Polar factorization and monotone rearrangement of vector-valued functions. (English) Zbl 0738.46011

Brenier, Yann

Commun. Pure Appl. Math. 44, No. 4, 375-417 (1991).

Author’s abstract: Given a probability space \((X,\mu)\) and a bounded domain \(\Omega\) in \(\mathbb{R}^ d\) equipped with the Lebesgue measure \(|\cdot|\) (normalized so that \(|\Omega|=1\)), it is shown (under additional technical assumptions on \(X\) and \(\Omega\)) that for every vector-valued function \(u\in L^ p(X,\mu;\mathbb{R}^ d)\) there is a unique “polar factorization” \(u=\nabla\psi\circ s\), where \(\psi\) is a convex function defined on \(\Omega\) and \(s\) is a measure-preserving mapping from \((X,\mu)\) into \((\Omega,|\cdot|)\), provided that \(u\) is nondegenerate, in the sense that \(\mu(u^{-1}(E))=0\) for each Lebesgue negligible subset \(E\) of \(\mathbb{R}^ d\).
Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of vector-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.

Reviewer: S.V.Kislyakov (St.Petersburg)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40	Spaces of vector- and operator-valued functions
28A99	Classical measure theory

Keywords:

convex function; measure-preserving mapping; polar factorization of real matrices; Helmholtz decomposition of vector fields; nondecreasing rearrangements of vector-valued functions; “Monge-Kantorovich” problem

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