A sharp form of the Marcinkiewicz Interpolation Theorem for Orlicz spaces

TL;DR

This paper extends the Marcinkiewicz Interpolation Theorem to include intermediate Orlicz spaces, providing a necessary and sufficient condition for quasilinear operators to be bounded between these spaces under certain endpoint estimates.

Contribution

It generalizes the classical theorem to Orlicz spaces, establishing a criterion for boundedness of operators with endpoint estimates between measure spaces.

Findings

Established a necessary and sufficient condition for operator boundedness.

Extended interpolation theorem to Orlicz spaces.

Provided a framework for analyzing operators with endpoint estimates.

Abstract

An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, $S(X,\mu)$, of all $\mu$-measurable simple functions on $\sigma$- finite measure space $(X,\mu)$ into $M(Y,\nu)$, the class of $\nu$-measurable functions on $\sigma$- finite measure space $(Y,\nu)$, and satisfies endpoint estimates of type: $1 < p< \infty$, $1 \leq r < \infty$, \begin{equation*} \lambda \, \nu \left( \left\lbrace y \in Y : |(Tf)(y)| > \lambda \right\rbrace \right)^{\frac{1}{p}} \leq C_{p,r} \left( \int_{\mathbb{R_+}} \mu \left( \left\lbrace x \in X : |(f)(x)| > t \right\rbrace \right)^{\frac{r}{p}} t^{r-1}dt \right)^{\frac{1}{r}}, \end{equation*} for all $f \in S(X,\mu)$ and $\lambda \in \mathbb{R_+}$; is bounded from an Orlicz space into another.