Decreasing rearrangements and doubly stochastic operators

Abstract

In this paper generalizations to measurable functions on a finite measure space ( X , Λ , μ ) (X,\Lambda ,\mu ) of some characterizations of the Hardy-Littlewood-Pólya preorder relation ≺ \prec are considered. Let ρ \rho be a saturated, Fatou function norm such that L ∞ ⊂ L ρ ⊂ L 1 {L^\infty } \subset {L^\rho } \subset {L^1} , and let L ρ {L^\rho } be universally rearrangement invariant. The following equivalence is shown to hold for all f ∈ L ρ f \in {L^\rho } iff ( X , Λ , μ ) (X,\Lambda ,\mu ) is nonatomic or discrete: g ≺ f g \prec f iff g is in the ρ \rho -closed convex hull of the set of all rearrangements of f. Finally, it is shown that g ≺ f ∈ L 1 g \prec f \in {L^1} iff g is the image of f by a doubly stochastic operator.