Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis

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Boundary properties of hypergenic-Cauchy type integrals in CliffordanalysisYonghong Xie a ba School of Mathematical Sciences, University of Science andTechnology of China, Hefei, Anhui Province 230026, P.R. Chinab College of Mathematics and Information Science, Hebei NormalUniversity, Shijiazhuang, Hebei Province 050024, P.R. ChinaPublished online: 10 Jan 2013.

To cite this article: Yonghong Xie (2013): Boundary properties of hypergenic-Cauchy typeintegrals in Clifford analysis, Complex Variables and Elliptic Equations: An International Journal,DOI:10.1080/17476933.2012.744403

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Complex Variables and Elliptic Equations2013, 1–17, iFirst

Boundary properties of hypergenic-Cauchy type integrals in

Clifford analysis

Yonghong Xieab*

aSchool of Mathematical Sciences, University of Science and Technology of China, Hefei,Anhui Province 230026, P.R. China; bCollege of Mathematics and Information Science,

Hebei Normal University, Shijiazhuang, Hebei Province 050024, P.R. China

Communicated by Y. Xu

(Received 29 June 2012; final version received 24 October 2012)

The aim of this article is to give some boundary properties of hypergenicquasi-Cauchy type integrals in Clifford analysis. First, the Plemelj formulafor hypergenic quasi-Cauchy type integrals is given by virtue of the Cauchyintegral formula for hypergenic functions. Then, the Privalov theorem forhypergenic quasi-Cauchy type integrals is obtained by virtue of the Plemeljformula.

Keywords: hypergenic functions; Plemelj formula; Privalov theorem; realClifford analysis

AMS Subject Classifications: 30B30; 31B10

1. Introduction

Clifford algebra is an associative and noncommutable algebra introduced byClifford [1]. It has many applications in physics [2]. In 1982, Brackx et al. [3]established the theoretical basis of Clifford analysis. Gilbert and Murray [4] studiedDirac operators and Clifford algebra in harmonic analysis. In 2009, Eriksson andOrelma [5] put forward k-hypergenic functions, which are the generalization ofmonogenic functions. Left k-hypergenic functions and right k-hypergenic functionsare the solutions to hyperbolic Dirac operators Hl

kf ðxÞ ¼ Df ðxÞ � kx0Q0f ðxÞ and

Hrkf ðxÞ ¼ f ðxÞD� k

x0Q00f ðxÞ, respectively. The definitions of hyperbolic Dirac oper-

ators are significative according to the Riesz point of view [5]. The vector form of ak-hypergenic function is the solution to the modified Riesz system or Hodge systemin PDE system [5].

This article gives Plemelj formula and Privalov theorem for hypergenic quasi-Cauchy type integrals, which play an important part in dealing with boundary valueproblems. This article generalizes and extends some results in [5–7].

*Email: [email protected]

ISSN 1747–6933 print/ISSN 1747–6941 online

� 2013 Taylor & Francis

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2. Preliminaries

See [5], let Clnþ1,0(R) be a 2nþ1-dimensional real Clifford algebra space and it hasidentity element eø¼ 1 and basis elements e0, e1, . . . , en; e0e1, . . . , en�1en; � � � ; e0e1�� en.For each integer i, j with 0� i, j� n and when i 6¼ j one has eiej¼�ejei and e2j ¼ 1.Any element in Clnþ1,0(R) has the form a ¼

PA

aAeA, eA ¼ e�1 � � � e�h or eø¼ 1, where

A¼ {�1, . . . ,�h} with 0��1<�2< � � �<�h� n and aA is a real number. The norm of

a2Clnþ1,0(R) is defined as jaj ¼�P

A

jaAj2�1

2

: If x¼x0e0þ x1e1þ � � � þxnen, it may be

observed that x2¼ jxj2 and when x 6¼ 0 the inverse of x is x�1 ¼ xjxj2:

Define be0 ¼ �e0, bej ¼ ej ð j ¼ 1, 2, . . . , nÞ and bab ¼babb: For any a2Clnþ1,0(R), wedefine ba ¼P

A

aAbeA, a0 ¼PA

aAe0A, where beA ¼ de�1,...,�h ¼ ce�1ce�1 � � �ce�h , e0A ¼ ð�1ÞjAjeA,

and jAj is the number of the elements in A.

Suppose Cln,0(R) is a 2n-dimensional real Clifford algebra space and it hasidentity element eø¼ 1 and basis elements e1, . . . , en; e1e2, . . . , en�1en; � � � ; e1e2 � � � en.

See [5], any element a2Clnþ1,0(R) may be uniquely decomposed as a¼ bþ e0c,where b, c2Cln,0(R). Using this decomposition we can define the mappingsP0: Clnþ1,0(R)!Cln,0(R) and Q0: Clnþ1,0(R)!Cln,0(R) by P0a¼ b, Q0a¼ c, whereb, c are called P0 part and Q0 part of a, respectively.

LEMMA 2.1: [5] For any a2Clnþ1,0(R), we have

P0a ¼1

2ðaþbaÞ, Q0a ¼

e02ða�baÞ: ð1Þ

3. Hypergenic-Cauchy type integrals

Let � be an open connected set in Rnþ1. The function f(x): �!Clnþ1,0(R) is denotedby f ðxÞ ¼

PA

fAðxÞeA, where fA(x)2R. The function f(x): �!Clnþ1,0(R) is said to be

continuous in � if and only if each component of f(x) is continuous in �. SupposeCr(�,Clnþ1,0(R))¼ff ðxÞj f ðxÞ : �! Clnþ1,0ðRÞ, f ðxÞ ¼

PA

fAðxÞeA,where fA(x) is

r-time continuously differentiable and r� 1 is an integer}.

For f(x)2C1(�, Clnþ1,0(R)), we introduce Dirac operators as follows:

Df ðxÞ ¼Xnj¼0

ej@f ðxÞ

@xj¼Xnj¼0

ejXA

@fAðxÞ

@xjeA,

f ðxÞD ¼Xnj¼0

@f ðxÞ

@xjej ¼

Xnj¼0

XA

@fAðxÞ

@xjeAej:

Let �1 be an open connected subset in Rnþ1n{(x0, x1, . . . , xn)jx0¼ 0}. For f(x)2C1(�1,Clnþ1,0(R)), we introduce hyperbolic Dirac operators as follows:

Hlkf ðxÞ ¼ Df ðxÞ �

k

x0Q0f ðxÞ, Hr

kf ðxÞ ¼ f ðxÞD�k

x0Q00f ðxÞ,

where k is an integer.

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Definition 3.1 f(x) is said to be a (left) monogenic function in � if f(x)2C1

(�,Clnþ1,0(R)) satisfies Df(x)¼ 0 in �. Similarly, f(x) is called a right monogenicfunction in � if f(x)2C1(�,Clnþ1,0(R)) and satisfies f(x)D¼ 0 in �.

Definition 3.2 f(x) is said to be a (left) k-hypergenic function in �1 if f(x)2C1

(�1,Clnþ1,0(R)) and satisfies Hlkf ðxÞ ¼ 0 in �1. It is a (left) monogenic function when

k¼ 0. It is called a (left) hypergenic function when k¼ n� 1. And Hln�1f ðxÞ is

abbreviated by Hlf(x).Similarly, f(x) is called a right k-hypergenic function in �1 if f(x)2C1(�1,

Clnþ1,0(R)) and satisfies Hrkf ðxÞ ¼ 0 in �1. It is a right monogenic function when

k¼ 0. It is called a right hypergenic function when k¼ n� 1. And Hrn�1f ðxÞ is

abbreviated by Hrf(x).

Definition 3.3 The function f(x): @�!Clnþ1,0(R) is said to be holder continuouson @� if there exists a positive constant M1 such that jf(x1)� f(x2)j �M1jx

1� x2j�

(0<�< 1) holds for any x1, x22 @�.

The set of all holder continuous functions from @� into Clnþ1,0 is denoted byH(�, @�,Clnþ1,0(R)).

Let

Rnþ1þ ¼ fy ¼ ð y0, y1, . . . , ynÞj y0 4 0g; Eðx, yÞ ¼

x� y

jx� yjnþ1jx�byjn�1,Fðx, yÞ ¼

bx� y

jx� yjn�1jx�byjnþ1, Að yÞ ¼2n�1yn�10

!nþ1:

Let d�ðxÞ ¼Pnj¼0

ð�1Þjejd �xj, where d �xj ¼ dx0V� � �V

dxj�1V

dxjþ1V� � �V

dxn. Then,dd�ðxÞ ¼Pnj¼0

ð�1Þjbejd �xj:

A Lyapunov surface S is a kind of surface satisfying the following threeconditions [8]:

(1) through each point in S, there is a tangent plane;(2) there is a real constant number d such that for any N02S, E is a ball with

radius d, centred at N0, and E is divided into two parts by S, the part ofS lying in the interior of E is denoted by S0, the other is in the exterior of S;and each straight line paralleling to the normal direction of S at N0 intersectsat one point;

(3) if the angle �(N1,N2) between outward normal vectors through N1, N2 isan acute angle and r is the distance between N1 and N2, then there aretwo numbers �, �(0�� 1, �> 0) independent of N1, N2 such that�(N1,N2)��r

�.

Definition 3.4 If � � Rnþ1þ is a domain, U � Rnþ1

þ is an open connected setsatisfying � � U, the boundary @� of � is a smooth compact oriented Lyapunovsurface, f(x)2H(�, @�,Clnþ1,0(R)), define

ðS½ f �Þð yÞ ¼ Að yÞ

Z@�

Eðx, yÞd�ðxÞ f ðxÞ �

Z@�

Fðx, yÞ dd�ðxÞdf ðxÞ� �ð2Þ

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as a hypergenic quasi-Cauchy type integral. (S[ f ])(y) is a normal integral when

y2Rnþ1þ n @�. (S[ f ])(y) is a singular integral when y2 @�.

Let B(y, �) be a ball with radius �, centred at y when y2 @�. @� is divided into

two parts by B(y, �). And the part of @� lying in the interior of B(y, �) is denoted by

��. If

lim�!0

Að yÞ

Z@�n��


Z@�n��

Fðx, yÞ dd�ðxÞdf ðxÞ� �¼ I,

then I is called the Cauchy principal value of singular integral and is denoted by

I ¼ P:V:ðS½ f �Þð yÞ ¼ Að yÞ

Z@�n��


Z@�n��

Fðx, yÞ dd�ðxÞdf ðxÞ� �:

LEMMA 3.5: [5] (Cauchy integral formula) If � � Rnþ1þ is a domain, U � Rnþ1

þ is an

open connected set satisfying � � U, and f(x)2C1(U,Clnþ1,0(R)) is a hypergenic

function, then for each y2�, we have

f ð yÞ ¼ Að yÞ

Z@�


Z@�

Fðx, yÞ dd�ðxÞdf ðxÞ� �: ð3Þ

LEMMA 3.6: [5] If � � Rnþ1þ is a domain and U � Rnþ1

þ is an open connected set

satisfying � � U, then for any f(x), g(x)2C1(U,Clnþ1,0(R)), we haveZ@�

1

xn�10

P0½ gðxÞd�ðxÞ f ðxÞ� ¼

Z�

1

xn�10

P0½ðHrgðxÞÞ f ðxÞ þ gðxÞðHlf ðxÞÞ�dx, ð4Þ

Z@�

Q0½ gðxÞd�ðxÞ f ðxÞ� ¼

Z�

Q0½ðHr�ðn�1ÞgðxÞÞ f ðxÞ þ gðxÞðHlf ðxÞÞ�dx: ð5Þ

LEMMA 3.7: [5] The function �1ðx, yÞ ¼ xn�10ðx�yÞ�1�ðx�byÞ�1jx�yjn�1jx�byjn�1 is left and right (n� 1)-

hypergenic on Rnþ1 n fy,byg with respect to x for each y2Rnþ1. And the function

�2ðx, yÞ ¼ðx�yÞ�1þðx�byÞ�1jx�yjn�1jx�byjn�1 is right �(n� 1)-hypergenic on Rnþ1 n fy,byg with respect to x

for each y2Rnþ1.

4. Plemelj formula for hypergenic-Cauchy type integrals

THEOREM 4.1 If � � Rnþ1þ is a domain, U � Rnþ1

þ is an open connected set satisfying

� � U, the boundary @� of � is a smooth compact oriented Lyapunov surface,

f(x)2C1(U,Clnþ1,0(R)) is a hypergenic function, then


Z@�

Eðx, yÞd�ðxÞ f ðxÞ � Fðx, yÞ dd�ðxÞdf ðxÞ� �¼

f ð yÞ, y2�,

0, y2Rnþ1þ n�:

�Proof If y2�, from Lemma 3.5, we draw the conclusion.

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If y2Rnþ1þ n�, from Lemmas 2.1 and 3.6, it can be deduced that


Z@�


Z@�

Fðx, yÞ dd�ðxÞdf ðxÞ� �¼

1

2Að yÞ

Z@�


Z@�

Fðx, yÞ dd�ðxÞdf ðxÞ��þ

Z@�

dEðx, yÞ dd�ðxÞdf ðxÞ � Z@�

dFðx, yÞd�ðxÞ f ðxÞ�þ e20

Z@�


Z@�

Fðx, yÞ dd�ðxÞdf ðxÞ��

Z@�

dEðx, yÞ dd�ðxÞdf ðxÞ þ Z@�

dFðx, yÞd�ðxÞ f ðxÞ��¼

1

2Að yÞ

Z@�

½Eðx, yÞ � dFðx, yÞ�d�ðxÞ f ðxÞ��þ

Z@�

dEðx, yÞ � Fðx, yÞ� dd�ðxÞdf ðxÞh iþ e20

Z@�

½Eðx, yÞ þ dFðx, yÞ�d�ðxÞ f ðxÞ��

Z@�

½ dEðx, yÞ þ Fðx, yÞ� dd�ðxÞdf ðxÞ��¼ Að yÞ P0

Z@�

½Eðx, yÞ � dFðx, yÞ�d�ðxÞ f ðxÞ� ��þe0Q0

Z@�

½Eðx, yÞ þ dFðx, yÞ�d�ðxÞ f ðxÞ� ��¼ Að yÞ

Z@�

1

xn�10

P0½�1ðx, yÞd�ðxÞ f ðxÞ� þ e0

Z@�

Q0½�2ðx, yÞd�ðxÞ f ðxÞ�

� �¼ Að yÞ

Z�

1

xn�10

P0½ðHr�1ðx, yÞÞ f ðxÞ þ�1ðx, yÞðH

lf ðxÞÞ�dx

�þe0

Z�

Q0½ðHr�ðn�1Þ�2ðx, yÞÞ f ðxÞ þ�2ðx, yÞðH

lf ðxÞÞ�dx

�:

From lemma 3.7 it follows that Hr�1ðx, yÞ ¼ 0, Hr�ðn�1Þ�2ðx, yÞ ¼ 0: As f(x)2C1

(U,Clnþ1,0(R)) is a hypergenic function, Hlf(x)¼ 0. Hence (S[ f ])(y)¼ 0 when

y2Rnþ1þ n�.

COROLLARY 4.2 If � � Rnþ1þ is a domain, U � Rnþ1


satisfying � � U, the boundary @� of � is a smooth compact oriented Lyapunov

surface, then

ðS½1�Þð yÞ ¼ Að yÞ

Z@�

Eðx, yÞd�ðxÞ �

Z@�

Fðx, yÞ dd�ðxÞ� �¼

1, y2�,

0, y2Rnþ1þ n�:

�


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THEOREM 4.3 If � � Rnþ1þ is a domain, U � Rnþ1

þ is an open connected set satisfying

� � U, the boundary @� of � is a smooth compact oriented Lyapunov surface, y2 @�,

for any f(x)2H(�, @�,Clnþ1,0(R)), then

P:V:ðS½ f �Þð yÞ ¼ Að yÞ

Z@�

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð yÞ�

��

Z@�

Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð yÞ�

�þ1

2f ð yÞ:

Proof Let B(y, �) be the ball with radius �, centred at y. @� is divided into two parts

by B(y, �). And the part of @� lying in the interior of B(y, �) is denoted by ��. It may

be observed that

P:V:ðS½ f �Þð yÞ ¼ lim�!0

Að yÞ

Z@�n��


Z@�n��

Fðx, yÞ dd�ðxÞdf ðxÞ� �� ¼ lim

�!0Að yÞ

Z@�n��


��

Z@�n��


�þ Að yÞ

Z@�n��

Eðx, yÞd�ðxÞ f ð yÞ � Fðx, yÞ dd�ðxÞf ð yÞ�� :

In the following Mj( j¼ 2, 3, . . . , 84) are all positive constants. See [8], we have

jd�ðxÞj ¼ jdSðxÞj �M2�n�1d�: ð6Þ

Suppose the diameter of @� is L. As f(x)2H(�, @�, Clnþ1,0(R)), by (6) we haveZ@�n��


�� M3

Z@�n��

jd�ðxÞj

jx� yjn��M4

Z L

�

��1d�,

which is convergent. Hence,

lim�!0

Z@�n��

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð yÞ� ¼

Z@�

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð yÞ�: ð7Þ

Moreover,Z@�n��


��

Z@�n��

jFðx, yÞjj dd�ðxÞjjdf ðxÞ � f ð yÞj

�M5

Z@�n��

jbx� yj

jx� yjn�1jx�byjnþ1jd�ðxÞj�M6

Z@�n��

1

jx� yjn�1jd�ðxÞj

�M7

Z L

�

1

�n�1�n�1d� ¼M7

Z L

�

d�,

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which is convergent. Hence,

lim�!0

Z@�n��

Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð yÞ� ¼

Z@�

Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð yÞ�:

Suppose that @Bð y, �ÞTðRnþ1þ ��Þ is denoted by Dout. From Corollary 4.2, it follows

that

f ð yÞ ¼ Að yÞ

Zð@�n��Þ

SDout

Eðx, yÞd�ðxÞ f ð yÞ � Fðx, yÞ dd�ðxÞf ð yÞ" #:

Moreover,

Að yÞ

ZDout

Eðx, yÞd�ðxÞ f ð yÞ �

ZDout

Fðx, yÞ dd�ðxÞf ð yÞ� �¼ Að yÞ

ZDout

x� y

jx� yjnþ1jx�byjn�1d�ðxÞ f ð yÞ �Z

Dout

bx� y

jx� yjn�1jx�byjnþ1 dd�ðxÞf ð yÞ� �

¼ Að yÞ

ZDout

1

jx� yjnjx�byjn�1dSðxÞ f ð yÞ �Z

Dout

ðbx� yÞðbx�byÞjx� yjnjx�byjnþ1dSðxÞ f ð yÞ

� �¼ Að yÞ

ZDout

1

�njx�byjn�1dSðxÞ f ð yÞ �Z

Dout

ðbx� yÞðbx�byÞjx� yjnjx�byjnþ1dSðxÞ f ð yÞ�:

When �! 0, we have x! y, consequently bx!by, and jx�byj ! j y�byj ¼ 2y0:Hence,

lim�!0

Að yÞ

Z@�n��

Eðx, yÞd�ðxÞ f ð yÞ � Fðx, yÞ dd�ðxÞf ð yÞ� �¼ f ð yÞ �

1

2f ð yÞ ¼

1

2f ð yÞ:

g

COROLLARY 4.4 If � � Rnþ1þ is a domain, U � Rnþ1



surface, y2 @�, then P:V:ðS½1�Þð yÞ ¼ 12:

LEMMA 4.5: [8] (Hile lemma) If x, y2Rnþ1(n� 2) and m(� 0) is an integer, then

x

jxjmþ2�

y

j yjmþ2

�� Pmðx, yÞ

jxjmþ1j yjmþ1jx� yj,

where

Pmðx, yÞ ¼

Pmk¼0

jxjm�kj yjk, m 6¼ 0,

1, m ¼ 0:

8<:For y0 ¼ ð y00, y

01, . . . , y0nÞ 2 @� and y ¼ ð y0, y1, . . . , ynÞ 2R

nþ1þ n @�, we have

ðS½ f �Þð yÞ ¼ �ð yÞ þ Að yÞ

Z@�

Eðx, yÞd�ðxÞ f ð y0Þ � Fðx, yÞ dd�ðxÞf ð y0Þ� �, ð8Þ


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where

�ð yÞ ¼ Að yÞ

Z@�

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð y0Þ� � Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð y0Þ�

� �:

THEOREM 4.6 If � � Rnþ1þ ðn � 2Þ is a domain, U � Rnþ1

þ is an open connected setsatisfying � � U, the boundary @� of � is a smooth compact oriented Lyapunovsurface, and f(x)2H(�, @�, Clnþ1,0(R)), then for each y02 @�, we have

limy! y0

y2Rnþ1þ n @�

�ð yÞ ¼ �ð y0Þ:

Proof First, we assume that y! y0 which is not along the direction of tangentplane at y02 @�. This means that the angle between the tangent plane of @� and yy0

is greater than 2�0(�0 is a positive constant), otherwise we call y! y0 along thedirection of tangent plane. See [8], in this case we have

x� y0

x� y

�� M8,y� y0

x� y

�� M8: ð9Þ

Let B(y0, �) be the ball with radius �0, centred at y0. @� is divided into two parts byB(y0, �). And the part of @� lying in the interior of B(y0, �) is denoted by ��. It may beobserved that

j�ð yÞ ��ð y0Þj � Að yÞ

Z@�

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð y0Þ�

��Að y0Þ

Z@�

Eðx, y0Þd�ðxÞ½ f ðxÞ � f ð y0Þ�

��þ Að yÞ

Z@�

Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð y0Þ�

��Að y0Þ

Z@�

Fðx, y0Þ dd�ðxÞ½df ðxÞ � f ð y0Þ�

�� J1 þ J2:

J1 � Að yÞ

Z@�

½Eðx, yÞ � Eðx, y0Þ�d�ðxÞ½ f ðxÞ � f ð y0Þ�

�� þ ðAð yÞ � Að y0ÞÞ

Z@�


�� M9

Z��

½Eðx, yÞ � Eðx, y0Þ�d�ðxÞ½ f ðxÞ � f ð y0Þ

��

� ��þ

Z@�n��

½Eðx, yÞ � Eðx, y0Þ�d�ðxÞ½ f ðxÞ � f ð y0Þ�

��)

þ ðAð yÞ � Að y0ÞÞ

Z@�


�� ¼M9½J3 þ J4� þ J5:

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Eðx, yÞ � Eðx, y0Þ�� ¼ x� y

jx� yjnþ1jx�byjn�1 � x� y0

jx� y0jnþ1jx� by0jn�1��

��

x� y

jx� yjnþ1jx�byjn�1 � x� y0

jx� y0jnþ1jx�byjn�1�� þ

x� y0

jx� y0jnþ1jx�byjn�1 � x� y0


��

1

jx�byjn�1 x� y

jx� yjnþ1�

x� y0

jx� y0jnþ1

�� þjx� y0j

jx� y0jnþ11

jx�byjn�1 � 1

jx� by0jn�1��

��:As 1

jx�byjn�1 is continuous on @�, there exists a positive constant M10 such that

1

jx�byjn�1 �M10: From Hile lemma and (9), it follows that

J3 �

Z��

M10

Xn�1k¼0

j y� y0j

jx� yjkþ1jx� y0jn�kþM11

1

jx� y0jn

" #d�ðxÞ½ f ðxÞ � f ð y0Þ��

�

Z��

M10

Xn�1k¼0

j y� y0j

jx� yj

jx� y0jk

jx� yjk1

jx� y0jnþM11

1

jx� y0jn

" #d�ðxÞ½ f ðxÞ � f ð y0Þ��

�

Z��

M121

jx� y0jnjd�ðxÞjM1jx� y0j� �M13

Z �

0

�n�1

�n��d� ¼M13

Z �

0

��1d�:

Hence, for any "> 0, there exists a �> 0, when �<�, we have J3<".Take a fixed � such that 0<�<�. When x2 @� \ ��, we have

jEðx, yÞ � Eðx, y0Þj

�1

jx�byjn�1Xn�1

k¼0

jx� yjn�1�kjx� y0jk

jx� yjnjx� y0jnj y� y0j þ

1

jx� y0jnjx� by0jn�1 � jx�byjn�1jx�byjn�1jx� by0jn�1

��

�M10

Xn�1k¼0

x� y0

x� y

�� kþ1 j y� y0j

jx� y0jnþ1þM14

j y� y0j

jx� y0jn:

Hence,

J4 �

Z L

�

M151

�nþ1��þ

1

�n��

�n�1d�j y� y0j:

When y! y0 we have J4<". As

lim�!0

Z@�n��

x� y0

jx� y0jnþ1jx� by0jn�1d�ðxÞ½ f ðxÞ � f ð y0Þ�

exists, there exists �1> 0(�1<�) such thatZ@�n��

x� y0

jx� y0jnþ1jx� by0jn�1d�ðxÞ½ f ðxÞ � f ð y0Þ�

��5M16


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when 0<�< �1. Hence, J5 �M17j yn�10 � ð y00Þ

n�1j �M18j y� y0j. When y! y0 we

have J5<". Hence, we have J1<" when y! y0. Similarly, we can prove J2<" wheny! y0. As � is independent of y0, �(y) is uniformly continuous on @�.

Next, we assume that y! y0 which is along the direction of tangent plane aty02 @�. When y! y0, we can choose x2 @� such that jx� yj! 0, jx� y0j! 0. As�(y) is uniformly continuous on @�, for any "> 0, when jx� yj, jx� y0j is littleenough we have j�ð yÞ ��ðxÞj �

"

2, j�ðxÞ ��ð y0Þj �

"

2: Hence, j�(y)��(y0)j �

j�(y)��(x)j þ j�(x)��(y0)j � ". g

Denote � by �þ,�S@� by �þ, Rnþ1 n� by ��, and ��

S@� by ��,

respectively.By Theorem 4.6, we can establish the Plemelj formula for hypergenic quasi-

Cauchy type integrals as follows.

THEOREM 4.7 (Plemelj formula) If � � Rnþ1þ ðn � 2Þ is a domain, U � Rnþ1

þ is anopen connected set satisfying � � U, the boundary @� of � is a smooth compactoriented Lyapunov surface, and f(x)2H(�, @�,Clnþ1,0(R)), then for each y02 @�,we have

ðS½ f �Þþð y0Þ ¼ P:V:ðS½ f �Þð y0Þ þ1

2f ð y0Þ

ðS½ f �Þ�ð y0Þ ¼ P:V:ðS½ f �Þð y0Þ �1

2f ð y0Þ

8><>: ð10Þ

or

ðS½ f �Þþð y0Þ � ðS½ f �Þ�ð y0Þ ¼ f ð y0Þ

ðS½ f �Þþð y0Þ þ ðS½ f �Þ�ð y0Þ ¼ 2P:V:ðS½ f �Þð y0Þ

(ð11Þ

Proof From (8), Theorem 4.6, Corollaries 4.2 and 4.4, it follows thatðS½ f �Þð yÞ !ðS½ f �Þþð y0Þ ¼ �ð y0Þ þ f ð y0Þ ¼ P:V:ðS½ f �Þð y0Þ þ 1

2 f ð y0Þ when y! (y0)þ. And ðS½ f �Þ

ð yÞ ! ðS½ f �Þ�ð y0Þ ¼ �ð y0Þ ¼ P:V:ðS½ f �Þð y0Þ � 12 f ð y

0Þ when y! (y0)�. g

Remark Plemelj formula for hypergenic quasi-Cauchy type integrals plays animportant part in dealing with boundary value problems of hypergenic functions.

5. Privalov theorem for hypergenic-Cauchy type integrals

THEOREM 5.1 (Privalov theorem) If � � Rnþ1þ ðn � 2Þ is a domain, U � Rnþ1

þ is anopen connected set satisfying � � U, the boundary @� of � is a smooth compactoriented Lyapunov surface, and f(x)2H(�, @�, Clnþ1,0(R)), then ðS½ f �Þð yÞ 2Hð�,�þ,Clnþ1,0ðRÞÞ and ðS½ f �Þð yÞ 2Hð�,��,Clnþ1,0ðRÞÞ, where (S[ f ])(y) means (S[ f ])�(y)when y2 @�.

Proof We only need to prove ðS½ f �Þð yÞ 2Hð�,�þ,Clnþ1,0ðRÞÞ and the otherconclusion can be proved in the same way.

Case 1 For any y1, y22 @�, we need to prove j(S[ f ])þ(y1)� (S[ f ])þ(y2)j �M19jy

1� y2j�.

First, for any y1, y22 @�, we prove j(S[ f ])�(y1)� (S[ f ])�(y2)j �M20jy1� y2j�.

For any y1, y22 @�, let jy2� y1j ¼ �, (0<�< 1, 6�< d, which is the d definedin Lyapunov surface [8]). Let B(y1, 3�) be the ball with radius 3�, centred at y1.

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Suppose that @�T

Bð y1, 3�Þ is denoted by �3�. From Theorem 4.7 and Corollary 4.2,it follows that

ðS½ f �Þ�ð y1Þ � ðS½ f �Þ�ð y2Þ�� ¼ P:V:ðS½ f �Þð y1Þ �

1

2f ð y1Þ

� �� P:V:ðS½ f �Þð y2Þ �

1

2f ð y2Þ

� �� ¼ Að y1Þ

Z@�

Eðx, y1Þd�ðxÞ½ f ðxÞ � f ð y1Þ� �

Z@�

Fðx, y1Þ dd�ðxÞ½df ðxÞ � f ð y1Þ

��

�� Að y2Þ

Z@�


Z@�


� ��M21

Z�3�


Z�3�


�� þ

Z�3�


Z�3�


�� þ Að y1Þ

Z@�n�3�

Eðx, y1Þd�ðxÞ½ f ðxÞ � f ð y1Þ� � Fðx, y1Þ dd�ðxÞ½df ðxÞ � f ð y1Þ

��

��

�Að y2Þ

Z@�n�3�

Eðx, y2Þd�ðxÞ½ f ðxÞ � f ð y2Þ� � Fðx, y2Þ dd�ðxÞ½df ðxÞ � f ð y2Þ�

� ��¼M21ðI1 þ I2Þ þ I3:

I1 �

Z�3�

M22jd�ðxÞj

jx� y1jn��jx� by1jn�1 þZ�3�

M23jd�ðxÞj

jx� y1jn�1jx� by1jn�

Z 3�

0

M24��1d�þ

Z 3�

0

M25d� �M26�� ¼M26j y

1 � y2j�

As jx� y2j � jx� y1j � jy1� y2j � 2�, similarly we can prove I2�M27jy1� y2j�.

I3 ¼ Að y1Þ

Z@�n�3�

½Eðx, y1Þ � Eðx, y2Þ�d�ðxÞ½ f ðxÞ � f ð y1Þ�

��þ Að y1Þ

Z@�n�3�


� Að y2Þ

Z@�n�3�


þ Að y2Þ

Z@�n�3�

½Fðx, y2Þ � Fðx, y1Þ� dd�ðxÞ½df ðxÞ � f ð y2Þ�

þ Að y2Þ

Z@�n�3�


�Að y1Þ

Z@�n�3�


��M21

Z@�n�3�

½Eðx, y1Þ � Eðx, y2Þ�d�ðxÞ½ f ðxÞ � f ð y1Þ�

��

(

þ

Z@�n�3�

½Fðx, y2Þ � Fðx, y1Þ� dd�ðxÞ½df ðxÞ � f ð y2Þ�

��


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þ Að y1Þ

Z@�n�3�


��(

�Að y2Þ

Z@�n�3�


��)

þ Að y2Þ

Z@�n�3�


��(

�Að y1Þ

Z@�n�3�


��)

¼M21½I4 þ I5� þ I6 þ I7:

As x2 @�n�3� and y1, y22 �3�, jx�y2

x�y1jmþ1 and jx�y

1

x�y2jmþ1ðm ¼ 0, 1, . . . , n� 1Þ are

continuous on @�n�3� which is compact. Hence, there exists a positive constant

M28 such that jx�y2

x�y1jmþ1 �M28, j

x�y1

x�y2jmþ1 �M28ðm ¼ 0, 1, . . . , n� 1Þ: From Hile

lemma and the above inequalities, it follows that

I4 �

Z@�n�3�

x� y1

jx� y1jnþ1jx� by1jn�1 � x� y2


��"

þx� y2

jx� y2jnþ1jx� by1jn�1 � x� y2


��#d�ðxÞjj f ðxÞ � f ð y1Þ��

�

Z@�n�3�

1

jx� by1jn�1 x� y1

jx� y1jnþ1�

x� y2

jx� y2jnþ1

�� jd�ðxÞjj f ðxÞ � f ð y1Þj

þ

Z@�n�3�

1

jx� y2jn1

jx� by1jn�1 � 1

jx� by2jn�1��

��jd�ðxÞjj f ðxÞ � f ð y1Þj

�

Z@�n�3�

M29

Xn�1k¼0

x� y2

x� y1

�� kþ1 j y1 � y2j

jx� y2jnþ1þM30

j y1 � y2j

jx� y2jn

" #jd�ðxÞjjx� y1j�

�

Z L

3�

M31jx� y1j�

jx� y2jnþ1þM32

jx� y1j�

jx� y2jn

� ��n�1d�j y1 � y2j

� M33

Z L

3�

��2d�þM34

Z L

3�

��1d�

� �j y1 � y2j �M35j y

1 � y2j�:

In the same way we can prove I5�M36jy1� y2j�.

I6 � ðAð y1Þ � Að y2ÞÞ

Z@�n�3�

x� y2

jx� y2jnþ1jx� by2jn�1d�ðxÞ f ðxÞ��

��þ Að y1Þ

Z@�n�3�

x� y2

jx� y2jnþ1jx� by2jn�1d�ðxÞ½ f ð y1Þ � f ð y2Þ�

��

þ ðAð y1Þ � Að y2ÞÞ

Z@�n�3�

x� y2

jx� y2jnþ1jx� by2jn�1d�ðxÞ f ð y2Þ��

��:

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As

lim�!0

Z@�n�3�

x� y2

jx� y2jnþ1jx� by2jn�1d�ðxÞexists, there exists a �2> 0 such that

Z@�n�3�

x� y2

jx� y2jnþ1jx� by2jn�1d�ðxÞ��

��5M37

when 0<�< �2. Hence, I6 �M38fjð y10Þ

n�1� ð y20Þ

n�1j þ j f ð y1Þ � f ð y2Þjþ jð y10Þ

n�1�

ð y20Þn�1jg �M39j y

1 � y2j�. In the same way we can prove I7�M40jy1� y2j�.

Consequently, I3�M41jy1� y2j�. Now, j(S[ f ])�(y1)� (S[ f ])�(y2)j �M20jy

1� y2j�.

Next, by (S[ f ])þ(y)¼ (S[ f ])�(y)þ f(y) it is easy to prove j(S[ f ])þ(y1)� (S[ f ])þ

(y2)j �M19jy1� y2j�.

Case 2 For any y2�, y02 @�, we need to prove j(S[ f ])(y)� (S[ f ])þ(y0)j �

M42jy� y0j�.As @� is compact, for any y2Rnþ1n@�, there exists at least a point y@�2 @� such

that j y@� � yj ¼ infz2 @�jz� yj, we call y@� is the nearest point of y on @�. If y! y0

then y@�! y0 and for any z2 @�, we have

jz� y@�j � 2jz� yj: ð12Þ

For any z2 @� and f(x)2H(�, @�,Clnþ1,0(R)), by (12) we have

j f ðzÞ � f ð y@�Þj �M12�jz� yj�: ð13Þ

For any y2�, y02 @�, let �¼ jy� y0j, we have

j y� y@�j � j y� y0j ¼ �: ð14Þ

jðS½ f �Þð yÞ � ðS½ f �Þþð y0Þj

� jðS½ f �Þð yÞ � ðS½ f �Þþð y@�Þj þ jðS½ f �Þþð y@�Þ � ðS½ f �Þþð y0Þj

¼ I8 þ I9:

By virtue of case 1 and (12), we have I9¼ j(S[ f ])þ(y@�)� (S[ f ])þ(y0)j �

M43jy@�� y0j��M44jy� y0j�.

We first assume that y! y@� which is not along the direction of tangent plane

at y@�. See [8], we have

x� y@�

x� y

�� M45: ð15Þ


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Let B(y@�, 3�) be the ball with radius 3�, centred at y@�. Suppose that @�T

Bð y@�, 3�Þis denoted by �3�. From Plemelj formula, Corollaries 4.2 and 4.4, it follows that

I8 ¼ jðS½ f �ÞðyÞ � ðS½ f �Þþðy@�Þj

� AðyÞ

Z@�

Eðx,yÞd�ðxÞ½ f ðxÞ � f ðy@�Þ� �

Z@�

Fðx,yÞ dd�ðxÞ½df ðxÞ � f ðy@�Þ

��

�� Aðy@�Þ

Z@�

Eðx,y@�Þd�ðxÞ½ f ðxÞ � f ðy@�Þ� �

Z@�

Fðx,y@�Þ dd�ðxÞ½df ðxÞ � f ðy@�Þ

��

��M46

Z�3�


Z�3�


��

� ��þ

Z�3�

Eðx,y@�Þd�ðxÞ½ f ðxÞ � f ðy@�Þ� �

Z�3�

Fðx,y@�Þ dd�ðxÞ½df ðxÞ � f ðy@�Þ

��

��þ AðyÞ

Z@�n�3�


Z@�n�3�


��

��

�Aðy@�Þ

Z@�n�3�

Eðx,y@�Þd�ðxÞ½ f ðxÞ � f ðy@�Þ�

��

Z@�n�3�

Fðx,y@�Þ dd�ðxÞ½df ðxÞ � f ðy@�Þ�

��M46½I10þ I11� þ I12:

It follows from (12) and (15) that

I10 �

Z�3�

Eðx, yÞd�ðxÞ½ f ðxÞ � f ð y@�Þ�

�� þ Z�3�

Fðx, yÞ dd�ðxÞ½df ðxÞ � f ð y@�Þ�

��

Z�3�

x� y

jx� yjnþ1jx�byjn�1d�ðxÞ½ f ðxÞ � f ð y@�Þ�

�� þ

Z�3�

bx� y

jx� yjn�1jx�byjnþ1 dd�ðxÞ½df ðxÞ � f ð y@�Þ�

��

Z�3�

x� y

jx� yjnþ1jx�byjn�1d�ðxÞ½ f ðxÞ � f ð y@�Þ�

�� þM47

Z�3�

jd�ðxÞj

jx� yjn�1jx�byjn�M48

Z�3�

1

jx� yjnjd�ðxÞjjx� y@�j� þM49

Z�3�

jd�ðxÞj

jx� yjn�1

�M50

Z�3�

1

jx� y@�jn��jd�ðxÞj þ

Z�3�

jd�ðxÞj

jx� y@�jn�1

� ��M51

Z 3�

0

��1d�þ

Z 3�

0

d�

�M52j y� y0j�:

Similarly, we can prove I11�M53jy� y0j�.

I12 � Að yÞ

Z@�n�3�

Eðx, yÞd�ðxÞ � Að y@�Þ

Z@�n�3�

Eðx, y@�Þd�ðxÞ

� �½ f ðxÞ � f ð y@�Þ�

��

þ Að yÞ

Z@�n�3�

Fðx, yÞ dd�ðxÞ � Að y@�Þ

Z@�n�3�

Fðx, y@�Þ dd�ðxÞ� �½df ðxÞ � f ð y@�Þ�

��

¼ I13 þ I14:

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From Hile lemma, (9), (7) and (14) it follows that

I13 � AðyÞ

Z@�n�3�

½Eðx,yÞ �Eðx,y@�Þ�d�ðxÞ½ f ðxÞ � f ðy@�Þ�

��

þ ðAðyÞ �Aðy@�ÞÞ

Z@�n�3�

Eðx,y@�Þd�ðxÞ½ f ðxÞ � f ðy@�Þ�

��

�

Z@�n�3�

M54

Xn�1k¼0

x� y@�

x� y

�� kþ1 jy� y@�j

jx� y@�jnþ1þM55

jy� y@�j

jx� y@�jn

" #jd�ðxÞjj f ðxÞ � f ðy@�Þj

þM56jyn�10 � ðy@�0 Þ

n�1j

�M57

Z@�n�3�

jy� y@�j

jx� y@�jnþ1��jd�ðxÞj þ

Z@�n�3�

jy� y@�j

jx� y@�jnjd�ðxÞj

� �þM58jy� y@�j

�M59

Z@�n�3�

jy� y0j

jx� y@�jnþ1��jd�ðxÞj þ

Z@�n�3�

jy� y0j

jx� y@�jn��jd�ðxÞj

� �þM60jy� y0j

�M61

Z L

3�

�n�1

�nþ1��þ�n�1

�n��

� �d�jy� y0j þM60jy� y0j �M62jy� y0j �M63jy� y0j�:

Similarly, we can prove I14�M64jy� y0j�. Now I12�M65jy� y0j�, i.e.I8�M66jy� y0j�.

If y! y@� which is along the direction of tangent plane at y@�, we can choose apoint y0 which is not on tangent plane. As j(S[ f ])(y)� (S[ f ])þ(y@�)j � j(S[ f ])(y)�(S[ f ])þ(y0)j þ j(S[ f ])(y0)� (S[ f ])þ(y@�)j, the proof can be finished by the case thaty! y@�, which is not along the direction of tangent plane at y@� and case 1. Hencej(S[ f ])(y)� (S[ f ])þ(y0)j �M42jy� y0j�. We finish the proof for case 2.

Case 3 For any y12�, y22�, we need to prove j(S[ f ])(y1)� (S[ f ])(y2)j �M67jy

1� y2j�.

As the linear segment of y1y2 and @� are compact, there exist ey2 y1y2 andfy@� 2 @� such that jfy@� �eyj ¼ infz2 @�, y2 y1y2

jz� yj: Let �¼ jy1� y2j, �0 ¼ jfy@� �eyj:(1) If �0¼ 0, then fy@� ¼ey2 @�: From case 2, it follows that

jðS½ f �Þð y1Þ � ðS½ f �Þð y2Þj

� jðS½ f �Þð y1Þ � ðS½ f �ÞþðeyÞj þ jðS½ f �ÞþðeyÞ � ðS½ f �Þð y2Þj�M68j y

1 �eyj� þM69j y2 �eyj� �M67j y

1 � y2j�:

(2) If �0> 0 and �� 0, then j y1 � fy@�j � 2�, j y2 � fy@�j � 2�: From case 2, we

have


� jðS½ f �Þð y1Þ � ðS½ f �Þþðfy@�Þj þ jðS½ f �Þþðfy@�Þ � ðS½ f �Þð y2Þj�M70j y

1 � fy@�j� þM71j y2 � fy@�j� �M67j y

1 � y2j�:

(3) If �0> 0 and �< �0, for any x2 @�, then jx� y1j � 2jx� y2j,jx� y2j � 2jx� y1j. Hence,

jx� fy@�j � 3jx� y1j, jx� fy@�j � 3jx� y2j: ð16Þ


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Let Bðfy@�, 3�Þ be a ball with radius 3�, centred at fy@�. Suppose that @�TBðfy@�, 3�Þ is

denoted by �3�. From Plemelj formula and Corollary 4.2, it follows that


� Að y1Þ

Z@�

Eðx, y1Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Z@�

Fðx, y1Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ�� Að y2Þ

Z@�

Eðx, y2Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Z@�

Fðx, y2Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ�� M72

Z�3�

Eðx, y1Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Z�3�

Fðx, y1Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ�� þ

Z�3�

Eðx, y2Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Z�3�

Fðx, y2Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ�� þ Að y1Þ

Z@�n�3�

Eðx, y1Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Fðx, y1Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ��Að y2Þ

Z@�n�3�

Eðx, y2Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ� � Fðx, y2Þ dd�ðxÞ½df ðxÞ � f ðfy@�Þ��M72½I15 þ I16� þ I17:

The proofs of the inequalities I15�M73jy1� y2j� and I16�M74jy

1� y2j� are

similar to the proof of I10.

I17 � Að y1Þ

Z@�n�3�

Eðx, y1Þd�ðxÞ � Að y2Þ

Z@�n�3�

Eðx, y2Þd�ðxÞ

� �½ f ðxÞ � f ðfy@�Þ��

��þ Að y1Þ

Z@�n�3�

Fðx, y1Þ dd�ðxÞ � Að y2Þ

Z@�n�3�

Fðx, y2Þ dd�ðxÞ� �½df ðxÞ � f ðfy@�Þ��

�� I18 þ I19:

From Hile lemma and (16), it follows that

I18 � Að y1Þ

Z@�n�3�

½Eðx, y1Þ � Eðx, y2Þ�d�ðxÞ½ f ðxÞ � f ðfy@�Þ��þ ðAð y2Þ � Að y1ÞÞ

Z@�n�3�

Eðx, y2Þd�ðxÞ½ f ðxÞ � f ðfy@�Þ��

�M75

Z@�n�3�

1

jx� by1jn�1Xn�1k¼0

jx� y1jn�1�kjx� y2jk

jx� y1jnjx� y2jnj y1 � y2j

"

þ1

jx� y2jnjx� by2jn�1 � jx� by1jn�1jx� by1jn�1jx� by2jn�1

��#jd�ðxÞjM1jx� fy@�j� þM76j y

20 � y10j

�

Z@�n�3�

M77

Xn�1k¼0

x� y2

x� y1

�� kþ1 j y1 � y2j

jx� y2jnþ1þM78

j y1 � y2j

jx� y2jn

" #jd�ðxÞjM1jx� fy@�j�

þM76j y1 � y2j

16 Yonghong Xie

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�

Z@�n�3�

M79j y1 � y2j

jx� y2jnþ1þM78

j y1 � y2j

jx� y2jn

� �jd�ðxÞjM1jx� fy@�j� þM76j y

1 � y2j

�

Z L

�3�

M80�n�1þ�

�nþ1þM81

�n�1þ�

�n

� �d�j y1 � y2j þM76j y

1 � y2j

�M82j y1 � y2j�

Similarly, we can prove I19�M83jy1� y2j�. Hence, j(S[ f ])(y1)� (S[ f ])(y2)j �

M67jy1� y2j�. We finish the proof for case 3. g

COROLLARY 5.2 If � � Rnþ1þ ðn52Þ is a domain, U � Rnþ1



surface, and f(y)2H(�, @�,Clnþ1,0(R)), then P.V.(S[ f ])(y)2H(�, @�, Clnþ1,0(R)).

Proof From the proof of case 1 in Theorem 5.1, it follows that j(S[ f ])�(y1)� (S[ f ])�

(y2)j �M20jy1� y2j�, j(S[ f ])þ(y1)� (S[ f ])þ(y2)j �M19jy1� y2j

�. By Theorem 4.7,

we have P:V:S½ f �Þð yÞ ¼ ðS½ f �Þþð yÞþðS½ f �Þ�ð yÞ

2 : It is easy to prove that

jP.V.(S[ f ])(y1)�P.V.(S[ f ])(y2)j �M84jy1� y2j�. g

Acknowledgements

This work was supported by the National Science Foundation of China (Grant No.11101139), Hebei Key Laboratory of Computational Mathematics and Applications, theScience Foundation of Hebei Province (A2010000351; A2010000346), the Science Foundationof Zhejiang Province (Y6090036; Y6100219) and the Foundation of Hebei Normal University(L2009y02).

References

[1] Clifford WK. Applications of Grassman’s extensive algebra. Am J Math. 1878;4:350–358.[2] Dirac PAM. The quantum theory of the electron. Proc Roy Soc A. 1928;117(778):610–624.[3] Brackx F, Delanghe R, Sommen F. Clifford Analysis. Research Notes in Mathematics,

Pitman Books Limits, 1982.[4] Gilbert R, Murray MA. Clifford algebra and Dirac operators in harmonic analysis.

Cambridge University Press, 1991.[5] Eriksson SL, Orelma H. Hyperbolic function theory in the Clifford algebra Clnþ1,0. Adv

Appl Clifford Algebra. 2009;19:283–301.[6] Qiao YY, Bernstein S, Eriksson SL, Ryan J. Function theory for Laplace and Dirac-Hodge

operators in hyperbolic space. J D’Analyse Math. 2006;98(1):43–64.[7] Yang HJ, Xie YH. The Privalov theorem of some singular integral operators in real

Clifford analysis. Numer Func Anal and Optim. 2011;32(2):189–211.[8] Huang S, Qiao YY, Wen GC. Real and complex Clifford analysis. New York: Springer

Press; 2006.


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Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis