The Marcinkievicz Interpolation Theorem for Rearrangement-Invariant Function Spaces and Applications

Abstract

The interpolation theorem of J. Marcinkievicz [17] states that any sublinear operator $T$ which is simultaneously of weak types ($p_1,q_1$) and ($p_2,q_2$) is also a bounded operator from the Lebesgue space $L_p(0,l),0<l\le \infty$, into itself, provided $p_2<p<p_1$. The aim of this paper is to generalize this theorem to the setting of rearrangement-invariant Banach function spaces, and thus to render the theorem available to a much larger range of applications.