  We establish that solving an optimal transportation problem in which the
source and target densities are defined on manifolds with different dimensions,
is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation,
introduced here for the first time. Under suitable topological conditions, we
also establish that solutions are smooth if and only if a local variant of the
same equation admits a smooth and uniformly elliptic solution. We show that
this local equation is elliptic, and $C^{2,\alpha}$ solutions can therefore be
bootstrapped to obtain higher regularity results, assuming smoothness of the
corresponding differential operator, which we prove under simplifying
assumptions. For one-dimensional targets, our sufficient criteria for
regularity of solutions to the resulting ODE are considerably less restrictive
than those required by earlier works.
