An Interpolation Theorem for Sublinear Operators on 77„ Spaces

The purpose of this paper is to extend an interpolation theorem for sublinear operators on Lp spaces, obtained by Calderon and Zygmund, to similar operators acting on 77p spaces, p>0 [l]. We begin by making a few definitions. Let T be a mapping defined on a class of functions that are analytic in the interior of the unit circle \w\ <1, taking values that are measurable functions on some measure space (M, n). We say that T is sublinear if (i) TF is uniquely defined if F=Fi+F2 and TFt and TF2 are defined; (ii) For any constant k, T(kF) is defined if TF is defined; (iii) | TO + F,)| :g | TTil+177^1, | T(kF)\ = \k\ ■ \T(F)\ almost everywhere. We shall also say that T is of type (a, b),a,b>0, if the domain of T includes 77„ and there exists a constant M, independent of F in 77OI such that ||rF||!lgi7||7'||a (the norm on the left is the Lb norm and the norm on the right is the 77„ norm). For the results in the theory of 77p spaces we shall use, we refer the reader to [2].