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On the interpolation constant for Orlicz spaces

Authors: Alexei Yu. Karlovich and Lech Maligranda
Journal: Proc. Amer. Math. Soc. 129 (2001), 2727-2739
MSC (1991): Primary 46B70, 46E30; Secondary 26D07
Published electronically: April 17, 2001
MathSciNet review: 1838797
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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 $C=1$.

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Additional Information

Alexei Yu. Karlovich
Affiliation: Department of Mathematics and Physics, South Ukrainian State Pedagogical University, Staroportofrankovskaya 26, 65020 Odessa, Ukraine
Address at time of publication: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

Lech Maligranda
Affiliation: Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden

Keywords: Orlicz spaces, interpolation constant, interpolation of operators, $K$-functional, convex function, concave function
Received by editor(s): January 24, 2000
Published electronically: April 17, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society