GENERALIZED HYPERCOMPLEX FUNCTION THEORY(l) - American

transactions of the american mathematical society Volume 195, 1974

GENERALIZEDHYPERCOMPLEXFUNCTIONTHEORY(l) BY

ROBERT P. GILBERT AND GERALD HILE ABSTRACT. lytic

function

Lipman

Bers and Ilya Vekua extended

by considering

two equations

the distributional

with two unknowns

and two independent

tions have come to be known as generalized sequently,

satisfy

Avron Douglis

(classically)

introduced

the principal

2r unknowns

and two independent

of equations

can be represented

variables.

by a single

are termed hyperanalytic

functions

by Douglis

studied

same way as Bers and Vekua extended

extended

class

of functions

1. The equation

tem of the first order in two independent

quirements

on the coefficients

the canonical

equation.

variables,

with

systems

Solutions

In this work, the class sense

functions.

of generalized

A. Douglis

these

Sub-

which

of 2r equations

algebra

"hypercomplex"

the analytic

and the algebra.

system

In Douglis'

of

solu-

functions.

of functions

in a distributional

as the class

These

analytic

and a class

functions.

is extended

systems

variables.

part of an elliptic

of an ana-

of elliptic

(or pseudo)

an algebra

of such equations

the concept

solutions

of

much in the

We refer to this

hyperanalytic

functions.

[l] showed that an elliptic

with sufficient

sys-

smoothness

of the first order terms, can be decomposed

re-

into

subsystems U0,X-V0,y

+ ---

"«O»

U0.y + V0,x + ---

(1.1)

uk,x-vk,y

=V

+ auk-l,x

+ buk-l,y

Uk,y + vk,x + avk-l,x

+ '"=Sk,

+ bvk- l.y + * ' ' = hk>

k = l, 2,.-.,r-l, where the dots represent

zero order terms.

Beltrami system if it is homogeneous showed that with a certain

Received

by the editors

and contains

commutative

December

He called

algebra

the system

a generalized

no terms of zero order.

He

such a system can be written in a

22, 1972 and, in revised

form, March 29, 1973.

AMS(MOS)subject classifications(1970). Primary 30A92,30A93,30A96,30A97, 35J45; Secondary35C15. Key words and phrases. and Vekua, 0)

elliptic

Hypercomplex

variables,

pseudo

analytic

functions

of Bers

systems.

This research

was supported

in part by the Air Force

Office of Scientific

Research

through AF-AFOSR Grants No. 71-2205A, and No. 74-2592. Copyright® 1974, AmericanMathematicalSociety

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R. P. GILBERT AND GERALD HILE

2 brief form. This algebra the multiplication

is generated

of this algebra

2r independent

i and e , subject

to

rules

i The elements

by the two elements

= —1,

ie = ei,

eT = 0.

are linear combinations

with real coefficients

of the

elements

ek, where e° = 1. A particular

iek,

k = 0, l,...,r-

element

(1.2)

1,

c of the algebra can be written

c = '¿

where each c, is a complex number.

ckek,

The conjugate

of c, c, is given by

r-1

c = Z- cke • k =0 If c0 = 0, then c does-not A hypercomplex

function

have a multiplicative

inverse

and is called

is a map from the plane into this algebra

nilpotent.

and has the form

r-1

(1.3)

wix, y) = £

u/fc(x, y)e*,

*=0

where each

w,

is complex

valued.

(1.4)

Then with the identification

wk - uk + ivk

the generalized

Beltrami

system

can be written

(1.5) where

%v = 0 2) is the differential

operator x

y

We will treat somewhat more general

is represented

y

systems

in this paper.

One such system

by

(1.6)

%u+Aw + Bw = 0,

where A and B ate hypercomplex

functions.

r-1

(1.7)

x

a =*=° z v*. Ak = M(pfc+ st) + Hitrr - qk),

With r-I

B- fe=0 s v*. Bk = ^k~sk}

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+ ^l(r* + 9fe^'

3

GENERALIZED HYPERANALYTIC FUNCTION THEORY (1.6) becomes

the system

"o,*-*0,y

+ Po"o

"0,y + i;O.x + r0"0

+?0V0

=0,

+S0"0

=°«

k

(1.8)

ukx - vkiy + auk_Ux + buk_l/ii + eb), we obtain

+ eb))= 0. Thus if we let iff =

3)ty= tJD(f,/ii + eb) and therefore

(4.1) and (4.2) are equivalent.

We now need some results

from Vekua [3] concerning

dz~2\dx + dy)' Vekua defined

Definition df/dz

ó^~2\o\

in a Sobolev sense

¿V'

these

operators

4.4.

Let / and g be complex functions

in ®, or (ii) g = df/dz

the operators

in the following

in L^i®).

in ®, if for all complex functions

(i)

ff(f*t+g(p)dxdy=0,

(ii)

ff(f7Ê+ ® **)**-*•

manner.

Then (i) g = rp in C[.(®),

The following theorem is then obtained [3, p. 72].

Theorem4.5. If f £ LJ^i®), df/dz £ L^.(®) for some p,p>l, exists

and is in L^.(®).

Theorem 4.6.

Thus fx and f

Let w e lL(®),

Then for each k, wk x and w,

then df/dz

exist and are in l£(®).

3)w = v, where v £ E?i®) exist in the Sobolev sense

for some p, p > 1.

and are in LJL(®).

Moreover, the following formulas hold:

2dw0/dz = vQ,

2dwk/dz+awk_y

x + bwk_y

=vk,

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k = 1, • • •, r - 1.

10

R. P. GILBERT AND GERALD HILE

Proof. Let iff be a complex function in C^.(®). By (4.2),

JJ [wiiffx + iiffy+ eiaiff)x + eibiff)) + vifAdxdy = 0. © y Equating

powers of c, we obtain the equations

JJ íw0^x + ^J ©

+ v0iff\dxdy= 0,

JJ ^wk^x + ^y* + wk-l^a^K ©

+ ^'y

+ v¿fi\dxdy = 0,

k = 1,•••,/■- 1. The first equation

states

2dwQ/dz=

vQ, and by Vekua's

theorem

tz>0 and wQ

exist and are in L?(®).

Setting

k = 1 in the second equation,

and using

aiff e Cli®), we obtain

}}[wAiffx + iiffy) - u>QtXaiff - w0 =5JïTTbvJ^)dxdy JJí+ebiz)

\l™J¿i+ebiOtit)-tiz)

tjiO -if-i+ebiO The interchange

iLv)iOi$+ Jyv)=

in ®.

if O ¿s hyperanalytic

in ®,v

£ L1i®), then O + /^n eL^®),

V.

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12

R. P. GILBERT AND GERALD HILE Proof.

theorem

If w e L]^. (®), v e L1i®), and v = iDzzzin ®, then by the previous

jXzz; - Lf ) = v —v = 0. By Theorem 4.8, 22z— Lti

is hyperanalytic

in ®.

The converse part of the corollary follows immediately from the previous theorem.

Theorem 5.3. Let % be a bounded domain. If v e Lpi®), 2< p< 00, then the function

w = L.f satisfies

(i) \w(z)\ < M(a, b, p, ®)\v, ®\p, z e C, (ii) \w(zy)-w(z2)\
rule"

(Ak[W[ + Bk¡w¡) = 0

and the A, ■ and B, , ate complex valued. and a "chain

rule"

for the operator

2).

Theorem 6.1. Let w,v e L1^®), with 5)w, % e Lp(®), 2 < p < ~. Then in ®,$(wv) Proof. we assume

= @w)v + w(§v). Since the product rule is easily it is true in these

cases.

verified

if either

Let ®, be a bounded

w ot v is in C1 (®), subdomain,

®, C @.

By Corollary 5.2, we have in ®,

«, = * + /ÄiCM, where /ö

0 and W are hyperanalytic

(3)v) ate in BaiC),

v = V+J9A%f),

in ® j, and, by Theorem

a= ip - 2)/p.

Therefore

inside

5.3, L. (d)w) and

®p

%m)=$i%)+W(iM +3)[/@ flw)•J9flv)]. Thus it suffices

to prove the product

cfoe CA®y), iffn e C~(®,),

rule for the product

J& (3V) • La (3)v).

Let

with „-» î)w in Lp(®y). Since Douglis showed

[1, p. 273] that Jçiftn e cH®j), we have

IJtÏz ©j W©^ +V^+

^©/t, •7.^))]A*=0.

By Theorem 5.3, J uniformly in C. Thus after taking limits, we have the result

Sty^tfM • 7Sl(^)] = S)»• J&l(Vv)+ J&1&w).%.

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16

R. P. GILBERT AND GERALD HILE Theorem

6.2.

Let w € L^i®),

î)w e Lpi®),

Niz) where wQ is complex and N nilpotent.

bounded domain ®#, and let f be a complex function

main containing

2(z) where

C is a complex constant,

and u> is a complex function continuous

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and bounded

GENERALIZEDHYPERANALYTICFUNCTIONTHEORY 072 the whole plane.

(Thus

w is bounded away from zero provided

17

it is not identi-

cally zero.) Following

is an analogous

Theorem

6.4.

result

for solutions

Let w be continuous

of (6.1).

and bounded

and satisfy

whole plane, where each Akl, Bk[ e Lp* (C)» 2 2, \w(z)\ Ju^wiz, O

MO - tiz))

2M4")- Hz)) = -Aiz)X^\z, We now define fundamental

i-Aiz)wiz,0-Biz)wiz,0)

O - Biz)X^\z, O-

kernels

associated

with fixed A, B in L^ÍC)

and the equation (7.1). Definition

7.5.

and B in L^ÍC),

The fundamental

kernels

ñ'

' and ii,

(7.8)

&lXz, O = X(1)(z,O + ¿X(z, 4").

(7.9)

n(2)(z, O - x(1)(z, O - ¿x(z,4:),

where X(

and X1 ' are the functions

Theorem 7.6.

associated

are

The fundamental

described

kernels

il'

in Theorem 7.4. ' an«? fl^2' satisfy:

il) For each C in C,in C-{£\

(7.10)

$ß{1\z, O + Aiz)&lKz, O + B(z)íí(2)(z,O = 0,

(7.11)

3)zíí(2\z, O + A(z)S2(2)(z, O + Biz)Sl{l)iz,O = o.

(2) For fixed Ç, and j = 1, 2,

l^'Hz.O^Odzl-1)

as \z\ -oo.

(3) As \z - ¿| - 0,

(7.12)

QaW)-.__i

KO - Kz)

= Oi\z-t\-2/p),

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with A

23

GENERALIZEDHYPERANALYTICFUNCTIONTHEORY

(7.13)

|«(2)(z,o\ = o(\z- c\-2/ p).

Proof.

Property

(1) is readily verified

from (1) of Theorem 7.4.

Property

(2)

follows from the relations

OÜV

r\

exp[6)(1>(Z)- co^KQ] + exP[a>