Lecture 9 ing

lecture 9

9.1 Chapter 4 Brenier’s Polar Factorization Theorem

9.1.1 Existence of mapping and polar factorization

Definition 9.1 (Rearrangement) Let \(m: (W, \lambda) \to (X, \mu)\) is a measurable function between two measurable spaces. Another function \(\tilde{m}: (W, \lambda) \to (X, \mu)\) is said to be a rearrangement of \(m\) if it satifies the following condition:

\((W, \lambda) \overset{m, \; \tilde{m}}{\longrightarrow} (X, \mu) \overset{F}{\longrightarrow} \mathbb{R}\)

for every measurable function \(F: X \to \mathbb{R}\) satifing \(F\circ m\in L^1(\rm{d}\lambda)\) ,we have

\[ F\circ \tilde{m} \in L^1( \rm{d}\lambda) \\ \begin{align} &\int_W F\circ m \rm{d}\lambda = \int_W F\circ \tilde{m} \rm{d}\lambda \quad \\ i.e. &\; m\# \lambda = \tilde{m}\# \lambda \\ i.e. &\; \forall A \subset X, \; \lambda (m^{-1}(A))= \lambda (\tilde{m}^{-1}(A)) \end{align} \]

  • The rearrangement means one can not tell the difference between \(m\) and \(\tilde{m}\) by only looking at their values.
  • If \(X = \mathbb{R}\), then \(\Vert m \Vert_p = \Vert \tilde{m}\Vert_p\). But \(\Vert \nabla m \Vert\) does not preserve.

Construction of rearrangement:

Definition 9.2 (Measure-preserving map) Let \(m: (W, \lambda)\) be a measurable space. A measurable function \(s: W \to W\) is said to be a measure-preserving map of \(m\) if \(s\#\lambda = \lambda\).

Then \(m\#s\) is a rearrangement of \(m\).

Example 9.1 Let \(\Omega\subset\mathbb{R}\) is open,

In the sequel,

  • The set of all measure-preserving maps on \((W, \lambda)\) will be denoted by \(S(W)\).
  • The group of diffeomorphisms \(s : \Omega \to \Omega\) with unit Jacobian will be denoted by \(SD(S)\)
  • The subgroup of diffeo-morphisms \(s: \Omega \to \Omega\) with \(\rm{det}(\nabla s) = 1\) will be denoted by G(S).

Proposition 9.1 (Measure-preserving maps and rearrangements) If \(s \in S(W), \; \tilde{m} = m \circ s\), then \(\tilde{m}\) is a rearrangement of \(m\).

Conversely, if \(\tilde{m}\) is a one-to-one rearrangement of \(m\), then \(\tilde{m}^{-1} \circ m \in S(W)\).

Proof. skip

Now we mainly consider the case where \(W \subset \mathbb{R}^n\) and \(X \subset \mathbb{R}^n\), particular class \(\mathcal{R} =\) gradient of convex functions.

Theorem 9.1 (Brenier's Polar Factorization Theorem) Let \(\Omega \subset \mathbb{R}^n\) is bounded and \(\vert \Omega \vert > 0\). Let \(\mathcal{R} = \{ g \in L^2(\Omega, \mathbb{R}^n) \vert \; \exists \;\text{convex function}\; \psi: \mathbb{R}^n \to \mathbb{R} \;\text{s.t.}\;g = \nabla \psi \vert_\Omega \}\).

Let \(h: \Omega \to \mathbb{R}^n\) be an \(L^2\) mapping satisfing the nondegeneracy condition:
for any small set \(N \subset \mathbb{R}^n: \vert h^{-1}(N) \vert = 0\)

Then, \(h\) exists a unique rearrangement in \(\mathcal{R}\), and exists a unique \(s \in S(\Omega)\) s.t. \(h = \nabla \psi \circ s\). Furthermore, \(s\) is the \(L^2\) projection of \(h\) onto \(S(\Omega)\).