On the closability of differential operators

Abstract

We discuss the closability of directional derivative operators with respect to a general Radon measure μ on $R^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}\left(R^d\right)$ to $L^p\left(\mu \right)$, for $1\le p\le \infty$. We also discuss the closability of the same operators from $L^q\left(\mu \right)$ to $L^p\left(\mu \right)$, 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 $L^q\left(\mu \right)$ to $L^p\left(\mu \right)$ 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.