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Cousin problem for biregular functions withvalues in a clifford algebraLe Hung Son aa Department of Mathematics, D.H.B.K., Institute of Technology, Hanoi,VietnamVersion of record first published: 29 May 2007.

To cite this article: Le Hung Son (1992): Cousin problem for biregular functions with values in a cliffordalgebra, Complex Variables, Theory and Application: An International Journal: An International Journal,20:1-4, 255-263

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Cousin Problem for Biregi mctions with Values in a Clifford Algebra

LE HUNG SON Department of Mathematics, D.H.B.K., Institute of Technology, Hanoi, Vietnam

The Cousin Problem is important in Theory of holomorphic functions of several complex variables. We study in this paper the additive Cousin problem for biregular functions with values in a Clifford algebra, w k h are a generalization of analyt~c iunctlons. It is shown that this problem is solvable for polydomains

AMS No. 30635, 32A05, 32A07, 47A60, 47A62 Communicated: Delanghe (Received May 11, 1990)

In the classical theory of holomorphic functions there is Mittag-Uffler Theorem, from which one can construct a meromorphic function by the given singularities. These results have been generalized in theory of holomorphic functions on several complex variables as the Additive Cousin Problem, which permits the construction of a meromorphic function from given (local) singularities. On the other hand many results from theory of holomorphic functions on several complex variables have been generalized for the so-called biregular functions with values in a Clifford Al- gebra (see [2, 31). In this paper we shall consider the Cousin problem for biregular functions with values in a Clifford algebra A and give a criterion for solvability of this problem.

2. PRELIMINARIES AND NOTATION

Let A be the universal Clifford algebra constructed over a real n-dimensional qua- dratic vector space V with orthonormal basis {el,. . . , en) (see [I]). A basis for A is given by

{ e ~ : A = (111, .. .,h,) E ,..,,, 1; 1 5 hl < . . . < h, 5 n )

where e$ = eo is the identity element. Multiplication in A is defined by the following rule for the basis elements

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256 L. H. SON

A n involution of A is given by

where FA = ( - ~ ) ( ~ ! ~ ) " ~ f ( " , ~ + ' ) e ~ , nA being the cardinality of A. The inner product on A is defined by

which turns A ifit:: ;: Banac?: r:?ge?:ra. 1~ the f(:!!o\.~!ing we &a!i dcfictt: afi (?per. ~nhyr: gf i;nc!i&3r. spacr En' . gk by <:,

where 1 5 m (, tz, 1 < k < n, m + k 5 n. We consider functions f defined on R and taking values in the Clifford Algebra A. These functions a re glven by the following form:

Next we introduce the generalized Cauchy-Riemann operators (see [3])

In what follows we shall consider the system

DEFINITION 1 A furzctiorz f E c l ( R ; A ) is called biregulur in 12 @ it is a solufion of (1). In fhe following we slzall denote the set of all hireg~ilur futzctiotzs irz R hy R(f2; A).

DEFINITION 2 A poitlt ( x , y ) t R"' x Fik is called a regular poitlt of Jutzction /. iJjr there exiszs urz open rzeigllbourlzood U of ( x , y ) in wlziclz f is h i r e M r . If for all open tzeiglzhour~zlood.~ of ( x , y ) rlze jiitzcliorz f is rzor biregnlar in whiclz rlzen rlze point ( x , y ) is called a sitzgulur poitlt of S

Let {U,} be an open covering of R. By Cousin-data for the covering {U,} of fi we mean 2 cG!!ec:ii;n Gf functions { h o p ) defined biregular iii the in:ciscc- tions U, r l U p ot pairs of open sets of the covering, and satisfy~ng the following

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conditions:

In this paper the following problem is considered:

PROBLEM I To find the functions h, defined and biregular in U, of the given cover- ing {U,} of fl such that

3. RUNGE'S THEOREM FOR BIREGULAR FUNCT!QNS

For solving Problem I we net-b some properties of biregular functions. In this sec- tlon we shall prove a theorem of Rmge's type.

Denoting by R(Wm x R ~ ) the set of all biregular functions Y(x,y) h i x E R m , y E W~ we get

THEOREM 1 (Runge's Theorem) Let R = Ql x R2 be a simply connected po do- i? muilz in W m x R ~ , where R1 is a domain in Wm(x) and Q2 is a domain in W (y). Further let f (x, y) be a biregular function in R and K be a co.wpact subset yf Q. Then

k f is uniformEy approximariie iri K byJfiLz"'tk~s in R(Wnl x W ).

Proof Let E > 0. We shall show that there is a function R(x,y) E R(Wm x f I k ) such that

I I f (x ,~ ) - R ( x , Y ) ~ ~ K < E . (*>

By enlarging K , we may assume that K = K1 x K2, KI c 01, K2 c fl2, and that each K; is simply connected. By fixed y E R2 the function f (x, y) is (left) monogenic on x E R1. Using the Cauchy's integral formula (see [I], p. 54) for x E K1 and y E a2 we have

where E(u - x ) = -(l/w,)((a - f ) / lu - xIm), and wm = 2 P I 2 ( 1 / I ' (m/2)) stands for the area of the unit sphere Sm-I in W m . For u E do l , x E K1 the function

- u - x

is bounded. I U - xim

It is easy to verify that the Riemann-sums which approximate the integral on the right of (4) do so uniformly on K1 x K2. Thus f is approximable uniformly on Kl x K2 by functions of the type

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258 L. H. SON

Hence we get

We need only prove that the functions

are approximable (uniformly) on Kl x K z by the functions in R(Wm x Rk). By every fixed u, E 8R1 the function (a, - I ) / ( u , - xirn is (left) monogenic (on variable x) in an open neighbourhood 0; of K1 such that K1 c 0; c R1. Because of the Runge- Theorem (see [I], p. 157 Corollary 18.14) this function is approximable uniformly m K, bj: :he f t~nct iom on 'R(Rn') , where R(W") is the set of all left monogenic tunctions P ( x ) for x E Wm.

By denoting C, = llf;(y)ll~, we get a function P!!x) E R(R"') such that

On the other hand fnr fked i the fmct im A(y ) = (I/c~i,). j(ui1yj is (right) mono- genic for y E Rz. Because of the same Runge's Theorem it is approximable uni- formly on K2 by the functions of R(H"), which means the set of all (right) mono- genic functions Q(y) for y E W k .

Thus for given ~/(2'+" CC;') > 0 there is Qi(y ) E 7Z(Wk) such that

From (7) and (8) it follows:

Denoting

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COUSIN PROBLEM FOR BIREGULAR FUNCTIONS 259

it is easy to show that R(x, y) E R(Rrn x R ~ ) . Further we have (because of (8))

From (6) and (10) follows (+), which completes the proof. rn

4. INHOMOGENEOUS SYSTEM

In this sectior! we cclnsidei the foilowing inhomngen system

for (x, y) E R = R1 x R2, where R is as in Tneorern 1. We consider first the equation

where g(x) E Cs(R1;A) is a fmction of x and (right) monogenic in y E Rz.

LEMMA 1 The equatim (12) lzar at least a solutioiz f (x,y) E CW(Rl; A), which is (right) rno~togenic for y E R2.

Proof (a) First we assume that

where Com(RI;A) is the set of all A-valued Cw-functions with compact support in 521; and g is (right) monogenic for y E Rz.

We define the function

where durn = dul A . . . A durn. It is easy to show that f E Cm(Rrn;A). Because of Theorem 9.5 (see [I]) f is a solution of equation (12). On the other hand we get

The Lemma is proved for g E CF(R1;A). (b) Now let g E Cc"(R1;A). We shall cover Rl by the covering {G,} with G, cc

G,+l cc . - . where the G, are simply connected domains. Next we define the func- tions g, E Com(R1; A) such that

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260 L. H. SON

and consider the system Dxf = gve

From step (a) of the proof it follows that system (15) is solvable in the class

Cm(R1; A).

It is now shown that we can find the solution f, of (15) in G,, such that

1 for v = 1 , 2 , ... Ilf"+l - f u l lo < 2,

a d fuDy = 0

Next define

Because of Runge's Theorem 1 there exists a function Rl(x,y) E R(Rm x Rn) such that

llf2-fi-R1lI<+ in GI.

Denoting: fi = f2 - R: 2 -

This means that the condition (16) is valid for v = 1. By a similar method we can construct all f,, v = 2,3,. . . for which the condition (16) is valid.

The sequence {f,) converges uniformly in every compact subset K of R1 to the function

CO

In every Gp the function f is the sum of finite number of functions from CO" and of a series of the monogenic functions (f,,, - f,), which converges uniformly in every compact subset. On the other hand it is

for v 2 p. Hence it follows that

and Dx f = lim,,,Dx f, = g for all x E Rl. Since g is (right) monogenic on y E 02, this means that if gDy = 0, then it follows from (14):

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Hence f, is monogenic in for every v. The function f (x,y) is the sum of a series of the (right) monogenic functions in y, which converges uniformly in every compact subset Kz c 0 2 . Hence f is (right) monogenic (see [I], p. 58; Theorem 9.11). Lemma 1 is proved.

Now we consider the system (11). Suppose that there exists a c2-solution f ( x , y ) of (11) in 0. Then we have

gDy = Dxh. (17)

This means that (17) is the necessary condition for solvability of systern (I?).

LEMMA 2 If the right hand side furzctions g and h satisfy the condition (17), then system ( 1 1 ) is solvable in class Cm(S2; A).

Fixoof We consider the first equation of (11)

Because o f Lemma 1 we get a solution fi(x.yj of this equation. ?:ow we sha!l find the solution (x,y) of system (11) in the fr~rm

f = fl + $.

Since

f D , = f iD, + $D, = h,

it follows that the function Q, is a solution of system

$Dy = h',

where h ' = h - flD,.

Further because of (17) we get

namely h' is monogenic for x E Ql. Because of Lemma 1 there exists a solution $ of system (19), which is monogenic

on x. Then f = fi + $ is a solution of system (11). Thus Lemma 2 is proved. . 5. SOLVING PROBLEM I

Let {Gi} be a subordinate partition of unity for the covering U = {U,) with $ j E C?(U,,). Further we define for every fixed ,6 the function

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L. H. SON

and

in U,nUp. This means that {Dxg,j defines a function g E Cm(R; A) and {g,Dyj defines a

function h E Cm(R; A). We consider now the inhnmogenr~us system:

At first we have

in every U,. Hence the necessary condition of type (17) is valid. Because of Lemma 2 the system (22) is solvable in class Cm(S1;A), namely there is a function h E Cm(Q; A) which satisfies system (22).

Defining k, := g, + It in every U, (24)

we get (because of Definition of g and h )

lzoDY = GODy + hDy =g,Dy - h = 0

in U,. This means that lz, E R(U,; A). From (24) it follows

l113-Iz, = (gp+h)-(g, + h ) = g p - g , = ha/,.

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Thus we get

THEOREM 2 Problem I is solvable for the domain R = Ql x R2 as in Theorem 1. In other words there exist the functions h, E R(U,; A) such thar

6. APPLICATIONS

Let f,(x,y) be the given biregular functions in U, with singularities such that

are biregular and have no singularities in U, n Up. we set

f hod J p - ,a?

then hmp satisfy assunlptiijns (2 ) and (3) and become Cousin-data for the covering {li,). Because of Theorem 2 there is k, E R(lJ,A) such that

Hence h, + f a = hp + fp in U, fl Up. This means that {h , + f a ) define a function f ( x , y ) in the whole of R, such that

is biregular in U,. Thus we get

COROLLARY Assume that ifc} satisfies the conditions (25). Then we can construct a biregular function f ( x , y ) in Q with singularities, such that f - f , biregular U,. In other words we can construct a ('global) biregular function with singularities in which has (local) given singularities.

ACKNOWLEDGMENTS

The author wishes to offer his thanks to Prof. Dr. N. Trudinger for his warm hos- pitality at the CMA of the Australian National University where his manuscript was prepared, and to Dorothy Nash and Mrs. Marilyn Gray for their untiring assistance.

References [I] E Bracks, R. Delanghe and E Sommen, Cliflord Analysis, Research Notes in Mathematics 76, Pitman

Books Ltd., London. [2] E Bracks and W. Pincket, Two Hartogs theorems for null-solutions of overdetermined system in

Euclidean space. Complex Variables: llteory and Applicafions 5 (1985), 205-222. [3] E Bracks and W. Pincket, The biregular functions of Clifford Analysis: some special topics, Clif-

ford Algebra and 7heir Applications in Mathematical Physics, NATO AS1 Series C: Mathematical and Physical Sciences, 183, 159-166.

141 L. Hormander, An Introduction to Complex Analysis in Several Variables, Princeton, NJ: Van Nos- trand, 1966.

[5] Le Hung Son, Cousinsches Problem beziiglich des Beltramwhen Differcntialg!eicbungssystem mehr- Beitrage zur Analysis 17 (1981), 49-55.

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