Optimal transport of signed fractal measures with dimensional distortion: a variational characterization

Abstract

We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\varepsilon \Phi(d_s(x) - d_t(y))$ -- where $\Phi$ is a fixed smooth strictly convex function and $d_s, d_t$ are the local Hausdorff dimensions of the fractal supports -- is added to the transport cost on inter-sign regions, with~$\varepsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1--H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\varepsilon}$ for every~$\varepsilon > 0$; coupled Monge--Amp\`ere equations with a distortion correction term, generalizing the classical Brenier--Caffarelli equation; a double Legendre--Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre--Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis.