On the closability of differential operators

TL;DR

This paper characterizes when directional derivative operators are closable with respect to general Radon measures, revealing that classical differential operators are closable only under absolute continuity conditions, and explores multilinear operators via metric currents.

Contribution

It provides a complete characterization of the vector fields for which directional derivatives are closable from Lipschitz functions to L^p spaces, and establishes conditions for closability of classical and multilinear differential operators.

Findings

Classical differential operators are closable only if the measure is absolutely continuous.

The paper characterizes closability of directional derivatives from Lipschitz functions to L^p spaces.

Necessary and sufficient conditions for closability of operators from L^q to L^p are provided.

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.