On the closability of differential operators

TL;DR

The paper addresses the problem of when directional derivative operators $T_v$ are closable with respect to a general Radon measure $\u02cmu$ on $\u02cR^d$. The authors establish a complete characterization: $T_v$ is closable from $\text{Lip}(\u02cR^d)$ to $L^p_w(\u02cmu)$ for $1\le p\le \infty$ if and only if $v(x)\in V(\u02cmu,x)$ for $\u02cmu$-a.e. $x$, and they extend the analysis to closability from $L^q(\u02cmu)$ to $L^p(\u02cmu)$ with partial criteria. A key corollary shows that classical differential operators like the gradient, divergence, and Jacobian determinant are closable from $L^q(\u02cmu)$ to $L^p(\u02cmu)$ only when $\u02cmu$ is absolutely continuous with respect to Lebesgue measure $\mathscr{L}^d$. The paper further develops the theory for multilinear differential operators and reformulates results in terms of metric currents, linking functional-analytic and geometric-measure-theoret perspectives and touching on the Flat Chain Conjecture. Overall, the work connects differentiability along measures to absolute continuity and provides a robust framework for analyzing closability in non-smooth settings.

Abstract

We discuss the closability of directional derivative operators with respect to a general Radon measure $μ$ 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 $L^p(μ)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(μ)$ to $L^p(μ)$, 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 $L^q(μ)$ to $L^p(μ)$ 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.

Theorems & Definitions (14)

• Theorem 1.3
• Corollary 1.5
• Corollary 1.6
• Theorem 2.3: (see AlbMar)
• Theorem 2.4
• Lemma 3.1
• Theorem 4.1
• Lemma 4.3
• Lemma 4.4
• Theorem 5.3