On the interpolation constant for subadditive operators in Orlicz spaces

Abstract

Let $1\le p<q\le \infty$ and let $T$ be a subadditive operator acting on $L^p$ and $L^q$. We prove that $T$ is bounded on the Orlicz space $L^{\varphi }$, where ${\varphi }^{-1}(u)=u^{1/p}\rho (u^{1/q-1/p})$ for some concave function $\rho$ and \[ \|T\|_{L^ϕ\to L^ϕ}\le C\max\{\|T\|_{L^p\to L^p},\|T\|_{L^q\to L^q}\}. \] The interpolation constant $C$, in general, is less than 4 and, in many cases, we can give much better estimates for $C$. In particular, if $p=1$ and $q=\infty$, then the classical Orlicz interpolation theorem holds for subadditive operators with the interpolation constant C=1. These results generalize our results for linear operators obtained in \cite{KM01}.