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%\title{\Large \bf On the Interpolation Constant for Orlicz Spaces}

\author{By\\[2ex]
{\bf Alexei Yu. Karlovich}$^*$, Odessa, and{\bf Lech~Maligranda}$^{**}$, 
Lule{\aa}}

\date{}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document} 

\vspace*{30mm}
\begin{center}
\Large
{\bfseries On the Interpolation Constant for Orlicz Spaces$^{*}$}
\\[15 mm]
\normalsize
by
\\[15 mm]
{\bf Alexei Yu. Karlovich}$^{a}$ and {\bf Lech Maligranda}$^{b}$
\\[10mm]
\noindent 
$^a$\begin{minipage}[t]{8cm}
Department of Mathematics and Physics \\
South Ukrainian Pedagogical University \\
Staroportofrankovskaya 26 \\
65020 Odessa, Ukraine \\ 
{\em e-mail}: karlik@paco.net \\
\end{minipage}
\\[5mm]
$^b$\begin{minipage}[t]{8cm}
Department of Mathematics \\
Lule{\aa} University of Technology \\
S\mbox-971\,87~Lule{\aa}, Sweden \\
{\em e-mail}: lech@sm.luth.se \\
\end{minipage}

\end{center}

\thispagestyle{empty}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}


\footnotetext[0]{$^{*}$~Research supported by a grant from the 
Royal Swedish Academy of Sciences for cooperation between Sweden 
and the former Soviet Union}


\newpage
{\bfseries \noindent Alexei Yu. Karlovich and Lech Maligranda},  
 {\em On the interpolation constant for Orlicz spaces}, 
 Lule\aa \ University of Technology, Department of Mathematics, Research 
 Report 1 (2000)
\vfill
{\bfseries \noindent 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 the 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 $C=1$.

\vfill
{\bfseries \noindent Subject Classification:} AMS (1991) Primary 46B70, 46E30; 
Secondary 26D07
\vfill
{\bfseries \noindent Key  words:} Orlicz spaces, interpolation constant, 
interpolation of operators, $K$-functional, convex function, concave function
\vfill
{\bfseries \noindent Note:} This report will be submitted for publication 
elsewhere
\vfill
{\bfseries \noindent ISSN:   1400-4003}
\vfill
{\bfseries \noindent Lule\aa \  University of Technology
\\Department of Mathematics
\\S-97187 Lule\aa, \ Sweden}

\newpage

%\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: 
%%%
%\begin{equation}\label{eq:RT-norm}
\[
\|T\|_{L^p\to L^q}\le C
\left(\|T\|_{L^{p_0}\to L^{q_0}}\right)^{1-\theta} 
\left(\|T\|_{L^{p_1}\to L^{q_1}}\right)^{\theta} \]
%\end{equation}
%%%
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: 

\vspace{0.3cm}
{\em
\noindent
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 $L^\psi$? }
\vspace{0.3cm}

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 let us 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 {\rm (\ref{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 %%%%
%\begin{equation}\label{eq:GP-norm}
\[
\|T\|_{L^\varphi\to L^\psi}\le C\max\Big\{ M_0, M_1\Big\}, \quad
M_i:=\|T\|_{L^{\varphi_i}\to L^{\psi_i}}, \quad i=0,1, \]
%\end{equation}
%%%
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 (see Section 6) 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 
$\varphi(u^{1/p})$.
Then, 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 $C=1$. Finally, Section 6 is reserved for some additional 
results and remarks.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\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 
({\it 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 non-increasing rearrangement of $x$ (see 
\cite{BeSh,kps,LT}).
The {\it 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 {\it Luxemburg norm} \[
\|x\|_\varphi:=\inf\left\{\lambda>0\ : \ I_\varphi(x/\lambda)\le 
1\right\} \]
and the {\it 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 $X$ 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 
${\cal A}(X_0,X_1)$ 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 {\it interpolation space} between $X_0$ 
and $X_1$ if every admissible operator $T\in{\cal A}(X_0,X_1)$ 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 $T\in{\cal A}(X_0,X_1)$, 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it Proof.}
If $x=x_0+x_1$ is any decomposition of $x\in X_0+X_1$, then \[
\frac{Tx}{M}=\frac{Tx_0}{M}+\frac{Tx_1}{M}, \]
and
\begin{eqnarray*}
&&
K_{p,q}\left(t,\frac{Tx}{M};X_0,X_1\right)\le 
\left\|\frac{Tx_0}{M}\right\|_{X_0}^p+t\left\|\frac{Tx_1}{M}\right\|_{X_1}^q 
\\
&&
\le
\left(\frac{M_0}{M}\right)^p\|x_0\|_{X_1}^p+ 
t\left(\frac{M_1}{M}\right)^q\|x_1\|_{X_1}^q\le 
\|x_0\|_{X_0}^p+t\|x_1\|_{X_1}^q.
\end{eqnarray*}
Taking the infimum over all decompositions we obtain our assertion. 
\rule{2mm}{2mm}
%%%%

Following \cite{Sparr}, we consider the functional  $K^*_{p,q}$ on the 
couple of Lebesgue spaces $(L^p,L^q)$ 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}\label{le:sparr}
{\rm (G. Sparr \cite[Lemma~5.1, Example~5.3]{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
\[
\gamma_{p,q}:=\inf\left\{\gamma>0\::\:
\inf_{\tiny\begin{array}{c}
x+y=\gamma, \\
x,y\ge 0
\end{array}}
\Big(x^p+y^q\Big)=1
\right\}
\]
is the Sparr constant, which can not 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 $\cal P$ the set of all quasi-concave functions 
$\rho:[0,\infty)\to[0,\infty)$ which are continuous, positive on 
${\bf R}_+:=(0,\infty)$ and such that 
\[
\rho(s)\le\max\left(1,\frac{s}{t}\right)\rho(t), 
\quad\mbox{for 
all}\quad s,t>0.
\]
Let $\widetilde{\cal P}$ denote the subset of all concave functions 
in ${\cal P}$. Note that if $\rho\in{\cal P}$, then $\tilde{\rho}$ 
defined by
\[
\tilde{\rho}(t):=\inf_{s>0}\left(1+\frac{t}{s}\right)\rho(s) \]
belongs to $\widetilde{\cal P}$ 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 $p$, $1\le 
p\le\infty$, i.e., $1/p+1/p'=1$ ($1/\infty$ means $0$). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{lemma}\label{le:convex-increasing} Let $1<p<\infty$ and 
$\varphi$ be a convex function on ${\bf R}_+$.
The function $u^{-p}\varphi(u)$ is increasing (decreasing) on ${\bf 
R}_+$ if and only if the function $u^{-p'}\varphi^*(u)$ is decreasing 
(increasing) on ${\bf R}_+$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it 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 using also 
the fact that for convex function $\varphi$ we have $\varphi^{**}=\varphi$. 
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

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 $\rho\in\widetilde{\cal P}$, 
\[
\varphi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p}) \quad\mbox{for all}\quad u>0.
\]
{\rm (a)} Then $\varphi$ is convex.

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

\noindent
{\rm (d)}
If $q=\infty$ and $\rho_*(t):=t\rho(1/t)$ satisfies $\rho_*({\bf 
R_+})={\bf R_+}$, then $\varphi(u)=\rho_*^{-1}(u)^p$ and 
$\psi(u)=\varphi(u^{1/p})$ is a convex function. \end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it 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 ${\bf R}_+$ and 
$\varphi^{-1}({\bf R}_+)={\bf R}_+$. 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) $h\in{\cal P}$, 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) $k\in{\cal P}$, 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 $\rho_*({\bf R}_+)={\bf R}_+$, we can see that 
$\varphi^{-1}({\bf R}_+)={\bf R}_+$ and $\varphi^{-1}$ is concave. 
Hence, $\varphi$ is 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. 
\rule{2mm}{2mm}

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

\vspace{0.3cm}
\noindent
{\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, $\rho\in\tilde{\cal P}$, 
but $h\in{\cal P}\setminus\widetilde{\cal P}$. 

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 $k\in\widetilde{\cal P}$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\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 
$h\in\widetilde{\cal P}$ 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
%%%%
\begin{equation}\label{eq:concave-represent-2} a_h:=\lim_{u\to 0+} 
h(u),
\quad
b_h:=\lim_{u\to \infty}\frac{h(u)}{u},
\end{equation}
%%%%
and $m:{\bf R}_+\to{\bf R}_+$ 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 announced estimations of norms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{theorem}\label{th:int-conc}
Let $1\le p<q\le\infty$ and $T\in{\cal A}(L^p,L^q)$. 

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

{\rm (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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it Proof.}
(a)
Due to Lemma~\ref{le:concave-represent}, $\rho$ 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{\bf 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{equation}\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
\end{equation}
%%%
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}
&&
\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)
=
\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)
=
\gamma_{p,q}\int_{\Omega}
\left[
\int_0^\infty\min(|x(s)|^p,t|x(s)|^q)dm(t) \right]
d\mu(s).
\label{eq:int-conc-4}
\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). \rule{2mm}{2mm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

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 $1\le 
p,q<\infty$.

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

{\rm (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,
\[
\gamma_{q,q}=2^{1-1/q},
\quad
\gamma_{1,q}=1+q^{1/(1-q)}-q^{q/(1-q)}.
\]

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

{\rm (d)} If $p\le q$, then
\[
2^{1-1/p}\le\gamma_{p,q}\le 2^{1-1/q}.
\]
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it 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},
\quad
\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 
$\gamma_{p,q}$ 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} \left\{
\begin{array}{l}
\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{array}
\right.
\end{equation}
%%%
taking into account that $\frac{\partial^2 F}{\partial x^2}>0$ (see 
(\ref{eq:sparr-constant-2})), one can found 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 $\gamma(p)$ 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, $\gamma_{p,q}$ 
increases in $p$. Property (c) is proved. 

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\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 $\rho\in\widetilde{\cal P}$. If 
$q=\infty$ assume in addition that $\rho_*({\bf R}_+)={\bf R}_+$, 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

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

{\rm (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 $1<p<q<\infty$.
 
{\rm (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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it 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 $h\in{\cal P}$ 
such that $\varphi(u)=u^qh(u^{p-q})$. From (\ref{eq:conc-maj}) we see 
that $\tilde{h}\in\widetilde{\cal P}$ 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}
%%%
By using Theorem~\ref{th:int-conc}(a) to the function $\psi(u)=u^q 
\tilde{h}(u^{p-q})$, 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})= 
\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 $T\in{\cal A}(L^p,L^q)$. 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, $T^*\in{\cal 
A}(L^{p'},L^{q'})$. Due to Lemma~\ref{le:properties}(c), there is a 
function $k\in{\cal P}$ 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 
completed.

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). \rule{2mm}{2mm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
\noindent
{\bf Remark 5.2.}
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).


\vspace{0.3cm}
\noindent
{\bf Remark 5.3.}
If $1\le p<q<\infty$ and $\varphi(u)=u^qh(u^{p-q})$, where 
$h\in\widetilde{\cal P}$, 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: \[
C\le (\gamma_{p,q})^{1/p}\le 2^{1/(q'p)}<2. \]

We illustrate Remark 5.3 with the following example:

\vspace{0.3cm}
\noindent
{\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 \[
h(u)=\sqrt{u}\left(1+\frac{1}{q-p}|\log u|\right). \]
One can prove that $h\in\widetilde{\cal P}$. 

\vspace{0.3cm}
\noindent
{\bf Remark 5.5.}
In connection to 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 $m:{\bf R}_+\to{\bf R}_+$, the interpolation 
constant $C$ is $1$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Complementary results and remarks} \setcounter{equation}{0}
1.
We present an explanation that the constant in 
Theorem~\ref{th:Maligranda} satisfies the estimate 
(\ref{eq:BK-constant}). In fact, if $\rho$ is a concave function, 
then the Calder\'on-Lozanovskii space (for the definition, see 
\cite[Section~4.3]{BK} or \cite[Ch.~15]{M89}) coincide with the 
Orlicz space $L^\psi$, and (cf. \cite[p.~179]{M89}) %%%
\begin{equation}\label{eq:identif}
\|x\|_{\rho(L^{\psi_0},L^{\psi_1})}
\le
\|x\|_{L^\psi}
\le
C_\psi\|x\|_{\rho(L^{\psi_0},L^{\psi_1})}. \end{equation}
%%%
Consider the concave functions $\rho_0(t):=a_\rho+b_\rho t$ and 
$\rho_1(t):=\rho(t)-\rho_0(t)$, where $a_\rho,b_\rho$ are defined by 
(\ref{eq:concave-represent-2}). From the proof of Theorem~4.3.11 
\cite{BK} one can see that %%%
\begin{eqnarray}
\|T\|_{\rho_0(L^{\varphi_0},L^{\varphi_1})\to\rho_0(L^{\psi_0},L^{\psi_1})} 
& \le & M,
\\
\|T\|_{\rho_1(L^{\varphi_0},L^{\varphi_1})\to\rho_1(L^{\psi_0},L^{\psi_1})} 
&\le &
(3+2\sqrt{2})M,
\end{eqnarray}
%%%
where $M:=\max\Big\{
\|T\|_{L^{\varphi_0}\to L^{\psi_0}},\|T\|_{L^{\varphi_1}\to 
L^{\psi_1}} \Big\}$.
On the other hand,
%%%
\begin{equation}\label{eq:identif2}
\rho_0(L^{\psi_0},L^{\psi_1})+\rho_1(L^{\psi_0},L^{\psi_1}) 
\stackrel{1}{\subset}
\rho(L^{\psi_0},L^{\psi_1})
\stackrel{2}{\subset}
\rho_0(L^{\psi_0},L^{\psi_1})+\rho_1(L^{\psi_0},L^{\psi_1}). 
\end{equation}
%%%
%%%%%%%%%%%%
Hence, our statement follows from (\ref{eq:identif}) -- 
(\ref{eq:identif2}). Moreover, if $a_\rho=b_\rho=0$, then even 
$C=(3+2\sqrt{2})C_\psi<6C_\psi$. 

2. All results of Sections~2 -- 5 are true both for real or complex 
cases because for Banach lattices we have equalities 
$\|x_i\|_{X_i}=\||x_i|\|_{X_i}, i=0,1$, which give \begin{eqnarray*}
K_{p,q}(t,x;X_0,X_1) &=& 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,\ 0\le 
x_0\in X_0, 0\le x_1\in X_1\Big\} \end{eqnarray*}
(cf. \cite[Proposition~3.1.15]{BK}, \cite[Lemma~15.3]{M89}). Hence, 
we have real-valued $x$ and in the decompositions of $|x|$ one can 
take non-negative elements $x_i\in X_i, i=0,1$. 

3. The estimate in Proposition~\ref{pr:K-bound} for bounded linear 
operators is essential in the proof of Theorem~\ref{th:int-conc}. 
This estimate is true even for bounded sublinear operators in Banach 
lattices because
\begin{eqnarray*}
K_{p,q}(t,x;X_0,X_1) &=&
\inf\Big\{\|x_0\|_{X_0}^p+t\|x_1\|_{X_1}^q \::\ |x|\le x_0+x_1,\ 0\le 
x_0\in X_0, 0\le x_1\in X_1\Big\} \end{eqnarray*}
for Banach lattices $X_0$ and $X_1$
(see \cite[p.~169]{M89}). This observation shows that in 
Theorem~\ref{th:int-conc}
and Theorem~\ref{th:main} (except the part obtained by the duality 
arguments) are true even for sublinear operators.

\vspace{0.3cm}
{\it 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, and to Professor Lars Erik Persson for reading through 
and comment our manuscript.

\newpage

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\end{document}