Interpolation und indexbedingungen auf rearrangement—invarianten funktionenräumen. I. Grundlagen und die K-methode

Abstract

In this paper the K-interpolation method of J. Peetre is built up for rearrangement invariant norms ϱ on (0, ∞). The spaces ( X 1, X 2) θ, ϱ; K (−∞ < θ < ∞), defined by the norm ∥ f∥ θ, ϱ; K = ϱ( t − θ K( t, f)), are shown to be intermediate spaces of the Banach spaces X 1 and X 2 if the condition α < θ ⩽ 1 upon the upper index α of ϱ is assumed. For these spaces an interpolation theorem of M. Riesz-Thorin-type as well as theorems of reiteration and stability are valid, again under certain conditions upon the indices of ϱ. These index-conditions, which turn out to be of central importance in the interpolation theory on rearrangement invariant spaces, are shown to be equivalent to a generalized Hardy-Littlewood inequality, which is established in the first part of the paper.