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Description
Inspired by Monge-Kantorovich optimal transport theory, we present a notion of tangency to a measure based on 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 with respect to the $W_1$-distance. For any Radon measure $\mu$ in $Rd$, a tangent bundle $T_\mu$ can be defined $\mu$-almost everywhere that provides a local projection and allows to construct a $\mu$-tangential gradient operator on Lipschitz functions that is weakly-star continuous and enjoys integration by parts.
Then we show that the tangency of a given velocity vector field $\sigma \in (L^1_\mu(Rd))^d$ 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 $\sigma$. In addition, we prove that $T_\mu$ coincides with the decomposability bundle introduced by Alberti and Marchese.