@article { ISI:000168998500027, title = {On the interpolation constant for Orlicz spaces}, journal = {Proceedings of the American Mathematical Society}, volume = {129}, number = {{9}}, year = {2001}, pages = {2727-2739}, publisher = {{AMER MATHEMATICAL SOC}}, type = {{Article}}, address = {{201 CHARLES ST, PROVIDENCE, RI 02940-2213 USA}}, 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 \(C=1\).

}, keywords = {concave function}, convex function, interpolation constant, interpolation of operators, K-functional, {Orlicz spaces}, issn = {{0002-9939}}, doi = {{10.1090/S0002-9939-01-06162-7}}, url = {http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-06162-7/}, author = {Karlovich, Alexei Yu. and Maligranda, Lech} }