Regularity of the positional penalization function in inter-sign optimal transport on real measures

Abstract

We study the Monge--Kantorovich optimal transport problem between two signed measures~$\hat{I}¼$ and~$\hat{I}½$ on convex compact subsets of~${\mathbb{R}}^d$, with a positional penalization function~$\hat{I}»(x,y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(\ddot{I}{€}^{++},\ddot{I}{€}^{+-},\ddot{I}{€}^{-+},\ddot{I}{€}^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(\hat{I}¼,\hat{I}½)$ is weakly-$\ast$ compact and non-empty if and only if $\hat{I}{¼}^{+}(X)=\hat{I}{½}^{+}(Y)$ and~$\hat{I}{¼}^{-}(X)=\hat{I}{½}^{-}(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$\hat{I}»$ at the intersection of inter-sign supports. The penalization~$\hat{I}»$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge--Ampère equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $\ddot{I}ƒ\det (D_{yx}^2\hat{I}›)e>0$. The classical Brenier equation is recovered in the limit~$\hat{I}»\to 0$.

Together with Bwo'nyahre et al. (2026), this completes a three-part framework for signed measure optimal transport: (1) existence, uniqueness, and fractal preservation; (2) local regularity, governing equations, and well-posedness; and (3) a variational characterization of dimensional distortion in signed fractal measures