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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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