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{\Large\bf
On the Interpolation Constant 
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for Sublinear Operators 
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in Orlicz Spaces$^{*}$}

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{\large Alexei Yu. Karlovich} and {Lech Maligranda}

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\footnotetext[0]{$^{*}$~Research supported by a grant from the 
Royal Swedish Academy of Sciences for cooperation between Sweden 
and the former Soviet Union (project 35124).}


\noindent
In this talk we deal with the interpolation 
of bounded sublinear operators
from Lebesgue spaces $L^p$ and $L^q$ into an Orlicz space $L^\varphi$, 
where 
\[
1\le p<q\le\infty
\quad\mbox{and}\quad
\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})
\] 
for some concave function $\rho:{\bf R}_+\to{\bf R}_+$. 
If $q=\infty$ assume in addition that $\rho_*({\bf R}_+)={\bf R}_+$, where 
$\rho_*(t):=t\rho(1/t)$. Our main aim is to get estimates
for the interpolation constant $C$.

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\noindent
{\bf Theorem.}
If a sublinear operator $T$ is bounded in Lebesgue spaces
$L^p$ and $L^q$, then it is bounded in the Orlicz space $L^\varphi$
(with both, the Luxemburg and the Orlicz norm) and
\[
\|T\|_{L^\varphi\to L^\varphi} 
\le C\max
\Big\{
\|T\|_{L^p\to L^p},
\|T\|_{L^q\to L^q} 
\Big\}, 
\]
with the interpolation constant $C$ satisfying

{\rm (a)}
$C\le 2^{1-1/p}<2$ for $1\le p<q= \infty$;

{\rm (b)}
$C\le
(2\gamma_{p,q})^{1/p} 
\le
2^{(2-1/q)/p}<4$
for $1\le p<q<\infty$,
where
\[
\gamma_{p,q}=\inf\left\{
x+\left(\frac{p}{q}x^{p-1}\right)^{1/(q-1)}\ : \ 
x^p+\left(\frac{p}{q}x^{p-1}\right)^{q/(q-1)} = 1 \right\}
\in[2^{1-1/p},2^{1-1/q}]
\]
is the constant introduced by G.~Sparr \cite[Lemma~5.1]{Sparr}.


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\bibitem{Sparr}
G.~Sparr,
{\em Interpolation of weighted $L_p$-spaces}, Studia Math. {\bf 62} 
(1978), 229 -- 271. 

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