Interpolation of Spaces Defined by the Level Function

Abstract

Suppose ⋋ is a regular, Borel measure on R and suppose that $\lambda ( - \infty ,x) < \infty {\text{ for all }}x \in {\text{R}}$. The Lebesgue spaces, $L_\lambda ^p,{\text{ for 1 }} \leqslant p \leqslant \infty$ will then contain non-trivial, non-increasing functions. Define

$$ {\left\| f \right\|_{p \downarrow \lambda }} = {\text{sup}}\left\{ {\smallint _{ - \infty }^\infty \left| f \right|gd\lambda :g \geqslant 0,g{\text{ non - increasing, }}{{\left\| g \right\|}_{p',\lambda }} \leqslant 1} \right\} $$

where p′ is defined by 1/p + 1/p′=1.

This paper is in final form and no version of it has been or will been or will be submitted elsewhere. Revised June 5, 1991.

Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.