Hindawi Publishing Corporation
Boundary Value Problems
Volume 2007, Article ID 61602, 9 pages
doi:10.1155/2007/61602
Research Article
Removable Singularities of ᐃ᐀-Differential Forms
and Quasiregular Mappings
Olli Martio, Vladimir Miklyukov, and Matti Vuorinen
Received 14 May 2006; Revised 6 September 2006; Accepted 20 September 2006
Recommended by Ugo Pietro Gianazza
A theorem on removable singularities of ᐃ᐀-differential forms is proved and applied to
quasiregular mappings.
Copyright © 2007 Olli Martio et al. This is an open access article distr ibuted under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Main theorem
We recall some fac t s on d i fferential forms and quasiregular mappings. Our notation is as
in [1]. Let ᏹ be a Riemannian manifold of the class C
3
,dimᏹ = n, without boundary.
Each differential form α can be written in terms of the local coordinates x
1
, ,x
n
as the
linear combination
α
=

1≤i
1

···i
k
∈ L
p
loc
(ᏹ)
is called weakly closed if for each differential form β,degβ
= k + 1, with compact support
suppβ
= {m ∈ ᏹ : β = 0} in ᏹ and with coefficients in the class W
1
q,loc
(ᏹ), 1/p+1/q = 1,
1
≤ p, q ≤∞,wehave

ᏹ
α,δβ∗
ᏹ
= 0. (1.2)
2 Boundary Value Problems
Here the operator
∗ and the exterior differentiation d define the codifferential operator
δ by the formula
δα
= (−1)
k
∗
−1
d ∗ α (1.3)

k

i=1


w
i


. (1.6)
Taking these into account and using the inequality between geometric and arithmetic
means

k

i=1


w
i



1/k
≤
1
k
k

i=1

w
= w
1
∧···∧w
k
, θ = θ
1
∧···∧θ
n−k
(1.9)
be simple weakly closed differential forms on ᏹ.
We say that the pair of forms (1.9) satisfies a ᐃ᐀-condition on ᏹ if there exist con-
stants ν
1
,ν
2
> 0 such that almost everywhere on ᏹ
ν
1
w
kp
≤w,∗θ, θ≤ν
2
w. (1.10)
Our main removability result for differential forms is the foll owing.
Theorem 1.1. Let ᏹ be a Riemannian C
3
-manifold, dimM = n ≥ 2,andletE ⊂ ᏹ be
acompactsetofp-capacity zero, 1
≤ p ≤ n.LetZ and θ be simple forms on ᏹ \ E of de-


θ satisfies the
ᐃ᐀-condition on ᏹ and their restrictions to ᏹ
\ E coincide with Z, θ,respectively.
2. p-capacity
First we recall some basic facts about condensers. Let D be an open set on ᏹ and let
A,B
⊂ D be such that A and B are compact in D and A ∩ B = ∅.Eachtriple(A,B;D)is
called a condenser on ᏹ.
We fix p
≥ 1. The p-capacity of the condenser (A,B; D)isdefinedby
cap
p
(A,B;D) = inf

D
|∇ϕ|
p
∗
ᏹ
, (2.1)
where the infimum is taken over the set of all continuous functions ϕ of class W
1
p,loc
(D)
such that ϕ
|
A
= 0, ϕ|
B

⊂ ᏹ is of p-capacity zero, if cap
p
(E,U;ᏹ) = 0forallopen
sets U
⊂ ᏹ such that E ∩ U = ∅.
We will need the following lemma.
Lemma 2.1. AsetE
⊂ ᏹ is of 1-capacity zero if and only i f
Ᏼ
n−1
(E) = 0. (2.3)
Proof. Fix ε>0andanopensetU
⊂ ᏹ such that cap
1
(E,U;M) = 0. Choose a smooth
function ϕ : ᏹ
→ [0,1] such that ϕ|
E
= 0, ϕ|
U
= 1, ∇ϕ = 0a.e.onᏹ \ (E ∪ U)and

ᏹ
|∇ϕ|∗
ᏹ
≤ ε. (2.4)
By the coarea formula we have

ᏹ
|∇ϕ|∗

n−1

G
t

≤
ε (2.6)
and there exist sets G
t
of ar bitrarily small (n − 1)-measure.
Since U is open it is possible only for the set E of (n
− 1)-measure zero. 
If a compact set E ⊂ ᏹ is of p-capacity zero, then E is of q-capacity zero for all
q
∈ [1, p]. By Lemma 2.1 we conclude that a set E of p-capacity zero, p ≥ 1, satisfies
Ᏼ
n−1
(E) = 0. In particular, such a set has n-measure zero.
3. Applications to quasiregular mappings
Let ᏹ and ᏺ be Riemannian manifolds of dimension n. It is convenient to use the follow-
ing definition [3, Section 14]. A continuous mapping F : ᏹ
→ ᏺ of the class W
1
n,loc
(ᏹ)is
called a quasiregular mapping if F satisfies


F


= (F
1
, ,F
n
):ᏹ → R
n
is a quasiregular mapping and 1 ≤ k<n, then the
pair of forms
w
= dF
1
∧···∧dF
k
, θ = dF
k+1
∧···∧dF
n
(3.2)
satisfies a ᐃ᐀-condition on ᏹ with the structure constants ν
1
= ν
1
(n,k,K), ν
2
= ν
2
(n,k,K),
and p
= n/k.
We point out some special cases of Theorem 1.1.

F
i
dF
1
∧ dF
2
∧···∧

dF
i
∧···∧dF
k
, (3.4)
where the symbol

dF
i
means that this factor is omitted and c
i
= const,

k
i
=1
c
i
= 1.
Then there exists a quasiregular mapping

F : D → R

= dF
1
∧···∧dF
k
. (3.5)
If we put
θ
= dF
k+1
∧···∧dF
n
, (3.6)
then by Lemma 3.1 the pair of forms w
= dZ and θ satisfies (1.10)onD \ E. Using
Theorem 1.1 wecanconcludethatformsZ and θ have extensions to D. Moreover for
an arbitrary subdomain D

, E ⊂ D

⊂⊂ D,itfollows

D

\E
J
F
(x) dx
1
···dx
n

θ
L
q
(D

\E)
,
(3.7)
where C
= const < ∞ [2, Section 1.7] and p = n/k, q = n/(n − k).
From this it is easy to see that the vector function F belongs to W
1
n,loc
in D and E is
removable for the quasiregular mapping F. Note that in the definition of a quasiregular
mapping continuity is not needed, see [4, Section 3, Chapter II]. This property has a local
characteranditsproofforsubdomainsof
R
n
implies its correctness for manifolds. 
The case k = 1 reduces to the well-known case, see Miklyukov [5].
Corollary 3.3. Let D
⊂ R
n
be a domain, and let E ⊂ D be a compact set of n-capacity zero.
Suppose that
F
=

F

= n we have the following result.
Corollary 3.4. Let D
⊂ R
n
be a domain, and let E ⊂ D be a compact set of Hausdorff
(n
− 1)-measure zero. Suppose that
F
=

F
1
,F
2
, ,F
n

: D \ E −→ R
n
(3.10)
is a quasiregular mapping such that
ess sup
x∈D\E
J
F
(x) < ∞. (3.11)
Then there exists a quasiregular mapping f
∗
: D → R
n

(D). Hence the corollary follows from Theorem 3.2. 
Remark 3.5. Observe that Corollary 3.4 has an easy alternative proof. Since J
F
(x)is
bounded and E is of (n
− 1)-measure zero, the quasiregularity of F implies that the de-
rivative of F belongs to L
∞
loc
(D)andF is a Lipschitz mapping in D \ E. This shows that
F can be extended to a Lipschitz mapping on D. It is clear that the extended mapping is
quasiregular in D.
Corollary 3.4 gives the following version of the well-known Painlev
´
etheorem.
Corollary 3.6. Let E
⊂ D ⊂ C be a compact set of linear measure zero. Let F : D \ E → C
be a holomorphic function. The set E is removable for F if and only if
sup
z∈K\E


F

(z)


< ∞, (3.13)
for each compact set K
⊂ D.

⊂ U
k
, U
k
⊂ D, ∩
∞
k=1
U
k
= E. (4.2)
Fix a nonnegative smooth function ψ : ᏹ
→ R,0≤ ψ ≤ 1, with a compact support and
ψ
≡ 1onD.Fixak = 1,2, and a smooth function ϕ : ᏹ → R,0≤ ϕ ≤ 1, with the
properties
ϕ
|
E
= 0, suppϕ ⊂ U
k
, ϕ = 1 ∀m ∈ ᏹ \ U
k
. (4.3)
The form ψ
p
ϕ
p
Z ∧ θ has a compact support in ᏹ \ E. This yields

ᏹ\E


ψ
p
ϕ
p

∧
Z ∧ θ. (4.5)
Observe that
dZ
∧ θ =dZ,∗θ∗
ᏹ
. (4.6)
The form θ is closed and, consequently, from (1.10)weget
ν
1

ᏹ\E
ψ
p
ϕ
p
dZ
kp
∗≤

ᏹ\E
ψ
p
ϕ

ᏹ\E


d

ψ
p
ϕ
p

∧
Z


|∗
θ|∗ .
(4.7)
But degθ
= n − k and by (1.8)wehave
|∗θ|=|θ|≤(n − k)
(n−k)/2
θ
n−k
. (4.8)
Thus from the second condition of (1.10), it follows that
ν
1

ᏹ\E
ψ

ν
2
.
By (1.11) there exists a constant 0 <M<
∞ such that


Z(m)


<M for a.e. in ᏹ \ E. (4.10)
Thus, we obtain
ν
1

ᏹ\E
ψ
p
ϕ
p
dZ
kp
∗≤ν
3
M

ᏹ\E


d

p−1
ψ
p
|∇ϕ|, (4.12)
ν
1

ᏹ\E
ψ
p
ϕ
p
dZ
kp
∗
≤
pν
3
M

ᏹ\E
ϕ
p
ψ
p−1
|∇ψ|dZ
p−1
∗ + pν
3
M

≥ 1.
For ε>0 this implies two estimates

ᏹ\E
ϕ
p
ψ
p−1
|∇ψ|dZ
n−k
∗
≤
n − k
kp
ε
kp/(k−n)

ᏹ\E
ϕ
p
ψ
p
dZ
kp
∗ +
ε
kp
kp

ᏹ\E

kp

ᏹ\E
ψ
p
|∇ϕ|
p
∗ .
(4.15)
Now from (4.13)itfollows
ν
1

ᏹ\E
ψ
p
ϕ
p
dZ
kp
∗
≤
C
1

ᏹ\E
ψ
p
ϕ
p

, C
2
= ν
3
M
ε
kp
k
. (4.17)
Choose ε
= ε
0
> 0suchthatC
1
= ν
1
/2. Then we obtain
1
2
ν
1

ᏹ\E
ψ
p
ϕ
p
dZ
kp
∗

ε
kp
0
k

U
k
\E
|∇ϕ|
p
∗ + ν
3
M
ε
kp
0
k

ᏹ\D
|∇ψ|
p
∗
(4.18)
and since 0
≤ ψ, ϕ ≤ 1,
1
2
ν
1


1

D\U
k
dZ
kp
∗≤ν
3
M
ε
kp
0
k
cap
p

E,U
k
;ᏹ

+ ν
3
M
ε
kp
0
k
cap
p
(D,ᏹ;ᏹ). (4.20)

D
dZ
kp
∗≤ν
3
M
ε
kp
0
k
cap
p
(D,ᏹ;ᏹ) (4.22)
because by Lemma 2.1 the set E is of (n
− 1)-measure zero.
Next by Lemma 2.1,thecoefficients of Z can be extended to W
1
p,loc
-functions in ᏹ.
This is due to the estimate (4.22) and to the ACL-property of W
1
p
-functions; note that the
ACL-property can be easily transformed to the manifold ᏹ since ᏹ is in the class C
3
.
Thus, Z canbeextendeduptosomeform

Z.Moreoverclearly,d
