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.
Categories: math.CA