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Polar factorization and monotone rearrangement of vector‐valued functions
Author(s) -
Brenier Yann
Publication year - 1991
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160440402
Subject(s) - mathematics , factorization , monotone polygon , lebesgue measure , bounded function , lebesgue integration , measure (data warehouse) , pure mathematics , standard probability space , function (biology) , space (punctuation) , probability measure , weierstrass factorization theorem , polar , vector space , domain (mathematical analysis) , discrete mathematics , mathematical analysis , geometry , linguistics , philosophy , physics , algorithm , database , evolutionary biology , astronomy , computer science , biology
Given a probability space ( X , μ) and a bounded domain Ω in ℝ d equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector‐valued function u ∈ L p ( X , μ; ℝ d ) there is a unique “polar factorization” u = ∇Ψ s , where Ψ is a convex function defined on Ω and s is a measure‐preserving mapping from ( X , μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ( u −1 ( E )) = 0 for each Lebesgue negligible subset E of ℝ d . 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. The Monge‐Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge‐Kantorovich” problem.