# Diameter of 2-distance graphs

S.H. Jafaria, S.R. Musawib

aFaculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box 36199-95161, Shahrood, Iran
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

bFaculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box 36199-95161, Shahrood, Iran
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

For a simple graph $G$, the $2$-distance graph, $D_{2}(G)$, is a graph with
the vertex set $V(G)$ and two vertices are adjacent if and only if their
distance is $2$ in the graph $G$. In this paper, for graphs $G$ with diameter
2, we show that $diam(D_{2}(G))$ can be any integer $t\geqslant 2$. For graphs
$G$ with $diam(G)\geqslant 3$, we prove that $\frac{1}{2}diam(G)\leqslant
diam(D_{2}(G))$ and this inequality is sharp. Also, for $diam(G)=3$, we prove
that $diam(D_{2}(G))\leqslant 5$ and this inequality is sharp.

Keywords: 2-distance graph, paths, diameter, connectivity
Mathematics Subject Classification : 05C12, 05C38, 05C40

##  1 introduction

In this paper, we only consider the undirected finite simple graphs $G=(V,E)$.
The distance between two vertices of $x,y\in V(G)$,
$\mathrm{d}(x,y)=\mathrm{d}_{G}(x,y)$, is the length of the shortest path
between them. The diameter of $G$, $\mathrm{diam}(G)$, is the maximum distance
between vertices of $G$. $2$-distance graph of a graph $G$, $D_{2}(G)$, is a
graph with the vertex set $V(G)$ and two vertices are adjacent if and only
their distance is $2$ in the graph $G$. For any $u,v\in V(D_{2}(G))$, we set
$\mathrm{d_{2}}(u,v)=\mathrm{d}_{D_{2}(G)}(u,v)$.

The idea of 2-distance graphs is a particular case of a general notion of
k-distance graphs, which was first studied by Harary et al. [6]. They
investigated the connectedness of 2-distance graph of some graphs. In the book
by Prisner [10], the dynamics of the k-distance operator was explored.
Furthermore, Boland et al. [3] extended the k-distance operator to a graph
invariant they call distance-n domination number. To solve one problem posed
by Prisner [10], Zelinka [12] constructed a graph that is r-periodic under the
2-distance operator for each positive even integer r. Prisner’s problem was
completely solved by Leonor in her master’s thesis [9], wherein she worked
with a graph construction different from that of Zelinka’s. Recently, Azimi
and Farrokhi [1] studied all graphs whose 2-distance graphs have maximum
degree 2. They also solved the problem of finding all graphs whose 2-distance
graphs are paths or cycles. More recently, the same authors [2] determined all
finite simple graphs with no square, diamond, or triangles with a common
vertex that are self 2-distance graphs (that is, graphs $G$ satisfying the
equation $D^{2}(G)=G$. They further showed the nonexistence of cubic self
2-distance graphs. In [5], Ching gives some characterizations of 2-distance
graphs and finds all graphs X such that $D_{2}(X)=kP_{2}$ or $K_{m}\cup
K_{n}$, where $k\geqslant 2$ is an integer, $P_{2}$ is the path of order $2$,
and $K_{m}$ is the complete graph of order $m\geqslant 1$. In [8] Khormali
gives some conditions for connectivity $D_{k}(G)$. For $D_{2}(G)$, he proves
that if $G$ has no odd cycle, then $D_{2}(G)$ is disconnected. In [7], the
authors characterized all graphs with connected 2-distance graph. Also, for
graphs with diameter 2, they proved that $D_{2}(G)$ is connected if and only
if $G$ has no spanning complete bipartite subgraphs.

In this paper, for graphs $G$ with diameter 2, we show that
$\mathrm{diam}(D_{2}(G))$ can be any integer $t\geqslant 2$. For graphs $G$
with $\mathrm{diam}(G)\geqslant 3$, we prove that
$\frac{1}{2}\mathrm{diam}(G)\leqslant\mathrm{diam}(D_{2}(G))$ and this
inequality is sharp. Also, for $\mathrm{diam}(G)=3$ we prove that
$\mathrm{diam}(D_{2}(G))\leqslant 5$ and this inequality is sharp.

##  2 Main results

In this section, we are going to discuss the diameters of 2-distance graphs.
If $G$ is disconnected, then $D_{2}(G)$ is disconnected too, thus we have to
consider the connected graphs. We are going to find a relation between
$\mathrm{diam}(G)$ and $\mathrm{diam}(D_{2}(G))$, and for this, we have to
partition all connected graphs in two distinct partites, the graphs with
diameter 2 and graphs with diameter greater than 2.

###  2.1 Graphs with diameter 2

The following observation shows that there is no relation between
$\mathrm{diam}(G)$ and $\mathrm{diam}(D_{2}(G))$.

######  Observation 2.1.

For any integer $n\geqslant 2$ there are some graphs $G$ with
$\mathrm{diam}(G)=2$ and $\mathrm{diam}(D_{2}(G))=n$.

######  Example 2.2.

For $n=2,3,4$ the following graphs $G_{2},G_{3},G_{4}$ satisfies the
conditions of Observation 2.1.

$D_{2}(G_{2})=C_{5}$$\mathrm{diam}(D_{2}(G_{2}))=2$$G_{2}=C_{5}$$\mathrm{diam}(G_{2})=2$$D_{2}(G_{3})$$\mathrm{diam}(D_{2}(G_{3}))=3$$G_{3}$$\mathrm{diam}(G_{3})=2$$D_{2}(G_{4})$$\mathrm{diam}(D_{2}(G_{4}))=4$$G_{4}=P_{5}$$\mathrm{diam}(G_{4})=2$

######  Example 2.3.

If $n\geqslant 4$ and the graph $G_{n}=P_{n+1}^{C}$ satisfies the conditions
of Observation 2.1, since $\mathrm{diam}(G)=2$,
$D_{2}(G)=(P_{n+1}^{C})^{C}=P_{n+1}$ and so $\mathrm{diam}(D_{2}(G))=n$.

###  2.2 A lower bound for diameter of 2-distance graphs

Here, we obtain a lower bound for the diameter of 2-distance graphs of the
graphs $G$ with $\mathrm{diam}(G)\geqslant 3$.

######  Lemma 2.4.

Let $G$ be a graph and $D_{2}(G)$ be connected. For any $u,v\in V(G)$ :

$\mathrm{d_{2}}(u,v)\geqslant\left\lceil\frac{1}{2}\mathrm{d}(u,v)\right\rceil$

###### Proof.

Let $\mathrm{d_{2}}(u,v)=k$ and $u=x_{0}x_{1}x_{2}\cdots x_{k}=v$ be a path in
$D_{2}(G)$. Now, there is a walk $u=x_{0}y_{1}x_{1}y_{2}x_{2}\cdots
y_{k}x_{k}=v$ in $G$ with length $2k$. Thus $2k\geqslant\mathrm{d}(u,v)$.
Consequently
$\mathrm{d_{2}}(u,v)\geqslant\left\lceil\frac{1}{2}\mathrm{d}(u,v)\right\rceil$.
∎

An immediate result from Lemma 2.4 is the following.

######  Theorem 2.5.

Suppose that $G$ be a graph and $D_{2}(G)$ be connected. Then
$\mathrm{diam}(D_{2}(G))\geqslant\left\lceil\frac{1}{2}\mathrm{diam}(G)\right\rceil$

The following examples show that the inequality of the above theorem is sharp.

######  Example 2.6.

In the following graph $G$, we have $\mathrm{diam}(G)=3$ and
$\mathrm{diam}(D_{2}(G))=2$.

$G$$D_{2}(G)$

######  Example 2.7.

Let $G_{t,3}$ be the following graph.

$u$$v$$w$$b_{0}$$a_{13}$$a_{14}$$a_{11}$$a_{16}$$a_{12}$$a_{15}$$b_{1}$$b_{2}$$a_{23}$$a_{24}$$a_{21}$$a_{26}$$a_{22}$$a_{25}$$\cdots$$b_{t-1}$$b_{t}$$a_{t3}$$a_{t4}$$a_{t1}$$a_{t6}$$a_{t2}$$a_{t5}$

Since $b_{0},b_{1},\cdots,b_{t}$ are cut-vertices,
$\mathrm{diam}(G_{t,3})=4t+3$. Also, $D_{2}(G_{t,3})$ is the following.

$u$$v$$w$$a_{14}$$a_{13}$$a_{16}$$a_{11}$$b_{0}$$a_{12}$$a_{15}$$a_{24}$$a_{23}$$a_{26}$$a_{21}$$b_{1}$$b_{2}$$a_{22}$$a_{25}$$\cdots$$a_{t4}$$a_{t3}$$a_{t6}$$a_{t1}$$b_{t-1}$$b_{t}$$a_{t2}$$a_{t5}$

It is easily seen that $\mathrm{d_{2}}(u,w)=2t+2$ and then
$\mathrm{diam}(D_{2}(G_{t,3}))=2t+2$.

Also, By setting $G_{t,2}=G_{t,3}-w$, $G_{t,1}=G_{t,3}-\\{v,w\\}$ and
$G_{t,0}=G_{t,3}-\\{u,v,w\\}$, we have $\mathrm{diam}(G_{t,i})=4t+i$ and
$\mathrm{diam}(D_{2}(G_{t,i}))=2t+\left\lceil\dfrac{i}{2}\right\rceil$.

###  2.3 An upper bound for the diameter of 2-distance graphs

In this subsection, we obtain an upper bound for the diameter of 2-distance
graphs of the graphs $G$ with $\mathrm{diam}(G)=3$.

######  Theorem 2.8.

If $\mathrm{diam}(G)=3$ then $\mathrm{diam}(D_{2}(G))\leqslant 5$.

###### Proof.

We prove this by contradiction.
Suppose that $G$ be a graph with $\mathrm{diam}(G)=3$ and
$a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}$ is a path with length $6$ in $D_{2}(G)$.

Claim 1: $\mathrm{d}(a_{1},a_{3})=1$

###### Proof.

We consider the following two possible cases.

Case 1: $\mathrm{d}(a_{1},a_{3})=3,\mathrm{d}(a_{3},a_{7})=1$

Let $a_{1}bca_{3}$ be a path in $G$. Since $ca_{3}a_{7}$ is a path in G,
$\mathrm{d}(c,a_{7})\leqslant 2$. If $\mathrm{d}(c,a_{7})=2$ then
$a_{1}ca_{7}$ is a path in $D_{2}(G)$, a contradiction. Thus
$\mathrm{d}(c,a_{7})=1$. If $\mathrm{d}(b,a_{7})=2$ then
$a_{1}a_{2}a_{3}ba_{7}$ is a path in $D_{2}(G)$, a contradiction. Thus
$\mathrm{d}(b,a_{7})=1$. Consequently, $\mathrm{d}(a_{1},a_{7})\leqslant 2$.
But, $\mathrm{d}(a_{1},a_{7})\neq 2$ and then $\mathrm{d}(a_{1},a_{7})=1$.
Thus, $a_{1}a_{7}a_{3}$ is a path in $G$ and $\mathrm{d}(a_{1},a_{3})\leqslant
2$, a contradiction.

Case 2: $\mathrm{d}(a_{1},a_{3})=3,\mathrm{d}(a_{3},a_{7})=3$.

Let $a_{1}bca_{3}$ and $a_{3}dea_{7}$ be paths in $G$. Since,
$\mathrm{d}(a_{1},a_{3})=3$, $b\neq d$, similarly $c\neq e$. Then we have the
following three subcases.

Subcase 1: $c=d$.

Then $a_{1}ca_{7}$ is a path in $D_{2}(G)$, a contradiction.

Subcase 2: $c\neq d$ and $b=e$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b=e$$c$$d$

If $\mathrm{d}(a_{1},d)=2$ then $a_{1}da_{7}$ is a path in $D_{2}(G)$, a
contradiction. Thus $\mathrm{d}(a_{1},d)=1$. Consequently,
$\mathrm{d}(a_{1},a_{3})\leqslant 2$, a contradiction.

Subcase 3: $c\neq d$ and $b\neq e$.
If $\mathrm{d}(c,d)=2$, then $a_{1}cda_{7}$ is a path in $D_{2}(G)$, a
contradiction. Then, we have the following.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b$$c$$d$$e$

If $\mathrm{d}(b,d)=2$ then $a_{1}a_{2}a_{3}bda_{7}$ is a path in $D_{2}(G)$,
a contradiction. Then $\mathrm{d}(b,d)=1$. If $\mathrm{d}(a_{1},d)=2$ then
$a_{1}da_{7}$ is a path in $D_{2}(G)$, a contradiction. Then
$\mathrm{d}(a_{1},d)=1$. Consequently, $\mathrm{d}(a_{1},a_{3})\leqslant 2$, a
contradiction. ∎

Therefore, $\mathrm{d}(a_{1},a_{3})=1$ and by symmetric
$\mathrm{d}(a_{5},a_{7})=1$

Claim 2: $\mathrm{d}(a_{3},a_{5})=1$.

###### Proof.

By contradiction, let $\mathrm{d}(a_{3},a_{5})=3$ and $a_{3}bca_{5}$ be a path
in $G$. Since $\mathrm{d}(b,a_{5})=\mathrm{d}(c,a_{3})=2$, $a_{1}\neq b,c$.
Similarly, $a_{7}\neq b,c$. Noe, we have the following

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b$$c$

If $\mathrm{d}(a_{1},b)=2$ then $a_{1}ba_{5}$ is a path in $D_{2}(G)$, a
contradiction. Thus $\mathrm{d}(a_{1},b)=1$ and by symmetry
$\mathrm{d}(c,a_{7})=1$.

If $\mathrm{d}(a_{1},c)=1$ then $\mathrm{d}(a_{1},a_{5})\leqslant 2$ and so
$\mathrm{d}(a_{1},a_{5})=1$. Consequently $\mathrm{d}(a_{3},a_{5})\leqslant
2$, a contradiction. Thus $\mathrm{d}(a_{1},c)=2$ and by symmetry
$\mathrm{d}(b,a_{7})=2$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b$$c$

Since $\mathrm{d}(a_{3},a_{4})=\mathrm{d}(a_{4},a_{5})=2$, $a_{4}\neq b,c$.

If $\mathrm{d}(b,a_{4})=2$ then $a_{7}ba_{4}$ is a path in $D_{2}(G)$, a
contradiction. Thus $\mathrm{d}(b,a_{4})=1\text{ or }3$. Let
$\mathrm{d}(b,a_{4})=3$ and $ba_{3}ua_{4}$ be a path in $G$. Since
$\mathrm{d}(b,a_{4})=3$, $u\neq a_{1},b$. If $\mathrm{d}(a_{1},u)=2$ then
$a_{1}uba_{7}$ is a path in $D_{2}(G)$, a contradiction. Thus
$\mathrm{d}(a_{1},u)=1$, $\mathrm{d}(a_{1},a_{4})\leqslant 2$ and so
$\mathrm{d}(a_{1},a_{4})=1$. Consequently, $\mathrm{d}(b,a_{4})\leqslant 2$, a
contradiction. Therefore, $\mathrm{d}(b,a_{4})=1$ and by symmetry
$\mathrm{d}(c,a_{4})=1$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b$$c$

Now, we have $\mathrm{d}(a_{1},a_{4})\leqslant 2$ and then
$\mathrm{d}(a_{1},a_{4})=1$. By symmetry $\mathrm{d}(a_{4},a_{7})=1$ and then
$\mathrm{d}(a_{1},a_{7})\leqslant 2$. Thus $\mathrm{d}(a_{1},a_{7})=1$ and so
$\mathrm{d}(a_{1},a_{5})=1$ which shows $\mathrm{d}(a_{3},a_{5})\leqslant 2$,
a contradiction. ∎

Therefore, $\mathrm{d}(a_{3},a_{5})=1$ and then
$\mathrm{d}(a_{1},a_{5})=\mathrm{d}(a_{3},a_{7})=\mathrm{d}(a_{1},a_{7})=1$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$

Claim 3: $\mathrm{d}(a_{2},a_{4})=1$

###### Proof.

By contradiction, let $\mathrm{d}(a_{2},a_{4})=3$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$b$$c$$u$

If $\mathrm{d}(u,a_{7})=1$ then $\mathrm{d}(a_{4},a_{7})\leqslant 2$. Thus
$\mathrm{d}(a_{4},a_{7})=1$. If $\mathrm{d}(c,a_{7})=2$ then $a_{2}ca_{7}$ is
a path in $D_{2}(G)$, a contradiction, and so $\mathrm{d}(c,a_{7})=1$. If
$\mathrm{d}(b,a_{7})=2$ then $a_{4}ba_{7}$ is a path in $D_{2}(G)$, a
contradiction, and so $\mathrm{d}(b,a_{7})=1$. Thus
$\mathrm{d}(a_{2},a_{7})\leqslant 2$ and $\mathrm{d}(a_{2},a_{7})=1$. Which
shows $\mathrm{d}(a_{2},a_{4})\leqslant 2$, a contradiction, and so Then
$\mathrm{d}(u,a_{7})=2$.

Now $\mathrm{d}(u,a_{1})\leqslant 2$. If $\mathrm{d}(u,a_{1})=2$ then
$a_{1}ua_{7}$ is a path, a contradiction. Thus $\mathrm{d}(u,a_{1})=1$.
Consequently, $\mathrm{d}(a_{1},a_{4})=1$. And also
$\mathrm{d}(a_{4},a_{7})=1$. Similarly, we have a contradiction. ∎

Thus, $\mathrm{d}(a_{2},a_{4})=1$, and by symmetry,
$\mathrm{d}(a_{4},a_{6})=1$ and then $\mathrm{d}(a_{2},a_{6})=1$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$

Claim 4: $\mathrm{d}(a_{1},a_{4})=1$.

###### Proof.

By contradiction, let $\mathrm{d}(a_{1},a_{4})=3$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$u$

If $\mathrm{d}(u,a_{6})=1$ then $\mathrm{d}(a_{1},a_{6})\leqslant 2$. Thus
$\mathrm{d}(a_{1},a_{6})=1$ and $\mathrm{d}(a_{1},a_{4})\leqslant 2$, a
contradiction. Therefore, $\mathrm{d}(u,a_{6})=2$. If $\mathrm{d}(u,a_{7})=2$
then $a_{7}ua_{7}$ is a path in $D_{2}(G)$, a contradiction. Thus
$\mathrm{d}(u,a_{7})=1$. Thus $\mathrm{d}(a_{2},a_{7})\leqslant 2$ and
$\mathrm{d}(a_{2},a_{7})=1$. Then, $\mathrm{d}(a_{4},a_{7})\leqslant 2$ and so
$\mathrm{d}(a_{4},a_{7})=1$. Which shows $\mathrm{d}(a_{1},a_{4})\leqslant 2$,
a contradiction. ∎

Hence, $\mathrm{d}(a_{1},a_{4})=1$, and by symmetry,
$\mathrm{d}(a_{4},a_{7})=1$ and then
$\mathrm{d}(a_{1},a_{6})=\mathrm{d}(a_{2},a_{7})=\mathrm{d}(a_{2},a_{5})=\mathrm{d}(a_{3},a_{6})=1$.
Thus, the induced subgraph generated by $A=\\{a_{1},a_{2},\cdots,a_{7}\\}$ is
the following.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$

The diameter of this subgraph is 2 and since $\mathrm{diam}(G)=3$, $G$ has a
vertex $b$ where $\mathrm{d}(b,A)=1$.

Claim 5: $|N(b)\cap A|\geqslant 4$.

###### Proof.

By contradiction, suppose that $|N(b)\cap A|\leqslant 3$. There are at least 4
vertices $a_{i_{1}},a_{i_{2}},a_{i_{3}},a_{i_{4}}$, where $1\leqslant
i_{1}\leqslant\cdots\leqslant i_{4}\leqslant 7$ such that
$\mathrm{d}(b,a_{i_{k}})=2$, $k=1,2,3,4$. Then $a_{i_{1}}ba_{i_{4}}$ is a path
in $D_{2}(G)$ while $\mathrm{d_{2}}(a_{i_{1}},a_{i_{4}})\geqslant 3$, a
contradiction. ∎

Let $B=N[A]$ and $u,v\in B\setminus A$. By Claim 5, $|N(u)\cap A|\geqslant 4$
and $|N(v)\cap A|\geqslant 4$ and then $|N(u)\cap N(v)|\geqslant 1$ and so
$\mathrm{d}(u,v)\leqslant 2$. Therfore $\mathrm{diam}(\langle B\rangle)=2$.
Since $\mathrm{diam}(G)=3$, $G$ has a vertex $c\in N[B]\setminus B$. The
vertex $c$ has an adjacent vertex $w\in B$ and since $|N(w)\cap A|\geqslant
4$, there are at least $4$ vertices $a_{i_{1}},a_{i_{2}},a_{i_{3}},a_{i_{4}}$,
where $1\leqslant i_{1}\leqslant\cdots\leqslant i_{4}\leqslant 7$ such that
$\mathrm{d}(c,a_{i_{k}})=2$, $k=1,2,3,4$. Then $a_{i_{1}}ba_{i_{4}}$ is a path
in $D_{2}(G)$, a contradiction. ∎

By Theorems 2.5 and 2.8 we have the following.

######  Corollary 2.9.

If $\mathrm{diam}(G)=3$ then $2\leqslant\mathrm{diam}(D_{2}(G))\leqslant 5$
and the inequalities are sharp.

###### Proof.

By Example 2.6 the first inequality is sharp. Now, the following graph $G$
shows that the second inequality is sharp too.

$G$$D_{2}(G)$

∎

##  3 Conclusions

In this paper, we focused on the diameter of the connected $2$-distance graph
of graphs. We found a sharp lower bound for the diameter of $2$-distance
graphs relative to the diameter of their initial graphs. Also, we showed that
$\mathrm{diam}(D_{2}(G))$ can be any integer $t\geqslant 2$ when
$\mathrm{diam}(G)=2$. But for $\mathrm{diam}(G)=3$, we obtained a sharp upper
bound for $\mathrm{diam}(D_{2}(G))$. The above achievements raise the
following questions.

######  Conjecture 3.1.

Let $\mathrm{diam}(G)=n\geqslant 3$. Then $\mathrm{diam}(D_{2}(G))\leqslant
n+2$.

Problem 3.2. Charactrize all graphs $G$ which $\mathrm{diam}(G)=2$ and
$\mathrm{diam}(D_{2}(G))=2\text{ or }3$.

In Theorem 2.5, we proved
$\mathrm{diam}(D_{2}(G))\geqslant\left\lceil\frac{1}{2}\mathrm{diam}(G)\right\rceil$
and in Example 2.7 we show this inequality is sharp.

Problem 3.3. Are there some graphs with $\mathrm{diam}(G)=n$ and
$\mathrm{diam}(D_{2}(G))=\left\lceil\frac{1}{2}\mathrm{diam}(G)\right\rceil+1$.

Declarations

Conflict of interest No potential conflict of interest was reported by the
authors.

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On the closability of differential operators

Giovanni Alberti, David Bate, Andrea Marchese

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 consider certain classes of multilinear differential operators.

We then discuss the closability of the same operators from $L^{q}(\mu)$ to
$L^{p}(\mu)$; we give necessary conditions 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.

Lastly, we rephrase our results on multilinear operators in terms of metric
currents.

Keywords: closable operators, directional derivative operators, Lipschitz
functions, Sobolev spaces, normal currents, metric currents.

MSC (2010): 26B05, 49Q15, 26A27.

##  1\. Introduction

One way of defining the Sobolev spaces $W^{1,p}_{0}(\Omega)$ for an open set
$\Omega$ in $\mathbb{R}^{d}$ is taking the completion of the space
$C^{1}_{c}(\Omega)$ of functions of class $C^{1}$ with compact support on
$\Omega$ with respect to the Sobolev norm $\|\cdot\|_{W^{1,p}}$.

This construction can be made more precise as follows: we consider the graph
of the gradient operator $\nabla:C^{1}_{c}(\Omega)\to
C^{0}_{c}(\Omega;\mathbb{R}^{d})$ as a subset of the product space
$L^{p}(\Omega)\times L^{p}(\Omega;\mathbb{R}^{d})$, we take its closure
$\Gamma$, and we show that $\Gamma$ is still a graph, that is, for every $u\in
L^{p}(\Omega)$ there exists _at most_ one $v\in L^{p}(\Omega;\mathbb{R}^{d})$
such that $(u,v)\in\Gamma$. We then consider the operator whose graph is
$\Gamma$: the domain is the Sobolev space $\smash{W^{1,p}_{0}(\Omega)}$ and
the operator is the gradient for Sobolev functions.111 Moreover
$\|u\|_{W^{1,p}}$ is the norm of $(u,\nabla u)$ in the product space
$L^{p}(\Omega)\times L^{p}(\Omega;\mathbb{R}^{d})$.

Note that the essential ingredient in this construction is that the closure of
the graph of the gradient is still a graph. The extension of this construction
to more general operators leads to the following abstract definition:

###### 0\. Closable operators.

Given $X,Y$ topological spaces, $D$ subset of $X$, and a map $T:D\to Y$, we
denote by $\Gamma$ the closure of the graph $\\{(x,T(x))\colon x\in D\\}$ in
$X\times Y$, and we say $T$ is _closable_ (from $X$ to $Y$) if $\Gamma$ is
also a graph, that is, for every $x\in X$ there exists _at most_ one $y\in Y$
such that $(x,y)\in\Gamma$.

In this paper we study the closability of certain first-order differential
operators. The spaces $X$, $D$ and $Y$ will be always linear spaces of
functions on $\mathbb{R}^{d}$, and we focus in particular on directional
derivative operators.

More precisely:

###### 0\. Functions spaces.

Through this paper $X$ is either of the following spaces:

  * •

$L^{q}(\mu)$ where $\mu$ is a Radon measure on $\mathbb{R}^{d}$ and $1\leq
q\leq\infty$;

  * •

the space $\mathrm{Lip}(\mathbb{R}^{d})$ of all Lipschitz functions
$u:\mathbb{R}^{d}\to\mathbb{R}$, endowed with the weak∗ topology of
$W^{1,\infty}$; in particular a sequence $(u_{n})$ converges to $u$ in
$\mathrm{Lip}(\mathbb{R}^{d})$ if and only if $u_{n}\to u$ uniformly and the
Lipschitz constants $\mathrm{Lip}(u_{n})$ are uniformly bounded.

Moreover $D$ is always the space $C^{1}_{c}(\mathbb{R}^{d})$, and $Y$ is
$L^{p}(\mu)$ with $1\leq p\leq\infty$.
We write $L^{p}_{w}$ for the $L^{p}$ space endowed with the weak topology,
except that, with a slight abuse of notation, both $L^{\infty}$ and
$L^{\infty}_{w}$ stand for $L^{\infty}$ endowed with the weak* topology (as
dual of $L^{1}$).

###### 0\. Directional derivative operator.

Let $v$ be a vector field on $\mathbb{R}^{d}$ which is Borel measurable. We
denote by $T_{v}$ the directional derivative operator on
$C^{1}_{c}(\mathbb{R}^{d})$ associated to $v$, that is,

$T_{v}u:=\frac{\partial u}{\partial v}\quad\text{for every $u\in
C^{1}_{c}(\mathbb{R}^{d})$.}$ (1.1)

Next theorem is our main result. The statement involves the notion of
decomposability bundle $V(\mu,\cdot)$ of a measure $\mu$: the precise
definition is given in §2.2; for the time being it suffices to know that
$V(\mu,x)$ is a linear subspace of $\mathbb{R}^{d}$ for every
$x\in\mathbb{R}^{d}$.

###### 1.1. Theorem.

Let $\mu$ be a Radon measure, let $v$ and $T_{v}$ be as above, and assume that
$v\in L^{p}(\mu)$ for some $p\in[1,\infty]$.

  1. (i)

If $v(x)\in V(\mu,x)$ for $\mu$-a.e. $x$, then every function
$u\in\mathrm{Lip}(\mathbb{R}^{d})$ is differentiable at $\mu$-a.e.
$x\in\mathbb{R}^{d}$ in the direction $v(x)$, and the linear operator
$\widetilde{T}_{v}:\mathrm{Lip}(\mathbb{R}^{d})\to L^{p}_{w}(\mu)$ defined by

$\widetilde{T}_{v}u(x):=\frac{\partial u}{\partial v}(x)\quad\text{for
$\mu$-a.e.~{}$x\in\mathbb{R}^{d}$}$ (1.2)

is a continuous extension of $T_{v}$.
It follows that $T_{v}$ is closable from $\mathrm{Lip}(\mathbb{R}^{d})$ to
$L^{p}_{w}(\mu)$.

  2. (ii)

Conversely, if $\\{x\colon v(x)\not\in V(\mu,x)\\}$ has positive
$\mu$-measure, then $T_{v}$, viewed as an operator from
$C^{1}_{c}(\mathbb{R}^{d})$ to $L^{p}_{w}(\mu)$, is not continuous at any
$u\in C^{1}_{c}(\mathbb{R}^{d})$. More precisely, for every $\varepsilon>0$
there exist a sequence $(u_{n})$ in $C^{1}_{c}(\mathbb{R}^{d})$ such that

     * •

$u_{n}\to u$ uniformly;

     * •

$\mathrm{Lip}(u_{n})\leq\mathrm{Lip}(u)+\varepsilon$ for every $n$;

     * •

$T_{v}u_{n}\to w$ in $L^{p}(\mu)$ for some $w\neq T_{v}u$.

It follows that $T_{v}$ is not closable from $\mathrm{Lip}(\mathbb{R}^{d})$ to
$L^{p}(\mu)$.

###### 1.2. Remarks.

. The following are immediate consequences of Theorem 1.1.

  1. (i)

If $v(x)\in V(\mu,x)$ for $\mu$-a.e. $x$, then $\tilde{T}_{v}$ is continuous
also as an operator from $\mathrm{Lip}(\mathbb{R}^{d})$ to $L^{p}(\mu)$. It
follows that $T_{v}$ is closable from $\mathrm{Lip}(\mathbb{R}^{d})$ to
$L^{p}(\mu)$.

  2. (ii)

If $\\{x\colon v(x)\not\in V(\mu,x)\\}$ has positive $\mu$-measure, then
$T_{v}$ is discontinuous everywhere also as an operator from
$C^{1}_{c}(\mathbb{R}^{d})$ to $L^{p}(\mu)$. Moreover $T_{v}$ is not closable
also from $\mathrm{Lip}(\mathbb{R}^{d})$ to $L^{p}_{w}(\mu)$ nor from
$L^{q}(\mu)$ to $L^{p}(\mu)$, for any $1\leq q\leq\infty$. The latter follows
from the fact that the functions $u_{n}$ can be chosen so that $u_{n}-u$ is
compactly supported, see also Remark 4.2.

From Theorem 1.1 and Remarks 1.2, we deduce the following corollaries:

###### 1.3. Corollary.

For $1\leq p\leq\infty$, the gradient operator is closable from
$\mathrm{Lip}(\mathbb{R}^{d})$ to $L^{p}(\mu)$ (resp. $L^{p}_{w}(\mu)$) if and
only if $\mu$ is absolutely continuous with respect to the Lebesgue measure
($\mu\ll\mathscr{L}^{d}$). The same holds for the divergence and the Jacobian
determinant.222 The Jacobian determinant of $u\in
C^{1}_{c}(\mathbb{R}^{d};\mathbb{R}^{d})$ is $Ju:=\det(\nabla u)$.

###### 1.4. Corollary.

The gradient operator is closable from $L^{q}(\mu)$ to $L^{p}(\mu)$ for some
$1\leq p\leq\infty$ and $1\leq q\leq\infty$ only if $\mu\ll\mathscr{L}^{d}$.

###### 1.5. Remarks.

  1. (i)

These results are naturally connected to the definition of Sobolev spaces in
weighted Euclidean spaces and have applications in the representation of low-
dimensional elastic structures, see [7]. Theorem 1.1 leads to a proof of the
chain rule for BV functions of [4] which can be adapted to finite dimensional
RCD spaces, see [8]. Corollary 1.4 answers a question posed by Fukushima, see
[6, Section 2.6] and [10].

  2. (ii)

We remind that the condition $\mu\ll\mathscr{L}^{d}$ alone is not sufficient
for the closability of the gradient operator from $L^{p}(\mu)$ to
$L^{p}(\mu)$. In [3, Theorem 2.2], the (absolutely continuous) measures $\mu$
on $\mathbb{R}$ with this property for $p=2$ are characterized, see also [12,
Theorem 3.1.6]. The question on the closability from $L^{p}(\mu)$ to
$L^{p}(\mu)$ of directional derivative operators is also interesting. The
condition given in point $(i)$ of Theorem 1.1 remains necessary but it is no
longer sufficient. In §4 we give some sufficient conditions on $\mu$. We don’t
know if these conditions are also necessary.

  3. (iii)

In §5 we discuss the closability of some more general multilinear operators
than the jacobian and we rephrase these results in terms of Ambrosio-Kirchheim
metric currents in $\mathbb{R}^{d}$.

###### 0\. Acknowledgements.

This research was initiated during visits of D.B. and A.M. at the Mathematics
Department in Pisa. The visits of A.M. were partly supported by INdAM-GNAMPA.
The research of D.B. is supported by the European Union’s Horizon 2020
Research and Innovation Programme, grant agreement no. 948021. The research of
G.A. and A.M. has been partially supported by the Italian Ministry of
University and Research via the PRIN projects 2017BTM7SN and 2017TEXA3H,
respectively.

##  2\. Notation and preliminary results

###  2.1. Classical currents

Recall that a _$k$ -dimensional current_ $T$ in $\mathbb{R}^{d}$ is a
continuous linear functional on the space of smooth and compactly supported
differential $k$-forms on $\mathbb{R}^{d}$. The _boundary_ of $T$, denoted by
$\partial T$, is the $k-1$-current defined via $\langle\partial
T,\omega\rangle:=\langle T,d\omega\rangle$ for every smooth and compactly
supported $k-1$-form $\omega$. The _mass_ of $T$, denoted by $\mathbb{M}(T)$,
is the supremum of $\langle T,\omega\rangle$ over all $k$-forms $\omega$ such
that $|\omega|\leq 1$ everywhere. A current $T$ is called _normal_ if both $T$
and $\partial T$ have finite mass.

By the Radon–Nikodým theorem, a $k$-dimensional current $T$ with finite mass
can be written in the form $T=\tau\mu$ where $\mu$ is a finite positive
measure and $\tau$ is a $k$-vector field in $L^{1}(\mu)$. In particular, the
action of $T$ on a smooth and compactly supported $k$-form $\omega$ is given
by

$\langle T,\omega\rangle=\int_{\mathbb{R}^{d}}\langle\omega(x),\tau(x)\rangle
d\mu(x)\;.$

Given a Lipschitz curve $\gamma:[0,1]\to\mathbb{R}^{d}$, we denote by
$[\gamma]$ the associated current, that is the current defined by

$\langle[\gamma],\omega\rangle=\int_{0}^{1}\langle\omega(\gamma(t)),\gamma^{\prime}(t)\rangle
dt\;,$

for every smooth and compactly supported $1$-form $\omega$. More information
on currents can be found in [11].

###  2.2. Decomposability bundle

We recall the definition of decomposability bundle of a Radon measure, see [1,
§2.6] and its relation with the differentiability properties of Lipschitz
functions. We prefer to give a definition which is different from the original
one, but it is equivalent and easier to state, see [1, §6.1 and Theorem 6.4].

######  2.3. Definition (Decomposability bundle).

Given a Radon measure $\mu$ on $\mathbb{R}^{d}$ its _decomposability bundle_
is a Borel map $V(\mu,\cdot)$ on $\mathbb{R}^{d}$ whose values are vector
subspaces of $\mathbb{R}^{d}$, defined as follows: a vector
$v\in\mathbb{R}^{d}$ belongs to $V(\mu,x)$ if and only if there exists a
1-dimensional normal current $N$ with $\partial N=0$ such that

$\lim_{r\to 0}\frac{\mathbb{M}((N-v\mu)\llcorner B(x,r))}{\mu(B(x,r))}=0.$

###### 2.4. Definition.

For $1<m\leq d$ we will denote $V_{m}(\mu,\cdot)$ the map, whose values are
vector subspaces of the space $\Lambda_{m}(\mathbb{R}^{d})$ of $m$-vectors in
$\mathbb{R}^{d}$, which is obtained by replacing 1-dimensional normal currents
with $m$-dimensional ones in the previous definition. For more details see §4
of [2].

######  2.5. Theorem (see Theorem 1.1 of [1]).

Let $\mu$ be a positive Radon measure on $\mathbb{R}^{d}$ and let
$V(\mu,\cdot)$ be the decomposability bundle of $\mu$. Then the following
statements hold:

  * (i)

Every Lipschitz function $f$ on $\mathbb{R}^{d}$ is differentiable at
$\mu$-a.e. $x$ with respect to the linear subspace $V(\mu,x)$. That is, there
exists a linear function from $V(\mu,x)$ to $\mathbb{R}$, denoted by
$d_{V}f(x)$, such that

$f(x+h)=f(x)+\langle d_{V}f(x),h\rangle+o(|h|)\quad\text{for $h\in
V(\mu,x)$.}$

  * (ii)

The previous statement is optimal in the sense that there exists a Lipschitz
function $f$ on $\mathbb{R}^{d}$ such that for $\mu$-a.e. $x$ and every
$v\notin V(\mu,x)$ the derivative of $f$ at $x$ in the direction $v$ does not
exist.

We conclude by recalling the converse of Rademacher’s theorem. Its combination
with Theorem 2.5 ensures that the decomposability bundle of a singular measure
$\mu$ is a proper subspace $\mu$-a.e.

######  2.6. Theorem (see Theorem 1.14 of [9]).

Let $\mu$ be a positive Radon measure on $\mathbb{R}^{d}$ such that every
Lipschitz function $f:\mathbb{R}^{d}\to\mathbb{R}$ is differentiable
$\mu$-a.e. Then $\mu\ll\mathscr{L}^{d}$.

###### 2.7. Corollary.

Let $\mu$ be a positive Radon measure on $\mathbb{R}^{d}$ such that
$V(\mu,x)=\mathbb{R}^{d}$ for $\mu$-a.e. $x$. Then $\mu\ll\mathscr{L}^{d}$.

##  3\. Closability of operators from ${\rm{Lip}}$ to $L^{p}$

  * Proof of Theorem 1.1: case $p=\infty$.

(i) By [1, Theorem 6.3] there exists a normal 1-current
$N=\tilde{v}\tilde{\mu}$ on $\mathbb{R}^{d}$ with $\partial N=0$ such that
$\tilde{v}\in L^{\infty}(\tilde{\mu})$ and $\tilde{v}$ and $\tilde{\mu}$ are
extensions of $v$ and $\mu$ in the following sense: $\tilde{\mu}=\mu+\sigma$
with $\sigma\perp\mu$ and $\tilde{v}(x)=v(x)$ for $\mu$-a.e. $x$. Let $u_{n}$
be a sequence such that $u_{n}\to u$ in $\mathrm{Lip}(\mathbb{R}^{d})$. Using
[1, Proposition 5.13] and the fact that $\partial N=0$, we have

$\partial(u_{n}N)=-d_{\tilde{v}}u_{n}\tilde{\mu},\quad\mbox{and}\quad\partial(uN)=-d_{\tilde{v}}u\tilde{\mu}.$
(3.1)

The uniform convergence $u_{n}\to u$ implies that $\mathbb{M}(u_{n}N-uN)\to 0$
and therefore the convergence is also in the sense of currents, and the same
holds for their boundaries, that is,

$\lim_{n\to\infty}\int_{\mathbb{R}^{d}}fd_{\tilde{v}}u_{n}d\tilde{\mu}=\int_{\mathbb{R}^{d}}fd_{\tilde{v}}ud\tilde{\mu},\quad\mbox{for
every $f\in C^{\infty}_{c}(\mathbb{R}^{d})$.}$ (3.2)

The density of $C^{\infty}_{c}(\mathbb{R}^{d})$ in $L^{1}(\tilde{\mu})$
implies that (3.2) holds for every $f\in L^{1}(\tilde{\mu})$ and therefore
$d_{\tilde{v}}u_{n}\to d_{\tilde{v}}u$ in $L^{\infty}_{w^{*}}(\tilde{\mu})$.
In particular $d_{v}u_{n}\to d_{v}u=\langle d_{V}u,v\rangle$ in
$L^{\infty}_{w^{*}}(\mu)$.

(ii) For every $w\in\mathbb{R}^{d}$ and $0<\theta<\frac{\pi}{2}$, let
$C(w,\theta)$ be the cone

$C(w,\theta)=\\{x\in\mathbb{R}^{d}:x\cdot w\geq|x|\cos\theta\\}.$

By assumption, there exists $w\in\mathbb{R}^{d}$ and
$0<\alpha<\beta<\frac{\pi}{2}$ such that the Borel set

$E:=\\{x\in\mathbb{R}^{d}:v(x)\in C(w,\alpha)\mbox{ and }V(\mu,x)\cap
C(w,\beta)=\\{0\\}\\},$

has positive $\mu$-measure. By [1, Lemma 7.5], there exists a compact set
$F\subset E$ with $\mu(F)>0$ and $F$ is $C(w,\beta)$-null, in the sense of [1,
§4.11]. By [1, Lemma 4.12], for every $n=1,2,\dots$ there exists
$f_{n}:\mathbb{R}^{d}\to\mathbb{R}$ smooth such that for every
$x\in\mathbb{R}^{d}$:

    * (a)

$0\leq f_{n}(x)\leq\frac{1}{n}$;

    * (b)

$0\leq d_{w}f_{n}(x)\leq 1$ and $d_{w}f_{n}(x)\equiv 1$ on $F$;

    * (c)

$|d_{W}f_{n}(x)|\leq\frac{1}{\tan\beta}$ where $W=w^{\perp}$, $d_{W}$ is the
restriction of the differential $d$ to $W$ and $|\cdot|$ is the operator norm.

By (b) and (c) there exists $L>0$ such that every $f_{n}$ is $L$-Lipschitz and
moreover there is a constant $C>0$ depending on $\alpha$ and $\beta$ such that
$d_{v(x)}f_{n}(x)\geq C$ for every $x\in F$. Clearly the functions
$u_{n}:=u+L^{-1}\varepsilon f_{n}$ satisfy the requirements. ∎

Before proving Theorem 1.1 for $p<\infty$ we state the following corollary of
Mazur’s lemma.

###### 3.1. Proposition.

Let $1\leq p<\infty$ and $T:C^{1}_{c}(\mathbb{R}^{d})\to L^{p}(\mu)$ be a
bounded linear operator. Let $u_{n}$ be a sequence of functions in
$C^{1}_{c}(\mathbb{R}^{d})$ such that $u_{n}\to u$ uniformly and $Tu_{n}\to w$
in $L^{p}_{w}(\mu)$. Then there is a sequence $\tilde{u}_{n}$ of convex
combinations of the elements of $u_{n}$ such that $\tilde{u}_{n}\to u$
uniformly and $T\tilde{u}_{n}\to w$ strongly in $L^{p}(\mu)$.

  * Proof.

For every $m\in\mathbb{N}$ we consider the set $A_{m}$ which is the $w$-closed
convex hull of the set $\\{Tu_{m},Tu_{m+1},\dots\\}$. By Mazur’s Lemma the set
$A_{m}$ coincides with the (strongly)-closed convex hull of the set
$\\{Tu_{m},Tu_{m+1},\dots\\}$. Since by assumption $w\in A_{m}$ for every $m$,
then for every $m$ there exists a sequence $(w^{m}_{n})_{n\in\mathbb{N}}$ of
convex combinations of the elements of the set $\\{Tu_{m},Tu_{m+1},\dots\\}$
such that $w^{m}_{n}\to w$ (strongly). For every $m$ and $n$ we denote by
$\tilde{u}_{n}^{m}$ the convex combination of the elements of the set
$\\{u_{m},u_{m+1},\dots\\}$ with the same coefficients as those used to obtain
$w_{n}^{m}$. Clearly the diagonal sequence $\tilde{u}_{n}:=\tilde{u}_{n}^{n}$
has the desired properties. In particular the fact that $\tilde{u}_{n}^{n}\to
u$ uniformly as $n\to\infty$ follows from the fact that the diameter of the
convex hull of the set $\\{u_{m},u_{m+1},\dots\\}$ (with respect to the
uniform norm) tends to zero as $m\to\infty$. ∎

In what follows we denote by $p^{\prime}$ the conjugate Hölder exponent of
$p$. Moreover, for every $v\in\mathbb{R}^{d}$ we denote

$\hat{v}:=\begin{cases}\frac{v}{|v|}\quad\mbox{if $v\neq 0$}\\\
0\quad\mbox{otherwise}.\end{cases}$ (3.3)

  * Proof of Theorem 1.1: case $p<\infty$.

Towards a proof of (i), we observe that the condition $v(x)\in V(\mu,x)$ is
$\mu$-a.e equivalent to the condition $\hat{v}(x)\in V(\mu,x)$. Therefore if
$v(x)\in V(\mu,x)$ for $\mu$-a.e. $x$, by the case $p=\infty$, $T_{\hat{v}}$
can be extended to the continuous operator
$\tilde{T}_{\hat{v}}:\mathrm{Lip}(\mathbb{R}^{d})\to L^{\infty}_{w^{*}}(\mu)$
given by $\tilde{T}_{\hat{v}}=\langle d_{V}u(x),\hat{v}(x)\rangle$. It follows
that the operator $\hat{T}_{v}:\mathrm{Lip}(\mathbb{R}^{d})\to L^{p}_{w}(\mu)$
given by

$\hat{T}_{v}(u):=|v|\tilde{T}_{\hat{v}}u=|v|\langle d_{V}u(x)\
,\hat{v}(x)\rangle=\langle d_{V}u(x)\ ,v(x)\rangle$

is a continuous extension of $T_{v}$. Indeed, assume $u_{n}\to u$ in
$\mathrm{Lip}$ and take $\phi\in L^{p^{\prime}}(\mu)$. Then we have

$\langle\hat{T}_{v}u_{n},\phi\rangle=\langle\tilde{T}_{\hat{v}}u_{n},|v|\phi\rangle.$

Since $|v|\phi\in L^{1}(\mu)$, by the continuity of
$\tilde{T}_{\hat{v}}:\mathrm{Lip}(\mathbb{R}^{d})\to L^{\infty}_{w^{*}}(\mu)$
we have

$\lim_{n\to\infty}\langle\tilde{T}_{\hat{v}}u_{n},|v|\phi\rangle=\langle\tilde{T}_{\hat{v}}u,|v|\phi\rangle=\langle\hat{T}_{v}u,\phi\rangle.$

Towards a proof of (ii), the possibility to replace $L^{\infty}_{w^{*}}(\mu)$
with $L^{p}_{w}(\mu)$ in the third bullet point can be proved with the same
sequence $u_{n}$ used in the case $p=\infty$. In order to replace
$L^{p}_{w}(\mu)$ with $L^{p}(\mu)$, fix $\varepsilon>0$ and a sequence $u_{n}$
of functions in $C^{1}_{c}(\mathbb{R}^{d})$ such that

    * –

$u_{n}\to u$ uniformly as $n\to\infty$;

    * –

$\mathrm{Lip}(u_{n})\leq\mathrm{Lip}(u)+\varepsilon$ for every
$n\in\mathbb{N}$;

    * –

$T_{v}u_{n}\to w$ in $L^{p}_{w}(\mu)$ with $w\neq T_{v}u$.

Applying Proposition 3.1 to such sequence we obtain a new sequence satisfying
the required properties. ∎

  * Proof of Corollary 1.3.

The closability of the gradient operator (as well as the closability of the
divergence and the Jacobian determinant) implies the closability of all the
partial derivatives operators. By point (ii) of Theorem 1.1, this implies that
$V(\mu,x)=\mathbb{R}^{d}$ for $\mu$-a.e. $x$. We deduce from Corollary 2.7
that $\mu\ll\mathscr{L}^{d}$. ∎

  * Proof of Corollary 1.4.

This follows immediately from Remark 1.2 (ii) and Corollary 2.7. ∎

##  4\. Closability of operators from $L^{q}$ to $L^{p}$

###### 4.1. Theorem.

Let $\mu$ be a Radon measure and let $v\in L^{p}(\mu,\mathbb{R}^{d})$. Let
$T_{v}:C^{1}_{c}(\mathbb{R}^{d})\to L^{p}(\mu)$ be the directional derivative
operator defined in (1.1). Assume there exists a real valued function $\alpha$
such that

  * •

$\alpha\neq 0\;\mu$-a.e.;

  * •

$\alpha\in L^{p^{\prime}}(\mu)$ and $\alpha v\in L^{q^{\prime}}(\mu)$;

  * •

$N:=\alpha v\mu$ is a normal 1-current.

Then $T_{v}$ is closable from $L^{q}(\mu)$ to $L^{p}(\mu)$.

  * Proof of Theorem 4.1.

Assume by contradiction that $T_{v}$ is not closable from $L^{q}(\mu)$ to
$L^{p}(\mu)$. Since $T_{v}$ is linear, this is equivalent to assuming that
$T_{v}$ is not closable ”at $u=0$”, that is, there is a sequences $u_{n}$ such
that $u_{n}\to 0$ in $L^{q}(\mu)$ and $T_{v}u_{n}\to w\neq 0\in L^{p}(\mu)$.
Denote $\tau:=\alpha v$ and let $\hat{\tau}$ be defined as in (3.3).

It follows from Smirnov’s theorem, see [16], that $N$ can be written as

    * (a)

$N=\int_{0}^{1}[\gamma_{s}]ds$, where $\gamma_{s}:[0,L_{s}]\to\mathbb{R}^{d}$
are 1-Lipschitz, nontrivial, and parametrized with constant speed;

    * (b)

$\|N\|=\int_{0}^{1}\|[\gamma_{s}]\|ds$;

    * (c)

$\frac{\gamma^{\prime}_{s}(t)}{|\gamma^{\prime}_{s}(t)|}=\hat{\tau}(\gamma_{s}(t))$
for a.e. $s$ and a.e. $t$.

The existence of such decomposition can be found in [14, Theorem 3.1] and in
particular the validity of (c) can be proved as in [1, Theorem 5.5 (ii)].

Since $u_{n}\to 0$ in $L^{q}(\mu)$ and $|\tau|\in L^{q^{\prime}}(\mu)$, then
$u_{n}|\tau|\to 0$ in $L^{1}(\mu)$ or equivalently $u_{n}\to 0$ in
$L^{1}(|\tau|\mu)=L^{1}(\|N\|)$. Hence by (b), denoting
$\mu_{s}:=\|[\gamma_{s}]\|$ for every $s\in[0,1]$, we have, up to
subsequences,

$u_{n}\to 0\quad\mbox{in $L^{1}(\mu_{s})$ for a.e. $s\in[0,1]$.}$ (4.1)

Since $T_{v}u_{n}\to w$ in $L^{p}(\mu)$ and $\alpha\in L^{p^{\prime}}(\mu)$,
then $\alpha T_{v}u_{n}\to\alpha w$ in $L^{1}(\mu)$ or equivalently
$\frac{\partial u_{n}}{\partial\hat{\tau}}\to{\rm{sign}}(\alpha)\frac{w}{|v|}$
in $L^{1}(|\tau|\mu)$. Hence by (b) we have, up to subsequences,

$\frac{\partial
u_{n}}{\partial\hat{\tau}}\to{\rm{sign}}(\alpha)\frac{w}{|v|}\quad\mbox{in
$L^{1}(\mu_{s})$ for a.e. $s\in[0,1]$.}$ (4.2)

By (4.1) and property (a) for a.e. $s\in[0,1]$, up to subsequences,

$u_{n}\circ\gamma_{s}\to 0\quad\mbox{in $L^{1}([0,L_{s}])$}.$ (4.3)

Moreover by property (c), for every $n$ and for a.e. $s,t$, up to
subsequences,

$(u_{n}\circ\gamma_{s})^{\prime}(t)=\frac{\partial
u_{n}}{\partial\hat{\tau}}(\gamma_{s}(t))|\gamma_{s}^{\prime}(t)|,$ (4.4)

so that, by (4.2) and (4.3), for a.e. $s$ we have that
$({\rm{sign}}(\alpha)\frac{w}{|v|})\circ\gamma_{s}|\gamma_{s}^{\prime}|=0$.
Since $\alpha\neq 0$ $\mu$-a.e., this contradicts the fact that $w\neq 0\in
L^{p}(\mu)$. ∎

###### 4.2. Remark.

Theorem 1.1, and Theorem 4.1 can be localized, namely they hold true also when
the function spaces are replaced by their local versions. This follows from
the fact that, by a standard cutoff argument, the functions $f_{n}$ in the
proof of Theorem 1.1 (ii) can be chosen to be compactly supported.

##  5\. Closability of multilinear operators and metric currents

In this section we extend point (i) of Theorem 1.1 to the setting of
multilinear operators and we rephrase some of the results of the paper in
terms of metric currents. Our remarks are mostly consequences of the fact that
the _continuity_ property in the definition of metric current is equivalent to
the closability of the operator $J_{v}$ of Theorem 5.1 for a suitable
$k$-vectorfield $v$ and a measure $\mu$.

###### 5.1. Theorem.

Let $v\in L^{\infty}(\mu,\Lambda_{k}(\mathbb{R}^{d}))$ and let
$J_{v}:C^{1}_{c}(\mathbb{R}^{d},\mathbb{R}^{k})\to L^{\infty}(\mu)$ be the
multilinear differential operator

$J_{v}:(u_{1},\dots,u_{k})\mapsto\langle du_{1}\wedge\dots\wedge
du_{k};v\rangle.$

If $v(x)\in V_{k}(\mu,x)$ for $\mu$-a.e. $x$, then $J_{v}$ can be extended to
a continuous operator
$\tilde{J}_{v}:\mathrm{Lip}(\mathbb{R}^{d},\mathbb{R}^{k})\to
L^{\infty}_{w^{*}}(\mu)$ and more precisely

$\tilde{J}_{v}(u_{1},\dots,u_{k})(x)=\langle d_{V}u_{1}(x)\wedge\dots\wedge
d_{V}u_{k}(x);v(x)\rangle,\quad\mbox{for $\mu$-a.e. $x$}$

where $V=V(\mu,\cdot)$. It follows that $J_{v}$ is closable from
$\mathrm{Lip}(\mathbb{R}^{d},\mathbb{R}^{k})$ to $L^{\infty}_{w^{*}}(\mu)$.

  * Proof.

The theorem can be proved verbatim as Theorem 1.1 (i), replacing the 1-current
$N$ with a $k$-current with the same properties, whose existence is proved by
combining Theorems 1.1 and 1.2 of [2]. ∎

We do not know whether the condition $v(x)\in V_{k}(\mu,x)$ for $\mu$-a.e. $x$
is necessary for the closability of $J_{v}$, see Remark 5.6.

###  5.2. Metric currents

Let $(X,d)$ be a complete metric space. We denote
$\mathscr{D}^{k}(X):=\mathrm{Lip}_{b}(X,\mathbb{R})\times\mathrm{Lip}(X,\mathbb{R})^{k}$,
where $\mathrm{Lip}_{b}(X,\mathbb{R})$ is the space of bounded Lipschitz
functions on $X$.

######  5.3. Definition (Metric current).

A multilinear functional $T:\mathscr{D}^{k}(X)\to\mathbb{R}$ is said to be a
$k$-dimensional _metric current_ if

  * (i)

_continuity_ : for every $f\in\mathrm{Lip}_{b}(X,\mathbb{R})$,
$(\pi_{1}^{n})_{n\in\mathbb{N}},\dots,(\pi_{k}^{n})_{n\in\mathbb{N}}\subset\mathrm{Lip}(X,\mathbb{R})$
converging pointwise to $\pi_{1},\dots,\pi_{k}$ and with
$\mathrm{Lip}(\pi_{i}^{n})\leq C$ for every $n$

$T(f,\pi_{1}^{n},\dots,\pi_{k}^{n})\to T(f,\pi_{1},\dots,\pi_{k});$

  * (ii)

_locality_ : if there exists $i\in\\{1,\dots,k\\}$ such that $\pi_{i}\equiv c$
on a neighbourhood of $supp(f)$ then $T(f,\pi_{1},\dots,\pi_{k})=0$;

  * (iii)

_finite mass_ : there exists a finite Radon measure $\mu$ such that

$|T(f,\pi_{1},\dots,\pi_{k})|\leq\mathrm{Lip}(\pi_{1})\cdots\mathrm{Lip}(\pi_{k})\int_{X}|f|d\mu.$

More information on metric currents can be found in [5] and [13].

We will now focus on the choice $X=\mathbb{R}^{d}$, endowed with the Euclidean
distance. We recall that to every $k$-dimensional metric current $T$ on
$\mathbb{R}^{d}$ with compact support one can associate a ”classical”
$k$-dimensional current $\tilde{T}$ as follows, see [5, Theorem 11.1].
Denoting $\Lambda(k,d)$ the set of multi-indices
$\alpha=(1\leq\alpha_{1}<\dots<\alpha_{k}\leq d)$ of length $k$ in
$\mathbb{R}^{d}$, one requires that for every smooth and compactly supported
$k$-form

$\omega=\sum_{\alpha\in\Lambda(k,d)}\omega_{\alpha}dx_{\alpha_{1}}\wedge\dots\wedge
dx_{\alpha_{k}}$

it holds

$\langle\tilde{T},\omega\rangle=\sum_{\alpha\in\Lambda(k,d)}T(\omega_{\alpha},x_{\alpha_{1}},\dots,x_{\alpha_{k}}).$
(5.1)

Conversely, to every flat chain $T$ with finite mass and compact support one
can associate a metric current $\hat{T}$ in such a way that the two maps are
one the inverse of the other, when restricted to normal currents, see also
[13, Theorem 5.5].

###### 5.4. Proposition.

Let $T$ be a $k$-dimensional metric current on $\mathbb{R}^{d}$ with compact
support. Then there exists a $k$-vector field $\tau$ and a positive measure
$\mu$ with ${\mbox{span}}(\tau(x))\subset V(\mu,x)$ for $\mu$-a.e.
$x\in\mathbb{R}^{d}$ such that, for every
$(f,\pi_{1},\dots,\pi_{k})\in\mathscr{D}^{k}(\mathbb{R}^{d})$,

$T(f,\pi_{1},\dots,\pi_{k})=\int_{\mathbb{R}^{d}}f\langle
d_{V}\pi_{1}\wedge\dots\wedge d_{V}\pi_{k},\tau\rangle d\mu.$ (5.2)

  * Proof.

Consider the positive measure $\mu$ and the unit $k$-vector field $\tau$ such
that $\tilde{T}=\tau\mu$. Assume by contradiction that there exists a Borel
set $E$ with $\mu(E)>0$ and a Borel vector field $w:E\to\mathbb{R}^{d}$ with
$w(x)\in{\mbox{span}}(\tau(x))$ such that $w(x)\not\in V(\mu,x)$ for every
$x\in E$.

By [1, Proposition 5.9 (v)] there exist $\alpha_{1},\dots,\alpha_{k-1}$ and a
Borel set $E^{\prime}\subset E$ with $\mu(E^{\prime})>0$ such that

$v(x):=\tau(x)\llcorner dx_{\alpha_{1}}\wedge\dots\wedge
dx_{\alpha_{k-1}}\not\in V(\mu,x),$

for every $x\in E^{\prime}$. Now consider the 1-dimensional metric current

$S:=T\llcorner dx_{\alpha_{1}}\wedge\dots\wedge dx_{\alpha_{k-1}}$

(see [5, Definition 2.5]). Since $\tilde{S}=\tilde{T}\llcorner
dx_{\alpha_{1}}\wedge\dots\wedge dx_{\alpha_{k-1}}$, then we can write
$\tilde{S}=(\tau\llcorner dx_{\alpha_{1}}\wedge\dots\wedge
dx_{\alpha_{k-1}})\mu$. Choose any $\alpha_{k}$ different from
$\alpha_{1},\dots,\alpha_{k-1}$ and let $u_{n}\to x_{\alpha_{k}}$ be as in
Theorem 1.1 (ii) for the vector field $v$. For every
$f\in\mathrm{Lip}_{b}(\mathbb{R}^{d})$ we have

$\begin{split}S(f,x_{\alpha_{k}})&=T(f,x_{\alpha_{1}},\dots,x_{\alpha_{k}})=\langle\tilde{T},fdx_{\alpha_{1}}\wedge\dots\wedge
dx_{\alpha_{k}}\rangle\\\ &=\int_{\mathbb{R}^{d}}\langle
fdx_{\alpha_{1}}\wedge\dots\wedge dx_{\alpha_{k}},\tau\rangle
d\mu=\int_{\mathbb{R}^{d}}f\langle dx_{\alpha_{k}},v(x)\rangle
d\mu\end{split}$ (5.3)

and similarly

$S(f,u_{n})=T(f,x_{\alpha_{1}},\dots,x_{\alpha_{k-1}},u_{n})=\int_{\mathbb{R}^{d}}f\langle
du_{n},v(x)\rangle d\mu.$ (5.4)

By the density of bounded Lipschitz functions in $L^{1}(\mu)$, the validity of
(5.3) and (5.4) may be extended to $f\in L^{1}(\mu)$. The continuity of metric
currents, see Definition 5.3 (i), implies that
$\lim_{n\to\infty}S(f,u_{n})=S(f,x_{\alpha_{k}})$. However, Theorem 1.1(ii)
implies that there exists $f\in L^{1}(\mu)$ such that the limit of the RHS of
(5.4) is different from the the RHS of (5.3), which is a contradiction.

The validity of

$T(f,\pi_{1},\ldots,\pi_{k})=\langle\tilde{T},f\wedge
d\pi_{1}\wedge\ldots\wedge d\pi_{k})$

for smooth $f,\pi_{1},\dots,\pi_{k}$ is shown in [13, Theorem 5.5].
Consequently, (5.2) holds in this case, with $d_{V}$ replaced by the full
derivative. Its extension to
$(f,\pi_{1},\dots,\pi_{k})\in\mathscr{D}^{k}(\mathbb{R}^{d})$ follows from [1,
Corollary 8.3]. ∎

###### 5.5. Proposition.

Let $T=\tau\mu$ be a classical $k$-dimensional current on $\mathbb{R}^{d}$
with compact support with ${\mbox{span}}(\tau(x))\subset V(\mu,x)$ for
$\mu$-a.e. $x\in\mathbb{R}^{d}$. Then the multilinear functional
$\hat{\hat{T}}$ defined, for every
$(f,\pi_{1},\dots,\pi_{k})\in\mathscr{D}^{k}(\mathbb{R}^{d})$, by

$\hat{\hat{T}}(f,\pi_{1},\dots,\pi_{k})=\int_{\mathbb{R}^{d}}f\langle
d_{V}\pi_{1}\wedge\dots\wedge d_{V}\pi_{k},\tau\rangle d\mu$ (5.5)

is separately continuous in each variable. If additionally $\tau(x)\in
V_{k}(\mu,x)$ for $\mu$-a.e. $x\in\mathbb{R}^{d}$, then $T$ is a classical
flat chain and therefore $\hat{\hat{T}}$ is a metric current and coincides
with $\hat{T}$.

  * Proof.

The continuity of $\hat{\hat{T}}$ in each variable is an immediate consequence
of Theorem 1.1 (i). If $\tau(x)\in V_{k}(\mu,x)$, then [2, Theorem 1.2]
implies that $T$ is a flat chain, and hence by [2, Theorem 1.2] there exists a
(classical) $k$-dimensional normal current $N$ and a Borel set $E$ such that
$T=N\llcorner E$. By Proposition 5.4 the metric current $\hat{N}$ can be
written as in (5.2) for a $k$-vector field $\sigma$ in place of $\tau$ and a
positive measure $\nu$ in place of $\mu$, so that, since $\tilde{\hat{N}}=N$
then $\sigma\nu=\tau\mu$ as vector measures. Since $\hat{T}=\hat{N}\llcorner
E$, then it follows that $\hat{T}=\hat{\hat{T}}$. ∎

###### 5.6. Remark.

The Ambrosio-Kirchheim flat chain conjecture can be equivalently reformulated
as follows. If $T$ is a metric current such that $\tilde{T}=\tau\mu$ then
$\tau\in V_{k}(\mu,\cdot)$.

The following corollary is a known fact, see [15, Theorem 1.6] and [9, Theorem
1.15]. We simply remark that it follows from the previous results.

###### 5.7. Corollary.

Let $T$ be either a 1-dimensional or a $d$-dimensional metric current on
$\mathbb{R}^{d}$. Then $\tilde{T}$ is a flat chain.

  * Proof.

For $k=1$, the result follows from Proposition 5.4 and Proposition 5.5. For
$k=d$, the result follows additionally from Corollary 2.7. ∎

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G.A.
Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127
Pisa, Italy
e-mail<EMAIL_ADDRESS>

D.B.
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4
7AL, UK
e-mail<EMAIL_ADDRESS>

A.M.
Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123
Povo, Italy
e-mail<EMAIL_ADDRESS>