On the interpolation constant for Orlicz spaces
- Citation:
- Karlovich, Alexei Yu., and Lech Maligranda. "On the interpolation constant for Orlicz spaces." Proceedings of the American Mathematical Society. 129 (2001): 2727-2739.
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 \(C=1\).
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