# Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method

Sara Ban (sban@math.uniri.hr), ORCID: 0000-0002-1837-8701
Sanja Rukavina (sanjar@math.uniri.hr), ORCID: 0000-0003-3365-7925
Faculty of Mathematics, University of Rijeka
Radmile Matej$\check{c}$ić 2, 51000 Rijeka, Croatia

###### Abstract

Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual
$\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the
largest possible minimum Euclidean weight for a given length. We introduce a
doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length
$n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this
method, we develop an algorithm to construct new extremal Type II
$\mathbb{Z}_{8}$-codes starting from an extremal Type II $\mathbb{Z}_{8}$-code
of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_{4}$-residue code and
length $24,32$ or $40$.
We construct at least ten new extremal Type II $\mathbb{Z}_{8}$-codes of
length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_{8}$-codes of
length $32$ of this type were not known before. Moreover, the binary residue
codes of the constructed extremal $\mathbb{Z}_{8}$-codes are optimal $[32,15]$
binary codes.

Keywords: Type II $\mathbb{Z}_{2k}$-code, extremal $\mathbb{Z}_{8}$-code,
residue code, doubling method
Mathematics Subject Classification: 94B05

##  1 Introduction

The discovery of good nonlinear binary codes arising via the Gray map from
$\mathbb{Z}_{4}$-linear codes motivated the study of codes over rings in
general (see [14]). Construction of unimodular lattices with large minimum
norm has motivated the construction of new self-dual $\mathbb{Z}_{2k}$-codes
with large minimum Euclidean weights (see, for example, [4, 10]). Especially,
$\mathbb{Z}_{8}$-codes have received attention by many researchers. For
instance, some construction methods for self-dual codes over $\mathbb{Z}_{8}$
for arbitrary length greater than $8$ are given in [1].

Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual
$\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the
largest possible minimum Euclidean weight for a given length. For lengths $8$
and $16,$ every Type II $\mathbb{Z}_{8}$-code is extremal (see [15]). In [13],
the previously known results on the existence of extremal Type II
$\mathbb{Z}_{8}$-codes for greater lengths are summarized: there are three
such codes of length $24$ (see [4, 11]), five codes of length $32,$ a large
number of codes of length $40$ and one such code of length $48$ (see [8, 9]),
up to equivalence. In addition, a new extremal Type II $\mathbb{Z}_{8}$-code
$D_{n}$ is constructed for each of the lengths $n\in\\{24,32,40\\}$ (see
[[13], Subsection 5.2]).

The doubling method for a construction of Type II $\mathbb{Z}_{4}$-codes is
introduced in [7]. It was used for the construction of extremal Type II
$\mathbb{Z}_{4}$-codes of length $32$ and $40$ in [2] and [3], respectively.
In this paper, we introduce a doubling method for constructing a Type II
$\mathbb{Z}_{2k}$-code of length $n$ from a known Type II
$\mathbb{Z}_{2k}$-code of length $n$. We also develop an algorithm that uses
this method to construct a Type II $\mathbb{Z}_{2^{m}}$-code of length $n$
from a known Type II $\mathbb{Z}_{2^{m}}$-code of length $n$. Finally, by
specifying the method for $m=3$ and $n\in\\{24,32,40\\},$ we obtain an
algorithm that is then used to construct at least $10$ new extremal Type II
$\mathbb{Z}_{8}$-codes of length $32$.

The paper is organized as follows. The next section gives definitions and
basic properties of codes over $\mathbb{Z}_{2k}$ that will be needed in our
work. In Section 3, we introduce the doubling method to construct new Type II
$\mathbb{Z}_{2k}$-codes starting from a Type II $\mathbb{Z}_{2k}$-code.
Especially, we consider the codes over $\mathbb{Z}_{2^{m}}.$ Finally, in the
last section, we present a method to construct new extremal Type II
$\mathbb{Z}_{8}$-codes starting from an extremal Type II $\mathbb{Z}_{8}$-code
of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_{4}$-residue code and
length $24,32$ or $40$. Using this method, we construct $68850$ extremal Type
II $\mathbb{Z}_{8}$-codes of type $(15,1,1)$ and length $32.$ We give the
weight distributions of the corresponding binary residue codes. With respect
to these weight distributions, all constructed extremal Type II
$\mathbb{Z}_{8}$-codes are divided into ten classes. For each of them we give
a generator matrix in the standard form for one class representative.

##  2 Preliminaries

For terms not defined in this paper and the basic facts of coding theory we
refer the reader to [5, 16, 18].

Let $\mathbb{Z}_{2k}$ denote the ring of integers modulo $2k.$ A linear code
$C$ of length $n$ over $\mathbb{Z}_{2k}$ (i.e., a $\mathbb{Z}_{2k}$-code) is
an additive subgroup of $\mathbb{Z}_{2k}^{n}.$ Two codes over
$\mathbb{Z}_{2k}$ are equivalent if one can be obtained from the other by
permuting the coordinates and (if necessary) changing the signs of certain
coordinates. Codes differing by only a permutation of coordinates are called
permutation-equivalent. An element of $C$ is called a codeword of $C.$ A
generator matrix of $C$ is a matrix whose rows generate $C.$

The dual code $C^{\bot}$ of $C$ is defined as

$C^{\bot}=\\{x\in\mathbb{Z}_{2k}^{n}\,|\,\left\langle x,y\right\rangle=0\
\text{for all}\ y\in C\\},$

where $\left\langle x,\\\
y\right\rangle=x_{1}y_{1}+x_{2}y_{2}+\dots+x_{n}y_{n}\ (\text{mod}\ 2k)$ for
$x=(x_{1},x_{2},\dots,x_{n})$ and $y=(y_{1},y_{2},\dots,y_{n}).$ The code $C$
is self-orthogonal when $C\subseteq C^{\bot}$ and self-dual if $C=C^{\bot}.$

The Euclidean weight of a codeword
$x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{Z}_{2k}^{n}$ is

$wt_{E}(x)=\sum_{i=1}^{n}\min\\{x_{i}^{2},(2k-x_{i})^{2}\\}.$

It holds

$wt_{E}(x+y)\equiv wt_{E}(x)+wt_{E}(y)+2\left\langle x,y\right\rangle\
(\text{mod}\ 4k)$ (1)

for all $x,y\in\mathbb{Z}_{2k}^{n}$ (see [4]). We denote the number of
coordinates $i$ (where $i=0,1,\dots,2k-1$) in a codeword
$x\in\mathbb{Z}_{2k}^{n}$ by $n_{i}(x).$

The minimum Euclidean weight $d_{E}$ of $C$ is the smallest Euclidean weight
among all nonzero codewords of $C.$ A self-dual $\mathbb{Z}_{2k}$-code is
called Type II if it has the property that every Euclidean weight is divisible
by $4k.$ Type II $\mathbb{Z}_{2k}$-codes are a remarkable class of self-dual
codes related to even unimodular lattices. There is a Type II
$\mathbb{Z}_{2k}$-code of length $n$ if and only if $n$ is divisible by eight
[4]. For Type II $\mathbb{Z}_{2k}$-codes $C$ of length $n,$ the upper bound on
the minimum Euclidean weight

$d_{E}(C)\leq 4k\left\lfloor\frac{n}{24}\right\rfloor+4k$ (2)

holds for $k=1$ and $2,$ and for $k\geq 3$ it holds under the assumption that
$\left\lfloor\frac{n}{24}\right\rfloor\leq k-2$ (see [4]). We say that a Type
II $\mathbb{Z}_{2k}$-code meeting (2) with equality is extremal.

If $C$ is a $\mathbb{Z}_{2^{m}}$-code, then the code $C^{(2^{k})}=\\{x\
(\text{mod}\ 2^{k})\,|\,x\in C\\}$, $1\leq k\leq m-1,$ is the
$\mathbb{Z}_{2^{k}}$-residue code of $C.$ Each code $C$ over
$\mathbb{Z}_{2^{m}}$ is permutation-equivalent to a code with a generator
matrix in standard form

$\left(\begin{array}[]{ccccccc}I_{k_{1}}&A_{1,2}&A_{1,3}&A_{1,4}&\cdots&\cdots&A_{1,m+1}\\\
0&2I_{k_{2}}&2A_{2,3}&2A_{2,4}&\cdots&\cdots&2A_{2,m+1}\\\
0&0&4I_{k_{3}}&4A_{3,4}&\cdots&\cdots&4A_{3,m+1}\\\
\vdots&\vdots&\vdots&\ddots&\ddots&&\vdots\\\
\vdots&\vdots&\vdots&\ddots&\ddots&&\vdots\\\
0&0&0&\cdots&0&2^{m-1}I_{k_{m}}&2^{m-1}A_{m,m+1}\end{array}\right),$

where the matrix $A_{i,j}$ has elements in $\mathbb{Z}_{2^{j-1}}.$ We say that
$C$ is of type $(k_{1},k_{2},k_{3},\dots,k_{m}).$ The code $C$ has
$\prod_{j=1}^{m}(2^{m-j+1})^{k_{j}}$ codewords.

In this work, we have used computer algebra systems GAP [17] and Magma [6].

##  3 Method of construction

In [7], the doubling method for a construction of Type II
$\mathbb{Z}_{4}$-codes is introduced. In the next theorem, we generalize
results from [7] and give the doubling method for a construction of Type II
$\mathbb{Z}_{2k}$-codes.

######  Theorem 3.1.

Let $k\geq 2.$ Let $C$ be a Type II $\mathbb{Z}_{2k}$-code of length $n$ and
let $n_{i}(x)$ denote the number of coordinates $i$ in
$x\in\mathbb{Z}_{2k}^{n}$. Let $ku\in\mathbb{Z}_{2k}^{n}\setminus C$ be a
codeword with all coordinates equal to $0$ or $k$ with the following property:
if $k$ is odd, $n_{k}(ku)$ is divisible by four, if $k$ is even and not
divisible by four, $n_{k}(ku)$ is even. Let $C_{0}=\\{v\in C\ |\ \left\langle
ku,v\right\rangle=0\\}.$ Then $\widetilde{C}=C_{0}\oplus\left\langle
ku\right\rangle$ is a Type II $\mathbb{Z}_{2k}$-code.

###### Proof.

The Euclidean weight $wt_{E}(ku)=n_{k}(ku)\cdot k^{2}$ is divisible by $4k$
and $\left\langle ku,ku\right\rangle=0.$ It follows from (1) that
$\widetilde{C}$ is self-orthogonal with all Euclidean weights divisible by
$4k.$

The codeword $ku\notin C,$ so there is a codeword $w\in C$ such that
$\left\langle ku,w\right\rangle=k.$ Suppose $\tilde{w}\in C\setminus C_{0}.$
Then $\left\langle ku,\tilde{w}\right\rangle=k.$ Therefore, $\tilde{w}\in
C_{0}+w$ and $C_{0}$ and $C_{0}+w$ are the only cosets of $C_{0}$ in $C.$
Since $|\widetilde{C}|=\left|C_{0}\right|\cdot 2=\left|C\right|,$
$\widetilde{C}$ is a Type II $\mathbb{Z}_{2k}$-code. ∎

When considering Type II $\mathbb{Z}_{2^{m}}$-codes, we can restrict the
possible choices for $ku=2^{m-1}u\in\mathbb{Z}_{2^{m}}^{n}\setminus C$.

######  Theorem 3.2.

Let $m\geq 2.$ Let $C$ be a Type II $\mathbb{Z}_{2^{m}}$-code of length $n$
and type $(k_{1},k_{2},\dots,k_{m}).$ The choice of
$2^{m-1}u\in\mathbb{Z}_{2^{m}}^{n}\setminus C$ in Theorem 3.1 can be limited
to codewords with zeroes on the first $k_{1}+k_{2}+\cdots+k_{m}$ coordinates.

###### Proof.

For every $2^{m-1}u\in\mathbb{Z}_{2^{m}}^{n}\setminus C$ satisfying the
conditions of Theorem 3.1, there exists a unique codeword $2^{m-1}v\in C$ with
all coordinates equal to $0$ or $2^{m-1}$ such that $2^{m-1}u$ coincides with
$2^{m-1}v$ on the first $k_{1}+k_{2}+\cdots+k_{m}$ entries. Then
$C_{0}\oplus\left\langle 2^{m-1}u\right\rangle=C_{0}\oplus\left\langle
2^{m-1}u-2^{m-1}v\right\rangle.$ ∎

Now, we generalize the statement of [[3], Theorem 5] to Type II
$\mathbb{Z}_{2^{m}}$-codes.

######  Theorem 3.3.

Let $m\geq 2.$ Let $C$ be a Type II $\mathbb{Z}_{2^{m}}$-code of length $n$
and type $(k_{1},k_{2},\dots,k_{m}).$ Let $G$ be a generator matrix of $C$ in
standard form and $G_{i}$ the $i^{th}$ row of $G.$ Let
$2^{m-1}u\in\mathbb{Z}_{2^{m}}^{n}\setminus C$ be a codeword with zeroes on
the first $k_{1}+k_{2}+\cdots+k_{m}$ coordinate positions such that
$n_{2^{m-1}}(2^{m-1}u)$ is even if $m=2.$ Let
$B=\\{G_{1},\dots,G_{k_{1}+k_{2}+\cdots+k_{m}}\\}.$ The following process
yields a generator matrix $\widetilde{G}$ of the $\mathbb{Z}_{2^{m}}$-code
$\widetilde{C}$ obtained from $C$ and $2^{m-1}u$ by the doubling method.

  1. Step 1: Let $B_{E}=\\{G_{i}\in B\,|\,\left\langle G_{i},2^{m-1}u\right\rangle=0\\}$ and $B_{O}=B\setminus B_{E}.$

  2. Step 2: Pick $G_{i}\in B_{O}$ arbitrarily. Define $B_{O}^{\prime}=\\{G_{i}+G_{j}\,|\,G_{j}\in B_{O}\\}.$

  3. Step 3: Let $\widetilde{G}$ be a matrix whose rows are the elements of the set $B_{O}^{\prime}\cup B_{E}\cup\\{2^{m-1}u\\}.$

The resultant code $\widetilde{C}$ is of type

$\left\\{\begin{array}[]{cc}(k_{1}-1,k_{2}+2),&\text{if}\ m=2,\\\
(k_{1}-1,k_{2}+1,k_{3}+1),&\text{if}\ m=3,\\\
(k_{1}-1,k_{2}+1,k_{3},\dots,k_{m-1},k_{m}+1),&\text{if}\ m\geq
4.\end{array}\right.$

The code $\widetilde{C}$ is independent of the choice of $G_{i}$ in Step 2.

###### Proof.

The set $B_{O}$ is not empty because $C$ is self-dual and $2^{m-1}u\notin C.$
Further, for all $t=k_{1}+1,\dots,k_{1}+k_{2}+\cdots+k_{m}$ it follows
$G_{t}\in B_{E}$. For $G_{i},G_{j}\in B_{O}$, we have

$\left\langle G_{i}+G_{j},2^{m-1}u\right\rangle=\left\langle
G_{i},2^{m-1}u\right\rangle+\left\langle G_{j},2^{m-1}u\right\rangle=0,$

and $G_{i}+G_{j}$ is an codeword with all even coordinates if and only if
$i=j$.

Note that $\left\langle B_{O}^{\prime}\cup B_{E}\right\rangle=\\{v\in
C\,|\,\left\langle 2^{m-1}u,v\right\rangle=0\\}$. It follows that
$\widetilde{C}$ is of type

$\left\\{\begin{array}[]{cc}(k_{1}-1,k_{2}+2),&\text{if}\ m=2,\\\
(k_{1}-1,k_{2}+1,k_{3}+1),&\text{if}\ m=3,\\\
(k_{1}-1,k_{2}+1,k_{3},\dots,k_{m-1},k_{m}+1),&\text{if}\ m\geq
4.\end{array}\right.$

The independence follows from the fact that

$G_{k}+G_{j}=(G_{i}+G_{k})+(G_{i}+G_{j})+(2^{m-1}-1)(G_{i}+G_{i}).$

∎

##  4 Construction of extremal Type II $\mathbb{Z}_{8}$-codes

Here we consider an extremal Type II $\mathbb{Z}_{8}$-code $C$ of length
$n\in\\{24,32,40\\}$ and type $(\frac{n}{2},0,0)$ which has an extremal
residue code $C^{(4)}$. Using the doubling method given in the previous
chapter, we developed an algorithm for a construction of extremal Type II
$\mathbb{Z}_{8}$-codes $\widetilde{C}$ of length $n$ and type
$(\frac{n}{2}-1,1,1).$

Note that
$wt_{E}(x)=n_{1}(x)+n_{7}(x)+4(n_{2}(x)+n_{6}(x))+9(n_{3}(x)+n_{5}(x))+16n_{4}(x),$
for $x\in\mathbb{Z}_{8}^{n}.$

######  Theorem 4.1.

Let $n\in\\{24,32,40\\}$. Denote by $S_{i}(w)$ the set of positions with the
element $i\in\mathbb{Z}_{8}$ in $w\in\mathbb{Z}_{8}^{n}.$ Let $C$ be an
extremal Type II $\mathbb{Z}_{8}$-code of length $n$ and type
$(k_{1},k_{2},k_{3})$ where $C^{(4)}$ is extremal. Suppose
$4u\in\mathbb{Z}_{8}^{n}$ is a codeword with all coordinates equal to $0$ or
$4$ such that $S_{4}(4u)\subseteq\\{k_{1}+k_{2}+k_{3}+1,\dots,n\\},$ where
$|S_{4}(4u)|\geq 2.$ If there is no codeword $v$ of $C$ that satisfies any of
the following conditions:

  1. 1.

$S_{3}(v)\cup S_{4}(v)\cup S_{5}(v)\subseteq S_{4}(4u)\subseteq S_{2}(v)\cup
S_{3}(v)\cup S_{4}(v)\cup S_{5}(v)\cup S_{6}(v)$ and

$wt_{E}(v\ (\text{mod}\ 4))=16,$

  2. 2.

$|S_{4}(4u)\setminus S_{4}(v)|+|S_{4}(v)\setminus S_{4}(4u)|=1$ and $wt_{E}(v\
(\text{mod}\ 4))=0,$

then the Type II $\mathbb{Z}_{8}$-code $\widetilde{C}$ generated by $4u$ and
$C$ using the doubling method is extremal. These choices of $4u$ are the only
candidates for the code $C$ in the doubling method which lead to an extremal
code.

###### Proof.

It follows from Theorem 3.2 that the choices of $4u$ can be limited to
codewords with zeroes on the first $k_{1}+k_{2}+k_{3}$ coordinates. Let us
assume that the code $\widetilde{C}$ is not extremal. Then it contains a
codeword of Euclidean weight $16$ of the form $w=v+4u,$ where $v\in C$ is such
a codeword that $\left\langle v,4u\right\rangle=0,$ and

$wt_{E}(w)=W+8(n_{3}(w)+n_{5}(w))+16n_{4}(w),$

where $W=n_{1}(w)+n_{7}(w)+n_{3}(w)+n_{5}(w)+4(n_{2}(w)+n_{6}(w)).$ It holds
$W=0$ or $W\geq 16,$ since $C^{(4)}$ is extremal. There are three cases to
consider.
Case 1: $W>16.$
Then $wt_{E}(w)\geq 32.$
Case 2: $W=16.$
For $wt_{E}(w)$ to be equal to $16,$ $n_{3}(w)=n_{4}(w)=n_{5}(w)=0,$ which is
impossible because of the first condition.
Case 3: $W=0.$
This condition implies that $w$ and $v$ have all coordinates equal to $0$ or
$4.$ Then, for $wt_{E}(w)=16n_{4}(w)$ to be equal to $16,$ $n_{4}(w)=1,$ which
is impossible because of the second condition.

The resulting choices for $4u$ are the only candidates for the code $C$ in the
doubling method, since the conditions of the theorem exclude all choices that
lead to a code $\widetilde{C}$ which is not extremal. ∎

For an extremal Type II $\mathbb{Z}_{8}$-code $C$ of length
$n\in\\{24,32,40\\}$ and type $(\frac{n}{2},0,0)$ which has an extremal
residue code $C^{(4)},$ the next algorithm returns all unsuitable candidates
$4u$, i.e., the candidates for which the application of the doubling method
leads to a Type II $\mathbb{Z}_{8}$-code $\widetilde{C}$ which is not
extremal. Thus, performing the given steps will find all possible candidates
$4u$ for code $C$ to produce a new extremal Type II $\mathbb{Z}_{8}$-code
$\widetilde{C}$ by the doubling method.

Algorithm C
Let $n\in\\{24,32,40\\}.$ Denote by $S_{i}(w)$ the set of positions with the
element $i\in\mathbb{Z}_{8}$ in $w\in\mathbb{Z}_{8}^{n}.$ Let $C$ be an
extremal Type II $\mathbb{Z}_{8}$-code of length $n$ and type
$(\frac{n}{2},0,0),$ where $C^{(4)}$ is extremal, with the generator matrix
$G=\left[\begin{tabular}[]{cc}$I_{\frac{n}{2}}$&$A$\end{tabular}\right]$ in
the standard form.

  * 1.

Let $v=(v_{1},\dots,v_{n})\in C^{(4)}$ be a codeword of Euclidean weight $16.$

  * 1.2.

Let $F_{v}=\\{v_{1},\dots,v_{\frac{n}{2}}\\},\ A_{v}=S_{2}(v)\cap F_{v}$ and
$B_{v}=S_{3}(v)\cap F_{v}.$

  * 1.3.

Repeat the following steps on all $A\subseteq A_{v}:$

    * 1.3.1.

Calculate $v^{\prime}=v+4s_{A}+4s_{B_{v}},$ where $s_{A}$ is the sum of rows
in the generator matrix $G$ of $C$ with row indices in $A$ and $s_{B_{v}}$ is
the sum of rows in $G$ with row indices in $B_{v}.$

    * 1.3.2.

Let

$O_{v^{\prime}}=(S_{2}(v^{\prime})\cup
S_{6}(v^{\prime}))\cap\left\\{\frac{n}{2}+1,\dots,n\right\\},$
$P_{v^{\prime}}=S_{4}(v^{\prime})\cap\left\\{\frac{n}{2}+1,\dots,n\right\\},$
$Q_{v^{\prime}}=(S_{3}(v^{\prime})\cup
S_{5}(v^{\prime}))\cap\left\\{\frac{n}{2}+1,\dots,n\right\\}.$

    * 1.3.3.

Let $\mathcal{B}$ be the collection of all sets

$B=O\cup P_{v^{\prime}}\cup Q_{v^{\prime}},\ O\subseteq O_{v^{\prime}},$

where $|B|\geq 2.$

  * 2.

For all $i\in\\{1,\dots,\frac{n}{2}\\},$ do the following.

    * 2.1.

Let $O_{i}=S_{4}(4G_{i})\cap\left\\{\frac{n}{2}+1,\dots,n\right\\},$ where
$G_{i}$ is the $i^{th}$ row of $G.$

    * 2.2.

Include all $O_{i}$ such that $|O_{i}|\geq 2$ in $\mathcal{B}.$

Our method of construction is based on the following theorem.

######  Theorem 4.2.

Let $n\in\\{24,32,40\\}.$ Denote by $S_{i}(w)$ the set of positions with the
element $i\in\mathbb{Z}_{8}$ in $w\in\mathbb{Z}_{8}^{n}.$ Let $C$ be an
extremal Type II $\mathbb{Z}_{8}$-code of length $n$ and type
$(\frac{n}{2},0,0),$ where $C^{(4)}$ is extremal. Furthermore, let
$\mathcal{S}$ be the collection of all
$S\subseteq\left\\{\frac{n}{2}+1,\dots,n\right\\}$ such that $|S|\geq 2.$ Then
$\mathcal{G}=\mathcal{S}\setminus\mathcal{B}$ is the set of all possible
$S_{4}(4u)$ for the code $C$ in the doubling method which lead to an extremal
Type II $\mathbb{Z}_{8}$-code $\widetilde{C}$ of length $n$ and type
$(\frac{n}{2}-1,1,1),$ where $\mathcal{B}$ is the set obtained by applying
Algorithm C.

###### Proof.

The first condition in Theorem 4.1 is checked in Step 1. of Algorithm C. Since
the condition requires that $S_{3}(v)\cup S_{4}(v)\cup S_{5}(v)\subseteq
S_{4}(4u),$ the coefficients of the rows of $G$ in the linear combination of
$v$ cannot be $3,4$ or $5.$ Step 1.3.1. generates all such codewords
$v^{\prime}$ with $wt_{E}(v^{\prime}\ (\text{mod}\ 4))=16.$ All subsets
$B=S_{4}(4u)$ satisfying the first condition are included in $\mathcal{B}$ in
Step 1.3.3.

The second condition of Theorem 4.1, implies that $v$ is one of the rows of
$G$ with coefficient $4.$ These codewords are considered in Step 2.

∎

###  4.1 New extremal Type II $\mathbb{Z}_{8}$-codes of length $32$

Six inequivalent extremal Type II $\mathbb{Z}_{8}$-codes of length $32$ are
known: $C_{8,32,i},\ i=1,\dots,5$ from [15] and $D_{32}$ from [13]. Since all
of them are of type $(16,0,0)$, we investigate the possibility of constructing
new extremal Type II $\mathbb{Z}_{8}$-codes of length $32$ by using the
introduced doubling method. The result of our analysis is given in Proposition
4.3.

######  Proposition 4.3.

There are at least $10$ inequivalent extremal Type II $\mathbb{Z}_{8}$-codes
of length $32$ and type $(15,1,1).$

###### Proof.

The extremal Type II $\mathbb{Z}_{8}$-codes $C_{8,32,1}$ and $C_{8,32,2}$ are
of type $(16,0,0)$ and length $32$ and have extremal $\mathbb{Z}_{4}$-residue
codes. So, we can apply Theorem 4.2. We applied Algorithm C and found $23067$
candidates $4u$ for a construction of extremal Type II $\mathbb{Z}_{8}$-codes
of type $(15,1,1)$ and length $32$ from $C_{8,32,1}$ by doubling method. Also,
we found $22818$ candidates $4u$ for a construction of extremal Type II
$\mathbb{Z}_{8}$-codes of type $(15,1,1)$ and length $32$ from $C_{8,32,2}$ by
doubling method.

The extremal Type II $\mathbb{Z}_{8}$-codes $C_{8,32,i},\ i=3,4,5$ are of type
$(16,0,0)$ and length $32$ and have $\mathbb{Z}_{4}$-residue codes of minimum
Euclidean weight $8.$ So, we cannot apply Theorem 4.2 to obtain new extremal
Type II $\mathbb{Z}_{8}$-codes from $C_{8,32,i},\ i=3,4,5$ using the doubling
method.

Further, the extremal Type II $\mathbb{Z}_{8}$-code $D_{32}$ is of type
$(16,0,0)$ and length $32$ and it has an extremal $\mathbb{Z}_{4}$-residue
code. So, we can apply Theorem 4.2. We applied Algorithm C and found $22965$
candidates $4u$ for a construction of extremal Type II $\mathbb{Z}_{8}$-codes
of type $(15,1,1)$ and length $32$ from $D_{32}$ by doubling method.

We use Theorem 3.3 to obtain the generator matrices for the $68850$
constructed extremal Type II $\mathbb{Z}_{8}$-codes of type $(15,1,1)$ and
length $32.$ Using Magma ([6]), we calculated the weight distributions of the
corresponding $68850$ binary residue codes and obtained that, with respect to
the weight distribution of their binary residue codes, all constructed
extremal Type II $\mathbb{Z}_{8}$-codes are distributed into $10$ classes. The
corresponding weight distributions are given in Table 1.

| $i$ | 0 | 8 | 12 | 16 | 20 | 24 | 32
---|---|---|---|---|---|---|---|---
${C_{1}}^{(2)}$ | $W_{i}$ | 1 | 316 | 6912 | 18310 | 6912 | 316 | 1
${C_{2}}^{(2)}$ | $W_{i}$ | 1 | 332 | 6848 | 18406 | 6848 | 332 | 1
${C_{3}}^{(2)}$ | $W_{i}$ | 1 | 337 | 6888 | 18259 | 7000 | 283 | 0
${C_{4}}^{(2)}$ | $W_{i}$ | 1 | 305 | 6952 | 18259 | 6936 | 315 | 0
${C_{5}}^{(2)}$ | $W_{i}$ | 1 | 308 | 6944 | 18262 | 6944 | 308 | 1
${C_{6}}^{(2)}$ | $W_{i}$ | 1 | 300 | 6976 | 18214 | 6976 | 300 | 1
${C_{7}}^{(2)}$ | $W_{i}$ | 1 | 364 | 6720 | 18598 | 6720 | 364 | 1
${C_{8}}^{(2)}$ | $W_{i}$ | 1 | 380 | 7168 | 17670 | 7168 | 380 | 1
${C_{9}}^{(2)}$ | $W_{i}$ | 1 | 324 | 6880 | 18358 | 6880 | 324 | 1
${C_{10}}^{(2)}$ | $W_{i}$ | 1 | 340 | 6816 | 18454 | 6816 | 340 | 1
Table 1: Weight distributions of the binary residue codes

For each of the obtained weight distribution classes we give the generator
matrix in standard form of one extremal Type II $\mathbb{Z}_{8}$-code of
length $32$ and type $(15,1,1)$, namely, the generator matrices for the
following class representatives:

$C_{1}={C_{8,32,1}}_{0}\oplus\left\langle 4u\right\rangle,\
S_{4}(4u)=\\{17,19,21,22\\},$ $C_{2}={C_{8,32,1}}_{0}\oplus\left\langle
4u\right\rangle,\ S_{4}(4u)=\\{17,18,20,21\\},$
$C_{3}={C_{8,32,1}}_{0}\oplus\left\langle 4u\right\rangle,\
S_{4}(4u)=\\{17,19,21\\},$ $C_{4}={C_{8,32,1}}_{0}\oplus\left\langle
4u\right\rangle,\ S_{4}(4u)=\\{17,19,20,21,22\\},$
$C_{5}={C_{8,32,1}}_{0}\oplus\left\langle 4u\right\rangle,\
S_{4}(4u)=\\{17,18,19,20\\},$ $C_{6}={C_{8,32,1}}_{0}\oplus\left\langle
4u\right\rangle,\ S_{4}(4u)=\\{17,18,19,20,21,22\\},$
$C_{7}={C_{8,32,1}}_{0}\oplus\left\langle 4u\right\rangle,\
S_{4}(4u)=\\{18,24,25,27\\},$ $C_{8}={C_{8,32,1}}_{0}\oplus\left\langle
4u\right\rangle,\ S_{4}(4u)=\\{20,25\\},$
$C_{9}={C_{8,32,2}}_{0}\oplus\left\langle 4u\right\rangle,\
S_{4}(4u)=\\{17,18,19,21\\},$ $C_{10}={C_{8,32,2}}_{0}\oplus\left\langle
4u\right\rangle,\ S_{4}(4u)=\\{17,21,24,25\\}.$

Those are, respectively:

$G_{1}=\left(\begin{array}[]{c}10000000000000003476716356020474\\\
01000000000000001703275645602047\\\ 00100000000000002574363514560204\\\
00010000000000010615011726731415\\\ 00001000000000001365303604145602\\\
00000100000000013130517745777155\\\ 00000010000000012611632063324043\\\
00000001000000011163700305167532\\\ 00000000100000003474020675235254\\\
00000000010000002347402047523525\\\ 00000000001000000234740234752352\\\
00000000000100013461057440750622\\\ 00000000000010011200362013622110\\\
00000000000001011426617330117347\\\ 00000000000000100100242336536347\\\
00000000000000022004046042646062\\\
00000000000000004040440000000000\end{array}\right),\
G_{2}=\left(\begin{array}[]{c}10000000000000003032756356020474\\\
01000000000000011003437173330306\\\ 00100000000000002574363514560204\\\
00010000000000003257436341456020\\\ 00001000000000001721343604145602\\\
00000100000000002532574360414560\\\ 00000010000000001657657406041456\\\
00000001000000010665127256332404\\\ 00000000100000013130222123763513\\\
00000000010000002347402047523525\\\ 00000000001000010770142562400611\\\
00000000000100012163276311123574\\\ 00000000000010010346541764075062\\\
00000000000001000464634753634752\\\ 00000000000000111142225713011734\\\
00000000000000020200404664264606\\\ 00000000000000004404400000000000\\\
\end{array}\right),$
$G_{3}=\left(\begin{array}[]{c}10000000000000010034371733303061\\\
01000000000000012301250222165434\\\ 00100000000000002574363514560204\\\
00010000000000003257436341456020\\\ 00001000000000001365343604145602\\\
00000100000000002532574360414560\\\ 00000010000000001213657406041456\\\
00000001000000011163740305167532\\\ 00000000100000003474020675235254\\\
00000000010000002347402047523525\\\ 00000000001000000234740234752352\\\
00000000000100002063074063475235\\\ 00000000000010011200362013622110\\\
00000000000001000020634753634752\\\ 00000000000000100100202336536347\\\
00000000000000022004046042646062\\\ 00000000000000004040400000000000\\\
\end{array}\right),\
G_{4}=\left(\begin{array}[]{c}10000000000000003472716356020474\\\
01000000000000012305210222165434\\\ 00100000000000002574363514560204\\\
00010000000000003257436341456020\\\ 00001000000000012767326261420277\\\
00000100000000013130517745777155\\\ 00000010000000001217617406041456\\\
00000001000000000561765720604145\\\ 00000000100000003474020675235254\\\
00000000010000013745425424006112\\\ 00000000001000000234740234752352\\\
00000000000100002067034063475235\\\ 00000000000010011200362013622110\\\
00000000000001011422617330117347\\\ 00000000000000100104242336536347\\\
00000000000000022004046042646062\\\ 00000000000000004044440000000000\\\
\end{array}\right),$
$G_{5}=\left(\begin{array}[]{c}10000000000000003072356356020474\\\
01000000000000011003437173330306\\\ 00100000000000002574363514560204\\\
00010000000000013753230677104367\\\ 00001000000000011465545132673141\\\
00000100000000002532574360414560\\\ 00000010000000011313051734577715\\\
00000001000000000161325720604145\\\ 00000000100000003474020675235254\\\
00000000010000002347402047523525\\\ 00000000001000010730542562400611\\\
00000000000100012163276311123574\\\ 00000000000010000602347436347523\\\
00000000000001000424234753634752\\\ 00000000000000111102625713011734\\\
00000000000000020200404664264606\\\ 00000000000000004444000000000000\\\
\end{array}\right),\
G_{6}=\left(\begin{array}[]{c}10000000000000003072716356020474\\\
01000000000000001307275645602047\\\ 00100000000000012270525042216543\\\
00010000000000003257436341456020\\\ 00001000000000001761303604145602\\\
00000100000000002532574360414560\\\ 00000010000000001617617406041456\\\
00000001000000010665127256332404\\\ 00000000100000003474020675235254\\\
00000000010000002347402047523525\\\ 00000000001000000234740234752352\\\
00000000000100002467034063475235\\\ 00000000000010010306501764075062\\\
00000000000001010120036201362211\\\ 00000000000000111102265713011734\\\
00000000000000020200404664264606\\\ 00000000000000004444440000000000\\\
\end{array}\right),$
$G_{7}=\left(\begin{array}[]{c}10000000000000000736356436520474\\\
01000000000000011641650322265434\\\ 00100000000000001274363114560204\\\
00010000000000013455451236631415\\\ 00001000000000001525743364045602\\\
00000100000000000232574170214560\\\ 00000010000000000553257246041456\\\
00000001000000002425325130604145\\\ 00000000100000011472043462110641\\\
00000000010000001247402747023525\\\ 00000000001000003034740664752352\\\
00000000000100000623474443275235\\\ 00000000000010003002347646347523\\\
00000000000001002460234433134752\\\ 00000000000000101440602576736347\\\
00000000000000022204046442046062\\\ 00000000000000004000000440400000\\\
\end{array}\right),\
G_{8}=\left(\begin{array}[]{c}10000000000000010576150634556733\\\
01000000000000013007437113330306\\\ 00100000000000010630165062216543\\\
00010000000000013313230677104367\\\ 00001000000000013465545112673141\\\
00000100000000002536574320414560\\\ 00000010000000013313051714577715\\\
00000001000000011661127206332404\\\ 00000000100000012534622173763513\\\
00000000010000013403204365251064\\\ 00000000001000012370542542400611\\\
00000000000100011163276321123574\\\ 00000000000010012746141744075062\\\
00000000000001010120036201362211\\\ 00000000000000102402023415363475\\\
00000000000000022204404604264606\\\ 00000000000000004004000000000000\\\
\end{array}\right),$
$G_{9}=\left(\begin{array}[]{c}10000000000000003076052565220712\\\
01000000000000003743645266522071\\\ 00100000000000010566173551245033\\\
00010000000000003637436417665220\\\ 00001000000000000723343601766522\\\
00000100000000002436374360176652\\\ 00000010000000013075046441402411\\\
00000001000000000164763736601766\\\ 00000000100000010041231556227000\\\
00000000010000003221702247363425\\\ 00000000001000002762570234736342\\\
00000000000100002350263252546347\\\ 00000000000010000663221746347363\\\
00000000000001001066322154634736\\\ 00000000000000103546232225463473\\\
00000000000000020024026046066460\\\ 00000000000000004440400000000000\\\
\end{array}\right),\
G_{10}=\left(\begin{array}[]{c}10000000000000003436052125220712\\\
01000000000000011653420430260330\\\ 00100000000000010204547300310546\\\
00010000000000003637436417665220\\\ 00001000000000012273526053424061\\\
00000100000000002436374360176652\\\ 00000010000000013153412630555124\\\
00000001000000012434146100347225\\\ 00000000100000002217022573634254\\\
00000000010000003221702247363425\\\ 00000000001000002322570674736342\\\
00000000000100002632617423473634\\\ 00000000000010012573404550005622\\\
00000000000001001066322154634736\\\ 00000000000000111016015037121732\\\
00000000000000020620046064204606\\\ 00000000000000004000400440000000\\\
\end{array}\right).$

So, we obatined at least $10$ new inequivalent extremal Type II
$\mathbb{Z}_{8}$-codes of length $32.$

∎

Proposition 4.3, together with results from [13] and [15], yields the
following theorem.

######  Theorem 4.4.

There are at least $16$ inequivalent extremal Type II $\mathbb{Z}_{8}$-codes
of length $32.$

######  Remark 4.5.

According to [12], binary $[32,15,8]$ codes are the optimal binary $[32,15]$
codes. Therefore, all constructed extremal Type II $\mathbb{Z}_{8}$-codes of
length $32$ have optimal binary residue codes.

Acknowledgement

This work has been supported by Croatian Science Foundation under the project
4571 and by the University of Rijeka under the Project uniri-iskusni-
prirod-23-62-3007.

## References

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2458498.67826987 | 20.60608 | 8.99 | 380
2458498.68327102 | 20.61570 | 8.97 | 380
2458498.68819888 | 20.61782 | 7.81 | 380
2458498.69307304 | 20.62464 | 7.45 | 380
2458498.70007611 | 20.61316 | 4.57 | 760
2458498.70964022 | 20.60040 | 4.56 | 760
2458498.71876645 | 20.59894 | 4.35 | 760
2458498.72840475 | 20.56458 | 4.39 | 760
2458498.73782029 | 20.55429 | 3.99 | 800
Table 8: The pipeline calculated RV values for observations in DS1, as well as their uncertainties and exposure times used. BJD [days] | RV [km/s] |  $\sigma_{\textrm{RV}}$ [m/s] | EXP [s]
---|---|---|---
2458546.70029843 | 20.81404 | 5.42 | 1200
2458546.71730722 | 20.77644 | 8.64 | 1200
2458546.73022202 | 20.74880 | 9.11 | 1200
2458546.74635059 | 20.74357 | 7.49 | 1200
2458546.75597151 | 20.74175 | 17.71 | 410
2458546.76164054 | 20.70465 | 18.3 | 410
2458546.76693892 | 20.74182 | 20.89 | 410
2458546.77253972 | 20.66661 | 15.99 | 410
2458546.77779036 | 20.65585 | 15.02 | 410
2458546.78315677 | 20.66390 | 15.14 | 410
2458546.78829173 | 20.57827 | 20.02 | 410
2458546.79395184 | 20.59739 | 18.90 | 410
2458546.79956324 | 20.55623 | 20.08 | 410
2458546.80473385 | 20.63033 | 17.92 | 410
2458546.81027239 | 20.62332 | 17.76 | 410
2458546.81558596 | 20.59751 | 16.65 | 410
2458546.82247144 | 20.61843 | 15.00 | 670
2458546.83106347 | 20.50721 | 13.37 | 670
2458546.83938853 | 20.57332 | 11.28 | 670
2458546.84775486 | 20.55209 | 15.56 | 670
2458546.85650149 | 20.51568 | 13.58 | 670
Table 9: Same as Table 8 but for DS2. BJD [days] | RV [km/s] |  $\sigma_{\textrm{RV}}$ [m/s] | EXP [s]
---|---|---|---
2458565.62926856 | 20.85050 | 4.19 | 830
2458565.63970318 | 20.83263 | 4.15 | 830
2458565.64988114 | 20.81295 | 3.92 | 830
2458565.66009538 | 20.79194 | 4.31 | 830
2458565.67041463 | 20.76472 | 4.20 | 830
2458565.68037855 | 20.77662 | 3.64 | 830
2458565.68812129 | 20.78292 | 8.50 | 380
2458565.69318121 | 20.76429 | 8.68 | 380
2458565.69819600 | 20.74261 | 8.36 | 380
2458565.70319448 | 20.72989 | 8.13 | 380
2458565.70815185 | 20.68067 | 8.44 | 380
2458565.71324232 | 20.66203 | 8.96 | 380
2458565.71823768 | 20.64335 | 8.30 | 380
2458565.72317414 | 20.61127 | 8.58 | 380
2458565.72822643 | 20.61737 | 8.65 | 380
2458565.73323846 | 20.61288 | 7.69 | 380
2458565.73821655 | 20.62037 | 7.40 | 380
2458565.74326025 | 20.61360 | 7.33 | 380
2458565.75020172 | 20.60935 | 3.91 | 720
2458565.75912765 | 20.59612 | 3.89 | 720
2458565.76806310 | 20.57339 | 3.99 | 720
2458565.77696335 | 20.56269 | 4.03 | 720
2458565.78279489 | 20.54291 | 15.37 | 180.0
Table 10: Same as Table 8 but for DS3. BJD [days] | RV [km/s] |  $\sigma_{\textrm{RV}}$ [m/s] | EXP [s]
---|---|---|---
2458860.65788891 | 20.80633 | 2.47 | 900
2458860.66873710 | 20.78622 | 2.19 | 900
2458860.68006353 | 20.77083 | 2.24 | 900
2458860.69152885 | 20.75252 | 2.05 | 900
2458860.70249466 | 20.73853 | 2.01 | 900
2458860.71155150 | 20.76021 | 3.06 | 380
2458860.71684916 | 20.75988 | 3.29 | 380
2458860.72216158 | 20.74915 | 2.98 | 380
2458860.72762300 | 20.72062 | 2.96 | 380
2458860.73294862 | 20.68917 | 3.05 | 380
2458860.73822609 | 20.65960 | 2.89 | 380
2458860.74354597 | 20.62669 | 2.89 | 380
2458860.74880799 | 20.60702 | 3.00 | 380
2458860.75419957 | 20.59131 | 2.93 | 380
2458860.75953740 | 20.59485 | 2.85 | 380
2458860.76491886 | 20.60629 | 2.80 | 380
2458860.77016483 | 20.60589 | 2.88 | 380
2458860.77767389 | 20.59486 | 1.95 | 750
2458860.78728462 | 20.57681 | 2.00 | 750
2458860.79690229 | 20.56175 | 2.08 | 750
2458860.80650307 | 20.54011 | 2.13 | 750
2458860.81598265 | 20.52403 | 2.11 | 750
Table 11: Same as Table 8 but for DS4.

##  Appendix B RM fit posteriors

Figure 16: Posterior probability distributions for the parameters of the RM
model fit to the RV variations measured for DS4. The vertical gray dashed
lines in the diagonal plots indicate the mean and $\pm 1\sigma$ levels of the
distributions and the red line shows the maximum of the fitted log-normal
models (blue curves).

##  Appendix C CRM RV values

Table 12: Radial velocity values calculated for the purpose of CRM analysis from DS4. Column headers indicate the centres of the wavelength bins, each created from 5 spectra orders. The data is plotted in Fig. 13, where the time stamps are the same as those given in table 11. 3852 Å | 3977 Å | 4111 Å | 4254 Å | 4407 Å | 4571 Å | 4748 Å | 4940 Å
---|---|---|---|---|---|---|---
$[$km/s$]$ | [km/s] | [km/s] | [km/s] | [km/s] | [km/s] | [km/s] | [km/s]
$21.160\pm 0.041$ | $21.013\pm 0.017$ | $20.949\pm 0.008$ | $20.960\pm 0.008$ | $21.079\pm 0.010$ | $21.084\pm 0.009$ | $21.004\pm 0.006$ | $21.025\pm 0.006$
$21.114\pm 0.041$ | $20.968\pm 0.015$ | $20.928\pm 0.012$ | $20.946\pm 0.006$ | $21.049\pm 0.011$ | $21.075\pm 0.012$ | $21.000\pm 0.006$ | $20.999\pm 0.004$
$21.211\pm 0.022$ | $20.951\pm 0.013$ | $20.909\pm 0.008$ | $20.918\pm 0.008$ | $21.019\pm 0.010$ | $21.067\pm 0.010$ | $20.979\pm 0.008$ | $20.982\pm 0.006$
$21.091\pm 0.030$ | $20.914\pm 0.014$ | $20.882\pm 0.007$ | $20.902\pm 0.007$ | $21.026\pm 0.010$ | $21.048\pm 0.012$ | $20.953\pm 0.008$ | $20.970\pm 0.005$
$21.030\pm 0.033$ | $20.915\pm 0.010$ | $20.882\pm 0.007$ | $20.897\pm 0.006$ | $21.006\pm 0.011$ | $21.026\pm 0.010$ | $20.936\pm 0.007$ | $20.950\pm 0.006$
$21.166\pm 0.030$ | $20.947\pm 0.023$ | $20.881\pm 0.014$ | $20.915\pm 0.009$ | $21.012\pm 0.012$ | $21.058\pm 0.013$ | $20.962\pm 0.007$ | $20.970\pm 0.007$
$21.074\pm 0.045$ | $20.945\pm 0.025$ | $20.905\pm 0.014$ | $20.914\pm 0.011$ | $21.022\pm 0.009$ | $21.050\pm 0.016$ | $20.959\pm 0.009$ | $20.973\pm 0.007$
$21.169\pm 0.048$ | $20.948\pm 0.018$ | $20.871\pm 0.014$ | $20.951\pm 0.009$ | $21.001\pm 0.011$ | $21.029\pm 0.011$ | $20.931\pm 0.007$ | $20.960\pm 0.008$
$21.069\pm 0.047$ | $20.905\pm 0.013$ | $20.858\pm 0.011$ | $20.870\pm 0.008$ | $20.996\pm 0.010$ | $21.021\pm 0.009$ | $20.921\pm 0.008$ | $20.944\pm 0.008$
$21.089\pm 0.053$ | $20.886\pm 0.022$ | $20.831\pm 0.013$ | $20.824\pm 0.009$ | $20.945\pm 0.010$ | $20.991\pm 0.011$ | $20.875\pm 0.007$ | $20.915\pm 0.007$
$21.044\pm 0.062$ | $20.822\pm 0.020$ | $20.793\pm 0.011$ | $20.813\pm 0.009$ | $20.926\pm 0.013$ | $20.957\pm 0.014$ | $20.855\pm 0.008$ | $20.884\pm 0.006$
$20.977\pm 0.036$ | $20.775\pm 0.021$ | $20.756\pm 0.012$ | $20.773\pm 0.012$ | $20.891\pm 0.013$ | $20.928\pm 0.014$ | $20.810\pm 0.007$ | $20.845\pm 0.008$
$20.943\pm 0.049$ | $20.818\pm 0.023$ | $20.749\pm 0.014$ | $20.782\pm 0.010$ | $20.866\pm 0.014$ | $20.889\pm 0.013$ | $20.792\pm 0.008$ | $20.814\pm 0.007$
$21.002\pm 0.053$ | $20.823\pm 0.022$ | $20.714\pm 0.012$ | $20.768\pm 0.008$ | $20.845\pm 0.011$ | $20.873\pm 0.008$ | $20.768\pm 0.008$ | $20.812\pm 0.007$
$20.967\pm 0.044$ | $20.785\pm 0.021$ | $20.752\pm 0.014$ | $20.738\pm 0.011$ | $20.874\pm 0.011$ | $20.894\pm 0.013$ | $20.795\pm 0.006$ | $20.813\pm 0.007$
$20.917\pm 0.052$ | $20.810\pm 0.021$ | $20.748\pm 0.013$ | $20.768\pm 0.010$ | $20.896\pm 0.013$ | $20.889\pm 0.013$ | $20.793\pm 0.008$ | $20.812\pm 0.004$
$20.968\pm 0.050$ | $20.762\pm 0.016$ | $20.743\pm 0.010$ | $20.757\pm 0.009$ | $20.878\pm 0.010$ | $20.906\pm 0.008$ | $20.807\pm 0.006$ | $20.825\pm 0.007$
$20.930\pm 0.037$ | $20.801\pm 0.015$ | $20.738\pm 0.009$ | $20.765\pm 0.006$ | $20.864\pm 0.010$ | $20.877\pm 0.009$ | $20.788\pm 0.004$ | $20.810\pm 0.006$
$20.871\pm 0.028$ | $20.759\pm 0.014$ | $20.717\pm 0.008$ | $20.742\pm 0.007$ | $20.852\pm 0.011$ | $20.861\pm 0.010$ | $20.777\pm 0.006$ | $20.804\pm 0.006$
$20.921\pm 0.027$ | $20.758\pm 0.014$ | $20.706\pm 0.008$ | $20.737\pm 0.006$ | $20.831\pm 0.010$ | $20.848\pm 0.010$ | $20.769\pm 0.006$ | $20.775\pm 0.004$
$20.928\pm 0.029$ | $20.716\pm 0.015$ | $20.682\pm 0.011$ | $20.702\pm 0.008$ | $20.789\pm 0.010$ | $20.823\pm 0.011$ | $20.745\pm 0.005$ | $20.750\pm 0.006$
$20.839\pm 0.036$ | $20.701\pm 0.014$ | $20.679\pm 0.011$ | $20.687\pm 0.009$ | $20.793\pm 0.011$ | $20.807\pm 0.012$ | $20.722\pm 0.006$ | $20.741\pm 0.005$
5148 Å | 5329 Å | 5571 Å | 5848 Å | 6190 Å | 6563 Å | 7389 Å |
$[$km/s$]$ | [km/s] | [km/s] | [km/s] | [km/s] | [km/s] | [km/s] |
$20.943\pm 0.007$ | $21.013\pm 0.007$ | $20.960\pm 0.010$ | $21.006\pm 0.007$ | $21.026\pm 0.009$ | $20.960\pm 0.008$ | $21.096\pm 0.015$ |
$20.925\pm 0.007$ | $20.993\pm 0.007$ | $20.937\pm 0.026$ | $20.967\pm 0.010$ | $21.022\pm 0.007$ | $20.951\pm 0.009$ | $21.101\pm 0.007$ |
$20.915\pm 0.006$ | $20.980\pm 0.006$ | $20.925\pm 0.009$ | $20.968\pm 0.007$ | $21.014\pm 0.008$ | $20.937\pm 0.010$ | $21.047\pm 0.008$ |
$20.906\pm 0.006$ | $20.965\pm 0.005$ | $20.908\pm 0.008$ | $20.948\pm 0.009$ | $20.969\pm 0.007$ | $20.925\pm 0.010$ | $21.048\pm 0.009$ |
$20.877\pm 0.006$ | $20.961\pm 0.006$ | $20.904\pm 0.007$ | $20.925\pm 0.010$ | $20.966\pm 0.007$ | $20.907\pm 0.008$ | $21.026\pm 0.011$ |
$20.896\pm 0.011$ | $20.978\pm 0.010$ | $20.936\pm 0.009$ | $20.957\pm 0.013$ | $20.990\pm 0.009$ | $20.930\pm 0.011$ | $21.062\pm 0.012$ |
$20.897\pm 0.008$ | $20.991\pm 0.011$ | $20.911\pm 0.009$ | $20.930\pm 0.013$ | $20.984\pm 0.010$ | $20.933\pm 0.016$ | $21.085\pm 0.016$ |
$20.893\pm 0.008$ | $20.970\pm 0.010$ | $20.914\pm 0.013$ | $20.949\pm 0.012$ | $20.966\pm 0.008$ | $20.908\pm 0.012$ | $21.032\pm 0.013$ |
$20.855\pm 0.007$ | $20.940\pm 0.009$ | $20.874\pm 0.010$ | $20.899\pm 0.012$ | $20.936\pm 0.007$ | $20.883\pm 0.012$ | $21.009\pm 0.015$ |
$20.829\pm 0.007$ | $20.899\pm 0.008$ | $20.852\pm 0.009$ | $20.905\pm 0.012$ | $20.918\pm 0.008$ | $20.837\pm 0.016$ | $20.972\pm 0.012$ |
$20.800\pm 0.006$ | $20.849\pm 0.011$ | $20.805\pm 0.008$ | $20.827\pm 0.010$ | $20.886\pm 0.010$ | $20.785\pm 0.011$ | $20.931\pm 0.016$ |
$20.746\pm 0.007$ | $20.844\pm 0.008$ | $20.780\pm 0.009$ | $20.798\pm 0.012$ | $20.874\pm 0.010$ | $20.821\pm 0.010$ | $20.923\pm 0.016$ |
$20.729\pm 0.008$ | $20.820\pm 0.009$ | $20.753\pm 0.008$ | $20.776\pm 0.013$ | $20.819\pm 0.007$ | $20.744\pm 0.016$ | $20.888\pm 0.010$ |
$20.731\pm 0.007$ | $20.788\pm 0.008$ | $20.727\pm 0.009$ | $20.759\pm 0.010$ | $20.806\pm 0.010$ | $20.750\pm 0.016$ | $20.874\pm 0.012$ |
$20.712\pm 0.007$ | $20.810\pm 0.011$ | $20.750\pm 0.012$ | $20.791\pm 0.013$ | $20.824\pm 0.007$ | $20.742\pm 0.011$ | $20.862\pm 0.014$ |
$20.746\pm 0.006$ | $20.812\pm 0.009$ | $20.764\pm 0.010$ | $20.792\pm 0.009$ | $20.820\pm 0.010$ | $20.780\pm 0.011$ | $20.912\pm 0.016$ |
$20.749\pm 0.007$ | $20.822\pm 0.010$ | $20.760\pm 0.010$ | $20.802\pm 0.010$ | $20.799\pm 0.010$ | $20.784\pm 0.013$ | $20.888\pm 0.014$ |
$20.732\pm 0.007$ | $20.806\pm 0.008$ | $20.745\pm 0.008$ | $20.777\pm 0.008$ | $20.814\pm 0.007$ | $20.760\pm 0.010$ | $20.846\pm 0.012$ |
$20.712\pm 0.006$ | $20.788\pm 0.007$ | $20.721\pm 0.008$ | $20.762\pm 0.010$ | $20.803\pm 0.008$ | $20.739\pm 0.008$ | $20.881\pm 0.012$ |
$20.691\pm 0.008$ | $20.756\pm 0.008$ | $20.726\pm 0.007$ | $20.745\pm 0.011$ | $20.795\pm 0.006$ | $20.721\pm 0.014$ | $20.879\pm 0.010$ |
$20.678\pm 0.005$ | $20.751\pm 0.007$ | $20.700\pm 0.008$ | $20.739\pm 0.006$ | $20.755\pm 0.005$ | $20.714\pm 0.010$ | $20.818\pm 0.015$ |
$20.659\pm 0.005$ | $20.730\pm 0.008$ | $20.688\pm 0.008$ | $20.711\pm 0.009$ | $20.759\pm 0.007$ | $20.691\pm 0.010$ | $20.845\pm 0.011$ |


# On the existence of distributional potentials

Jürgen Voigt

###### Abstract

We present proofs for the existence of distributional potentials
$F\in\mathcal{D}^{\prime}(\Omega)$ for distributional vector fields
$G\in\mathcal{D}^{\prime}(\Omega)^{n}$, i.e. $\operatorname{grad}F=G$, where
$\Omega$ is an open subset of $\mathbb{R}\nonscript\hskip 0.29999pt^{n}$. The
hypothesis in these proofs is the compatibility condition
$\partial_{j}G_{k}=\partial_{k}G_{j}$ for all $j,k\in\\{1,\dots,n\\}$, if
$\Omega$ is simply connected, and a stronger condition in the general case. A
key tool in our treatment is the Bogovskiĭ formula, assigning vector fields
$v\in\mathcal{D}(\Omega)^{n}$ satisfying $\operatorname{div}v=\varphi$ to
functions $\varphi\in\mathcal{D}(\Omega)$ with
$\int\varphi(x)\mathclose{}\,\mathrm{d}x=0$. The results are applied to
properties of Hilbert spaces of functions occurring in the treatment of the
Stokes operator and the Navier–Stokes equations.

MSC 2010: 46F10, 46E35

Keywords: Distribution, Poincaré’s lemma, de Rham’s theorem, Bogovskiĭ
formula, Stokes operator

## Introduction

The most elementary version of Poincaré’s lemma is the statement that, given a
$C^{1}$-vector field $v\colon\mathbb{R}\nonscript\hskip
0.29999pt^{n}\to\mathbb{K}\nonscript\hskip 0.29999pt^{n}$ satisfying
$\partial_{j}v_{k}=\partial_{k}v_{j}$ for all $j,k=1,\dots,n$, there exists a
potential $p\in C^{2}(\mathbb{R}\nonscript\hskip 0.29999pt^{n})$ for $v$, i.e.
$\operatorname{grad}p=v$. The main issue of this paper is to present proofs of
the following two distributional versions of this kind of existence theorem.

######  0.1 Theorem.

(‘de Rham style’) Let $\Omega\subseteq\mathbb{R}\nonscript\hskip
0.29999pt^{n}$ be open and connected. Let
$G=(G_{1},\ldots,G_{n})\in\mathcal{D}^{\prime}(\Omega)^{n}$, and suppose that

$\langle G,\varphi\rangle=0\qquad(\varphi\in\mathcal{D}(\Omega)^{n},\
\operatorname{div}\varphi=0).$ (0.1)

Then there exists a distribution $F\in\mathcal{D}^{\prime}(\Omega)$ such that
$\operatorname{grad}F=G$. If $n\geqslant 2$ and $G$ has compact support, then
$F$ can be chosen with compact support.

######  0.2 Theorem.

(‘Poincaré style’) Let $\Omega\subseteq\mathbb{R}\nonscript\hskip
0.29999pt^{n}$ be open and simply connected. Let
$G\in\mathcal{D}^{\prime}(\Omega)^{n}$ be such that
$\partial_{j}G_{k}=\partial_{k}G_{j}$ for all $j,k\in\\{1,\dots,n\\}$. Then
there exists $F\in\mathcal{D}^{\prime}(\Omega)$ such that
$\operatorname{grad}F=G$.

Proofs of these theorems have been provided by S. Mardare [10, Theorems 4.1
and 2.1]. In his paper he explains that “to this day, there is no proof, to
the best knowledge of the author”, of Theorem 0.2 in the existing literature;
and concerning Theorem 0.1, the proof given in [12, § 22, Theorem 17’] is said
to require “an important prerequisite about chains and flows on differential
manifolds”. For more motivation and background we refer to [10]. A proof of
the local part of Theorem 0.2, on the basis of de Rham’s regularisation, is
given in [11, Corollary 3.6(i)].

The author was intrigued by these theorems as well as motivated by related
problems connected with the Stokes operator, and this led to proofs of the
theorems which are quite different from those presented in [11, 10]. Let us
comment briefly on some differences between Mardare’s and our proofs of
Theorem 0.2, which is Theorem 2.1 in [10]. Given
$G\in\mathcal{D}^{\prime}(\Omega)^{n}$ as in Theorem 0.2, Mardare first
carries out the local part of the proof on sets
$\omega=\prod_{j=1}^{n}(a_{j},b_{j})$ by presenting – following Schwartz [13,
chap. II, § 6] – a formula for the solution
$F\in\mathcal{D}^{\prime}(\omega)$. Our approach is to deal with the local
part on bounded open sets that are star-shaped with respect to an open ball
and to approximate $G$ by $C^{\infty}$-vector fields satisfying the
compatibility conditions. In both proofs, the global part is achieved by a
homotopy argument; our proof is arranged in a way to permit the application of
the divergence theorem on the parameter set $[0,1]^{2}$ of the homotopy.

The proofs of Theorems 0.1 and 0.2 are the content of Sections 1 and 2,
respectively. In the remaining part of the paper we treat applications of the
theorems stated above to Sobolev space versions of the theorems. In these
results the task is finding potentials in $L_{2}(\Omega)$ for suitable vector
fields in $H^{-1}(\Omega)^{n}$. Here, the author was motivated, amongst
others, by the recent paper [1].

##  1 Proof of de Rham’s theorem

In the proof of Theorem 0.1 we will need the Bogovskiĭ operator which we
discuss next. Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be
an open set, $\Omega$ star-shaped with respect to every point of an open ball
$B(x^{0},r_{0})\subseteq\Omega$. Let $\rho\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})_{+}\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\\{}\varphi\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega){;}\;\varphi\geqslant
0\bigr{\\}}$ with $\operatorname{spt}\rho\subseteq B(x^{0},r_{0})$ and
$\int\rho(x)\mathclose{}\,\mathrm{d}x=1$. For $\varphi\in\mathcal{D}(\Omega)$
we define $B\varphi\in\mathcal{D}(\Omega)^{n}$ by

$B\varphi(x)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\Omega}\varphi(y)(x-y)\int_{1}^{\infty}\rho(y+r(x-y))r^{n-1}\mathclose{}\,\mathrm{d}r\mathclose{}\,\mathrm{d}y\qquad(x\in\Omega).$

It is not too difficult to show that indeed $B\varphi$ belongs to
$C^{\infty}(\Omega)^{n}$, and

$\operatorname{spt}B\varphi\subseteq\bigl{\\{}\lambda
z_{1}+(1-\lambda)z_{2}{;}\;z_{1}\in\operatorname{spt}\varphi,\
z_{2}\in\operatorname{spt}\rho,\ 0\leqslant\lambda\leqslant 1\bigr{\\}},$
(1.1)

a compact subset of $\Omega$.

######  1.1 Remark.

The linear mapping $B\colon\mathcal{D}(\Omega)\to\mathcal{D}(\Omega)^{n}$ is
the Bogovskiĭ operator; we refer to [3] for its first appearance. It has the
important property that $\int_{\Omega}\varphi(x)\mathclose{}\,\mathrm{d}x=0$
implies that $\operatorname{div}\mkern-1.5muB\varphi=\varphi$; see Remark 2.3
below. This property will also be used in the less explicit version that for
any $\varphi\in\mathcal{D}(\Omega)$ with
$\int\mkern-1.5mu\varphi(x)\mathclose{}\,\mathrm{d}x=0$ there exists a vector
field $\Phi\in\mathcal{D}(\Omega)^{n}$ with the property
$\operatorname{div}\Phi=\varphi$. (The reader should be aware of the fact that
the Bogovskiĭ operator depends on the function $\rho$; so the use of the
definite article ‘the’ might be somewhat misleading.)

######  1.2 Lemma.

The operator $B\colon\mathcal{D}(\Omega)\to\mathcal{D}(\Omega)^{n}$ is
continuous with respect to the standard topologies.

###### Proof.

The function $B\varphi$ can be rewritten as

$B\varphi(x)=\int_{z\in\mathbb{R}\nonscript\hskip
0.20999pt^{n}}\varphi(x-z)\frac{z}{|z|^{n}}\int_{0}^{\infty}\rho\bigl{(}x+s\frac{z}{|z|}\bigr{)}(s+|z|)^{n-1}\mathclose{}\,\mathrm{d}s\mathclose{}\,\mathrm{d}z$

(where $\rho$ and $\varphi$ are considered as functions in
$\mathcal{D}(\mathbb{R}\nonscript\hskip 0.29999pt^{n})$). This shows that
derivatives of $B\varphi$ can be estimated by derivatives of $\varphi$ and
$\rho$ of the same and lower order. This fact together with the support
property (1.1) shows the assertion. ∎

###### Proof of Theorem 0.1.

(i) In this part of the proof we suppose that $\Omega$ is star-shaped with
respect to an open ball $B(x^{0},r)\subseteq\Omega$. Let
$\rho\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})_{+}$, $\operatorname{spt}\rho\subseteq B(x^{0},r)$ and
$\int\rho(x)\mathclose{}\,\mathrm{d}x=1$, and let
$B\colon\mathcal{D}(\Omega)\to\mathcal{D}(\Omega)^{n}$ be the corresponding
Bogovskiĭ operator. For $\varphi\in\mathcal{D}(\Omega)$ we define

$\langle F,\varphi\rangle\mathrel{\mathop{\mathchar 58\relax}}=-\langle
G,B\varphi\rangle.$

Then Lemma 1.2 implies that $F\in\mathcal{D}^{\prime}(\Omega)$.

In order to show that $\operatorname{grad}F=G$ let
$\varphi\in\mathcal{D}(\Omega)$, $j\in\\{1,\dots,n\\}$. Then
$\int_{\Omega}\partial_{j}\varphi(x)\mathclose{}\,\mathrm{d}x=0$, hence
$\operatorname{div}(B\partial_{j}\varphi-\varphi
e_{j})=\partial_{j}\varphi-\partial_{j}\varphi=0$ (where $e_{j}$ denotes the
$j$-th unit vector), and hypothesis (0.1) implies

$\langle G_{j},\varphi\rangle=\langle G,\varphi e_{j}\rangle=\langle
G,B\partial_{j}\varphi\rangle=-\langle
F,\partial_{j}\varphi\rangle=\langle\partial_{j}F,\varphi\rangle.$

This shows that $G=\operatorname{grad}F$.

(ii) For the proof of the general case let
$\bigl{(}B(x^{k},r_{k})\bigr{)}_{k\in N}$ be a countable covering of $\Omega$
by open balls contained in $\Omega$, with $1\in
N\subseteq\mathbb{N}\nonscript\hskip 0.29999pt$. To ease notation we put
$\Omega_{k}\mathrel{\mathop{\mathchar 58\relax}}=B(x^{k},r_{k})$ ($k\in N$).
For each $k\in N$ let $\rho_{k}\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})_{+}$ with $\operatorname{spt}\rho_{k}\subseteq\Omega_{k}$,
$\int\rho_{k}(x)\mathclose{}\,\mathrm{d}x=1$, and let
$B_{k}\colon\mathcal{D}(\Omega_{k})\to\mathcal{D}(\Omega_{k})^{n}$ be the
corresponding Bogovskiĭ operator.

For $k\in N$ and some $c_{k}\in\mathbb{K}\nonscript\hskip 0.29999pt$ we define
$F_{k}\in\mathcal{D}^{\prime}(\Omega_{k})$ by

$\langle F_{k},\varphi\rangle\mathrel{\mathop{\mathchar
58\relax}}=c_{k}{\textstyle\int}\varphi(x)\mathclose{}\,\mathrm{d}x-\langle
G,B_{k}\varphi\rangle\qquad(\varphi\in\mathcal{D}(\Omega_{k})).$

Then part (i) shows that $\operatorname{grad}F_{k}=G$ on $\Omega_{k}$. We are
going to show that the constants $c_{k}$ can be chosen such that the family
$(F_{k})_{k\in N}$ of distributions is consistent.

For $k\in N$ we observe that there exists a function
$\Phi_{k}\in\mathcal{D}(\Omega)^{n}$ such that
$\operatorname{div}\Phi_{k}=\rho_{1}-\rho_{k}$; this can be seen by connecting
$\Omega_{1}$ with $\Omega_{k}$ by a finite chain of consecutively intersecting
open balls and applying Remark 1.1 repeatedly. With this function we choose
$c_{k}$ in such a way that $\left\langle
F_{k},\rho_{k}\right\rangle=\left\langle G,\Phi_{k}\right\rangle$; then

$\left\langle F_{k},\varphi\right\rangle=\left\langle
F_{k},\rho_{k}+\operatorname{div}B_{k}(\varphi-\rho_{k})\right\rangle=\left\langle
G,\Phi_{k}-B_{k}(\varphi-\rho_{k})\right\rangle$

for all $\varphi\in\mathcal{D}(\Omega_{k})$ with
$\int\varphi(x)\mathclose{}\,\mathrm{d}x=1$.

In order to show that the family $(F_{k})_{k\in N}$ is consistent, let $k,l\in
N$ be such that $\Omega_{k}\cap\Omega_{l}\neq\varnothing$, and let
$\varphi\in\mathcal{D}(\Omega)$ with
$\operatorname{spt}\varphi\subseteq\Omega_{k}\cap\Omega_{l}$,
$\int\varphi(x)\mathclose{}\,\mathrm{d}x=1$. Then

$\left\langle F_{k},\varphi\right\rangle-\left\langle
F_{l},\varphi\right\rangle=\left\langle
G,\Phi_{k}-B_{k}(\varphi-\rho_{k})-\Phi_{l}+B_{l}(\varphi-\rho_{l})\right\rangle=0,$

because the divergence of the function to which $G$ is applied turns out to be
$(\rho_{1}-\rho_{k})-(\varphi-\rho_{k})-(\rho_{1}-\rho_{l})+(\varphi-\rho_{l})=0$.
It follows that $\left\langle F_{k},\varphi\right\rangle=\left\langle
F_{l},\varphi\right\rangle$ for all
$\varphi\in\mathcal{D}(\Omega_{k}\cap\Omega_{l})$.

We conclude that the family $(F_{k})_{k\in N}$ composes to a distribution
$F\in\mathcal{D}^{\prime}(\Omega)$ satisfying $\operatorname{grad}F=G$.

(iii) Now suppose that $n\geqslant 2$ and that $G$ has compact support. There
exists $\psi\in\mathcal{D}(\Omega)$ with $\psi=1$ in a neighbourhood of the
support of $G$. Then

$\langle\widehat{G},\varphi\rangle\mathrel{\mathop{\mathchar
58\relax}}=\langle
G,\psi\varphi\rangle\qquad(\varphi\in\mathcal{D}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})^{n})$

defines an extension of $G$ to $\mathcal{D}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})^{n}$, satisfying
$\partial_{j}\widehat{G}_{k}=\partial_{k}\widehat{G}_{j}$ for all
$j,k\in\\{1,\dots,n\\}$. According to part (ii) above, there exists
$\widehat{F}\in\mathcal{D}^{\prime}(\mathbb{R}\nonscript\hskip 0.29999pt^{n})$
such that $\operatorname{grad}\widehat{F}=\widehat{G}$. Denote by
$\Omega_{\infty}$ the unbounded component of $\mathbb{R}\nonscript\hskip
0.29999pt^{n}\setminus\operatorname{spt}G$. Then
$\operatorname{grad}\widehat{F}=0$ on the connected open set
$\Omega_{\infty}$; hence there exists $c\in\mathbb{K}\nonscript\hskip
0.29999pt$ such that $\widehat{F}=c$ on $\Omega_{\infty}$. Then the
restriction of $\widehat{F}-c$ to $\mathcal{D}(\Omega)$ is as asserted. ∎

##  2 Proof of the distributional version of Poincaré’s lemma

We start with a property that will be needed in the proof of Theorem 0.2.

######  2.1 Lemma.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be open. Let
$v\in C^{1}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$ be a vector
field satisfying

$\partial_{j}v_{k}=\partial_{k}v_{j}\qquad(j,k=1,\dots,n).$

Let $\gamma,\widetilde{\gamma}\colon[0,1]\to\Omega$ be $C^{1}$-paths with
$\gamma(0)=\widetilde{\gamma}(0)=\mathrel{\mathop{\mathchar 58\relax}}x^{0}$,
$\gamma^{\prime}(0)=\widetilde{\gamma}^{\prime}(0)=0$,
$\gamma(1)=\widetilde{\gamma}(1)=\mathrel{\mathop{\mathchar 58\relax}}x^{1}$,
$\gamma^{\prime}(1)=\widetilde{\gamma}^{\prime}(1)=0$. Suppose that
$\Gamma\colon[0,1]^{2}\to\Omega$ is an FEP-homotopy between $\gamma$ and
$\widetilde{\gamma}$, i.e., $\Gamma$ is continuous, $\Gamma(\cdot,0)=\gamma$,
$\Gamma(\cdot,1)=\widetilde{\gamma}$, $\Gamma(0,\cdot)=x^{0}$,
$\Gamma(1,\cdot)=x^{1}$. (FEP stands for ‘fixed end points’.) Then

$\int_{\gamma}v\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}v(\gamma(s))\cdot\gamma^{\prime}(s)\mathclose{}\,\mathrm{d}s=\int_{0}^{1}v(\widetilde{\gamma}(s))\cdot\widetilde{\gamma}^{\prime}(s)\mathclose{}\,\mathrm{d}s=\int_{\widetilde{\gamma}}v.$

###### Proof.

(i) In this step we suppose additionally that $\Gamma$ is twice continuously
differentiable. We define the vector field
$w\colon[0,1]^{2}\to\mathbb{K}\nonscript\hskip 0.29999pt^{2}$,

$w(s,t)\mathrel{\mathop{\mathchar
58\relax}}=\bigl{(}v(\Gamma(s,t))\cdot\frac{\partial}{\partial
t}\Gamma(s,t),-v(\Gamma(s,t))\cdot\frac{\partial}{\partial
s}\Gamma(s,t)\bigr{)}.$

Then, using the hypothesis on $v$, one easily obtains $\operatorname{div}w=0$;
hence the divergence theorem yields
$\int_{\partial[0,1]^{2}}w(s,t)\cdot\nu(s,t)\mathclose{}\,\mathrm{d}\sigma(s,t)=0$,
where $\nu$ is the outer unit normal and $\sigma$ the surface measure. The
integrals over the lines with $s=0$ and $s=1$ vanish, because
$\frac{\partial}{\partial t}\Gamma(s,t)=0$ for $s=0$, $s=1$ and all
$t\in[0,1]$. Hence the remaining integrals yield

$\int_{0}^{1}v(\Gamma(s,0))\cdot\frac{\partial}{\partial
s}\Gamma(s,0)\mathclose{}\,\mathrm{d}s-\int_{0}^{1}v(\Gamma(s,1))\cdot\frac{\partial}{\partial
s}\Gamma(s,1)\mathclose{}\,\mathrm{d}s=0,$

which is just the asserted equality.

(ii) In order to apply step (i) by smoothing the given homotopy $\Gamma$ we
first ‘contract and extend’ it as follows. We define the continuous function
$\alpha\colon[-1/4,5/4]\to[0,1]$ by $\alpha\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{[-1/4,1/4]}\mathrel{\mathop{\mathchar 58\relax}}=0$,
$\alpha\vphantom{f}\mskip 1.0mu\vrule\mskip
2.0mu_{[3/4,5/4]}\mathrel{\mathop{\mathchar 58\relax}}=1$,
$\alpha\vphantom{f}\mskip 1.0mu\vrule\mskip 2.0mu_{[1/4,3/4]}$ affine linear,
and then we put $\widetilde{\Gamma}\colon[-1/4,5/4]^{2}\to\Omega$,
$\widetilde{\Gamma}(s,t)\mathrel{\mathop{\mathchar
58\relax}}=\Gamma(\alpha(s),\alpha(t))$. It is easy to see that then

$\int_{\widetilde{\Gamma}(\cdot,0)\vphantom{f}\mskip 1.0mu\vrule\mskip
2.0mu_{[0,1]}}v=\int_{\gamma}v\qquad\text{and}\qquad\int_{\widetilde{\Gamma}(\cdot,1)\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{[0,1]}}v=\int_{\widetilde{\gamma}}v.$ (2.1)

Observe that

$\widetilde{\Gamma}(s,\cdot)\vphantom{f}\mskip 1.0mu\vrule\mskip
2.0mu_{[-1/4,1/4]}=\widetilde{\Gamma}(s,0),\quad\widetilde{\Gamma}(s,\cdot)\vphantom{f}\mskip
1.0mu\vrule\mskip
2.0mu_{[3/4,5/4]}=\widetilde{\Gamma}(s,1)\qquad(s\in[-1/4,,5/4])$

and

$\widetilde{\Gamma}\vphantom{f}\mskip 1.0mu\vrule\mskip
2.0mu_{[-1/4,1/4]\times[-1/4,5/4]}=x^{0},\quad\widetilde{\Gamma}\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{[3/4,5/4]\times[-1/4,5/4]}=x^{1}.$

Now let $(\rho_{k})_{k\in\mathbb{N}\nonscript\hskip 0.20999pt}$ be a sequence
in ${\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt)_{+}$, $\operatorname{spt}\rho_{k}\subseteq[-1/k,1/k]$,
$\int\rho_{k}(x)\mathclose{}\,\mathrm{d}x=1$ for all
$k\in\mathbb{N}\nonscript\hskip 0.29999pt$, and put

$\widetilde{\Gamma}_{k}(s,t)\mathrel{\mathop{\mathchar
58\relax}}=\int_{-1/4}^{5/4}\int_{-1/4}^{5/4}\widetilde{\Gamma}(s^{\prime},t^{\prime})\rho_{k}(s-s^{\prime})\rho_{k}(t-t^{\prime})\mathclose{}\,\mathrm{d}s^{\prime}\mathclose{}\,\mathrm{d}t^{\prime}\qquad(s,t\in[0,1],\
k\geqslant 4).$

The properties mentioned above imply that
$\widetilde{\Gamma}_{k}(\cdot,j)=\widetilde{\Gamma}(\cdot,j)*\rho_{k}$ and
$\widetilde{\Gamma}_{k}(j,\cdot)=\gamma(j)$ on $[0,1]$ for $j=0,1$ and all
$k\geqslant 4$. Using the uniform continuity of $\widetilde{\Gamma}$ one shows
that there exists $k_{0}\geqslant 4$ such that
$\widetilde{\Gamma}_{k}([0,1]^{2})\subseteq\Omega$ for all $k\geqslant k_{0}$.

Having established these properties, we can apply part (i) of the proof to
conclude that

$\int_{\widetilde{\Gamma}_{k}(\cdot,0)}v=\int_{\widetilde{\Gamma}_{k}(\cdot,1)}v\qquad(k\geqslant
k_{0}).$ (2.2)

From
$\widetilde{\Gamma}_{k}(\cdot,j)=\widetilde{\Gamma}(\cdot,j)*\rho_{k}\to\widetilde{\Gamma}(\cdot,j)$
in $C^{1}([0,1];\mathbb{R}\nonscript\hskip 0.29999pt^{n})$ ($k\to\infty$) it
follows that

$\lim_{k\to\infty}\int_{\widetilde{\Gamma}_{k}(\cdot,j)}v=\int_{\widetilde{\Gamma}(\cdot,j)\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{[0,1]}}v\qquad(j=0,1).$ (2.3)

Combining (2.2), (2.3) and (2.1) we obtain the assertion of the lemma. ∎

###### Proof of Theorem 0.2.

(i) In this part we prove the theorem for the case that $\Omega$ is bounded
and star-shaped with respect to an open ball $B(x^{0},r)\subseteq\Omega$. Let
$\rho\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)_{+}$ with
$\operatorname{spt}\rho\subseteq B(x^{0},r)$ and
$\int\rho(x)\mathclose{}\,\mathrm{d}x=1$, and let $B$ be the corresponding
Bogovskiĭ operator. The transpose of
$B\colon\mathcal{D}(\Omega)\to\mathcal{D}(\Omega)^{n}$ is the linear mapping
$B^{t}\colon\mathcal{D}^{\prime}(\Omega)^{n}\to\mathcal{D}^{\prime}(\Omega)$,
given by

$\langle B^{t}G,\varphi\rangle\mathrel{\mathop{\mathchar 58\relax}}=\langle
G,B\varphi\rangle\qquad(\varphi\in\mathcal{D}(\Omega),\
G\in\mathcal{D}^{\prime}(\Omega)^{n}),$

and $B^{t}$ is continuous with respect to the standard topologies.

The restriction of $-B^{t}$ to $C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ is the operator $A\colon
C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})\to
C^{\infty}(\Omega)$, given by

$Av(x)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\Omega}\rho(y)\int_{0}^{1}v(sy+(1-s)x)\cdot(x-y)\mathclose{}\,\mathrm{d}s\mathclose{}\,\mathrm{d}y;$
(2.4)

see Remark 2.3. It has the property that $\operatorname{grad}Av=v$ for all

$v\in C^{\infty}_{\gamma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})\mathrel{\mathop{\mathchar 58\relax}}=\bigl{\\{}u\in
C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n}){;}\;\partial_{j}u_{k}=\partial_{k}u_{j}\
(j,k=1,\dots,n)\bigr{\\}}.$

Now let $G\in\mathcal{D}^{\prime}(\Omega)^{n}$ with
$\partial_{j}G_{k}=\partial_{k}G_{j}$ for all $j,k\in\\{1,\dots,n\\}$. We show
that $G$ can be approximated by vector fields $v\in
C^{\infty}_{\gamma}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. Without
loss of generality we assume that $x^{0}$ (the centre of the ball mentioned
initially) is the origin. In a first step we introduce distributions
$G^{\lambda}$ for $\lambda\in(1,\infty)$, defined on $\lambda\Omega$ by

$\langle G^{\lambda},\varphi\rangle\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\langle}G,\varphi\bigl{(}\tfrac{1}{\lambda}\,\cdot\,\bigr{)}\bigr{\rangle}\qquad(\varphi\in\mathcal{D}(\lambda\Omega)^{n}).$

It is easy to see that $G^{\lambda}$ satisfies the compatibility condition and
that for all $\varphi\in\mathcal{D}(\Omega)^{n}$ one obtains
$\bigl{\langle}G^{\lambda},\varphi\bigr{\rangle}\to\langle G,\varphi\rangle$
as $\lambda\to 1$.

In order to approximate $G^{\lambda}$ from
$C^{\infty}_{\gamma}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$ we fix
$\lambda>1$ and observe that $\lambda\Omega$ is an open neighbourhood of the
compact set $\overline{\Omega}$, due to the strict star-shapedness of
$\Omega$. Let $(\rho_{l})_{l\in\mathbb{N}\nonscript\hskip 0.20999pt}$ be a
$\delta$-sequence in ${\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})_{+}$. Then, for large $l$, the function

$x\mapsto v^{\lambda,l}(x)\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\langle}G^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}=\bigl{(}\bigl{\langle}G_{k}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}\bigr{)}_{k=1,\dots,n}$

is defined on $\Omega$ and belongs to
$C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. For all
$j,k=1,\dots,n$ one obtains

$\displaystyle\partial_{j,x}\bigl{\langle}G_{k}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}$
$\displaystyle=\bigl{\langle}G_{k}^{\lambda},\partial_{j,x}\rho_{l}(\cdot-x)\bigr{\rangle}=-\bigl{\langle}G_{k}^{\lambda},\partial_{j}\rho_{l}(\cdot-x)\bigr{\rangle}=\bigl{\langle}\partial_{j}G_{k}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}$
$\displaystyle=\bigl{\langle}\partial_{k}G_{j}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}=\cdots=\partial_{k,x}\bigl{\langle}G_{j}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle};$

hence $v^{\lambda,l}\in C^{\infty}_{\gamma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$.

In order to keep notation simple we compute the convergence $v^{\lambda,l}\to
G^{\lambda}$ in $\mathcal{D}^{\prime}(\Omega)^{n}$ componentwise (where we use
the same symbol for the function $v^{\lambda,l}$ and for the regular
distribution generated by this function). Thus, let
$\varphi\in\mathcal{D}(\Omega)$, $k\in\\{1,\dots,n\\}$. Then

$\displaystyle\bigl{\langle}v_{k}^{\lambda,l},\varphi\bigr{\rangle}$
$\displaystyle=\int
v_{k}^{\lambda,l}(x)\varphi(x)\mathclose{}\,\mathrm{d}x=\int\bigl{\langle}G_{k}^{\lambda},\rho_{l}(\cdot-x)\bigr{\rangle}\varphi(x)\mathclose{}\,\mathrm{d}x$
$\displaystyle=\bigl{\langle}G_{k}^{\lambda},\int\rho_{l}(\cdot-x)\varphi(x)\mathclose{}\,\mathrm{d}x\bigr{\rangle}=\bigl{\langle}G_{k}^{\lambda},\rho_{l}*\varphi\bigr{\rangle}.$

Now, $\rho_{l}*\varphi\to\varphi$ in $\mathcal{D}(\lambda\Omega)$, hence
$v_{k}^{\lambda,l}\to G_{k}^{\lambda}$ ($l\to\infty$) in the weak topology
$\sigma(\mathcal{D}^{\prime}(\Omega),\mathcal{D}(\Omega))$.

To complete this step we first recall from above that

$-\operatorname{grad}B^{t}v^{\lambda,l}=\operatorname{grad}Av^{\lambda,l}=v^{\lambda,l}.$

Letting $l\to\infty$ we obtain
$-\operatorname{grad}B^{t}G^{\lambda}=G^{\lambda}$, and then letting
$\lambda\to 1$ we finally get $-\operatorname{grad}B^{t}G=G$. This shows that
$F\mathrel{\mathop{\mathchar 58\relax}}=-B^{t}G$ has the asserted property.

(ii) For the general case, let $(B(x^{m},r_{m}))_{m\in N}$ be a countable
covering of $\Omega$ by open balls, with $1\in
N\subseteq\mathbb{N}\nonscript\hskip 0.29999pt$; for brevity we put
$\Omega_{m}\mathrel{\mathop{\mathchar 58\relax}}=B(x^{m},r_{m})$ ($m\in N$).
From part (i) we conclude that for each $m\in N$ there exists
$F_{m}\in\mathcal{D}^{\prime}(\Omega_{m})$ such that
$\operatorname{grad}F_{m}=G$ on $\Omega_{m}$.

Now it remains to show that there exist constants
$c_{m}\in\mathbb{K}\nonscript\hskip 0.29999pt$ such that $(c_{m}+F_{m})_{m\in
N}$ is a consistent family of distributions. Without restriction we assume
that $N=\\{1,\dots,m_{0}\\}$ or $N=\mathbb{N}\nonscript\hskip 0.29999pt$, and
that $\Omega_{m}\cap\bigcup_{k=1}^{m-1}\Omega_{k}\neq\varnothing$ for all
$m\in N\setminus\\{1\\}$. Put $c_{1}\mathrel{\mathop{\mathchar 58\relax}}=0$.

Let $m^{\prime}\in N\setminus\\{1\\}$ and assume that
$c_{2},\dots,c_{m^{\prime}-1}$ are found such that the family
$(c_{m}+F_{m})_{m=1,\dots,m^{\prime}-1}$ is consistent; without restriction
$c_{2}=\dots=c_{m^{\prime}-1}=0$. Denote by
$\widehat{F}_{m^{\prime}-1}\in\mathcal{D}^{\prime}(\bigcup_{m=1}^{m^{\prime}-1}\Omega_{m})$
the composed distribution satisfying $\widehat{F}_{m^{\prime}-1}=F_{m}$ on
$\Omega_{m}$ for all $m=1,\dots,m^{\prime}-1$. There exists
$m\in\\{1,\dots,m^{\prime}-1\\}$ such that
$\Omega_{m^{\prime}}\cap\Omega_{m}\neq\varnothing$, and
$\operatorname{grad}F_{m^{\prime}}=G=\operatorname{grad}F_{m}$ on
$\Omega_{m^{\prime}}\cap\Omega_{m}$ implies that there exists
$c_{m^{\prime}}\in\mathbb{K}\nonscript\hskip 0.29999pt$ such that
$c_{m^{\prime}}+F_{m^{\prime}}=F_{m}$ on $\Omega_{m^{\prime}}\cap\Omega_{m}$;
without restriction $c_{m^{\prime}}=0$. In order to make sure that
$F_{m^{\prime}}$ is consistent with $\widehat{F}_{m^{\prime}-1}$ we have to
show that $F_{m^{\prime}}$ is consistent with $F_{\widetilde{m}}$ for all
$\widetilde{m}\in\\{1,\dots,m^{\prime}-1\\}\setminus\\{m\\}$. Thus, let
$\widetilde{m}\in\\{1,\dots,m^{\prime}-1\\}\setminus\\{m\\}$ be such that
$\Omega_{m^{\prime}}\cap\Omega_{\widetilde{m}}\neq\varnothing$. Then as before
there exists $\widetilde{c}_{m^{\prime}}\in\mathbb{K}\nonscript\hskip
0.29999pt$ such that
$\widetilde{c}_{m^{\prime}}+F_{m^{\prime}}=F_{\widetilde{m}}$ on
$\Omega_{m^{\prime}}\cap\Omega_{\widetilde{m}}$. There exists a continuously
differentiable path $\gamma\colon[0,1]\to\Omega_{m}\cup\Omega_{m^{\prime}}$
from $x^{m}$ to $x^{m^{\prime}}$ satisfying
$\gamma^{\prime}(0)=\gamma^{\prime}(1)=0$, and there exists a continuously
differentiable path
$\widetilde{\gamma}\colon[0,1]\to\bigcup_{k=1}^{m^{\prime}}\Omega_{k}$ from
$x^{m}$ to $x^{m^{\prime}}$ satisfying
$\widetilde{\gamma}([0,1/2])\subseteq\bigcup_{k=1}^{m^{\prime}-1}\Omega_{k}$,
$\widetilde{\gamma}(1/2)=x^{\widetilde{m}}$,
$\widetilde{\gamma}([1/2,1])\subseteq\Omega_{\widetilde{m}}\cup\Omega_{m^{\prime}}$,
and $\widetilde{\gamma}^{\prime}(0)=\widetilde{\gamma}^{\prime}(1)=0$. The
simple connectedness of $\Omega$ implies that there exists an FEP-homotopy
$\Gamma\colon[0,1]^{2}\to\Omega$ between $\gamma$ and $\widetilde{\gamma}$. As
$\Gamma([0,1]^{2})$ is compact, there exists $r>0$ such that
$\Gamma([0,1]^{2})+B(0,2r)\subseteq\Omega$. Let $\rho\in{\displaystyle
C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\mathbb{R}\nonscript\hskip
0.29999pt^{n})_{+}$, $\operatorname{spt}\rho\subseteq B(0,r)$,
$\int\mkern-5.5mu\rho(x)\mathclose{}\,\mathrm{d}x=1$, and put

$v(x)\mathrel{\mathop{\mathchar 58\relax}}=\langle
G,\rho(\cdot-x)\rangle\qquad(x\in\Gamma([0,1]^{2})+B(0,r)).$

Then $v\in C^{\infty}(\Gamma([0,1]^{2})+B(0,r);\mathbb{K}\nonscript\hskip
0.29999pt^{n})$, and the hypothesis on $G$ implies that
$\partial_{j}v_{k}=\partial_{k}v_{j}$ for all $j,k=1,\dots,n$. Applying Lemma
2.1 with the open set $\Gamma([0,1]^{2})+B(0,r)$ we conclude that

$\int_{\gamma}\mkern-5.5muv=\int_{\widetilde{\gamma}}\mkern-5.5muv$ (2.5)

The distributions $F_{m}$, $F_{m^{\prime}}$ are consistent; call
$\check{F}\in\mathcal{D}^{\prime}(\Omega_{m}\cup\Omega_{m^{\prime}})$ the
combined distribution. Then $\operatorname{grad}\check{F}=G$ on
$\Omega_{m}\cup\Omega_{m^{\prime}}$, and

$\displaystyle\langle G,\rho(\cdot-\gamma(s))\rangle\cdot\gamma^{\prime}(s)$
$\displaystyle=\langle\operatorname{grad}\check{F},\rho(\cdot-\gamma(s))\rangle\cdot\gamma^{\prime}(s)$
$\displaystyle=-\langle\check{F},\operatorname{grad}\rho(\cdot-\gamma(s))\rangle\cdot\gamma^{\prime}(s)=\frac{\mathclose{}\,\mathrm{d}}{\mathclose{}\,\mathrm{d}s}\langle\check{F},\rho(\cdot-\gamma(s)\rangle,$
$\displaystyle\int_{\gamma}\mkern-5.5muv$
$\displaystyle=\int_{0}^{1}\mkern-1.5mu\langle
G,\rho(\cdot-\gamma(s))\rangle\cdot\gamma^{\prime}(s)\mathclose{}\,\mathrm{d}t=\langle\check{F},\rho(\cdot-x^{m^{\prime}})\rangle-\langle\check{F},\rho(\cdot-x^{m})\rangle$
$\displaystyle=\langle
F_{m^{\prime}},\rho(\cdot-x^{m^{\prime}})\rangle-\langle
F_{m},\rho(\cdot-x^{m})\rangle$ $\displaystyle=\langle
F_{m^{\prime}},\rho(\cdot-x^{m^{\prime}})\rangle-\langle\widehat{F}_{m^{\prime}-1},\rho(\cdot-x^{m})\rangle.$

By the same token,

$\displaystyle\int_{\widetilde{\gamma}}\mkern-5.5muv$
$\displaystyle=\int_{0}^{1}\mkern-1.5mu\langle
G,\rho(\cdot-\widetilde{\gamma}(s))\rangle\cdot\widetilde{\gamma}^{\prime}(s)\mathclose{}\,\mathrm{d}s$
$\displaystyle=\int_{0}^{1/2}\mkern-1.5mu\langle
G,\rho(\cdot-\widetilde{\gamma}(s))\rangle\cdot\widetilde{\gamma}^{\prime}(s)\mathclose{}\,\mathrm{d}s+\int_{1/2}^{1}\mkern-1.5mu\langle
G,\rho(\cdot-\widetilde{\gamma}(s))\rangle\cdot\widetilde{\gamma}^{\prime}(s)\mathclose{}\,\mathrm{d}s$
$\displaystyle=\langle\widehat{F}_{m^{\prime}-1},\rho(\cdot-x^{\widetilde{m}})\rangle-\langle\widehat{F}_{m^{\prime}-1},\rho(\cdot-x^{m})\rangle$
$\displaystyle\qquad\qquad\qquad+\langle\widetilde{c}_{m^{\prime}}+F_{m^{\prime}},\rho(\cdot-x^{m^{\prime}})\rangle-\langle
F_{\widetilde{m}},\rho(\cdot-x^{\widetilde{m}})\rangle$
$\displaystyle=\widetilde{c}_{m^{\prime}}+\langle
F_{m^{\prime}},\rho(\cdot-x^{m^{\prime}})\rangle-\langle\widehat{F}_{m^{\prime}-1},\rho(\cdot-x^{m})\rangle.$

From (2.5) we obtain $\widetilde{c}_{m^{\prime}}=0$, and this shows that the
family $(F_{m})_{m\in\\{1,\dots,m^{\prime}\\}}$ is consistent; hence by
induction, the family $(F_{m})_{m\in N}$ is consistent. ∎

######  2.2 Remark.

We note that Theorem 0.2 implies classical versions of Poincaré’s lemma: If
$\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ is open and simply
connected, and $g\in C^{l}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$
(for some $l\in\mathbb{N}\nonscript\hskip 0.29999pt_{0}$) is a vector field
satisfying

$\partial_{j}g_{k}=\partial_{k}g_{j}\qquad(j,k=1,\dots,n),$

then there exists $f\in C^{l+1}(\Omega)$ with $\operatorname{grad}f=g$.
Indeed, the distributional solution $F$ of $\operatorname{grad}F=g$ obtained
by Theorem 0.2 is automatically a regular distribution generated by a
$C^{l+1}$-function.

######  2.3 Remark.

On the Bogovskiĭ operator $B\colon{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)\to{\displaystyle
C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ and the operator $A\colon
C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})\to
C^{\infty}(\Omega)$ from (2.4).

Let $\Omega$ and $\rho$ be as at the beginning of Section 1. It is a standard
exercise of calculus that $\operatorname{grad}Av=v$ for all $v\in
C_{\gamma}^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$.
Substitution of variables yields
$\int(Av)\varphi\mathclose{}\,\mathrm{d}x=-\int v\cdot
B\varphi\mathclose{}\,\mathrm{d}x$ for all $v\in
C^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$,
$\varphi\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)$.

Let $\varphi,\psi\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)$,
$\int\varphi\mathclose{}\,\mathrm{d}x=0$. Note that
$\operatorname{grad}\psi\in
C_{\gamma}^{\infty}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$,
therefore $\operatorname{grad}(A\operatorname{grad}\psi-\psi)=0$; hence
$A\operatorname{grad}\psi-\psi$ is constant,
$\int(A\operatorname{grad}\psi-\psi)\varphi\mathclose{}\,\mathrm{d}x=0$. This
implies

$\int\psi\operatorname{div}\mkern-1.5muB\varphi\mathclose{}\,\mathrm{d}x=-\int\operatorname{grad}\psi\cdot
B\varphi\mathclose{}\,\mathrm{d}x=\int(A\operatorname{grad}\psi)\varphi\mathclose{}\,\mathrm{d}x=\int\psi\varphi\mathclose{}\,\mathrm{d}x.$

As this equality holds for all $\psi\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)$ one obtains
$\operatorname{div}\mkern-1.5muB\varphi=\varphi$.

##  3 The ‘coarse’ and ‘simplified’ versions of de Rahm’s theorem

In this section we treat the existence of potentials in a Hilbert space
context. Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a
connected bounded open set. We define the Sobolev spaces

$\displaystyle H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\\{}u\in H^{1}_{0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n}){;}\;\operatorname{div}u=0\bigr{\\}},$ $\displaystyle
H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\overline{C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;{\mathbb{K}\nonscript\hskip
0.29999pt^{n}})}^{H^{1}_{0}},$

where $C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\\{}\varphi\in{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n}){;}\;\operatorname{div}\varphi=0\bigr{\\}}$. For a subspace
$V\subseteq H^{1}_{0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$ we
define the polar

$V^{\textstyle\circ}\mathrel{\mathop{\mathchar 58\relax}}=\bigl{\\{}g\in
H^{-1}(\Omega)^{n}{;}\;\langle g,\varphi\rangle_{H^{-1},H^{1}_{0}}=0\
(\varphi\in V)\bigr{\\}},$

where $H^{-1}(\Omega)$ is the anti-dual space of $H^{1}_{0}(\Omega)$ in the
Gelfand triple $H^{1}_{0}(\Omega)\subseteq L_{2}(\Omega)\subseteq
H^{-1}(\Omega)$.

We adopt the terminology in the title of this section from [1] (formerly used
already in [8, Chap. I, § 2]), where it is defined that the “coarse version of
de Rham’s theorem” holds (for $\Omega$) if

$\displaystyle\text{for all }g\in
H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}\text{ there exists }f\in L_{2}(\Omega)\text{
with }g=\operatorname{grad}f,$ ($\mathrm{H}_{0}$)

and the “simplified version of de Rham’s theorem” holds if

$\displaystyle\text{for all }g\in
C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}\text{ there exists }f\in L_{2}(\Omega)\text{
with }g=\operatorname{grad}f.$ ($\mathrm{H}_{\textnormal{c}}$)

Note that property ($\mathrm{H}_{\textnormal{c}}$) could have been stated
equivalently with $H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ instead of
$C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$, because these spaces have the same polar. Also note that in
both of these properties the existence of $f$ could have been stated with the
additional property that $f\in L_{2}^{0}(\Omega)\mathrel{\mathop{\mathchar
58\relax}}=\bigl{\\{}g\in L_{2}(\Omega){;}\;\int
g(x)\mathclose{}\,\mathrm{d}x=0\bigr{\\}}$.

From $H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}\subseteq
H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$ it is clear that property
($\mathrm{H}_{\textnormal{c}}$) implies ($\mathrm{H}_{0}$). It is shown in [1,
Theorem 4.1] that for $\Omega$ with Lipschitz boundary, “J. L. Lions’ lemma”
together with ($\mathrm{H}_{0}$) implies ($\mathrm{H}_{\textnormal{c}}$); see
also Remark 3.6. We are going to show that ($\mathrm{H}_{\textnormal{c}}$) is
‘hereditary’ (Proposition 3.3), that ($\mathrm{H}_{0}$) holds and implies
($\mathrm{H}_{\textnormal{c}}$) for strictly star-shaped sets (Remark 3.4) and
that ($\mathrm{H}_{\textnormal{c}}$) holds for sets with Lipschitz boundary
(Corollary 3.5).

######  3.1 Remark.

Let $\operatorname{grad}$ denote the operator $\operatorname{grad}\colon
L_{2}(\Omega)\to H^{-1}(\Omega)^{n}$, $f\mapsto\operatorname{grad}f$. Then
$\operatorname{\rm ran}(\operatorname{grad})\subseteq
H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}\subseteq
H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$
($=C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$), and ($\mathrm{H}_{0}$) is equivalent to
$\operatorname{\rm
ran}(\operatorname{grad})=H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$, whereas ($\mathrm{H}_{\textnormal{c}}$) is
equivalent to $\operatorname{\rm
ran}(\operatorname{grad})=H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$. This implies that
($\mathrm{H}_{\textnormal{c}}$) is equivalent to ($\mathrm{H}_{0}$) together
with $H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})=H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$. Note that the last equality is equivalent to the denseness of
$C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ in $H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$.

We start our investigation of these properties by a proof of the equivalence
of property ($\mathrm{H}_{\textnormal{c}}$) and “J. L. Lions’ lemma”; see [1,
Theorem 4.1].

######  3.2 Theorem.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a connected
bounded open set. Then property ($\mathrm{H}_{\textnormal{c}}$) for $\Omega$
is equivalent to the property that for all $F\in\mathcal{D}^{\prime}(\Omega)$
with $\operatorname{grad}F\in H^{-1}(\Omega)^{n}$ there exists $f\in
L_{2}(\Omega)$ with $F=f$ (as distributions).

###### Proof.

For the proof of the necessity let $F\in\mathcal{D}^{\prime}(\Omega)$ such
that $\operatorname{grad}F\in H^{-1}(\Omega)^{n}$. Then
$\operatorname{grad}F\in
C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$; hence ($\mathrm{H}_{\textnormal{c}}$)
implies that there exists $f\in L_{2}(\Omega)$ with
$\operatorname{grad}f=\operatorname{grad}F$, and because $\Omega$ is connected
there exists $c\in\mathbb{K}\nonscript\hskip 0.29999pt$ such that $F$ is
generated by the function $f+c\in L_{2}(\Omega)$.

For the sufficiency let $g\in
C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$. Then Theorem 0.1 implies that there exists
$F\in\mathcal{D}^{\prime}(\Omega)$ such that $\operatorname{grad}F=g\in
H^{-1}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$; hence there exists
$f\in L_{2}(\Omega)$ with $F=f$,
$g=\operatorname{grad}F=\operatorname{grad}f$. ∎

The following proposition shows a hereditary property of
($\mathrm{H}_{\textnormal{c}}$).

######  3.3 Proposition.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a connected
bounded open set, and assume that there exists a (finite) covering
$(\Omega_{j})_{j=1,\dots,m}$ of $\Omega$ by connected open sets
$\Omega_{j}\subseteq\Omega$ with property ($\mathrm{H}_{\textnormal{c}}$).
Then property ($\mathrm{H}_{\textnormal{c}}$) holds for $\Omega$.

###### Proof.

Let $g\in H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$. Then clearly
$g^{j}\mathrel{\mathop{\mathchar 58\relax}}=g\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{H^{1}_{0}(\Omega_{j};\mathbb{K}\nonscript\hskip
0.20999pt^{n})}\in H^{1}_{\sigma,0}(\Omega_{j};\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$, and the hypothesis implies that for all
$j=1,\dots,m$ there exists $f_{j}\in L_{2}(\Omega_{j})$ such that
$g^{j}=\operatorname{grad}f_{j}$. Note that each function $f_{j}$ is only
determined up to a constant, and we have to ‘glue together’ suitable versions
of these functions.

We apply Theorem 0.1 and obtain a distribution
$F\in\mathcal{D}^{\prime}(\Omega)$ such that $g=\operatorname{grad}F$. This
implies that for each $j\in\\{1,\dots,m\\}$ one has
$\operatorname{grad}F=\operatorname{grad}f_{j}$ on $\Omega_{j}$, and because
$\Omega_{j}$ is connected, there exists $c_{j}\in\mathbb{K}\nonscript\hskip
0.29999pt$ such that $F=f_{j}+c_{j}\mathbf{1}_{\Omega_{j}}$ on $\Omega_{j}$.
This implies that the family
$(f_{j}+c_{j}\mathbf{1}_{\Omega_{j}})_{j=1,\dots,m}$ of $L_{2}$-functions is
consistent; hence there exists $f\in L_{2}(\Omega)$ with $f\vphantom{f}\mskip
1.0mu\vrule\mskip 2.0mu_{\Omega_{j}}=f_{j}+c_{j}\mathbf{1}_{\Omega_{j}}$ for
all $j=1,\dots,m$, $\operatorname{grad}f=\operatorname{grad}F=g$. ∎

Next we turn to more concrete sufficient conditions for the validity of
($\mathrm{H}_{\textnormal{c}}$). The fundamental source of these conditions is
an important fact from [3, Lemma 1], stated in part (a) of the following
remark.

######  3.4 Remark.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a bounded
open set which is star-shaped with respect to an open ball $B(x^{0},r)$.

(a) Then the corresponding Bogovskiĭ operator $B\colon{\displaystyle C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega)\to{\displaystyle
C_{\raise
0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ possesses a (unique) bounded linear extension $B\colon
L_{2}(\Omega)\to H^{1}_{0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$.
(The paper [3] does not contain a proof of this fact, but rather the hint that
for the proof one should use [5]; this hint has been executed in [4, Theorem
2.4], [7, Section III.3].) From
$\operatorname{div}\mkern-1.5muB\varphi=\varphi$ for $\varphi\in{\displaystyle
C_{\raise 0.60275pt\hbox{$\scriptstyle{\textnormal{c}}$}}^{\infty}}\cap
L_{2}^{0}(\Omega)$ it follows that $\operatorname{div}\mkern-1.5muBf=f$ for
all $f\in L_{2}^{0}(\Omega)$.

(b) It is easy to see that the bounded operators $\operatorname{div}\colon
H^{1}_{0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})\to L_{2}(\Omega)$
and $\operatorname{grad}\colon L_{2}(\Omega)\to H^{-1}(\Omega)^{n}$ are
negative adjoints of each other, and therefore $\operatorname{\rm
ran}(\operatorname{grad})^{\textstyle\circ}=\ker(\operatorname{div})$ – a
well-established relation for operators on Hilbert spaces – implies
$\overline{\operatorname{\rm
ran}(\operatorname{grad})}=\ker(\operatorname{div})^{\textstyle\circ}$. As
$\operatorname{\rm ran}(\operatorname{div})=L_{2}^{0}(\Omega)$ is closed, the
closed range theorem implies that $\operatorname{\rm
ran}(\operatorname{grad})$ is closed; hence $\operatorname{\rm
ran}(\operatorname{grad})=\ker(\operatorname{div})^{\textstyle\circ}=H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$. This shows that ($\mathrm{H}_{0}$) is
satisfied.

(c) We now show that even ($\mathrm{H}_{\textnormal{c}}$) is satisfied; this
follows if we show that
$C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ is dense in
$H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. Without
restriction $x^{0}=0$. Let $\varphi\in
H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. Then the
extension of $\varphi$ to $\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ by zero
belongs to $H^{1}_{0,\sigma}(\mathbb{R}\nonscript\hskip
0.29999pt^{n};\mathbb{K}\nonscript\hskip 0.29999pt^{n})$; we keep the notation
$\varphi$ for the extension. For $\lambda\in(1,\infty)$ we note that
$\varphi(\lambda\,\cdot)\in H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ has compact support contained in
$\lambda^{-1}\overline{\Omega}\subseteq\Omega$. Regularisation shows that
$\varphi(\lambda\,\cdot)$ is approximated by functions in
$C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$; hence $\varphi(\lambda\,\cdot)\in
H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. Taking
$\lambda\to 1$ we then obtain $\varphi\in
H^{1}_{\sigma,0}(\Omega;\mathbb{K}\nonscript\hskip 0.29999pt^{n})$. (This
reasoning has also been used in [14, proof of Lemma 9, p. 31].)

######  3.5 Corollary.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a connected
bounded open set, and assume that there exists a covering
$(\Omega_{j})_{j=1,\dots,m}$ of $\Omega$ by open sets
$\Omega_{j}\subseteq\Omega$, where each $\Omega_{j}$ is star-shaped with
respect to an open ball. Then property ($\mathrm{H}_{\textnormal{c}}$) holds
for $\Omega$.

These assertions apply to any connected bounded open set with Lipschitz
boundary.

###### Proof.

From Remark 3.4(c) we know that the sets $\Omega_{j}$ have property
($\mathrm{H}_{\textnormal{c}}$). Hence Proposition 3.3 implies the assertion.

If $\Omega$ has Lipschitz boundary, then it is not difficult to see that for
all $x\in\overline{\Omega}$ there exists an open neighbourhood $U_{x}$ such
that $U_{x}\cap\Omega$ is star-shaped with respect to the points of a ball in
$U_{x}\cap\Omega$. This is obvious for $x\in\Omega$, and for
$x\in\partial\Omega$ it results from the Lipschitz property of
$\partial\Omega$. The compactness of $\overline{\Omega}$ implies that there
exists a finite open covering $(\Omega_{j})_{j=1,\ldots,m}$ of $\Omega$ as
required in the first part of the corollary. (See also [1, Theorem 2.1].) ∎

######  3.6 Remark.

In [1, Theorem 3.1] an ‘equivalence result’ is established between various
classical results, e.g. the (classical) J. L. Lions lemma, the J. Nečas
inequality, property ($\mathrm{H}_{0}$) and the property that
$\operatorname{div}\colon H^{1}_{0,\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})\to L_{2}^{0}(\Omega)$ is surjective, for bounded open sets with
Lipschitz boundary. Each of these results by itself is of a notoriously high
technical level, and in [1, Section 2] several sources for the results are
mentioned.

A treatment of part of these properties, for the case that $\Omega$ has
$C^{1}$-boundary, can be found in [14, pp. 26–32]. Related problems are
treated in a more general setting in [2]. For further information we also
mention [6, 8, 15].

##  4 A ‘weak version’ of Poincaré’s lemma

In the previous section we have seen how the distributional version of de
Rham’s theorem can be applied in the $L_{2}$-context. Here we present an
application of the distributional version of Poincaré’s lemma.

######  4.1 Theorem.

Let $\Omega\subseteq\mathbb{R}\nonscript\hskip 0.29999pt^{n}$ be a simply
connected bounded open set satisfying property
($\mathrm{H}_{\textnormal{c}}$). Let $g\in H^{-1}(\Omega)^{n}$ be such that
$\partial_{j}g_{k}=\partial_{k}g_{j}$ for all $j,k=1,\dots,n$ (with
derivatives in the sense of distributions). Then there exists $f\in
L_{2}(\Omega)$ such that $\operatorname{grad}f=g$.

###### Proof.

Theorem 0.2 implies that there exists $F\in\mathcal{D}^{\prime}(\Omega)$ such
that $\operatorname{grad}F=g$. This shows that for all $\varphi\in
C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})$ one obtains

$\langle
g,\varphi\rangle_{H^{-1},H^{1}_{0}}=\langle\operatorname{grad}F,\varphi\rangle=-\langle
F,\operatorname{div}\varphi\rangle=0,$

i.e. $g\in
C^{\infty}_{{\textnormal{c}},\sigma}(\Omega;\mathbb{K}\nonscript\hskip
0.29999pt^{n})^{\textstyle\circ}$. Now property
($\mathrm{H}_{\textnormal{c}}$) implies that there exists $f\in L_{2}(\Omega)$
such that $\operatorname{grad}f=g$. ∎

Recall from Corollary 3.5 that Theorem 4.1 applies to the case that $\Omega$
is a simply connected bounded open set with Lipschitz boundary. For this case,
proofs of Theorem 4.1 are contained in [9, proof of Theorem 2.1], [1, Proof of
Theorem 4.1], without the use of our Theorem 0.2.

Acknowledgement. The author is grateful to M. Kunzinger (Vienna) for pointing
out reference [11].

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Jürgen Voigt
Technische Universität Dresden
Fakultät Mathematik
01062 Dresden, Germany
<EMAIL_ADDRESS>