An interpolation theorem in symmetric function $F$-spaces

Abstract:

It is well known that every separable or perfect symmetric Banach function space $X$ is an interpolation space between ${L^1}$ and ${L^\infty }$ (see [1] and [4]). In this paper we prove that every symmetric function $F$-space is interpolation between ${L^0}$ and ${L^\infty }$, where ${L^0}$ is the space of all measurable functions whose support has finite measure. Moreover, for any function $f \in {L^0} + {L^\infty }$ the norm ${\left \| f \right \|_{{L^0}}} + {L^\infty }$ is computed in the terms of the nonincreasing rearrangement function ${f^ * }$ of $f$ as well as in terms of its distribution function ${d_f}$.