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

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.

### 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
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### References:

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