Interpolation of weighted $L^1$ spaces — a new proof of the Sedaev-Semenov theorem

Abstract

A new simpler proof is given of the theorem of Sedaev-Semenov that the couple $(L^{1}_{w_{0}},L^{1}_{w_{1}})$ of weighted $L^{1}$ spaces on an arbitrary measure space is a Calderón couple, i.e., all interpolation spaces with respect to this couple can be described in terms of the $K$-functional. This theorem has other important consequences. It is a component in an alternative proof of the Brudnyi-Krugljak $K$-divisibility theorem. Also, as shown by Dmitriev, it leads readily to a proof of Sparr's more general result that $(L^{p}_{w_{0}},L^{q}_{w_{1}})$ is a Calderón couple.