<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Maligranda, Lech</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the interpolation constant for Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Proceedings of the American Mathematical Society</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">concave function}</style></keyword><keyword><style  face="normal" font="default" size="100%">convex function</style></keyword><keyword><style  face="normal" font="default" size="100%">interpolation constant</style></keyword><keyword><style  face="normal" font="default" size="100%">interpolation of operators</style></keyword><keyword><style  face="normal" font="default" size="100%">K-functional</style></keyword><keyword><style  face="normal" font="default" size="100%">{Orlicz spaces</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2001</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-06162-7/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{9}</style></number><publisher><style face="normal" font="default" size="100%">{AMER MATHEMATICAL SOC}</style></publisher><pub-location><style face="normal" font="default" size="100%">{201 CHARLES ST, PROVIDENCE, RI 02940-2213 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">129</style></volume><pages><style face="normal" font="default" size="100%">2727-2739</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In this paper we deal with the interpolation from Lebesgue spaces \(L^p\) and \(L^q\), into an Orlicz space \(L^\varphi\), where \( 1 \le p &amp;lt; q \le \infty \) 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,&lt;br /&gt;
\[&lt;br /&gt;
\|T\|_{L^\varphi\to L^\varphi}&lt;br /&gt;
\le C\max\Big\{&lt;br /&gt;
\|T\|_{L^p\to L^p},&lt;br /&gt;
\|T\|_{L^q\to L^q}&lt;br /&gt;
\Big\},&lt;br /&gt;
\]&lt;br /&gt;
where the interpolation constant \(C\) depends only on \(p\) and \(q\). We give estimates for \(C\), which imply \(C&amp;lt;4\). Moreover, if either \( 1 &amp;lt; p &amp;lt; q \le 2\) or \( 2 \le p &amp;lt; q &amp;lt; \infty\), then \(C&amp;lt; 2\). 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 \(C=1\).&lt;/p&gt;
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