Polar factorization and monotone rearrangement of vector‐valued functions

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<jats:title>Abstract</jats:title><jats:p>Given a probability space (<jats:italic>X</jats:italic>, μ) and a bounded domain Ω in ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup> equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on <jats:italic>X</jats:italic> and Ω) that for every vector‐valued function u ∈ L<jats:sup>p</jats:sup> (<jats:italic>X</jats:italic>, μ; ℝ<jats:sup>d</jats:sup>) there is a unique “polar factorization” <jats:italic>u</jats:italic> = ∇Ψ<jats:italic>s</jats:italic>, where Ψ is a convex function defined on Ω and <jats:italic>s</jats:italic> is a measure‐preserving mapping from (<jats:italic>X</jats:italic>, μ) into (Ω, |·|), provided that <jats:italic>u</jats:italic> is nondegenerate, in the sense that μ(<jats:italic>u</jats:italic><jats:sup>−1</jats:sup>(<jats:italic>E</jats:italic>)) = 0 for each Lebesgue negligible subset <jats:italic>E</jats:italic> of ℝ<jats:sup>d</jats:sup>.</jats:p><jats:p>Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real‐valued functions are unified.</jats:p><jats:p>The Monge‐Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge‐Kantorovich” problem.</jats:p>

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