Indices of Function Spaces and their Relationship to Interpolation

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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 Lr 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 Lr 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.

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