Calculus of Variations and Geometric Measure Theory

G. Alberti - D. Bate - A. Marchese

On the closability of differential operators

created by marchese on 06 Jul 2023
modified by alberti on 09 May 2025

[BibTeX]

Published Paper

Inserted: 6 jul 2023
Last Updated: 9 may 2025

Journal: J. Funct. Anal.
Volume: 289
Number: 7
Year: 2025
Doi: 10.1016/j.jfa.2025.11102

ArXiv: 2311.08058 PDF

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 $\rm{Lip}(\mathbb{R}^d)$ 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.

Keywords: Lipschitz functions, metric currents, closable operators, directional derivative operators


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