On the closability of differential operators

We discuss the closability of directional derivative operators with respect to a general Radon measure μ on Rd; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions Lip(Rd) to Lp(μ), for 1≤p≤∞. We also discuss the closability of the same operators from Lq(μ) to Lp(μ), 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 Lq(μ) to Lp(μ) only if μ 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.