2006
DOI: 10.1016/j.jfa.2005.08.010
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Rearrangement inequalities for functionals with monotone integrands

Abstract: The inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F (u 1 , . . . , u m ) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives * i * j F for all i = j . This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and R… Show more

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Cited by 84 publications
(60 citation statements)
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“…The symmetric decreasing rearrangment does precisely this, moving weight onto a monotone set without changing the level sets of the u i . See for example Burchard-Hajaiej [5] and the references therein.…”
Section: Orientable Systems Compatible Cost and Sub-modular Functionsmentioning
confidence: 99%
“…The symmetric decreasing rearrangment does precisely this, moving weight onto a monotone set without changing the level sets of the u i . See for example Burchard-Hajaiej [5] and the references therein.…”
Section: Orientable Systems Compatible Cost and Sub-modular Functionsmentioning
confidence: 99%
“…A multivariate proof, based on the distributional transform, has been recently presented in [174] (compare also with [144,Lemma 3.2]). Another possible proof can be also derived (for positive random variables) from [14]. Sklar's Theorem on more abstract spaces has been given in [178].…”
Section: Sklar's Theoremmentioning
confidence: 99%
“…For this rearrangement inequality, started with the work of Lieb [11], we refer for instance to [4].…”
Section: Proofmentioning
confidence: 99%