Indices of Function Spaces and their Relationship to Interpolation

Source abstract

A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions ( p, p ) and ( q,q ), then T maps L r continuously into itself for any r with p < r < q . In a recent paper ( 5 ), as part of a more general theorem, Calderόn has characterized the spaces X which can replace L r in the conclusion of this theorem, independent of the operator T . The conditions which X must satisfy are phrased in terms of an operator S (σ) which acts on the rearrangements of the functions in X . One of Calderόn's results implies that if X is a function space in the sense of Luxemburg ( 9 ), then X must be a rearrangement-invariant space.