On the interpolation constant for subadditive operators in Orlicz spaces

TL;DR

This paper establishes bounds for subadditive operators acting on Orlicz spaces, generalizing classical interpolation results and providing explicit estimates for the interpolation constant, especially in the case of $p=1$ and $q=\infty$.

Contribution

It extends the classical Orlicz interpolation theorem to subadditive operators, providing explicit bounds for the interpolation constant and generalizing previous linear operator results.

Findings

The operator $T$ is bounded on $L^\phi$ with a controlled norm.

The interpolation constant $C$ is generally less than 4, often much smaller.

In the case $p=1$, $q=\infty$, the classical theorem holds with $C=1$.

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 $L^\phi$, where $\phi^{-1}(u)=u^{1/p}\rho(u^{1/q-1/p})$ for some concave function $\rho$ and \[ \|T\|_{L^\phi\to L^\phi}\le C\max\{\|T\|_{L^p\to L^p},\|T\|_{L^q\to L^q}\}. \] 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}.