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arXiv · 2311.08058

On the closability of differential operators

Giovanni Alberti·David Bate·Andrea Marchese
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Abstract

We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}(\mathbb{R}^d)$ 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.

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Giovanni Alberti, David Bate, Andrea Marchese. 2023-11-14. On the closability of differential operators. https://doi.org/10.1016/j.jfa.2025.11102

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