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

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 $\epsilon \mathrm{\Phi }(d_s(x)-d_t(y))$ – where $\mathrm{\Phi }$ 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~$\epsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1–H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\epsilon }$ for every~$\epsilon >0$; coupled Monge–Ampère 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.