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% Author Package file for use with AMS-LaTeX 1.2
\controldates{17-APR-2001,17-APR-2001,17-APR-2001,17-APR-2001}
 
\documentclass{proc-l}
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\dateposted{April 17, 2001}
\PII{S 0002-9939(01)06162-7}
\copyrightinfo{2001}{American Mathematical Society}

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\begin{document}
\title[The interpolation constant for Orlicz spaces]{On the
interpolation  constant for Orlicz spaces}
\author{Alexei Yu. Karlovich}
\address{Department of Mathematics and Physics, South Ukrainian State
Pedagogical University, Staroportofrankovskaya 26, 65020
Odessa, Ukraine}
\email{karlik@paco.net}
\curraddr{Departamento de Matem\'atica, Instituto Superior T\'ecnico, Av.
Rovisco Pais 1, 1049-001, Lisbon, Portugal}
\email{akarlov@math.ist.utl.pt}

\author{Lech Maligranda}
\address{Department of Mathematics, Lule\aa\ University of Technology,
971 87 Lule\aa, Sweden}
\email{lech@sm.luth.se}

\subjclass{Primary 46B70, 46E30; Secondary 26D07}

\date{January 24, 2000.}
\keywords{Orlicz spaces, interpolation constant, interpolation of operators,
$K$-functional, convex function, concave function}
\commby{Jonathan M. Borwein}

\begin{abstract}
In this paper we deal with the interpolation from Lebesgue spaces
$L^p$ and $L^q$, into an Orlicz space $L^\varphi$,
where $1\le p<q\le\infty$ and $\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})$
for some concave function $\rho$, with  special attention to the
interpolation constant $C$. For a bounded linear operator $T$ in
$L^p$ and $L^q$, we prove modular inequalities, which allow us to
get the estimate for both the Orlicz norm
and the Luxemburg norm,
\[
\|T\|_{L^\varphi\to L^\varphi}
\le C\max\Big\{
\|T\|_{L^p\to L^p},
\|T\|_{L^q\to L^q}
\Big\},
\]
where the interpolation constant $C$ depends only on $p$ and $q$. We
give estimates for $C$, which imply $C<4$. Moreover, if either $1<
p<q\le 2$ or $2\le p<q<\infty$, then $C< 2$. If $q=\infty$, then
$C\le 2^{1-1/p}$, and, in particular, for the case $p=1$ this gives
the classical Orlicz interpolation theorem with the constant $C=1$.
\end{abstract}

\maketitle

\section{Introduction}
\setcounter{equation}{0}
The classical Riesz-Thorin interpolation theorem says that
$(L^p,L^q)$ are interpolation spaces for linear operators between
$(L^{p_0},L^{q_0})$ and $(L^{p_1},L^{q_1})$, where
%%%
\begin{equation}\label{eq:RT-condition}
\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}, \quad
\frac{1}{q}=\frac{1-\theta}{q_0}+\frac{\theta}{q_1},
\end{equation}
%%%
with the estimation of the norm
%%%
\[
\|T\|_{L^p\to L^q}\le C
\left(\|T\|_{L^{p_0}\to L^{q_0}}\right)^{1-\theta}
\left(\|T\|_{L^{p_1}\to L^{q_1}}\right)^{\theta} \]
%%%
where $C\le\sqrt{2}$. The constant $C$ is $1$ when either the spaces
are complex or the spaces are real and $p_i \le q_i, i=0,1$ (see, e.g.,
\cite[Section~1.7]{BK}).

After the Riesz-Thorin interpolation theorem several results have
been proved about the interpolation of Orlicz spaces. The problem was
the following:
{\em
if $T$ is any bounded linear operator from $L^{\varphi_i}$ into
$L^{\psi_i}, i=0,1$, then under what conditions on $\varphi$ and
$\psi$
is it true that $T$ is also bounded from $L^\varphi$ to $L^\psi$? }

The assumption corresponding to (\ref{eq:RT-condition}) which appeares
naturally here is
%%%
\begin{equation}\label{eq:GP-condition}
\varphi^{-1}=\varphi_0^{-1}
\rho\left(\frac{\varphi_1^{-1}}{\varphi_0^{-1}}\right), \quad
\psi^{-1}=\psi_0^{-1}
\rho\left(\frac{\psi_1^{-1}}{\psi_0^{-1}}\right), \end{equation}
%%%
for some concave function $\rho$.

The first result on interpolation of Orlicz spaces appeared in the
case when $\rho(t)=t^\theta, 0\le\theta\le 1$, and we mention
here Ya.~B.~Rutickii (1963), A.~P.~Calder\'on (1964), M.~M.~Rao
(1966) as the precursors of such results. The constants found by
Ya.~B.~Rutickii were $4$ in the complex case and $8$ in the real
case. The corresponding constants in two other papers are $2$ and $4$,
respectively. Theorems with general concave $\rho$ (and the
interpolation theorems for, in fact, the Calder\'on-Lozanovskii
construction) were done by several authors; among others, we mention
V.~I.~Ovchinnikov (1976, 1984), J.~Gustavsson-J.~Peetre (1977),
E.~I.~Berezhnoi (1980), V.~A.~Shestakov (1981), E.~I.~Pustylnik
(1983), P.~Nilsson (1985) and L.~Maligranda (1985, 1989). For
the precise references and the proofs, see \cite{M89} and also
\cite{BK,GuPeetre,KMP,Ovch76,Ovch84}.

Only some of these results on interpolation of Orlicz (or
Cald\'eron-Lozanovskii)
spaces take care about the estimate of the operator norm.
One known result is the following:
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:Maligranda}
If \eqref{eq:GP-condition} holds for some concave function
$\rho$, then $(L^\varphi,L^\psi)$ are interpolation spaces for linear
operators between $(L^{\varphi_0},L^{\psi_0})$ and
$(L^{\varphi_1},L^{\psi_1})$, and
%%%%
\[
\|T\|_{L^\varphi\to L^\psi}\le C\max\big\{ M_0, M_1\big\}, \quad
M_i:=\|T\|_{L^{\varphi_i}\to L^{\psi_i}}, \quad i=0,1,
\]
%%%
where $C\le 26$ and all norms in Orlicz spaces are the Luxemburg
norms.
\end{theorem}

The proof of this statement can be found in \cite[Theorem~14.2]{M89}.
The careful analysis of the proof shows that, in fact,
%%%%
\begin{equation}\label{eq:BK-constant}
C\le 2(3+2\sqrt{2})C_\psi< 12C_\psi,
\mbox{ where }
C_\psi=\sup_{t>0}\frac{\psi^{-1}(2t)}{\psi^{-1}(t)}\le 2.
\end{equation}

In the general case we have only the fairly rough estimate of the norm
$C< 24$, but for the ``diagonal case'' and Lebesgue spaces, i.e., when
$L^{\varphi_0}=L^{\psi_0}=L^p$ and $L^{\varphi_1}=L^{\psi_1}=L^q$, one
can obtain more precise estimates.
Moreover, for most of the operators, we have information
about boundedness between $L^p$ spaces and we would like to get
estimates in more general spaces, for example, in Orlicz spaces. Our
problem here starts with
the bounded linear operators $T$ from
$L^{p}$ into $L^{p}$ which are also bounded from $L^{q}$ into $L^{q}$
(with $1\le p<q\le \infty$) and we want to have boundedness of $T$
from the Orlicz space $L^\varphi$ into itself
with the best possible estimate of the norm
%%%
\begin{equation}\label{eq:norm}
\|T\|_{L^\varphi\to L^\varphi}\le C\max\big\{ M_0, M_1\big\}, \quad
M_0:=\|T\|_{L^{p}\to L^{p}},\
M_1:=\|T\|_{L^{q}\to L^{q}},
\end{equation}
%%%
where $\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p})$ for some concave
function $\rho$.

Besides the above-mentioned authors working in the general case, there
were also others working either with weak type operators or
the diagonal case and Lebesgue spaces.
I.~B.~Simonenko \cite{Simonenko}, among others, showed in 1964 that
if
\begin{equation}\label{eq:Simonenko}
1\le p<
a_\varphi:=\inf_{u>0}\frac{u\varphi'(u)}{\varphi(u)} \le
b_\varphi:=\sup_{u>0}\frac{u\varphi'(u)}{\varphi(u)} <q<\infty,
\end{equation}
then the Orlicz space $L^\varphi$ is an interpolation space between
$L^p$ and $L^q$ but the constant $C$ in the estimation of the norm
can be large: $C\approx \Big((a_\varphi-p)(q-b_\varphi)\Big)^{-1}$.
B.~W.~Boyd \cite{Boyd} extended in 1967 this theorem to
rearrangement-invariant spaces and his constant in the estimation of
the norm increases to infinity when one of the Boyd indices of the
space is going either to $p$ or to $q$.

Let us also mention that interpolation theorems of Marcinkiewicz type
(operators are of weak type, i.e., maps $L^p$ into weak $L^p$ and
$L^q$ into weak $L^q$) in Orlicz spaces were done by A.~Zygmund
(1956), A.~Torchinsky (1976), A.~Cianchi (1998) (see \cite{Cianchi}
and the references given there)
and in symmetric spaces by
E.~M.~Semenov (1968), D.~W.~Boyd (1969), M.~Zippin (1971),
S.~G.~Krein and E.~M.~Semenov
(1973) (see \cite{kps,M85} and references given there). Observe that
the constants in these theorems are still large.

The paper is organized as follows. Section 2, called preliminaries,
contains necessary definitions and some auxiliary results
from the theory of Orlicz spaces and interpolation theory. In Section
3 we consider useful properties of concave and convex functions on
$[0,\infty)$. In particular, the equivalence between representations
$\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p})$ and
$\varphi(u)=u^{p}\rho(u^{q-p})$ is proved. In Section 4 we prove modular
estimates of bounded linear operators in $L^p$ and $L^q$ spaces which
are the keys for the estimates in the norms, both the Luxemburg one
and the Orlicz one.
In the case $q<\infty$, the idea of the proof goes back to J.~Peetre
\cite{Peetre70}. In the proof we use essentially the exact estimation
of the Sparr functional $K_{{p,q}}^*$ \cite{Sparr}. In the case
$q=\infty$ our proof is based on the Kr\'ee formula, the
Hardy-Littlewood-P\'olya majorization theorem and the convexity of
$\varphi(u^{1/p})$.
Finally, in Section 5, we put all our pieces of results
together and prove our main Theorem~\ref{th:main}, which shows that
the interpolation constant $C$ in the estimate (\ref{eq:norm}) is always
less than 4.
Moreover, if either $1<p<q\le 2$ or $2\le p<q<\infty$, then $C<2$. If
$q=\infty$, then $C\le 2^{1-1/p}$, and, in particular, for the case $p=1$
this gives the classical Orlicz interpolation theorem with the constant
$C=1$.

\section{Preliminaries}
\setcounter{equation}{0}
Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Let
$\varphi:[0,\infty)\to[0,\infty]$ be a convex nondecreasing function
such that $\varphi(0)=0$ and $\lim\limits_{u\to 0+}\varphi(u)=0$
but not identically zero or infinity on $(0,\infty)$. For a
measurable real or complex-valued function $x$, define a functional
(\emph{modular})
%%%
\begin{equation}\label{eq:modular}
I_\varphi(x):=\int_\Omega\varphi(|x(s)|)d\mu(s)=
\int_0^\infty\varphi(x^*(t))dt,
\end{equation}
%%%
where $x^*$ is the nonincreasing rearrangement of $x$ (see
\cite{BeSh,kps,LT}).
The \emph{Orlicz space} $L^\varphi=L^\varphi(\Omega,\mu)$ is the space
of all equivalence classes of measurable functions on $\Omega$ such
that $I_\varphi(\lambda x)<\infty$
for some $\lambda=\lambda(x)>0$. This space is a Banach space with
two norms: the \emph{Luxemburg norm} \[
\|x\|_\varphi:=\inf\left\{\lambda>0\ : \ I_\varphi(x/\lambda)\le
1\right\} \]
and the \emph{Orlicz norm} (in the Amemiya form) \[
\|x\|_\varphi^0:=\inf_{k>0}\frac{1}{k}\Big(1+I_\varphi(kx)\Big). \]
It is well known that $\|x\|_\varphi\le \|x\|_\varphi^0\le
2\|x\|_\varphi$, and $\|x\|_\varphi\le 1$ if and only if
$I_\varphi(x)\le 1$ (cf. \cite{KR}). The Orlicz space $L^\varphi$
with each of the above two norms is a rearrangement-invariant space
(= symmetric space with the Fatou property) (see \cite{BeSh,kps}). If
$\varphi$ satisfies the $\Delta_2$-condition, then the dual of the
Orlicz space $L^\varphi$ is an Orlicz space $L^{\varphi^*}$ generated
by the conjugate function $\varphi^*$, defined by \[
\varphi^*(u):=\sup_{v > 0} \Big(uv-\varphi(v)\Big). \]
Moreover, $(\|\cdot\|_\varphi)^*=\|\cdot\|_{\varphi^*}^0$ and
$(\|\cdot\|_\varphi^0)^*=\|\cdot\|_{\varphi^*}$.

Let $(X_0,X_1)$ be a couple of Banach spaces and let $X$ be an
intermediate Banach space between $X_0$ and $X_1$, that is, $X_0\cap
X_1\subset X\subset X_0+X_1$ with continuous imbeddings ($X_0\cap
X_1, X_0+X_1$ are standard spaces; see \cite{BeSh,BL,BK,kps,LT}). By
${\mathcal A}(X_0,X_1)$ we denote the class of all admissible operators,
i.e., linear operators $T:X_0+X_1\to X_0+X_1$ which restriction to
$X_i$ is bounded from $X_i$ into itself for $i=0,1$. We denote
\[
M:=\max\big\{M_0,M_1\big\}, \quad
\mbox{where}\quad
M_i:=\| T|_{X_i} \|_{X_i\to X_i},
\quad i=0,1.
\]
The space $X$ is said to be an \emph{interpolation space} between $X_0$
and $X_1$ if every admissible operator $T\in{\mathcal A}(X_0,X_1)$ maps
$X$ into itself and
\[
\|T\|_{X\to X}\le C\max \big\{M_0,M_1\big\}
\]
for some $C>0$.

For $0<p,q<\infty, t>0$ and $x\in X_0+X_1$, we define a functional
$K_{p,q}$
known as the Peetre $L$-functional (Peetre \cite{Peetre70}; see also
\cite[Th.~5.2.2]{BL}, \cite[Definition~3.1.22]{BK}):
 \[
K_{p,q}(t,x;X_0,X_1):=\inf\Big\{\|x_0\|_{X_0}^p+t\|x_1\|_{X_1}^q \::\
x=x_0+x_1,\ x_0\in X_0, x_1\in X_1\Big\}.
\]
%%%%%%
In the case $p=q=1$ this
is the classical Peetre $K$-functional, which we
shortly denote by $K(t,x;X_0,X_1)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:K-bound}
If $T\in{\mathcal A}(X_0,X_1)$, then
\[
K_{p,q}\left(t,\frac{Tx}{M};X_0,X_1\right) \le
K_{p,q}(t,x;X_0,X_1)
\quad\mbox{for all}\quad t>0.
\]
\end{proposition}

The proof is standard.

Following \cite{Sparr}, we consider the functional $K^*_{p,q}$ on the
couple of Lebesgue spaces defined by
%%%
\[
K^*_{p,q}(t,x;L^p,L^q)
:=
\int_\Omega
\min\Big(|x(s)|^p,t|x(s)|^q\Big)\,d\mu(s). \]
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}[G. Sparr {\cite[Lemma~5.1, Example~5.3]{Sparr}}]\label{le:sparr}
Suppose $1\le p
<q<\infty$. If $x,y\in L^p+L^q$ and \[
K_{p,q}(t,x;L^p,L^q)\le
K_{p,q}(t,y;L^p,L^q)\quad \mbox{for all}\quad t>0, \]
then
\[
K^*_{p,q}(t,x;L^p,L^q)\le
\gamma_{p,q}
K^*_{p,q}(t,y;L^p,L^q)\quad\mbox{for all}\quad t>0, \]
where
\[
\gamma_{p,q}:=\inf\left\{\gamma>0\::\:
\inf_{\begin{array}{c}
x+y=\gamma, \\
x,y\ge 0
\end{array}}
\Big(x^p+y^q\Big)=1
\right\}
\]
is the Sparr constant, which cannot be replaced by any smaller
constant, and which satisfies the inequalities $1<\gamma_{p,q}<2$.
\end{lemma}

\section{Some properties of concave and convex functions}
\setcounter{equation}{0}
We need some properties of concave and convex functions. More
information about concave and convex functions and their properties
can be found in \cite[Ch.~3]{BK}, \cite{KR}, \cite[Ch.~2,
Section~1]{kps}.

We denote by $\mathcal P$ the set of all quasi-concave functions
$\rho:[0,\infty)\to[0,\infty)$ which are continuous, positive on
$\mathbf{R}_+:=(0,\infty)$ and such that
$
\rho(st)\le\max\big\{1,s\big\}\rho(t)$ for all $s,t>0$.
Let $\widetilde{\mathcal P}$ denote the subset of all concave functions
in ${\mathcal P}$. Note that if $\rho\in{\mathcal P}$, then $\tilde{\rho}$
defined by
\[
\tilde{\rho}(t):=\inf_{s>0}\left(1+\frac{t}{s}\right)\rho(s) \]
belongs to $\widetilde{\mathcal P}$ and
%%%
\begin{equation}\label{eq:conc-maj}
\rho(t)\le \tilde{\rho}(t) \le 2\rho(t)
\quad\mbox{for all}\quad t>0.
\end{equation}

Later on $p'$ will always denote the conjugate number to $p$, $1\le
p\le\infty$, that is, $1/p+1/p'=1$ ($1/\infty$ means $0$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:convex-increasing} Let $1<p<\infty$ and
$\varphi$ be a convex function on $\mathbf{R}_+$.
The function $u^{-p}\varphi(u)$ is increasing (decreasing) on $\mathbf{R}
_+$ if and only if the function $u^{-p'}\varphi^*(u)$ is decreasing
(increasing) on $\mathbf{R}_+$.
\end{lemma}

\begin{proof}[Proof \rm(cf. {\cite[Lemma~6.1.4]{kok98}})]
Let $u^{-p}\varphi(u)$ be increasing.
If $0<u_1\le u_2$, then
%%%
\begin{eqnarray*}
\varphi^*(u_2)
&=&
\sup_{v>0}\Big(vu_2-\varphi(v)\Big)
=
\sup_{v>0}
\left(
\left(
\frac{u_2}{u_1}
\right)^{p'-1}vu_2 -
\varphi\left(
\left(\frac{u_2}{u_1}\right)^{p'-1}v
\right)
\right)
\\
&\le&
\sup_{v>0}\left(\left(\frac{u_2}{u_1}\right)^{p'}vu_1-
\left(\frac{u_2}{u_1}\right)^{(p'-1)p}\varphi(v)\right) \\
&=&
\left(\frac{u_2}{u_1}\right)^{p'}
\sup_{v>0}\Big(vu_1-\varphi(v)\Big)
=
\left(\frac{u_2}{u_1}\right)^{p'}\varphi^*(u_1), \end{eqnarray*}
which means that $u^{-p'}\varphi^*(u)$ is decreasing.
%%%
The remaining implications can easily be proved by also using the
fact that for convex function $\varphi$ we have $\varphi^{**}=\varphi$.
\end{proof}

Now we derive relations between some representations
of $\varphi$ and $\varphi^{-1}$.

\begin{lemma}\label{le:properties}
Suppose that $1\le p<q\le\infty$ and, for some $\rho\in\widetilde{\mathcal P}$,
\[
\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p}) \quad\mbox{for all}\quad u>0.
\]

\textup{(a)} Then $\varphi$ is convex.

\textup{(b)}
If $q<\infty$, then there exists a function $h\in{\mathcal P}$ such
that \[
\varphi(u)=u^q h(u^{p-q})\quad\mbox{for all}\quad u>0. \]

\textup{(c)}
If $1<p<q<\infty$, then there exists a function $k\in{\mathcal P}$
such that
\[
\varphi^*(u)=u^{p'}k(u^{q'-p'})\quad\mbox{for all}\quad u>0. \]

\textup{(d)}
If $q=\infty$ and $\rho_*(t):=t\rho(1/t)$ satisfies $\rho_*(
\mathbf{R}_+)={\mathbf{R}_+}$, then $\varphi(u)=\rho_*^{-1}(u)^p$ and
$\psi(u)=\varphi(u^{1/p})$ is a convex function. \end{lemma}

\begin{proof} Proposition (a) follows from \cite[Lemma~14.2]{M89}.

(b) Since $\rho(t)$ is increasing and $\rho(t)/t$ is decreasing, for
all $s,t>0$ we have
\[
\min\big\{s^{1/p},s^{1/q}\big\}\varphi^{-1}(t) \le
\varphi^{-1}(st)
\le
\max\big\{s^{1/p},s^{1/q}\big\}\varphi^{-1}(t). \]
Hence, $\varphi^{-1}$ is strictly increasing on $\mathbf{R}_+$ and
$\varphi^{-1}(\mathbf{R}_+)=\mathbf{R}_+$. It is easy to see that the
following statements are equivalent:

(i)
$\varphi^{-1}(st)
\le
\max\big\{s^{1/p},s^{1/q}\big\}\varphi^{-1}(t)$ for all $s,t>0$;

(ii)
$\varphi(st)
\le
\max\big\{s^{p},s^{q}\big\}\varphi(t)$ for all $s,t>0$;

(iii) $u^{-p}\varphi(u)$ is increasing and $u^{-q}\varphi(u)$ is
decreasing;

(iv) $h\in{\mathcal P}$, where $h$ is given by
$\varphi(u)=u^qh(u^{p-q})$.

Now we prove (c).
If $1<p<q<\infty$, then, by Lemma~\ref{le:convex-increasing}, (iii)
is equivalent to each of the following properties:

(v) $u^{-p'}\varphi^*(u)$ is decreasing and $u^{-q'}\varphi^*(u)$ is
increasing;

(vi)
$\varphi^*(st)
\le
\max\big\{s^{p'},s^{q'}\big\}\varphi^*(t)$ for all $s,t>0$;

(vii) $k\in{\mathcal P}$, where $k$ is given by
$\varphi^*(u)=u^{p'}k(u^{q'-p'})$.

Let us prove (d). First observe that $\rho_*$ is concave (see, e.g.,
\cite[Lemma~8.7]{M89}).
By the assumption $\rho_*(\mathbf{R}_+)=\mathbf{R}_+$, we can see that
$\varphi^{-1}(\mathbf{R}_+)=\mathbf{R}_+$ and $\varphi^{-1}$ is concave.
Hence, $\varphi$ is a finite valued convex function vanishing only at
zero, $\varphi(u)=\rho_*^{-1}(u)^p$. We show that even
$\psi(u)=\varphi(u^{1/p})$ is convex. Since $\rho_*$ is concave,
$\varphi_0:=\rho_*^{-1}$ is convex. Hence,
\[
\varphi_0(u)=\varphi_0^{**}(u)=\sup_{v\ge 0}
\Big(uv-\varphi_0^*(v)\Big). \]
To prove that
\[
\varphi(u^{1/p})=\varphi_0(u^{1/p})^p=\sup_{v>0}
\Big(u^{1/p}v-\varphi_0^*(v)\Big)^p
\]
is convex, it is enough to show that
$f(u):=\Big(u^{1/p}v-\varphi_0^*(v)\Big)^p$ is a convex function.
Since
\[
f'(u)=\Big(u^{1/p}v-\varphi_0^*(v)\Big)^{p-1}u^{1/p-1}v =
\left(\frac{u^{1/p}v-\varphi_0^*(v)}{u^{1/p}}\right)^{p-1}v=:g(u)^{p-1}v
\]
and $g(u)$ is increasing, it follows that $f$ is convex.
\end{proof}

Note that the previous lemma guarantees only that $h,k\in{\mathcal P}$
(and $h$ and $k$ need not necessarily to be concave).

\begin{exa}
%{\bf Example 3.3.}
If $1\le p<q<\infty$ and $\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p})$
with $\rho(t)=\min\big\{1,t\big\}$, then $\varphi(u)=u^qh(u^{p-q})$
with $h(t)=\max\big\{1,t\big\}$. Obviously, $\rho\in\tilde{\mathcal P}$,
but $h\in{\mathcal P}\setminus\widetilde{\mathcal P}$.
In particular, if $p=2$ and $q=3$, then
$\varphi^*(u)=u^{3/2}k(u^{2-3/2})$ with
\[
k(t)=\left\{
\begin{array}{ll}
t/4,    & 0\le t\le\sqrt{2},\\
t^{-1}-t^{-3}, & \sqrt{2}\le t\le\sqrt{3},\\ 2\cdot 3^{-3/2}&
t\ge\sqrt{3}.
\end{array}
\right.
\]
One can prove that $k\in\widetilde{\mathcal P}$.
\end{exa}

\section{Modular estimates}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the proof of the first modular estimate, we need the following
representation of concave functions, which goes back to J.~Peetre
\cite{Peetre66} (see also \cite[Lemma~5.4.3]{BL}).

\begin{lemma}\label{le:concave-represent} Every function
$h\in\widetilde{\mathcal P}$ can be represented in the following form:
%%%
\begin{equation}\label{eq:concave-represent-1}
h(u)=a_h+b_hu+\int_0^\infty\min(u,t)\,dm(t), \quad\mbox{for all}\quad
u>0,
\end{equation}
%%%
where
%%%%
\[
a_h:=\lim_{u\to 0+} h(u),
\quad
b_h:=\lim_{u\to \infty}\frac{h(u)}{u},
\]
%%%%
and $m:\mathbf{R}_+\to\mathbf{R}_+$ is a nondecreasing function (in fact,
$m(t)=-h'(t)$).
\end{lemma}


Now we are ready to prove some modular inequalities, which are the keys
for the above-mentioned estimations of norms.

\begin{theorem}\label{th:int-conc}
Let $1\le p<q\le\infty$ and $T\in{\mathcal A}(L^p,L^q)$.

\textup{(a)} If $q<\infty$ and $\varphi(u)=u^q h(u^{p-q})$ for some
$h\in\widetilde{\mathcal P}$
($\varphi$ not necessarily convex), then \[
I_\varphi\left(\frac{Tx}{M}\right)\le \gamma_{p,q}I_\varphi(x)
\quad\mbox{for all}\quad x\in L^p\cap L^q. \]

\textup{(b)} If $q=\infty$ and $\psi(u)=\varphi(u^{1/p})$ is convex,
then \[
I_\varphi\left(\frac{Tx}{2^{1-1/p}M}\right)\le I_\varphi(x)
\quad\mbox{for all}\quad x\in L^p+ L^\infty. \]
\end{theorem}

\begin{proof} (a)
Due to Lemma~\ref{le:concave-represent}, $h$ can be represented in
the form
(\ref{eq:concave-represent-1}). Hence,
%%%%%
\begin{equation}\label{eq:int-conc-0}
\varphi(u)=u^qh(u^{p-q})=a_h u^q+b_h u^p+
\int_0^\infty\min(u^p,tu^q)\, dm(t),
\quad u\in\mathbf{R}_+.
\end{equation}
Consequently,
%%%
\begin{eqnarray}
&&
I_\varphi\left(\frac{Tx}{M}\right)
=
\int_\Omega\varphi\left(\frac{|Tx(s)|}{M}\right)\,d\mu(s) =
a_h\left\|\frac{Tx}{M}\right\|_q^q+
b_h\left\|\frac{Tx}{M}\right\|_p^p
\\
\nonumber
&&
+
\int_\Omega\left[
\int_0^\infty\min\left(
\left(\frac{|Tx(s)|}{M}\right)^p,t\left(\frac{|Tx(s)|}{M}\right)^q
\right)\,dm(t)
\right]\,d\mu(s).
\end{eqnarray}
%%%
Since the operator $T$ is bounded in $L^p$ and $L^q$, we get
%%%
\begin{eqnarray}
\label{eq:int-conc-2}
a_h\left\|\frac{Tx}{M}\right\|_q^q+b_h\left\|\frac{Tx}{M}\right\|_p^p
&\le&
a_h\left(\frac{M_1}{M}\right)^q \|x\|_q^q +
b_h\left(\frac{M_0}{M}\right)^p \|x\|_p^p
\\
&\le&
a_h\|x\|_q^q +b_h\|x\|_p^p
\nonumber
\end{eqnarray}
%%%
and, according to Proposition~\ref{pr:K-bound},
\[
K_{p,q}\left(t,\frac{Tx}{M};L^p,L^q\right) \le
K_{p,q}(t,x;L^p,L^q)
\quad\mbox{for all}\quad t>0.
\]
By using Sparr's Lemma~\ref{le:sparr}
we obtain
%%%
\[
K^*_{p,q}\left(t,\frac{Tx}{M};L^p,L^q\right) \le
\gamma_{p,q}
K^*_{p,q}(t,x;L^p,L^q)
\quad\mbox{for all}\quad t>0.
\]
Hence, by the Fubini theorem and in view of the definition of
$K^*_{p,q}$, we get
%%%
\begin{eqnarray}
\label{eq:int-conc-4}
&&
\int_{\Omega}\left[\int_0^\infty
\min\left(
\left(\frac{|Tx(s)|}{M}\right)^p,t\left(\frac{|Tx(s)|}{M}\right)^q
\right)
d\,m(t)\right]
d\mu(s)
\\
\nonumber
&&
=
\int_0^\infty K^*_{p,q}\left(t,\frac{Tx}{M};L^p,L^q\right)dm(t)
\\
\nonumber
&&
\le
\gamma_{p,q}
\int_0^\infty
K^*_{p,q}(t,x;L^p,L^q) dm(t)
\\
\nonumber
&&
=
\gamma_{p,q}\int_{\Omega}
\left[
\int_0^\infty\min(|x(s)|^p,t|x(s)|^q)dm(t) \right]
d\mu(s).
\end{eqnarray}
%%%
Combining (\ref{eq:int-conc-2})--(\ref{eq:int-conc-4}) and taking
into account that $\gamma_{p,q}> 1$ we obtain
%%%
\begin{eqnarray*}
&&
I_\varphi\left(\frac{Tx}{M}\right)
\le
a_h\|x\|_q^q +b_h\|x\|_p^p+
\gamma_{p,q}
\int_\Omega\left[
\int_0^\infty\min(|x(s)|^p,t|x(s)|^q)\,dm(t)\right]\,d\mu(s)
\\
&&
\le
\gamma_{p,q}\left(
a_h\|x\|_q^q +b_h\|x\|_p^p+
\int_\Omega\left[
\int_0^\infty\min(|x(s)|^p,t|x(s)|^q)\,dm(t)\right]\,d\mu(s) \right)
\\
&&
=
\gamma_{p,q}\int_\Omega\varphi(|x(s)|)\,d\mu(s)=\gamma_{p,q}I_\varphi(x).
\end{eqnarray*}

(b)
For all $x\in L^p+L^\infty$ and $t>0$, according to the Kr\'ee formula
(see \cite[Theorem~5.2.1]{BL}),
%%%
\begin{equation}\label{eq:mod-infty-2}
\left(
\int_0^t x^*(s)^p ds
\right)^{1/p}
\le
K(t^{1/p}, x; L^p,L^\infty)
\le
2^{1-1/p}
\left(
\int_0^t x^*(s)^p ds
\right)^{1/p}.
\end{equation}
%%%
The constant $2^{1-1/p}$ cannot be improved (see \cite{Bergh}).
Due to Proposition~\ref{pr:K-bound},
%%%
\begin{equation}\label{eq:mod-infty-3}
K\left(t,\frac{Tx}{M};L^p,L^\infty\right) \le
K(t,x;L^p,L^\infty)
\quad\mbox{for all}\quad t>0.
\end{equation}
%%%
From (\ref{eq:mod-infty-2}) and (\ref{eq:mod-infty-3}) it follows
that
\[
\int_0^t\left(\frac{(Tx)^*(s)}{2^{1-1/p}M}\right)^pds \le
\int_0^t x^*(s)^p\,ds
\quad\mbox{for all}\quad t>0.
\]
Since $\psi(u)=\varphi(u^{1/p})$ is convex, by the
Hardy-Littlewood-P\'olya majorization theorem (see, e.g.,
\cite[p.~88]{BeSh}),
%%%
\begin{eqnarray*}
&&
\int_0^\infty
\varphi\left(\frac{(Tx)^*(s)}{2^{1-1/p}M} \right)ds
=
\int_0^\infty
\psi\left(
\left[
\frac{(Tx)^*(s)}{2^{1-1/p}M}
\right]^p
\right)ds
\\
&&
\le
\int_0^\infty \psi(x^*(s)^p)\,ds =
\int_0^\infty \varphi(x^*(s))\,ds.
\end{eqnarray*}
Since the modular $I_\varphi$ is rearrangement-invariant (see
(\ref{eq:modular})), the latter inequality gives (b).
\end{proof}

The method of the proof of part (a) is due to J.~Peetre
\cite{Peetre70}. In this proof the estimation of the functional
$K_{p,q}^{*}$ (``$K_{p,q}^{*}$-monotonicity'' property) is very essential.
G.~Sparr \cite{Sparr} proved that $\gamma_{p,q}$ is the
best possible constant in the estimation of the functional
$K_{p,q}^{*}$ (cf. Lemma \ref{le:sparr}). He also proved that
$1<\gamma_{p,q}<2$ for $1< p,q<\infty$. Now we give more precise
information about this constant.

\begin{proposition}\label{pr:sparr-constant} Suppose $1\le
p,q<\infty$.

\textup{(a)} Then $\gamma_{p,q}=\gamma_{q,p}$ and $\gamma_{1,1}=1$.

\textup{(b)} If $q>1$, then
%%%
\begin{equation}\label{eq:sparr-constant-1} \gamma_{p,q}=\inf\left\{
x+\left(\frac{p}{q}x^{p-1}\right)^{1/(q-1)}\ : \
x^p+\left(\frac{p}{q}x^{p-1}\right)^{q/(q-1)} = 1 \right\}.
\end{equation}
%%%
In particular,
\[
\gamma_{q,q}=2^{1-1/q},
\quad
\gamma_{1,q}=1+q^{1/(1-q)}-q^{q/(1-q)}.
\]

\textup{(c)} $\gamma_{p,q}$ continuously increases in $p$ and $q$.

\textup{(d)} If $p\le q$, then
$2^{1-1/p}\le\gamma_{p,q}\le 2^{1-1/q}$.
\end{proposition}

\begin{proof} Property (a) is obvious. Suppose $q>1$ and rewrite $\gamma_{p,q}$ in
the form \[
\gamma_{p,q}=\left\{\gamma>0\ : \
\min_{0\le x\le\gamma} F(\gamma,x,p,q)=1\right\}, \]
where $F(\gamma,x,p,q)=x^p+(\gamma-x)^q$ and
$\gamma\in(0,2),x\in[0,\gamma], p\in[1,\infty),q\in(1,\infty)$.
Obviously, for $x\in(0,\gamma)$,
%%%
\begin{equation}\label{eq:sparr-constant-2}
\frac{\partial
F}{\partial x}=
px^{p-1}-q(\gamma-x)^{q-1},
\frac{\partial^2 F}{\partial x^2}=
p(p-1)x^{p-2}+q(q-1)(\gamma-x)^{q-2}>0.
\end{equation}
%%%
Hence, $F(\gamma,x,p,q)$ is strictly convex in $x$ and it has a
unique minimum on $[0,\gamma]$ at a point
$x(\gamma,p,q)\in(0,\gamma)$, which is the solution of the equation
\[
\frac{\partial F}{\partial x}=
px^{p-1}-q(\gamma-x)^{q-1}=0.
\]
Clearly,
%%%
\begin{equation}\label{eq:sparr-constant-3}
0<\gamma-1<x(\gamma,p,q)<1,
\end{equation}
%%%
and
\[
\gamma=x(\gamma,p,q)+\left(\frac{p}{q}x(\gamma,p,q)^{p-1}\right)^{1/(q-1)},
\]
\[
\min_{0\le x\le\gamma} F(\gamma,x,p,q)=
x(\gamma,p,q)^p+\left(\frac{p}{q}x(\gamma,p,q)^{p-1}\right)^{q/(q-1)}.
\]
So, (\ref{eq:sparr-constant-1}) is proved. Using
(\ref{eq:sparr-constant-1}), one can easy calculate $\gamma_{1,q}$
and $\gamma_{q,q}$. The proof of (b) is finished.

Let us prove (c). Due to (a), it is sufficient to prove that
$\gamma_{p,q}$ continuously increases in $p$. Consider $\gamma=\gamma(p)$ such
that \[
\min_{0\le x\le\gamma} F(\gamma,x,p,q)=1. \]
From conditions
%%%
\begin{equation}\label{eq:sparr-constant-4}\begin{cases}
\min\limits_{0\le x\le\gamma} F(\gamma,x,p,q)-1=x^p+(\gamma-x)^q-1=0,
\\
\frac{\partial F}{\partial x}(\gamma,x,p,q)=
px^{p-1}-q(\gamma-x)^{q-1}=0,
\end{cases}
\end{equation}
%%%
taking into account that $\frac{\partial^2 F}{\partial x^2}>0$ (see
(\ref{eq:sparr-constant-2})), one can find the derivative of the
implicit function $\gamma(p)$:
\[
\frac{d\gamma}{dp}=-\frac{x^p\log x}{q(\gamma-x)^{q-1}}. \]
Taking into account (\ref{eq:sparr-constant-3}), we see that
$\frac{d\gamma}{dp}>0$ for all $\gamma(p)$ satisfying
(\ref{eq:sparr-constant-4}). Hence, such $\gamma(p)$ continuously increases in
$p$.

On the other hand, $\gamma_{p,q}$ is the smallest $\gamma(p)$
satisfying (\ref{eq:sparr-constant-4}) (note that conditions
(\ref{eq:sparr-constant-4}) depend on $p$). Thus, $\gamma_{p,q}$
continuously increases in $p$. Property (c) is proved.

Property (d) follows from the monotonicity of $\gamma_{p,q}$: \[
2^{1-1/p}=\gamma_{p,p}\le \gamma_{p,q}\le \gamma_{q,q}=2^{1-1/q}. \
\qed
\]
\renewcommand{\qed}{} \end{proof}

\section{The main interpolation theorem}
\setcounter{equation}{0}
Our main result reads:

\begin{theorem}\label{th:main}
Suppose $1\le p<q\le\infty$ and $\rho\in\widetilde{\mathcal P}$. If
$q=\infty$ assume in addition that $\rho_*(\mathbf{R}_+)=\mathbf{R}_+$, where
$\rho_*(t):=t\rho(1/t)$.
If $\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p})$, then the Orlicz space
$L^\varphi$ (with both the Luxemburg and the Orlicz norm) is an
interpolation space for linear operators between $L^p$ and $L^q$, and
\[
\|T\|_{L^\varphi\to L^\varphi}\le C\max
\big\{
\|T\|_{L^p\to L^p} ,
\|T\|_{L^q\to L^q}
\big\},
\]
where

\textup{(a)} $C\le 2\gamma_{1,q}=2(1+q^{1/(q-1)}-q^{q/(q-1)})\le 2^{2-1/q} < 4,$
when $1=p<q < \infty$.

\textup{(b)} $C\le
\min\left\{
(2\gamma_{p,q})^{1/p},(2\gamma_{q',p'})^{1/q'} \right\}
\le
2^{1/(pq')+\min\{1/p,1/q'\}}<4,$
when \\ $1<p<q<\infty$.

\textup{(c)} $C\le 2^{1-1/p}<2$,
when $1\le p<q= \infty$.

\noindent
In particular, if either $1<p<q\le 2$ or $2\le p<q\le \infty$, then $C<2$.
\end{theorem}

\begin{proof} First observe that the function $\varphi$ is convex, due to
Lemma~\ref{le:properties}(a). Hence, $\varphi$ generates an Orlicz
space $L^\varphi$.

Let $1\le p<q<\infty$.
By Lemma~\ref{le:properties}(b), there is a function $h\in{\mathcal P}$
such that $\varphi(u)=u^qh(u^{p-q})$. From (\ref{eq:conc-maj}) we see
that $\tilde{h}\in\widetilde{\mathcal P}$ and
%%%
\begin{equation}\label{eq:int-1}
\varphi(u)
\le
u^q\tilde{h}(u^{p-q})
\le 2\varphi(u)
\quad\mbox{for all } u>0.
\end{equation}
%%%
Applying Theorem~\ref{th:int-conc}(a) to the function $\psi(u)=u^q
\tilde{h}(u^{p-q})$ and taking into account (\ref{eq:int-1}), we obtain
%%%
\begin{equation}\label{eq:int-2}
I_\varphi\left(\frac{Tx}{M}\right)
\le
I_\psi\left(\frac{Tx}{M}\right)
\le
\gamma_{p,q}I_\psi(x)
\le
2
\gamma_{p,q}I_\varphi(x)
\quad \mbox{for all}\quad x\in L^p\cap L^q. \end{equation}
%%%%
Note that if $A>0$, then
%%%%
\begin{eqnarray}\label{eq:int-3}
\varphi(Au)=(Au)^qh((Au)^{p-q})
&\le&
A^qu^q\max\Big\{1,A^{p-q}\Big\}h(u^{p-q})
\\
\nonumber
&=&
\max\Big\{A^p,A^q\Big\}\varphi(u).
\end{eqnarray}
%%%%
In particular, $\varphi$ satisfies the $\Delta_2$-condition for all
$u\ge 0$. If $A=(2\gamma_{p,q})^{-1/p}$, then from
(\ref{eq:int-2}) and (\ref{eq:int-3}) we conclude \[
I_\varphi\left(\frac{Tx}{(2\gamma_{p,q})^{1/p}M}\right) \le
\frac{1}{2\gamma_{p,q}}
I_\varphi\left(\frac{Tx}{M}\right)
\le I_\varphi(x)
\quad \mbox{for all}\quad x\in L^p\cap L^q, \]
which gives
%%%
\begin{equation}\label{eq:OL-norm}
\|Tx\|_\varphi\le (2\gamma_{p,q})^{1/p}M\|x\|_\varphi, \quad
\|Tx\|^0_\varphi\le (2\gamma_{p,q})^{1/p}M\|x\|^0_\varphi
\end{equation}
%%%
for all $x\in L^p\cap L^q$. Since
$\varphi$ satisfies the $\Delta_2$-condition for all $u\ge 0$, it
follows that $L^p\cap L^q$ is dense in $L^\varphi$ (see
\cite[Ch.~2]{KR} for the case of $N$-functions and a finite measure,
in the general case this result can be obtained analogously). Hence,
(\ref{eq:OL-norm}) is true for all $x\in L^\varphi$. This fact and
Proposition~\ref{pr:sparr-constant}(d) show that
%%%
\begin{equation}\label{eq:int-4}
C\le (2\gamma_{p,q})^{1/p}\le 2^{(2-1/q)/p}< 4.
\end{equation}

Now we will prove the second estimate in (b) by using duality
arguments. Suppose $1<p<q<\infty$ and $T\in{\mathcal A}(L^p,L^q)$. Then
$T$ maps $L^p+L^q$ into itself, but $T$ maps also $L^p\cap L^q$ into
itself (see, e.g., \cite[Ch.~1, Lemma~4.1]{kps}). Since $L^p\cap L^q$
is dense in $L^p$ and $L^q$, then $(L^p\cap L^q)^*=L^{p'}+L^{q'}$
(see, e.g., \cite[Ch.~1, Th.~3.1]{kps}). Therefore, $T^*\in{\mathcal
A}(L^{p'},L^{q'})$. Due to Lemma~\ref{le:properties}(c), there is a
function $k\in{\mathcal P}$ such that $\varphi^*(u)=u^{p'}k(u^{q'-p'})$
for all $u>0$. As above one can prove that
%%%
\begin{equation}\label{eq:OL-norm-conjugate} \|T^*x\|_{\varphi^*}\le
(2\gamma_{q',p'})^{1/q'}M\|x\|_{\varphi^*}, \quad
\|T^*x\|^0_{\varphi^*}\le
(2\gamma_{q',p'})^{1/q'}M\|x\|^0_{\varphi^*} \end{equation}
%%%
for all $x\in L^{p'}\cap L^{q'}$,
and $\varphi^*$ satisfies the $\Delta_2$-condition for all $u\ge 0$.
Consequently,
(\ref{eq:OL-norm-conjugate}) holds for all $x\in L^{\varphi^*}$.
Taking into account duality of the Orlicz and Luxemburg norms and
Proposition~\ref{pr:sparr-constant}, this gives
%%%
\begin{equation}\label{eq:int-5}
C\le (2\gamma_{q',p'})^{1/q'}\le 2^{(2-1/{p'})/{q'}}< 4.
\end{equation}
%%%
Since
\[
\min\Big\{
2^{(2-1/q)/p},
2^{(2-1/{p'})/{q'}}
\Big\}
=
2^{1/(pq')+\min\{1/p,1/q'\}},
\]
we obtain from (\ref{eq:int-4}) and (\ref{eq:int-5}) that $C<2$ in
the cases $1<p<q\le 2$ or $2\le p<q<\infty$. The proof of (b) is
complete.

Let $1\le p<q=\infty$.
In view of Lemma~\ref{le:properties}(d), the function
$\psi(u)=\varphi(u^{1/p})$ is convex. Hence, by
Theorem~\ref{th:int-conc}(b), we obtain the modular estimate
\[
I_\varphi\left(\frac{Tx}{2^{1-1/p}M}\right)\le I_\varphi(x)
\quad\mbox{for all}\quad x\in L^\varphi, \]
which implies
%%%
\[
\|Tx\|_\varphi\le 2^{1-1/p}M\|x\|_\varphi, \quad
\|Tx\|^0_\varphi\le 2^{1-1/p}M\|x\|^0_\varphi \]
%%%
for all $x\in L^\varphi$. This gives (c).
\end{proof}

%{\bf Remark 5.2.}
\begin{rem}
Obviously Theorem \ref{th:main} (a), (b) can, in fact, be reformulated
in terms of Simonenko indices (\ref{eq:Simonenko}): if
\[
1\le p \le a_{\varphi}\le b_\varphi\le q<\infty,
\]
then the Orlicz space $L^\varphi$ is an interpolation space between
$L^p$ and $L^q$ with the same interpolation constant as in Theorem
\ref{th:main} (a), (b).
\end{rem}

%{\bf Remark 5.3.}
\begin{rem}
If $1\le p<q<\infty$ and $\varphi(u)=u^qh(u^{p-q})$, where
$h\in\widetilde{\mathcal P}$, then from the proof of the above theorem
it follows that $L^\varphi$ is an interpolation space between $L^p$ and
$L^q$, and we have a better estimate of the interpolation constant
\[
C\le (\gamma_{p,q})^{1/p}\le 2^{1/(q'p)}<2. \]
\end{rem}

We illustrate Remark 5.3 with the following example:

\begin{exa}
%{\bf Example 5.4.}
If $r\ge (3+\sqrt{5})/2$, then $\varphi(u)=u^r(1+|\log u|)$ is
convex. For every $p$ and $q$ such that $1\le p<r<q<\infty$ and
$r-p=q-r$, we have $\varphi(u)=u^qh(u^{p-q})$ with
$h(u)=\sqrt{u}\left(1+(q-p)^{-1}|\log u|\right)$.
One can prove that $h\in\widetilde{\mathcal P}$.
\end{exa}

%{\bf Remark 5.5.}
\begin{rem}
In connection with Theorem~\ref{th:main}(c) let us mention that for the
case $p=1$ and $q=\infty$ it coincides with the well-known Orlicz
interpolation theorem. More precisely, Orlicz proved it in 1934 with
certain constant $C>1$ but from the Calder\'on-Mitjagin interpolation
theorem it follows with the constant $1$ (see \cite[Ch.~2, Th.~4.9]{kps};
cf. also \cite{M89prime} for the direct proof). Moreover, G.~G.~Lorentz
and T.~Shimogaki \cite[Theorem~7]{LorShi} observed that for the
function \[
\varphi(u)=\int_0^u(u-t)^pdm(t)
\]
with increasing function $m:\mathbf{R}_+\to\mathbf{R}_+$, the interpolation
constant $C$ is $1$.
\end{rem}

\section*{Acknowledgements}
This paper was done during a visit of the first author at Lule\r{a}
University of Technology in November-December 1999. The research was
supported by a grant from the Royal Swedish Academy of Sciences for
cooperation between Sweden and the former Soviet Union (project
35124). We are grateful to both institutions for support and
hospitality.


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