On Sudakov's type decomposition of transference plans with norm costs

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $\mathrm{\mid }\cdot {\mathrm{\mid }}_{D^{\ast }}$ $$\min \{\int \mathrm{\mid }\mathtt{T}(x)-x{\mathrm{\mid }}_{D^{\ast }}d\mu (x),\mathtt{T}:{\mathbb{R}}^d\to {\mathbb{R}}^d,\nu ={\mathtt{T}}_{\mathrm{\#}}\mu \},$$ with $\mu$, $\nu$ probability measures in ${\mathbb{R}}^d$ and $\mu$ absolutely continuous w.r.t. ${\mathcal{L}}^d$. The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in $Z_{\mathfrak{a}}\times {\mathbb{R}}^d$, where ${Z_\mathfrak a}{\mathfrak a\in\mathfrak A} \subset \mathbb R^d$ are disjoint regions such that the construction of an optimal map $\mathtt T\mathfrak a : Z_\mathfrak a \to \mathbb R^d$issimplerthanintheoriginalproblem,andthentoobtain$\mathtt T$bypiecingtogetherthemaps$\mathtt T_\mathfrak a$. In this paper we show how the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set $Z_{\mathfrak{a}}$ and then in ${\mathbb{R}}^d$. The strategy is sufficiently powerful to be applied to other optimal transportation problems.