On the closability of differential operators

We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ on $R^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $Lip(R^d)$ to $L^p(\mu)$, for $1 \leq p \leq \infty$. We also discuss the closability of the same operators from $L^q(\mu)$ to $L^p(\mu)$, and give necessary and sufficient conditions for closability, but we do not have an exact characterization. As a corollary we obtain that classical differential operators such as gradient, divergence and Jacobian determinant are closable from $L^q(\mu)$ to $L^p(\mu)$ only if $\mu$ is absolutely continuous with respect to the Lebesgue measure. We finally consider the closability of a certain class of multilinear differential operators; these results are then rephrased in terms of metric currents. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).