Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces

Abstract

Fix the indices $\alpha$ and $\beta$, $1<\alpha<\beta<\infty$, and suppose $\varrho$ is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set $X$ which are measurable with respect to a~$\sigma$-finite measure $\mu$ on it. Set

$$M(\gamma,X):=\{f\colon X\to\mathbb R \text{ with } \sup_{\lambda>0}\lambda \mu(\{x\in X: |f(x)|>\lambda\})^{\frac1{\gamma}}<\infty\},$$

$\gamma=\alpha,\beta$. In this paper we obtain, as a special case, simple criteria to guarantee that a linear operator $T$ satisfies $T\colon L_{\varrho}(X)\to L_{\varrho}(X)$, whenever $T\colon M(\alpha,X)\to M(\alpha, X)$ and $T\colon M(\beta,X)\to M(\beta, X)$.