Decreasing rearrangements and doubly stochastic operators
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- by Peter W. Day PDF
- Trans. Amer. Math. Soc. 178 (1973), 383-392 Request permission
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.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 178 (1973), 383-392
- MSC: Primary 47B99; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0318962-8
- MathSciNet review: 0318962