 number of a Sobolev embedding in the one-dimensional case}, Preprint.


 \bibitem{EHS} W.~D.~Evans, D.~J.~Harris and Y.~Sait\=o, {\em On the
 approximation numbers of Sobolev embeddings on irregular domains and trees},
 Preprint.


 \bibitem{EHa} W.~D.~Evans and D.~J.~Harris, {\em Sobolev
 embeddings for generalized ridged domains}, Proc. London Math.
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 \bibitem{EHb} W.~D.~Evans and D.~J.~Harris, {\em Fractals, trees
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 \bibitem{Lin} P.~Lindqvist, {\em Some remarkable sine and cosine functions},
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\end{thebibliography}



\end{document}