The Monge-Kantorovich tangent bundle to a measure

Abstract

We study the tangent bundle $T_\mu$ to a Radon measure $\mu$ in Euclidean space, introduced by Bouchitt\'e, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $\sigma$ is $\mu$-tangent when the divergence of $\sigma \mu$ lies in the Arens-Eells space. The tangent bundle $T_\mu$ defined $\mu$-almost everywhere provides a local projection that allows to construct a $\mu$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $\sigma \in L^1_\mu(\mathbb{R}^d)$. Specifically, we show that the tangential condition $\sigma \in T_\mu$ holding $\mu$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \left((id+h \sigma)_{\#} \mu - \mu\right)$ as $h \to 0$, and the differentiability of Lipschitz functions along $\sigma$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_{\mu}$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_\mu$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.