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On the interpolation constant for Orlicz spaces
Authors:Alexei Yu Karlovich  Lech Maligranda
Institution:Department of Mathematics and Physics, South Ukrainian State Pedagogical University, Staroportofrankovskaya 26, 65020 Odessa, Ukraine ; Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden
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,

\begin{displaymath}\Vert T\Vert _{L^\varphi\to L^\varphi} \le C\max\Big\{ \Vert T\Vert _{L^p\to L^p}, \Vert T\Vert _{L^q\to L^q} \Big\}, \end{displaymath}

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 $C=1$.

Keywords:Orlicz spaces  interpolation constant  interpolation of operators  $K$-functional  convex function  concave function
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