POLAR FACTORIZATION AND MONOTONE REARRANGEMENT OF VECTOR-VALUED FUNCTIONS

Given a probability space (X, mu) and a bounded domain OMEGA in R(d) equipped with the Lebesgue measure \.\ (normalized so that \OMEGA\ = 1), it is shown (under additional technical assumptions on X and OMEGA) that for every vector-valued function u member-of L(p) (X, mu; R(d)) there is a unique "polar factorization" u = nabla-psi.s, where psi is a convex function defined on OMEGA and s is a measure-preserving mapping from (X, mu) into (OMEGA, \.\), provided that u is nondegenerate, in the sense that mu-(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-Ampere equation is involved in the polar factorization and the proof relies on the study of an appropriate "Monge-Kantorovich" problem.