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Some applications of integratedintegral operators and Cauchy–Pompeiurepresentations in Clifford analysisLe Hung Son aa Department of Applied Mathematics & Informatics, HanoiUniversity of Technology, Dai Co Viet Road 1, 1000 Hanoi, VietnamVersion of record first published: 15 Apr 2008.

To cite this article: Le Hung Son (2008): Some applications of integrated integral operators andCauchy–Pompeiu representations in Clifford analysis , Complex Variables and Elliptic Equations: AnInternational Journal, 53:5, 391-400

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Complex Variables and Elliptic EquationsVol. 53, No. 5, May 2008, 391–400

Some applications of integrated integral

operators and Cauchy–Pompeiurepresentations in Clifford analysisy

LE HUNG SON*

Department of Applied Mathematics & Informatics, Hanoi Universityof Technology, Dai Co Viet Road 1, 1000 Hanoi, Vietnam

Communicated by O. Celebi

(Received in final form 21 January 2005)

By using the higher-order Cauchy–Pompeiu formula for functions taking values in aClifford algebra, the class of generalized regular functions is defined. Some properties, suchas the Cauchy integral representation, the Cauchy theorem, Momera’s theorem, total analyti-city and extension theorem of Hartog’s type from theory of regular functions in Cliffordanalysis, are generalized for those functions.

Keywords: Cauchy–Pompeiu representation; Regular functions; Bianalytic functions;Total analyticity; Multi-biregular functions

AMS Subject Classifications: 35C15; 30620; 30635; 44A15; 46E20

1. Introduction

The theory of regular functions and biregular functions taking values in a Cliffordalgebra was studied in [2–4]. It is a natural generalization to higher dimensionsof the theory of holomorphic functions in one complex variable.

Based on the paper of Begehr (see [1]) the Pompeiu integral operators in the complexcase are presented and some higher-order Cauchy–Pompeiu representation formulaeare given. Hence the Cauchy–Pompeiu representation for functions taking valuesin a Clifford algebra is obtained.

Applying these results of Begehr we define the concept of bianalytic functionsin Clifford analysis; and obtain the integral representation formula for bianalyticfunctions. Then the total analyticity theorem for these functions is proved.

*Email: sonlehung2003@yahoo.comyDedicated to Professor H. Begehr on his 65th Birthday.

Complex Variables and Elliptic Equations

ISSN 1747-6933 print/ISSN 1747-6941 online � 2008 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/17476930600604109

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Furthermore, the multi-biregular functions taking values in a Clifford algebraare defined. This is a generation of biregular functions considered in [3] and [4].Finally, the extension theorem of Hartog’s-type is proved for the multi-regularfunctions.

2. Bianalytic functions taking values in a Clifford algebra

Let A be a Clifford algebra generated by the basic elements e1, e2, . . . , em andintroduced with a product, which satisfies the conditions

e21 ¼ e1 ¼ 1; e2j ¼ �e1 ¼ �1; eiej þ ejei ¼ 0; i, j ¼ 1, . . . ,m:

An element z 2 Cm can be identified with the element z ¼ z1e1 þ � � � þ zmem 2 A.

A function w 2 CkðD,AÞ can be understood as a transformation

w : D � Cm!A:

Then, w ¼P

�2N w�e�, w� 2 CkðDÞ.Where wA is complex valued functions, N ¼ f1, . . . ,mg, � ¼ ð�1, . . . ,�kÞ � N

(see [2]), e� ¼ e�1 , . . . , e�k .Suppose that wðzÞ 2 CkðD;AÞ \ Ck�1ðD;AÞ for k � 1 then we have the following

generated Cauchy–Pompeiu formula (see [1]):

wðxÞ ¼Xk�1�¼0

1

wm

Z@D

ð� � zÞ � ð� � zþ � � zÞ�

2��!j� � zjmd�!ð�Þ@�wð�Þ

�1

wm

ZD

ð� � zÞðz� � þ z� �Þk�1

2k�1ðk� 1Þ!j� � zjm@kwð�ÞdVð�Þ ð1Þ

where @ is the generalized Cauchy–Riemann operator

@ :¼Xmj¼1

ej@

@zj

Definition 1 A function w 2 CkðD,AÞ is called bianalytic in D if

@kwðzÞ ¼ 0: ð2Þ

We denote the set of bianalytic functions in D by BkRðD;AÞ.It is BkRðD;AÞ ¼ Ker @k.

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From (1) we get

LEMMA 1 If w 2 BRðD;AÞ \ Ck�1ð �D;AÞ then

wðzÞ ¼1

wm

Xk�1�¼0

ð�1Þ�Z@D

ð�1 � z1Þ�ð� � zÞ

2��!j� � zjmd�� for z 2 D: ð3Þ

Proof From (1) it follows

wðzÞ ¼Xk�1�¼0

1

!m

Z@D

ð� � zÞð� � zþ � � zÞ�

2��!j� � zjmd��@

�wð�Þ

where

� � z ¼ e1ð�1 � z1Þ þXmj¼2

ejð�j � zjÞ

� � z ¼ e1ð�1 � z1Þ �Xmj¼2

ejð�j � zjÞ

ð� � zÞ þ ð� � zÞ ¼ 2e1ð�1 � z1Þ:

Hence we get (3).

Remark From (3) it follows

wðzÞ ¼1

!m

Z@D

� � z

j� � zjmd�!

�wð�Þ þ1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@D

ð�1 � z1Þ�ð� � zÞ

j� � zjmd��@

�wð�Þ: ð3aÞ

Hence

@wðzÞ ¼1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@D

@hð�1 � z1Þ

� � � z

j� � zjm

id��@

�wð�Þ:

But

@ ð�1 � z1Þ� �� z

j�� zjm

� �¼ e1

@

@z1ð�1 � z1Þ

� �� z

j�� zjm

� �þXmj¼2

ð�1 � z1Þ�ej

@

@zj

�� z

j�� zjm

!

¼ e1 ��ð�1 � z1Þ��1 �� z

j�� zjm

� �þ e1 þ ð�1 � z1Þ

�Xmj¼1

ej@

@zj

� �� z

j�� zjm

���ð�1 � z1Þ

��1 �� z

j�� zjme1 þ ð�1 � z1Þ

�@�� z

j�� zjm

¼ ��ð�1 � z1Þ��1 �� z

j�� zjme1:

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Thus we have

@wðzÞ ¼1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@D

��ð�1 � z1Þ��1e1

� � z

j� � zjmd��@

�wð�Þ:

@wðzÞ ¼1

!m

Xk�1�¼1

ð�1Þ��1

ð�� 1Þ!

Z@D

ð�1 � z1Þ��1 � � z

j� � zjmd��@

�wð�Þ:

ð4Þ

3. Some properties of BRðD;AÞ

3.1. The Cauchy theorem

If w 2 BkRðD;AÞ, where D is a bounded domain in Rm with smooth boundary @D.

Then for each subdomain D1 � D with smooth boundary @D1 there is

Z@D1

d��@k�1w ¼ 0: ð5Þ

Proof Put w1 ¼ @k�1w, @w1 ¼ @

kw ¼ 0. Then w1 is (left) regular and (5) follows fromthe Cauchy theorem for (left) regular function (see [2]).

3.2. Momera’s theorem

THEOREM 2 (Momera’s) If w 2 CkðD;AÞ and

Z�

d��@k�1w ¼ 0 ð6Þ

for all surface � contained in the simply connected domain D; then

w 2 BkRðD;AÞ:

Proof Put w1 ¼ @k�1w. (6) states that

Z�

d��@w1 ¼ 0 for all surfaces � � D: ð7Þ

From Momera’s theorem for regular functions it follows that w1 2 RðD;AÞ. Hence

@w1 ¼ @kw ¼ 0: Q:e:d

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3.3. Total analyticity

THEOREM 3 Suppose that f 2 BkRðD;AÞ then f is (real ) analytic in all variablesz1, . . . , zn.

Proof Take arbitrarily a point a 2 D, choose r ¼ ðr1, . . . , rmÞ with rj > 0, j ¼ 1, . . . ,msuch that

�Bmða, rÞ :¼ fjzj � ajj < rj, j ¼ 1, . . . ,mg � D:

From (3a) it follows that

fðzÞ ¼1

!m

Z@Bm

� � z

j� � zjmd�� fð�Þ

þ1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@Bm

ð�1 � z1Þ� � � z

j� � zjmd�m@

�fð�Þ ð8Þ

for all z 2 B0m.

If z 2 B0m, � 2 @Bm, we have

� � z

j� � zjm¼X1s¼0

1

s!

X‘1,..., ‘s

ðz‘1 � a‘1 Þ � � � ðz‘s � a‘sÞ�ð0Þ‘1,..., ‘s

ð9Þ

where f‘1, . . . , ‘sg � f1, . . . ,mg; and the right-hand side series in (9) converges normallyfor z 2 B0

mða, ðffiffiffi2p� 1ÞrÞ (see [2,3])

�‘1,..., ‘s ¼ ð�1Þs@�‘1� � � @�‘s Eð�Þ ¼ @z‘1� � � @z‘s Eð� � zÞ

���z¼0: ð10Þ

By similar way, we get

ð�1 � z1Þ� � � z

j� � zjm¼X1s¼0

1

s!

X‘1...‘s

ðz‘1 � a‘1 Þ � � � ðz‘s � a‘s Þ�ð�Þ‘1...‘s

�ð�Þ‘1...‘s¼ @z‘1 � � � @z‘s

hð�1 � z1Þ

� � � z

j� � zjm

i���z¼0:

ð11Þ

From (8)–(10) it follows

f ðzÞ ¼1

!m

Z@Bm

X1s¼0

1

s!

X‘1...‘s

ðz‘1 � a‘1 Þ � � � ðz‘s � a‘sÞ�ð0Þ‘1...‘s

d�� fð�Þ

þ1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@Bm

X1s¼0

1

s!

X‘1...‘s

ðz‘1 � a‘1 Þ � � � ðz‘s � a‘sÞ�ð�Þ‘1...‘s

d��@�f ð�Þ

fðzÞ ¼X1s¼0

1

s!

X‘1...‘s

ðz‘1 � a‘1Þ � � � ðz‘s � a‘s ÞC�1...�s

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with

C�1...�s ¼1

!m

Z@Bm

�ð0Þ‘1...‘sd�� f ð�Þ þ

1

!m

Xk�1�¼1

ð�1Þ�

�!

Z@Bm

�ð�Þ�1...�s d��@�fð�Þ: ð12Þ

The right-hand side series of (12) converges normally in a sufficiently small neighbour-hood Bmða, r

0Þ of a, such that Bmða, r0Þ � D. a is an arbitrary point of D. Thus the

theorem is proved. Q.e.d.

COROLLARY If w 2 BkRðD;AÞ and if there exists an non-empty open subset � of Dsuch that w¼ 0 on �, then w � 0 in D (the Uniqueness theorem).

4. Multi-biregular functions

Let � be a domain of RnðyÞ y ¼ ð y1, . . . , ynÞ ðn � mÞ, then y can be identified

with y ¼Pn

j¼1 ejyj 2 A.Introducing the differential operator

@y :¼Xnj¼1

ej@

@yj: ð13Þ

Consider the function

f ðx, yÞ : D�� � RmðxÞ �R

nðyÞ ! A

then

f ðx, yÞ ¼X��N

e�f�ðx, yÞ: ð14Þ

Definition 2 A function f ðx, yÞ 2 CkðD��;AÞ is said to be ðk, ‘Þ-biregular function if

@ kx f ðx, yÞ ¼ f ðx, yÞ@‘y ð15Þ

where the operator @ kx acts from the left side and @ ‘y acts from the right side of f ðx, yÞ.Denote the set of all ðk, ‘Þ-biregular functions defined in D�� and taking values

in Clifford algebra A by Bk, ‘RðD��;AÞ.

Integral representation

Suppose that wðx, yÞ 2 Bk, ‘RðD��;AÞ then by fixed y 2 �, from (3) it follows that

wðx, yÞ ¼1

!m

Xk�1�¼0

ð�1Þ�Z@D

ð�1 � x1Þ�

2��!

� � x

j� � xjmd��@

�� wð�, yÞ

wð�, yÞ ¼1

!n

X‘�1�¼0

ð�1Þ�Z@�

wð�, �Þ@�� d���� y

j�� yj‘ð�1 � y1Þ

�

2��!:

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Hence

wðx, yÞ ¼1

!m

1

!n

Xk�1�¼0

X‘�1�¼0

ð�1Þ�þ�

2�þ��!�!

�

�

Z@D�@�

ð�1 � x1Þ� � � x

j� � xjmd��@

�� wð�, �Þ@

�� d�!

��� y

j�� yj‘ð�1 � y1Þ

�

�: ð16Þ

THEOREM 4 If w 2 Bk, ‘RðD��,AÞ then w can be represented by integral formula (16)for ðx, yÞ 2 D0 ��0.

Remark From Theorem 3 it follows that if w 2 Bk, ‘RðD��,AÞ then wðx, yÞ is (real)analytic for x1, . . . , xm by each fixed y ¼ y0 2 �, and w is (real) analytic for y1, . . . , ynby each fixed x0 2 D.

In the following we prove the global real-analyticity of w.

THEOREM 5 If w 2 Bk, ‘RðD��;AÞ then w is real-analytic in D�� with respect toall ðx1, . . . ,xm, y1, . . . , ynÞ.

Proof Take arbitrarily a point ðx0, y0Þ 2 D��. Choose rm; rn > 0 such that�Bmðx

0, rmÞ � �Bnðy0, rnÞ � D��, where �Bm and �Bn are closed balls in R

m and Rn

respectively. Using (10) for x 2 B0m, � 2 @Bn, we get

ð�1 � x1Þ� 1

!m

� � x

j� � xjm¼X1s¼0

1

s!

X‘1...‘s

ðx‘1 � x0‘1Þ � � � ðx‘s � x0‘s Þ�ð�Þ‘1...‘s

ð11aÞ

where

�ð�Þ‘1...‘s¼

@

@x‘1� � �

@

@x‘s

1

!mð�1 � x1Þ

� � � x

j� � xjm

���x¼0: ð10aÞ

Analogously we get

1

!n

�� y

j�� yjnð�1 � y1Þ

�¼X1r¼0

1

r!

Xr1...rt

ðyr1 � y0r1 Þ � � � ðyrt � y0rt Þ�ð�Þr1...rt

ð11bÞ

where

�ð�Þr1...rt¼

@

@yr1� � �

@

@yrt

1

!nð�1 � y1Þ

� �� y

j�� yjn

���y¼0: ð10bÞ

The series in (11a) converges normally in B0mðx

0, ðffiffiffi2p� 1ÞrmÞ and the series in (11b)

converges normally in B 0n ðy

0, ðffiffiffi2p� 1ÞrmÞ (see [2,3]).

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From (16), (11a) and (11b) it follows

wðx, yÞ ¼Xk�1�¼0

X‘�1�¼0

ð�1Þ�þ�

2�þ��!�!

Z@D�@�

X1s¼0

1

s!

X‘1...‘s

ðx‘1 � x0‘1 Þ � � � ðx‘s � x0‘sÞ�ð�Þ‘1...‘s

d��

� @�� wð�, �Þ@�� d��

X1r¼0

1

r!

Xr1...rt

ðyr1 � y0r1Þ � � � ðyrt � y0rt Þ�ð�Þr1...rt

: ð17Þ

Put

C�, �‘1,..., ‘s, r1,..., rt¼ð�1Þ�þ�

2�þ��!�!

Z@D�@�

�ð�Þ‘1...‘sd��@

�� wð�, �Þ@

���ð�Þr1...rt

: ð18Þ

We have

wðx, yÞ ¼Xk�1�¼0

X‘�1�¼0

X1s, r¼0

h 1

s!r!Cð�, �Þ‘1,..., ‘s, r1,..., rt

� ðx‘1�x0‘1Þ � � � ðx‘s�x

0‘sÞðyr1�y

0r1Þ � � � ðyrt�y

0rtÞ

ið19Þ

or

wðx, yÞ ¼X1s, r¼0

Xk�1�¼0

X‘�1�¼0

Cð�, �Þ‘1,..., ‘s, r1,..., rt

ðx‘1 � x0‘1 Þ � � � ðx‘s�x0‘sÞð yr1 � y0r1 Þ � � � ð yrt�y

0rtÞ ð20Þ

the series in (20) converges normally for

ðx, yÞ 2 B0mðx

0, ðffiffiffi2p� 1ÞrmÞ � B0

nðy0, ð

ffiffiffi2p� 1ÞrnÞ:

As direct application of the total analyticity for the functions in Bk, ‘RðD��Þ we getthe uniqueness theorem for the ðk, ‘Þ-biregular functions. Therefore, we can discussthe extension of the ðk, ‘Þ-biregular function, and the Hartog’s extensiontheorems for those functions. In the following, we understand (the extension) of agiven function w 2 Bk, ‘RðD,�Þ to a more larger domain eD� e�, whereeD � D,e� � � is the function ew 2 Bk, ‘ðeD� e�Þ such that

ew D�� ¼ w:j

Because of the Uniqueness theorem such extension ew of w (if it exists) is unique.

THEOREM 6 (Hartog’s extension theorem) Suppose that � is an open neighbourhoodof @ðD��Þ, w is a given function of Bk, ‘RðD��AÞ then there exsits a unique

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function ew 2 Bk, ‘RððD��Þ [�,AÞ such that

ewj� � w: ð21Þ

Proof Let � be an open neighbourhood of @ðD��Þ and w be a given functionbelonging to Bk, ‘Rð�Þ.

Consider

ewðx, yÞ :¼1

!m

Xk�1�¼0

ð�1Þ�Z@D

ð�1 � x1Þ�ð� � xÞ

2��!j� � xjmd��@

�wð�, yÞ ð22Þ

ewðx, yÞ is well defined for each ðx, yÞ 2 D��.It is

ewðx, yÞ@‘y ¼ 1

!m

Xk�1�¼0

ð�1Þ�Z@D

ð�1 � x1Þ�ð� � xÞ

2��!j� � xjmd��@

�� wð�, yÞ@‘y

h i¼ 0 ð23Þ

By similar methods as in (4) we get

@ pxgwðx, yÞ ¼ 1

!m

Xk�1�¼p

ð�1Þð��pÞ

ð�� pÞ!

Z@D

ð�1 � x1Þ��p � � x

j� � xjmd��@

�wð�, yÞ:

Specially

@ k�1x ewðx, yÞ ¼ 1

!m

Z@D

� � x

j� � xjmd��@

k�1wð�, yÞ:

Hence

@ kxewðx, yÞ ¼ 1

!m

Z@D

@x� � x

j� � xjm

� �d��@

k�1wð�, yÞ ¼ 0 ð24Þ

(23) and (24) state that

ewðx, yÞ 2 Bk, ‘RððD��Þ [�,AÞ:

If y is sufficiently closed to @�, then ðx, yÞ 2 � and ew ¼ w. From the Uniquenesstheorem for Bk, ‘RððD��Þ [�,AÞ it follows that

ew � w 2 �:

Thus ew is the extension of w. Q.e.d

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References

[1] Begehr, H., 2002, Integral representations in complex, hypercomplex and Clifford analysis.Integral Transforms and Special Functions, 13, 223–241.

[2] Brackx, F., Delaughe, R. and Sommen, F., 1982, Clifford Analysis (London: Pitman).[3] Brackx, F. and Pincket, W., 1984, A Bochner–Martinelli formula for the biregular functions of Clifford

analysis. Complex Variables, 4, 39–48.[4] Brackx, F. and Pincket, W., 1985, Two Hartogs theorems for nullsolutions of overdetermined systems

in Euclidean space. Complex Variables, 4, 205–222.

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