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Cauchy integral formula and Plemeljformula of bihypermonogenic functionsin real Clifford analysisXiaoli Bian a b , Sirkka-Liisa Eriksson c , Junxia Li a & Yuying Qiao aa College of Mathematics and Information, Hebei NormalUniversity, Shijiazhuang, 050016, P. R. Chinab Department of Mathematics and Physics, Tianjin University ofTechnology and Education, Tianjin, 300222, P. R. Chinac Department of Mathematics, Tampere University of Technology,P.O. Box 553, FI-33101 Tampere, FinlandPublished online: 18 Sep 2009.

To cite this article: Xiaoli Bian , Sirkka-Liisa Eriksson , Junxia Li & Yuying Qiao (2009): Cauchyintegral formula and Plemelj formula of bihypermonogenic functions in real Clifford analysis,Complex Variables and Elliptic Equations: An International Journal, 54:10, 957-976

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Complex Variables and Elliptic EquationsVol. 54, No. 10, October 2009, 957–976

Cauchy integral formula and Plemelj formula of

bihypermonogenic functions in real Clifford analysis

Xiaoli Bianab*, Sirkka-Liisa Erikssonc, Junxia Lia and Yuying Qiaoa

aCollege of Mathematics and Information, Hebei Normal University, Shijiazhuang050016, P. R. China; bDepartment of Mathematics and Physics, Tianjin University ofTechnology and Education, Tianjin 300222, P. R. China; cDepartment of Mathematics,

Tampere University of Technology, P.O. Box 553, FI-33101 Tampere, Finland

Communicated by H. Begehr

(Received 18 October 2008; final version received 21 July 2009)

In the first part of this article, we give the definition of bihypermonogenicfunctions in Clifford analysis. Using the idea of quasi-permutation,introduced by Sha Huang [Quasi-permutations and generalized regularfunctions in real Clifford analysis, J. Sys. Sci. and Math. Sci 18 (1998),pp. 380–384], we state an equivalent condition for bihypermonogenicity. Inthe second part, we discuss the Cauchy integral formula and Plemeljformula for the bihypermonogenic functions in real Clifford analysis.

Keywords: Clifford analysis; Cauchy integral formula; bihypermonogenicfunction; Plemelj formula

AMS Subject Classifications: 30G30; 34B05; 31B10

1. Introduction

Clifford algebra is an associative and a noncommutative algebraic structure that wasintroduced in the middle of the 1800s. Clifford analysis is an important branch ofmodern analysis that studies functions defined on R

nþ1 with the values in a Cliffordalgebra [1]. Clifford analysis possesses an important theoretical and applicable valuein many fields. Since 1987, Wen, Huang, Qiao, Ryan etc., have done a lot of work onboundary value problems and the properties of monogenic functions in real Cliffordanalysis [2–5]. In Clifford analysis, let D ¼

Pni¼0 ei

@@xi. A solution of the equation

Df¼ 0 is called a monogenic function. Monogenic functions are a generalization ofanalytic functions in complex analysis and their properties have been extensivelystudied (see e.g. [1]). Hengartner [6] and Leutwiler [7,8] researched hypermonogenicfunctions of R

3 [9–11]. Eriksson and Leutwiler [10] introduced hypermonogenicfunctions in Clifford analysis, researched some of its properties and discussed the

*Corresponding author. Email: bianxiaoli@yeah.net

ISSN 1747–6933 print/ISSN 1747–6941 online

� 2009 Taylor & Francis

DOI: 10.1080/17476930903197207

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integral representation for hypermonogenic functions. Qiao [12] discussed theboundary value problems of hypermonogenic functions. In recent years, Zhang andDu [13] investigated Riemann boundary value problems and singular integralequations in Clifford analysis. They obtained Laurent expansion and residuetheorems in universal Clifford analysis [14]. In real and complex Clifford analysis,Huang [15] introduced biregular functions and researched some of their properties.In this article, we discuss the Cauchy integral formula and Plemelj formula forbihypermonogenic functions. Using the idea of quasi-permutation, created byHuang [16], we give an equivalent definition of bihypermonogenic functions.

2. Preliminaries

2.1. Clifford algebra C‘0,n

Let C‘0,n be the universal Clifford algebra generated by the elements e1, . . . , ensatisfying the relation eiejþ ejei¼�2�ij, where �ij is the usual Kronecker delta. ThenC‘0,n has its basis e0¼ 1, e1, . . . , en; e1e2, . . . , en�1en, . . . , e1, . . . , en. An arbitraryelement of the basis may be written as

eA ¼ ei1,..., ik ¼ ei1ei2 , . . . , eik ,

where A¼ {i1, i2 , . . . , ik|1� i15 i25 � � �5 ik� n} or A¼; and e;¼ e0. Hence the realClifford algebra C‘0,n is formed by the elements presented as x¼

PA xAeA, in which

xA are real numbers. The real vector space Rnþ1 is identified with the set of elements

x ¼ x0e0 þ � � � þ xnen:

2.2. A general assumption of functions

Let ��Rnþ1 be an open connected set. We consider functions f, defined in � with

values in C‘0,n. These functions have the representation

f ðxÞ ¼XA

fAðxÞeA,

where the functions fA are real-valued.

2.3. Dirac operator

Let ��Rnþ1 be an open connected set. We usually assume that a function f is a

C1-function, defined in � with values in C‘0,n. The Dirac operators are defined

according to

Dl f ¼Xni¼0

ei@f

@xi, Drf ¼

Xni¼0

@f

@xiei,

Dl f ¼Xni¼0

ei@f

@xi, Drf ¼

Xni¼0

@f

@xiei:

A function f is called left monogenic in � if Dl f(x)¼ 0 in �, and correspondingly,right monogenic in � if Drf(x)¼ 0 in �. Left monogenic functions are usually called

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briefly monogenic functions. When there is no possible confusion, we denote Dinstead of Dl. Simply by calculating note that

DD ¼ DD ¼ D:

2.4. An involution and a decomposition in C‘0,n

The main involution 0 : C‘0,n!C‘0,n is defined by

x 0 ¼XA

xAð�1ÞjAjeA,

where usually jAj is the number of elements in the set A� {1 , . . ., n}. Clearly, we havee 00 ¼ e0 ¼ 1 and e 0i ¼ �ei, if i¼ 1, . . . , n. Moreover, the mapping 0 is an isomorphism,since

ðabÞ 0 ¼ a 0b 0:

The involution b: C‘0,n!C‘0,n is defined by ben ¼ �en,bei ¼ ei for i¼ 0, . . . , n� 1 andbab ¼babb.Recall that any element x 2 C‘0,n may be uniquely decomposed as x¼ bþ cen for

b, c 2 C‘0,n�1 (the Clifford algebra generated by e1, . . . , en�1). Using this decompo-sition, we define the mappings P : C‘0,n!C‘0,n�1 and Q : C‘0,n!C‘0,n�1 by Px¼ band Qx¼ c. Note that if x ¼

PA xAeA 2C‘0,n, then

Px ¼Xn 62 A

xAeA, Qx ¼Xn2A

xAeAnfng:

2.5. Hypermonogenic functions

We consider the modified Dirac operators Ml, Mr, Mland M

rin C‘0,n defined by

Mlf ðxÞ ¼ Dl f ðxÞ þ n� 1ð ÞQ 0f

xn,

Mr f ðxÞ ¼ Dr f ðxÞ þ n� 1ð ÞQf

xn,

Mlf ðxÞ ¼ Dl f ðxÞ � n� 1ð Þ

Q0f

xn,

Mrf ðxÞ ¼ Dr f ðxÞ � n� 1ð Þ

Qf

xn,

where Q0f¼ (Qf )0. Let � � Rnþ1þ ¼ fx ¼ ðx0, x1, . . . , xnÞ j xn 4 0g. A mapping

f :�!C‘0,n is called left hypermonogenic (briefly by hypermonogenic) if f 2 C1(�)and Mlf¼ 0 for any x 2 �. A right hypermonogenic function is defined similarly.Hypermonogenic functions were introduced in [15].

2.6. The first- and third-class quasi-permutation in C‘0,n

Let A¼ {h1, h2, . . . , hk}, 1� h15 h25 � � �5 hk� n, where hi 2 N for i¼ 1, 2, . . . , k.If m is a natural number satisfying 1�m� n, we define the arrangement mA by

mA ¼Anfmg, if m2A;

A [ fmg ¼ fg1, . . . , gkþ1g, if m 6 2A,

�

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where gi 2 A[ {m} and 1� g15 g25 � � �5 gkþ1� n. We call mA the first-class quasi-

permutation for the arrangement A[ {m}. We define m mf g ¼ ; and m; ¼ mf g. If

there are p distinct natural numbers in the set A\ [1,m], then the integer

�mA ¼ ð�1Þp

is called the sign of the first-class quasi-permutation mA. If m¼ 0, we

define mA ¼ A [ f0g and �mA ¼ 1: The motivation for the definition of the sign

function �mA comes from the property

emeA ¼ �mAemA: ð2:1Þ

Other properties of the first-class quasi-permutation are:

(1) if mA ¼ B, then mB ¼ A and(2) if mA ¼ B, then �mA ¼ ��mB:

Let j be a natural number and D¼ {h1, h2, . . . , hs}, 05 h15 h25 � � �5 hs� n,

where hp are natural numbers for any p 2 {1, 2, . . . , s}. The arrangement

(h1, h2, . . . , hs, j ) denoted by Dj is called the third-class quasi-permutation of

arrangement jD, and the integer

"Dj ¼ð�1Þs, 8p2 ½1, s�, hp 6¼ j,

ð�1Þs�1, 9p2 ½1, s�, hp ¼ j,

( ), ð2:2Þ

"0D ¼ 1

is called the sign of the third-class quasi-permutation.If j, D are as stated above, then

ejeD ¼ "DjeDej:

2.7. Holder continuous

Let �1�Rm and �2�R

k be open connected and m and k be natural numbers

satisfying 1�m� n and 1� k� n. Assume that � is a positive constant satisfying

05�5 1. A function

�ðu, �Þ ¼XA

�Aðu, �ÞeA, ðu, �Þ 2 @�1 � @�2,

is called Holder continuous with exponent � on @�1� @�2, if

j�ðu1, �1Þ � �ðu2, �2Þj � Gjðu1, �1Þ � ðu2, �2Þj�,

for any (u1, �1), (u2, �2)2 @�1� @�2, where G is a positive constant and

jðu1, �1Þ � ðu2, �2Þj ¼ ½ju1 � u2j2 þ j�1 � �2j

2�12:

Denote by H(@�1� @�2, �) the set of Holder continuous functions with exponent �on @�1� @�2. Define the norm in H(@�1� @�2, �) as

k f k� ¼ f, @�1 � @�2

�� ��0þ f, @�1 � @�2

�� ��H,�

,

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where

f, @�1 � @�2

�� ��H,�¼ supðui ,�iÞ 2 @�1�@�2ðu1, v1Þ6¼ðu2, �2 Þ

j f ðu1, v1Þ � f ðu2, v2Þj

jðu1, v1Þ � ðu2, v2Þj�,

f, @�1 � @�2

�� ��0¼ maxðu,�Þ 2 @�1�@�2

j f ðu, �Þj, f2Hð@�1 � @�2,�Þ:

3. The equivalent condition of bihypermonogenic functions

Denote by �¼�1��2 an open connected set in the Euclidean space Rmþ1�R

kþ1,

1�m� n, 1� k� n. Define a set FðrÞ� to consist all functions

f ðx, yÞ ¼X

A�f1,...,mg

XB�fmþ1,...,mþkg

fA,Bðx, yÞeAeB,

with values in C‘0,kþm for which fA,B(x, y) 2 Cr(�).

Definition 1 Let f2Fð1Þ� . A function f (x, y) is called bihypermonogenic on �, if

Mlx f ðx, yÞ ¼ 0,

Mry f ðx, yÞ ¼ 0,

(

for any (x, y) 2 �, where

Q 0xf ðx, yÞ ¼Xm2A

XB

fA,Bðx, yÞe0AnfmgeB

¼Xm2A

XB

�1ð ÞjAnfmgj

fA,Bðx, yÞeAnfmgeB

Mlx f x, yð Þ ¼

Xmi¼0

ei@f

@xix, yð Þ þ m� 1ð Þ

Q 0x f ðx, yÞð Þ

xm

is the left modified Dirac operator in C‘0,m calculated with respect to x 2 Rmþ1 and

Mry f x, yð Þ ¼

@f

@y0x, yð Þ þ

Xki¼1

@f

@yix, yð Þeiþm þ k� 1ð Þ

Qy f ðx, yÞð Þ

yk

is the right modified Dirac operator in the Clifford algebra generated by

emþ1, . . . , emþk, calculated with respect to y 2 Rkþ1.

LEMMA 1 If j is a fixed natural number satisfying j 2 {1, . . . , n}, the sets �¼ {eA j

A� {1, . . . , n}} and � ¼�ejA j A � 1, . . . , nf g

�are equal.

Proof Assume that j 2 {1, . . . , n}. It is obvious that �� �, since jA � 1, . . . , nf g. Let

eA 2 �. In the case j 2 A, setting B¼A\{ j}, we have ejB ¼ eA. In the second case

j 6 2 A, we have ejB ¼ eA for B ¼ ASf jg. Hence the sets are equal.

THEOREM 1 If

f ðx, yÞ ¼X

A�f1,...,mg

XB�fmþ1,...,mþkg

fA,Bðx, yÞeAeB,

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where fA,B 2 Cr(�) and x 2 Rmþ1, y 2 R

kþ1, then the f(x, y) is a bihypermonogenic

function if and only if it satisfies the system

@fA,Bðx, yÞ

@x0�Xmj¼1

�jA@fjA,Bðx, yÞ

@xj¼ 0, if m2A:

@fA,Bðx, yÞ

@x0�Xmj¼1

�jA@fjA,B@xjðx, yÞ þ

m� 1

xmð�1ÞjAjfmA,Bðx, yÞ ¼ 0, if m 6 2A:

@fA,Bðx, yÞ

@y0�Xki¼1

"BðiþmÞ�ðiþmÞB@fA,ðiþmÞB

@yiðx, yÞ þ

k� 1

ykfA,BðmþkÞ

ðx, yÞ ¼ 0,

if mþ k 6 2B:

@fA,Bðx, yÞ

@y0�Xki¼1

"BðiþmÞ�ðiþmÞB@f

A,ðiþmÞBðx, yÞ

@yi¼ 0, if mþ k2B:

Proof Assume that f (x, y) is bihypermonogenic.

Dlx f ðx, yÞ ¼

XA,B

eAeB@fA,Bðx, yÞ

@x0þXA,B

Xmj¼1

ejeAeB@fA,Bðx, yÞ

@xj

¼XA,B

eAeB@fA,Bðx, yÞ

@x0þXB

XA

Xmj¼1

�jAejAeB@fA,Bðx, yÞ

@xj:

Denoting jA ¼ C, we infer A ¼ jC and �jA ¼ ��jC, which implies

Dlx f ðx, yÞ ¼

XA,B

eAeB@fA,Bðx, yÞ

@x0�XB

XC

Xmj¼1

�jCeCeB@fjC,Bðx, yÞ

@xj: ð3:1Þ

Substituting Q-operator in C‘0,m to Qx we obtain

Q 0x f ðx, yÞ ¼Xm2A

XB

fA,Bðx, yÞe0AnfmgeB

¼Xm2A

XB

�1ð ÞjAnfmgj

fA,Bðx, yÞeAnfmgeB:

Denoting D¼A\{m}, we infer A ¼ mD and

Q 0x f ðx, yÞ ¼Xm62D

XB

ð�1ÞjDjfmD,Bðx, yÞeDeB: ð3:2Þ

Combining (3.1) and (3.2) and applying Lemma 1 we deduce

Mlx f ðx, yÞ¼D

lx f ðx, yÞ þ

m� 1

xmQ 0x f ðx, yÞ

¼XA,B

eAeB

@fA,Bðx, yÞ

@x0�Xmj¼1

�jA@fjC,Bðx, yÞ

@xj

!

þm� 1

xm

Xm 62A

XB

ð�1ÞjAjeAeB fmA,Bðx, yÞ:

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Collecting the coefficients of eAeB, we obtain that Mlx f ðx, yÞ ¼ 0 is equivalent to the

system

@fA,Bðx, yÞ

@x0�Xmj¼1

�jA@fjA,Bðx, yÞ

@xj¼ 0, if m2A,

@fA,Bðx, yÞ

@x0�Xmj¼1

�jA@fjA,Bðx, yÞ

@xjþm� 1

xmð�1ÞjAjfmA,Bðx, yÞ ¼ 0, if m 6 2A:

Similarly, we obtain the equivalent equations for Mry f ðx, yÞ ¼ 0. This completes the

proof.The preceding theorem gives the relation between the system

Mlx f ðx, yÞ ¼ 0

Mry f ðx, yÞ ¼ 0

(in Clifford analysis and the system that has 2n real partial differential equation.

4. The Cauchy integral formula of bihypermonogenic functions

We recall the integral formula for hypermonogenic functions.

LEMMA 2 [11] Let � be an open subset of Rnþ1þ ¼ fx ¼ ðx0, x1, . . . , xnÞjxn 4 0g and

K an (nþ 1)-chain satisfying K � �. If f is hypermonogenic in � and y2K, then

f ð yÞ ¼2n�1yn�1n

!nþ1

Z@K

Enðx, yÞd�ðxÞ f ðxÞ �

Z@K

Fnðx, yÞ dd�ðxÞdf ðxÞ� �,

where

Enðx, yÞ ¼ðx� yÞ�1

jx� yjn�1jx�by jn�1,Fnðx, yÞ ¼

ðbx� yÞ�1

jx� yjn�1jx�by jn�1,!nþ1 is the surface measure of the unit ball in R

nþ1 and

d� ¼Xni¼0

�1ð Þieidx0 ^ � � � ^ dxi�1 ^ dxiþ1 ^ � � � ^ dxn:

Corresponding integral formula for bihypermonogenic functions is stated next.

THEOREM 2 Let �0 and �00 be open subsets of Rmþ1\{xm� 0} and R

kþ1\{yk� 0},respectively. Suppose that �1 and �2 satisfy �1 � � 0 and �2 � �00, respectively. If’(x, y) is a bihypermonogenic function in �0 ��00, x2�1, y2�2, then

’ðx, yÞ ¼ �

Z@�1�@�2

Emðu, xÞd�mðuÞ’ðu, vÞd�kðvÞEkðv, yÞ

��

Z@�1�@�2

Emðu, xÞd�mðuÞ g’ðu, vÞ gd�kðvÞFkðv, yÞ

��

Z@�1�@�2

Fmðu, xÞ dd�mðuÞ d’ðu, vÞd�kðvÞEkðv, yÞ

þ�

Z@�1�@�2

Fmðu, xÞ dd�mðuÞ gd’ðu, vÞ gd�kðvÞFkðv, yÞ,

ð4:1Þ

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where

� ¼2m�1xm�1m 2k�1yk�1k

!mþ1!kþ1,

Elðu, xÞ ¼ðu� xÞ�1

ju� xjl�1ju�bxjl�1, u, x2Rmþ1,

Flðu, xÞ ¼ðbu� xÞ�1

ju� xjl�1ju�bxjl�1, u, x2Rmþ1,

v ¼ v0 þ v1emþ1 þ � � � þ vkemþk

y ¼ y0 þ y1emþ1 þ � � � þ ykemþk

Elðv, yÞ ¼ðv� yÞ�1

jv� yjl�1jv�eyjl�1,Flðv,yÞ ¼

ðev� yÞ�1

jv� yjk�1jv�eyjk�1d�m¼

Xmi¼0

�1ð Þieidx0 ^ � � � ^ dxi�1 ^ dxiþ1 ^ � � � ^ dxm

d�k¼dy1 ^ dy2 ^ � � � ^ dykþXki¼1

�1ð Þieiþmdy0 ^ dy1 ^ � � � ^ dyi�1 ^ dyiþ1 ^ � � � ^ dyk

and the involutions b and e are defined by

bei ¼ ei, i2 0, 1, . . . ,mþ kf gn mf g, bem ¼ �em, bab ¼babb,eei ¼ ei, i2 0, 1, . . . ,mþ k� 1f g, gemþk ¼ �emþk, eab ¼eaeb:Proof The right-hand side of (4.1) may be written as

�

Z@�1

Emðu,xÞd�mðuÞ

Z@�2

’ðu, vÞd�kðvÞEkðv, yÞ � g’ðu, vÞ gd�kðvÞFkðv, yÞh i� �

� �

Z@�1

Fmðu, xÞ dd�mðuÞ Z@�2

d’ðu, vÞd�kðvÞEkðv, yÞ �gd’ðu, vÞ gd�kðvÞFkðv, yÞ

� �:

Since ’(u, v) is a bihypermonogenic function, i.e Mlu’ðu, vÞ ¼ 0, Mr

v’ðu, vÞ ¼ 0, we

obtain

2k�1yk�1k

!kþ1

Z@�2

’ðu, vÞd�kðvÞEkðv, yÞ � g’ðu, vÞ gd�kðvÞFkðv, yÞh i

¼ ’ðu, yÞ:

Noting that the function d’ðu, vÞ is also hypermonogenic with respect to v for any u,

we have

2k�1yk�1k

!kþ1

Z@�2

d’ðu, vÞd�kðvÞEkðv, yÞ �gd’ðu, vÞ gd�kðvÞFkðv, yÞ

¼ d’ðu, yÞ:

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Hence, the right-hand side of (4.1) equals to

2m�1xm�1m

!mþ1

Z@�1

Emðu, xÞd�mðuÞ’ðu, yÞ �2m�1xm�1m

!mþ1

Z@�1

Fmðu, xÞdd�mðuÞ d’ðu, yÞ:Since Ml

u’ðu, yÞ ¼ 0, the result is obtained, completing the proof.

5. Plemelj formula of bihypermonogenic functions

We consider singular integrals for bihyperharmonic functions and principal values.The singular integral is defined as follows.

Definition 2 The integral

�ðt1, t2Þ ¼�

Z@�1�@�2

Emðu, t1Þd�mðuÞ’ðu, vÞd�kðvÞEkðv, t2Þ

��

Z@�1�@�2

Emðu, t1Þd�mðuÞ g’ðu, vÞ gd�kðvÞFkðv, t2Þ

��

Z@�1�@�2

Fmðu, t1Þ dd�mðuÞ d’ðu, vÞd�kðvÞEkðv, t2Þ

þ�

Z@�1�@�2

Fmðu, t1Þ dd�mðuÞ gd’ðu, vÞ gd�kðvÞFkðv, t2Þ,

is called a singular integral on @�1� @�2, where �, Em(u, t1), Ek(v, t2), Fm(u, t1) andFk(v, t1) are given in Theorem 2.

Definition 3 Let �4 0 be a constant and ��¼B(t1, �)�B(t2, �), Bi(ti, �), (i¼ 1, 2), areballs with the centre at ti and the radius �4 0. Denote

��ðt1, t2Þ ¼�

Z@�1�@�2n��

Emðu, t1Þd�mðuÞ’ðu, vÞd�kðvÞEkðv, t2Þ

��

Z@�1�@�2n��

Emðu, t1Þd�mðuÞ g’ðu, vÞg�kðvÞFkðv, t2Þ

��

Z@�1�@�2n��

Fmðu, t1Þ dd�mðÞ d’ðu, vÞd�kðvÞEkðv, t2Þ

þ�

Z@�1�@�2n��

Fmðu, t1Þ dd�mðuÞ gd’ðu, vÞg�kðvÞFkðv, t2Þ:

If lim�!0 ��ðt1, t2Þ ¼ I, then I is called the Cauchy principal value of a singularintegral, denoted by I¼�(t1, t2).

We use the following two lemmas.

LEMMA 3 [12] Let �, K, @K be as in Lemma 2, then

2n�1yn�1n

!nþ1

Z@K

Enðx, yÞd�ðxÞ �

Z@K

Fnðx, yÞ dd�ðxÞ� �

¼1, y2K,

0, y2Rnþ1�Kþ :

(

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LEMMA 4 [12] Let �, K and @K be as in Lemma 2 and y2 @K, then

2n�1yn�1n

!nþ1

Z@K

Enðx, yÞd�ðxÞ �

Z@K

Fnðx, yÞ dd�ðxÞ� �¼

1

2:

In order to derive the Plemelj formula of bihypermonogenic function, we will give

the following results (Theorems 3–7), which are a direct generalization of some

results from the theory of biregular functions [15].

We prove that for Holder continuous functions the Cauchy principal value exists.

THEOREM 3 If ’(u, v)2H(@�1� @�2,�), then there exists the Cauchy principal value

of singular integrals and

�ðt1, t2Þ ¼ �14’ðt1, t2Þ þ X1ðt1, t2Þ þ X2ðt1, t2Þ þ X3ðt1, t2Þ þ X4ðt1, t2Þ

þ 14 ðP1’þ P2’Þ þ

14 ðQ1’þQ2’Þ,

where

X1ðt1, t2Þ ¼ �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 1ðu, vÞd�k vð ÞEkðv, t2Þ,

X2ðt1, t2Þ ¼ ��

Z@�1�@�2

Emðu, t1d�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þ,

X3ðt1, t2Þ ¼ ��

Z@�1�@�2

Fmðu, t1Þ dd�m uð Þ 3ðu, vÞd�k vð ÞEkðv, t2Þ,

X4ðt1, t2Þ ¼ �

Z@�1�@�2

Fmðu, t1Þ dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, t2Þ,

1ðu, vÞ ¼ ’ðu, vÞ � ’ðu, t2Þ � ’ðt1, vÞ þ ’ðt1, t2Þ,

2ðu, vÞ ¼ g’ðu, vÞ � ’ðu, t2Þ � g’ðt1, vÞ þ ’ðt1, t2Þ, 3ðu, vÞ ¼ d’ðu, vÞ � d’ðu, t2Þ � ’ðt1, vÞ þ ’ðt1, t2Þ, 4ðu, vÞ ¼

gd’ðu, vÞ � d’ðu, t2Þ � g’ðt1, vÞ þ ’ðt1, t2Þ,and

P1’ ¼ 2�1

Z@�1

Emðu, t1Þd�m uð Þ’ðu, t2Þ,

P2’ ¼ 2�2

Z@�2

’ðt1, vÞd�k vð ÞEkðv, t2Þ,

Q1’ ¼ �2�1

Z@�1

Fmðu, t1Þ dd�m uð Þ d’ðu, t2Þ,Q2’ ¼ �2�2

Z@�2

g’ðt1, �Þ gd�k vð ÞFkðv, t2Þ,

� ¼ �1�2, �1 ¼2m�1xm�1

m

!mþ1, �2 ¼

2k�1yk�1k

!kþ1:

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Proof Denote

�� ¼X4i¼1

Ai þX4i¼1

Bi þX4i¼1

Ci þX4i¼1

Di,

where

A1 ¼ �

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, t2Þ,

A2 ¼ �

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ 1ðu, vÞd�k vð ÞEkðv, t2Þ,

A3 ¼ �

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ ’ðu, t2Þ � ’ðt1, t2Þ½ �d�k vð ÞEkðv, t2Þ,

A4 ¼ �

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ ’ðt1, vÞ � ’ðt1, t2Þ½ �d�k vð ÞEkðv, t2Þ,

B1 ¼ ��

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ’ðt1, t2Þ gd�k vð ÞFkðv, t2Þ,

B2 ¼ ��

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þ,

B3 ¼ ��

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ ’ðu, t2Þ � ’ðt1, t2Þ½ � gd�k vð ÞFkðv, t2Þ,

B4 ¼ ��

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ½ g’ðt1, �Þ � ’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ,

C1 ¼ ��

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, t2Þ,

C2 ¼ ��

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ 3ðu, vÞd�k vð ÞEkðv, t2Þ,

C3 ¼ ��

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ½ d’ðu, t2Þ � ’ðt1, t2Þ�d�k vð ÞEkðv, t2Þ,

C4 ¼ ��

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ½’ðt1, vÞ � ’ðt1, t2Þ�d�k vð ÞEkðv, t2Þ,

D1 ¼ �

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ’ðt1, t2Þ gd�k vð ÞFkðv, t2Þ,

D2 ¼ �

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, t2Þ,

D3 ¼ �

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ½ d’ðu, t2Þ � ’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ,

D4 ¼ �

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ½ g’ðt1, vÞ � ’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ:

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From

A1 þ B1 þ C1 þD1 ¼ �

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, t2Þ

��

Z@�1�@�2n��

Emðu, t1Þd�m uð Þ’ðt1, t2Þ dd�k vð ÞFkðv, t2Þ

��

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, t2Þ

þ �

Z@�1�@�2n��

Fmðu, t1Þ dd�m uð Þ’ðt1, t2Þ gd�k vð ÞFkðv, t2Þ,

and from Lemma 3, we infer

A1 þ B1 þ C1 þD1 ¼1

2�1

Z@�1nBðt1, �Þ

Emðu, t1Þd�m uð Þ’ðt1, t2Þ

�1

2�1

Z@�1nBðt1, �Þ

Fmðu, t1Þ dd�m uð Þ’ðt1, t2Þ þ R,

where R is the rest integral over (@�1\B(t1, �))� @�2. When �! 0, we obtain

A1 þ B1 þ C1 þD1!14 ’ðt1, t2Þ:

Since ’(u, �)2H(@�1� @�2,�), j 1(u, v)j �A0ju� t1j�, j 1(u, v)j �A0jv� t2j

�,

where A0¼ 2k’k�, then

j 1ðu, vÞj � A0ju� t1j�2jv� t2j

�2 : ð5:1Þ

By (2.11) in Chapter I of [15] and Equation (4.1), there exists an M1 such that

Emðu, t1Þd�m uð Þ 1ðu, vÞd�k vð ÞEkðv, t2Þ�� �� �M1A0

�2�1

01 d01�2�1

02 d02,

where 0i (i¼ 1, 2) be as in Chapter I of [15], so �! 0, A2!X1(t1, t2).Denoting again the rest integral over (@�1\B(t1, �))� @�2 by R and applying

A3 þ B3 ¼�12

Z@�1nBðt1, �Þ

Emðu, t1Þd�m uð Þ½’ðu, t2Þ � ’ðt1, t2Þ� þ R,

jEmðu, t1Þd�m uð Þ½’ðu, t2Þ � ’ðt1, t2Þ�j �MA0��101 d01,

we obtain when �! 0, A3 þ B3!�12

R@�1

Emðu, t1Þd�m(u)[’(u, t2)� ’(t1, t2)].Similarly, we can get when �! 0, A4 þ C4!

�22

R@�2½’ðt1, vÞ�

’ðt1, t2Þ�d�k vð ÞEkðv, t2Þ:From

jEmðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þj �MA0��101 d01d02,

we can get when �! 0, B2!X2(t1, t2).By

B4 þD4 ¼ ��22

Z@�2nB t2, �ð Þ

½ g’ðt1, vÞ � ’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ,

j½ g’ðt1, vÞ � ’ðt1, t2Þ�d g�k vð ÞFkðv, t2Þj � F3d02

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and denoting the rest integral over @�1� (@�2\B(t2, �)) we obtain when �! 0,

B4 þD4!��22

R@�2½ g’ðt1, vÞ � ’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ:

Similarly, we can get when �! 0, C3 þD3!��12

R@�1

Fmðu, t1Þ dd�m uð Þ

½ d’ð, t2Þ � ’ðt1, t2Þ�:From

jFmðu, t1Þ dd�m uð Þ 3ðu, vÞd�k vð ÞEkðv, t2Þj �M4A0d01��102 d02

we can get when �! 0, C2!��R@�1�@�2

Fmðu, t1Þ dd�m uð Þ 3ðu, vÞd�k vð ÞEkðv, t2Þ:Since

jFmðu, t1Þ dd�m uð Þ 4ðu, vÞÞ gd�k vð ÞFkðv, t2Þj �M5d01d02

we can get when �! 0, D2! �R@�1�@�2

Fmðu, t1Þ dd�m uð Þ 4ðu, vÞÞ gd�k vð ÞFkðv, t2Þ:Summarizing the above discussion shows that

�ðt1, t2Þ ¼1

4’ðt1, t2Þ þ �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 1ðu, vÞd�k vð ÞEkðv, t2Þ

��

Z@�1�@�2

Emðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þ

��

Z@�1�@�2

Fmðu, t1Þ dd�m uð Þ 3ðu, vÞd�k vð ÞEkðv, t2Þ

þ�

Z@�1�@�2

Fmðu, t1ðu, t1Þ dd�0 uð Þ 4ðu, vÞd�k vð ÞFkðv, t2Þ

þ�12

Z@�1

Emðu, t1Þd�m uð Þ½’ðu, t2Þ � ’ðt1, t2Þ�

þ�22

Z@�2

½’ðt1, vÞ � ’ðt1, t2Þ�d�k vð ÞEkðv, t2Þ

��22

Z@�2

½ g’ðt1, vÞ�’ðt1, t2Þ� gd�k vð ÞFkðv, t2Þ

��12

Z@�1

Fmðu, t1Þ dd�m uð Þ½ d’ðu, t2Þ � ’ðt1, t2Þ�¼

1

4’ðt1, t2Þ þ X1ðt1, t2Þ þ X2ðt1, t2Þ þ X3ðt1, t2Þ þ X4ðt1, t2Þ

þ�12

Z@�1

Emðu, t1Þd�m uð Þ’ðu, t2Þ ��12

Z@�1

Emðu, t1Þd�m uð Þ’ðt1, t2Þ

þ�22

Z@�2

’ðt1, vÞd�k vð ÞEkðv, t2Þ ��22

Z@�2

’ðt1, t2Þd�k vð ÞEkðv, t2Þ

��22

Z@�2

g’ðt1, vÞ gd�k vð ÞEkðv, t2Þ þ�22

Z@�2

’ðt1, t2Þ gd�k vð ÞEkðv, t2Þ

��12

Z@�1

Emðu, t1Þ dd�m uð Þ d’ð, t2Þ þ �12

Z@�1

Emðu, t1Þ dd�m uð Þ’ðt1, t2Þ

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¼1

4’ðt1, t2Þ þ X1ðt1, t2Þ þ X2ðt1, t2Þ þ X3ðt1, t2Þ þ X4ðt1, t2Þ

þ�12

Z@�1

Emðu, t1Þd�k vð Þ’ðu, t2Þ �1

4’ðt1, t2Þ

þ�22

Z@�2

’ðt1, vÞd�k vð ÞEkðv, t2Þ �1

4’ðt1, t2Þ

��22

Z@�2

g’ t1, vð Þd g�k vð ÞFkðv, t2Þ ��12

Z@�1

Fmðu, t1Þ dd�m uð Þ d’ðu, t2Þ¼ �

1

4’ðt1, t2Þ þ X1ðt1, t2Þ þ X2ðt1, t2Þ þ X3ðt1, t2Þ þ X4ðt1, t2Þ

þ1

4ðP1’þ P2’Þ þ

1

4ðQ1’þQ2’Þ:

This completes the proof.

LEMMA 5 [17] Suppose that t, x2Rn, n(�2) and m(�0) are integers, then

x

jxjmþ2�

t

jtjmþ2

���� ���� � Pmðx, tÞ

jxjmþ1jtjmþ1jx� tj,

where

Pmðx, tÞ ¼

Pmk¼0 jxj

m�kjtjk, m 6¼ 0,1, m ¼ 0:

�LEMMA 6 Let Em(u, x), Em(u, t1) be the same as in Theorem 2. If u2�1, x2�2,

t1 2�2, then there exists a constant M such that

jEmðu, xÞ � Emðu, t1Þj �MXm�1i¼0

���� u� t1u� x

����i���� x� t1u� x

����þ jx� t1j

" #ju� t1j

�m: ð5:2Þ

Proof We compute

jEmðu,xÞ � Emðu, t1Þj

¼ðu� xÞ�1

ju� xjm�1ju�bxjm�1 � ðu� t1Þ�1

ju� t1jm�1ju�bt1jm�1

����������

¼u� x

ju� xjmþ1ju�bxjm�1 � u� t1

ju� t1jmþ1ju�bxjm�1

����þ

� t1

j� t1jmþ1j�bxjm�1 � � t1

j� t1jmþ1j�bt1jm�1

�����

1

ju�bxjm�1 u� x

ju� xjmþ1�

u� t1

ju� t1jmþ1

���� ����þj� t1j

ju� t1jmþ1

1

ju�bxjm�1 � 1

ju�bt1jm�1���� ����:

Since 1

ju�bxjm�1 , 1

ju�bxjm�1ju�bt1jm�1 andPm�2

i¼0 ju�bxjm�2�iju�bt1ji are bounded by Lemma

5, there exist, M1 and M2, such that

jEmðu, xÞ � Emðu, t1Þj �M1

Pm�1i¼0 ju� xjm�1�iju� t1j

i

ju� xjmju� t1jm jx� t1j þM2

jx� t1j

ju� t1jm :

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Let M¼max{M1,M2}, then the result can be obtained.

THEOREM 4 If ’(u, v)2H(@�1� @�2, �), (x, y) 6 2 @�1� @�2, then

limðx,yÞ!ðt1, t2Þ

X1ðx, yÞ ¼ X1ðt1, t2Þ, ðt1, t2Þ 2 @�1 � @�2:

Proof Denote

X1ðx, yÞ � X1ðt1, t2Þ

¼�

Z@�1�@�2

½Emðu, xÞ � Emðu, t1Þ�d�m uð Þ 1ðu, vÞd�k vð ÞEkðv, t2Þ

þ �

Z@�1�@�2

½Emðu, xÞ � Emðu, t1Þ�d�m uð Þ 1ðu, vÞd�k vð Þ½Ekðv, yÞ�Ekðv, t2Þ�

þ �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 1ðu, vÞd�k vð Þ½Ekðv, yÞ�Ekðv, t2Þ�

¼L1ð@�1 � @�2Þ þ L2ð@�1 � @�2Þ þ L3ð@�1 � @�2Þ,

and suppose that 6�5 di, i¼ 1, 2, �4 0, O((t1, t2), �) is the �-neighbourhood of (t1, t2)with the center at point (t1, t2) and radius �. Setting @�i1 ¼ ð@�iÞ

T½Oððt1, t2Þ, �Þ�,

ði ¼ 1, 2Þ, @�i2 ¼ ð@�iÞnð@�i1Þ, ði ¼ 1, 2Þ, we have

Ljð@�1 � @�2Þ ¼ Ljð@�11 � @�21Þ þ Ljð@�11 � @�22Þ

þLjð@�12 � @�21Þ þ Ljð@�12 � @�22Þ, j ¼ 1, 2, 3:

It is easy to see that we only need to prove the result when x, y are approaching t1, t2along the line that are not in the tangent plane of @�1 at t1, @�2 at t2, respectively. Ifwe take an angle between the direction of x! t1 and the tangent plane of @�1 at t1,which is greater than 2�0, and similarly take an angle between the direction of y! t2and the tangent plane of @�2 at t2, which is greater than 2�0, then���� u� t1

u� x

���� �M,

���� x� t1u� x

���� �M,

���� v� t2v� y

���� �M,

���� y� t2v� y

���� �M: ð5:3Þ

Here M¼M(�0) and by (5.1)–(5.3), we can get

jL1ð@�11 � @�21Þj � A3��, jL1ð@�12 � @�21Þj � A4�

�2, jL1ð@�11 � @�22Þj � A5�

�2:

Now we consider L1(@�12� @�22) and by Lemma 6, we can get

jEmðu, xÞ � Emðu, t1Þj �M

"Xm�1i¼0

���� u� t1u� x

����iþ1 jx� t1j

ju� t1jmþ1þjx� t1j

ju� t1jm

#:

By (5.1) and (5.3), we have jL1(@�12� @�22j �A6jx� t1j. Hence if (x, y)! (t1, t2),then L1(@�1� @�2)! 0.

Similarly, if (x, y)! (t1, t2), then L2(@�1� @�2)! 0 and L3(@�1� @�2)! 0.Hence limðx,yÞ!ðt1, t2Þ X1ðx, yÞ ¼ X1ðt1, t2Þ, ðt1, t2Þ 2 @�1 � @�2:

THEOREM 5 If ’(u, �)2H(@�1� @�2, �), (x, y) 6 2 @�1� @�2, then

limðx,yÞ!ðt1, t2Þ

X2ðx, yÞ ¼ X2ðt1, t2Þ, ðt1, t2Þ 2 @�1 � @�2:

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Proof We compute

� X2ðx, yÞ þ X2ðt1, t2Þ

¼ �

Z@�1�@�2

Emðu, xÞd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, yÞ

� �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þ

¼ �

Z@�1�@�2

½Emðu, xÞ � Emðu, t1Þ�d�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, yÞ

þ �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, yÞ

� �

Z@�1�@�2

Emðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, t2Þ

¼ L1ð@�1 � @�2Þ þ L2ð@�1 � @�2Þ � L3ð@�1 � @�2Þ:

Since 2ðu, vÞ ¼ g’ðu, vÞ � ’ðu, t2Þ � g’ðt1, vÞ þ ’ðt1, t2Þ, we can get

j g 2ðu, vÞj � A2��: ð5:4Þ

Applying (5.2)–(5.4), we have

j½Emðu, xÞ � Emðu, t1Þ�d�m uð Þ 2ðu, vÞ gd�k vð ÞMkðv, yÞj � C1ð�1þ�01 d01Þd02:

Hence jL1(@�11� @�21)j �A7��þ1, jL1(@�11� @�22)j �A8�

� and jL1(@�12� @�21)j �

A9�. Again by (5.2)–(5.4), we have

jL1ð@�12 � @�22Þj � A10jx� t1j:

Hence if (x, y)! (t1, t2), then L1(@�1� @�2)! 0.Using

jEmðu, t1Þd�m uð Þ 2ðu, vÞ gd�k vð ÞFkðv, yÞj � C2ð�1þ�01 d01Þd02,

we infer jL2(@�11� @�21)j �A11��þ1, jL2(@�11� @�22)j �A12�

�, jL2(@�12� @�21)j �A13�.Similarly, we obtain the inequalities jL3(@�11� @�21)j �A14�

�þ1,

jL3(@�11� @�22)j �A15�� and jL3(@�12� @�21)j �A16�. Since

jFkðv, yÞ � Fkðv, t2Þj

¼ðb�� yÞ�1

j�� yjk�1j��byjk�1 � ðb�� t2Þ�1

j�� t2jk�1j��bt2jk�1

����������

¼ðb�� yÞ�1

j�� yjk�1j��byjk�1 � ðb�� yÞ�1

j��byjk�1j�� t2jk�1þ

ðb�� yÞ�1

j��byjk�1j�� t2jk�1

������

ðb�� t2Þ�1

j�� t2jk�1j��bt2jk�1 j � jðb�� yÞ�1j

j��byjk�1 1

j�� yjk�1�

1

j�� t2jk�1

���� ���� þ

1

j�� t2jk�1

ðb�� yÞ�1

j��byjk�1 � ðb�� t2Þ�1

j��bt2jk�1�����

�����,

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applying Lemma 5 we infer

jFkðv, yÞ � Fkðv, t2Þj � Fj y� t2j

j�� yjk�1j�� t2jk�1

Xk�2i¼0

jv�yjk�2�ijjv�t2ji

þ1

j�� t2jk�1

Xk�1i¼0

j��byjk�1�ij��byjk j��bt2ji

j��bt2jk j y� t2j: ð5:5Þ

By (5.3)–(5.5), we have jL2(@�12� @�22)�L3(@�12� @�22)j �C3jy� t2j.

Summarizing the above discussion we conclude that if (x, y)! (t1, t2), then

L2(@�1� @�2)�L3(@�1� @�2)! 0. Hence limðx,yÞ!ðt1, t2Þ X2ðx, yÞ ¼ X2ðt1, t2Þ,

ðt1, t2Þ 2 @�1 � @�2:Similarly, it is easy to prove the following theorem.

THEOREM 6 If ’(,�)2H(@�1� @�2, �), (x, y) 6 2 @�1� @�2, then

limðx,yÞ!ðt1, t2Þ

X3ðx, yÞ ¼ X3ðt1, t2Þ, ðt1, t2Þ 2 @�1 � @�2:

THEOREM 7 If ’(,�)2H(@�1� @�2,�), (x, y) 6 2 @�1� @�2, then

limðx,yÞ!ðt1, t2Þ

X4ðx, yÞ ¼ X4ðt1, t2Þ, ðt1, t2Þ 2 @�1 � @�2:

Proof We compute

X4ðx, yÞ � X4ðt1, t2Þ ¼ �

Z@�1�@�2

Fmðu, xÞ dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, yÞ

��

Z@�1�@�2

Fmðu, t1Þ dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, t2Þ

¼ L1ð@�1 � @�2Þ � L2ð@�1 � @�2Þ:

By

jFmðu, xÞ dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, yÞj � Fd01d02,

it follows that

jL1ð@�11 � @�21Þj � A17�2,

jL1ð@�11 � @�22Þj � A18�,

jL1ð@�12 � @�21Þj � A19�:

Similarly, we have jL2(@�11� @�21)j �A20�2, jL2(@�11� @�22)j �A21�, jL2(@�12�

@�21)j �A22�.

jL1ð@�12 � @�22Þ � L2ð@�12 � @�22Þj

¼

����� Z@�12�@�22

½Fmðu, xÞ � Fmðu, t1Þ� dd�m uð Þ 4ðu, vÞ gd�k vð ÞFkðv, yÞ

þ �

Z@�12�@�22

Fmðu, t1Þ dd�m uð Þ 4ðu, vÞ gd�k vð Þ½Fkðv, yÞ�Fkðv, t2Þ�

����,

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by (5.3) and (5.5), we can get

jL1ð@�12 � @�22Þ � L2ð@�12 � @�22Þj � C4jx� t1j þ C5j y� t2j:

Hence when (x, y)! (t1, t2), L1(@�1� @�2)�L2(@�1� @�2)! 0.Summarizing the above discussion, we can obtain the result.

THEOREM 8 If ’(u, v)2H(@�1� @�2,�), then

�þþðt1, t2Þ ¼14 ½’ðt1, t2Þ þ P1ð’Þ þ P2ð’Þ þQ1ð’Þ þQ2ð’Þ þ P3ð’Þ�,

�þ�ðt1, t2Þ ¼14 ½�’ðt1, t2Þ � P1ð’Þ þ P2ð’Þ �Q1ð’Þ þQ2ð’Þ þ P3ð’Þ�,

��þðt1, t2Þ ¼14 ½�’ðt1, t2Þ þ P1ð’Þ � P2ð’Þ þQ1ð’Þ �Q2ð’Þ þ P3ð’Þ�,

���ðt1, t2Þ ¼14 ½’ðt1, t2Þ � P1ð’Þ � P2ð’Þ �Q1ð’Þ �Q2ð’Þ þ P3ð’Þ�,

where (t1, t2)2 @�1� @�2, P3(’)¼ 4�(t1, t2).

Proof We first write (4.1) as

�ðx, yÞ ¼ X1ðx, yÞ þ X2ðx, yÞ þ X3ðx, yÞ þ X4ðx, yÞ

þ �

Z@�1�@�2

Emðu, xÞd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, yÞ

� �

Z@�1�@�2

Emðu, xÞd�m uð Þ’ðt1, t2Þ gd�k vð ÞFkðv, yÞ

� �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ’ðt1, t2Þd�k vð ÞEkðv, yÞ

þ �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ’ðt1, t2Þ gd�k vð ÞFkðv, yÞ

þ �

Z@�1�@�2

Emðu, xÞd�m uð Þ½’ðu, t2Þ � ’ðt1, t2Þ�d�k vð ÞEkðv, yÞ

� �

Z@�1�@�2

Emðu, xÞd�m uð Þ½’ðu, t2Þ � ’ðt1, t2Þ� gd�k vð ÞFkðv, yÞ

þ �

Z@�1�@�2

Emðu, xÞd�m uð Þ½’ðt1, vÞ � ’ðt1, t2Þ�d�k vð ÞEkðv, yÞ

� �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ½’ðt1, vÞ � ’ðt1, t2Þ�d�k vð ÞEkðv, yÞ

� �

Z@�1�@�2

Emðu, xÞd�m uð Þ½ g’ðt1, vÞ�’ðt1, t2Þ� gd�k vð ÞFkðv, yÞ

þ �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ½ g’ðt1, vÞ�’ðt1, t2Þ� gd�k vð ÞFkðv, yÞ

� �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ½ d’ðu, t2Þ � ’ðt1, t2Þ�d�k vð ÞEkðv, yÞ

þ �

Z@�1�@�2

Fm u, xð Þ dd�m uð Þ½ d’ðu, t2Þ � ’ðt1, t2Þ� gd�k vð ÞFkðv, yÞ:

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Applying Theorems 4 and 7 and Lemma 3, we obtain

�þþðt1,t2Þ

¼X1ðt1, t2ÞþX2ðt1, t2ÞþX3ðt1, t2ÞþX4ðt1, t2Þþ’ðt1,t2Þ

þ�1

Z@�1

Emðu,t1Þd�m uð Þ½’ðu,t2Þ�’ðt1, t2Þ�þ�2

Z@�2

½’ðt1,vÞ�’ðt1,t2Þ�d�k vð ÞEkðv,t2Þ

��2

Z@�2

½ g’ðt1,�Þ�’ðt1, t2Þ� gd�k vð ÞMkðv, t2Þ��1

Z@�1

Fm u,t1ð Þ dd�m uð Þ½ d’ð,t2Þ�’ðt1, t2Þ�¼X1ðt1, t2ÞþX2ðt1, t2ÞþX3ðt1, t2ÞþX4ðt1, t2Þþ’ðt1,t2Þ

þ�1

Z@�1

Emðu,t1Þd�m uð Þ’ðu, t2Þ��1

Z@�1

Emðu,t1Þd�m uð Þ’ðt1,t2Þ

þ�2

Z@�2

’ðt1,vÞd�k vð ÞEkðv,t2Þ��2

Z@�2

’ðt1, t2Þd�k vð ÞEkðv, t2Þ

� �2

Z@�2

g’ðt1, vÞ gd�k vð ÞFkðv, t2Þ þ �2

Z@�2

’ðt1, t2Þ gd�k vð ÞFkðv, t2Þ

� �1

Z@�1

Fm u, t1ð Þ dd�m uð Þ d’ðv, t2Þ þ �1 Z@�1

Fm u, t1ð Þ dd�m uð Þ’ðt1, t2Þ

¼ X1ðt1, t2Þ þ X2ðt1, t2Þ þ X3ðt1, t2Þ þ X4ðt1, t2Þ þ ’ðt1, t2Þ

þ �1

Z@�1

Emðu, t1Þd�m uð Þ’ðu, t2Þ �’ðt1, t2Þ

2þ �2

Z@�2

’ðt1, vÞd�k vð ÞEkðv, t2Þ

�’ðt1, t2Þ

2� �2

Z@�2

g’ðt1, vÞ gd�k vð ÞMkðv, t2Þ � �1

Z@�1

Fm u, t1ð Þ dd�m uð Þ d’ðu, t2Þ:From Theorem 3, it follows that

X4i¼1

Xiðt1, t2Þ ¼ �ðt1, t2Þ þ’ðt1, t2Þ

4�1

4ðP1’þ P2’Þ �

1

4ðQ1’þQ2’Þ,

which implies the result for �þþ(t1, t2). Similarly, we may prove that �þ�(t1, t2),��þ(t1, t2), �

��(t1, t2). This completes the proof.

COROLLARY 8 If ’(u, v)2H(@�1� @�2,�), (t1, t2)2 @�1� @�2, then

�þþðt1, t2Þ � �þ�ðt1, t2Þ � �

�þðt1, t2Þ þ ���ðt1, t2Þ ¼ ’ðt1, t2Þ,

�þþðt1, t2Þ � �þ�ðt1, t2Þ þ �

�þðt1, t2Þ � ���ðt1, t2Þ ¼ P1’þQ1’,

�þþðt1, t2Þ þ �þ�ðt1, t2Þ � �

�þðt1, t2Þ � ���ðt1, t2Þ ¼ P2’þQ2’,

�þþðt1, t2Þ þ �þ�ðt1, t2Þ þ �

�þðt1, t2Þ þ ���ðt1, t2Þ ¼ P3’:

Acknowledgements

Research of Y. Qiao was supported by the National Natural Science Fundamental of China(No. 10771049) and the National Natural Science Fundamental of Hebei (No. A2007000225).

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