Decreasing rearrangements and doubly stochastic operators

Abstract:

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