Products of white noise functionals and associated derivations

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J. Korean Math. Soc. 35 (1998), No. 3, pp. 559-574

PRODUCTS OF WHITE NOISE FUNCTIONALS AND ASSOCIATED DERIVATIONS

DONG MYUNG CHUNG1, TAE Su CHUNG2 AND UN Cm JI³

ABSTRACT. Let the Gel'fand triple (E){J C (L2) C (E)~ be the framework of white noise distribution theory constructed by Kon- dratiev and Streit. A new class of continuous multiplicative products on (E){3 is introduced and associated continuous derivations on (E){3 are discussed. Algebraic characterizations of first order differential operators on (E){3 are proved. Some applications are also discussed.

1. Introduction

We take, as a framework of white noise distribution theory, a Gel'fand triple

(1.1) o~.B <1,

which was constructed in [18], where E* is the space of tempered dis- tributions and JL is the standard Gaussian measure associated with a Gel'fand triple E C H c E*. In particular, if.B = 0, (1.1) becomes (E) C (£2) C (E)*, which was constructed in [19]. The white noise distribution theory is an infinite dimensional analogue of the Schwartz distribution theory in which the role of Lebesgue measure on jRn is replaced by the Gaussian measure JL onE*. This theory was initiated by Hida [11] and has been considerably developed with applications to

Received February 24, 1998.

1991 Mathematics Subject Classification: 46F25.

Key words and phrases: white noise, derivation, first order differential operator, Cauchy problem, Poisson type equation.

lResearch supported by BSRI, 97-1412.

2Research partially supported by Korea Research Foundation 1997.

3Research partially supported by Global Analysis Research Center.

This is an invited paper to the International Conference on Probability Theory and its Applications.

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560 Dong Myung Chung, Tae Su Chung and Un Cig Ji

stochastic analysis, Feynman path integral, infinite dimensional har- monic analysis, quantum probability and Cauchy problems in infinite dimensions and so on, see e.g., [lJ-[9J, [12]-[15], [22J-[26], [30].

It is well known [13,24J that the Wiener and Wick products are continuous multiplicative operations on(E){3 and so(E){3 becomes a topological algebra under the Wiener and Wick products. Furthermore, we note that the differential operator D_y is a first order differential operator as well as a continuous derivation on (E){3. Motivated by this point of view, Obata [29J determined all the continuous derivations on (E) with respect to the Wiener products and then showed that they are first order differential operators with variable coefficients. In [1J,

Chung and Chung showed that the results in [29] can be extended to the Wick product case. Recently, Chung and Chung has introduced a class {01'" E C} of multiplicative products on (E) including the Wiener and Wick products and then have showed that with each ^01' and an (E)-valued distribution q> on lR, we can associate a first order differential operator with variable coefficient <I>, which is indeed a continuous derivation on (E) with respect to^01"

In this paper, we shall see that 30,m(l'i:)+N is similar to N, i.e., there exists a linear homeomorphism 9K, E GL((E){3(m» such that 30,m(l'i:)+^N =9;;1N9K, (Corollary 3.7), where30,m(l'i:) and N are the integral kernel operator with kernel distribution I'i: E (E~m)* and the number operator, respectively. Next, we define a productOK,associated

with 9^K" and prove that a first order differential operator with variable

coefficient associated withÔK, is indeed a continuous derivation on(E){3 with respect to ÔK, (Theorem 5.4), and then 30,m(l'i:)+N is. a continuous derivation on (E){3 with respect to the product ÔK,' Finally, as applications, we shall study the eigenvalue problem, Cauchy problem and Poisson type equation associated with 30,m(l'i:) +^N.

2. Preliminaries

Let H be the real Hilbert space of square-integrable functions on lR with norm I .10. Let S(IR) be the Schwarz space consisting of rapidly decreasing COO-functions. Then we have a Gel'fand triple:

(2.1) E ==S(lR) cHc S/(lR) == E*,

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eE ^E,

Products of white noise functionals and associated derivations 561

where E* is the space of the tempered distributions. Note that the Gel'fand triple (2.1) is reconstructed by using a positive self-adjoint operator A = 1+t² - d?- /dt² onH with Hilbert-Schmidt inverse (see [13], [24], [28]). Infact, E is a nuclear space equipped with the Hilber- tian norms lelp = IAPelo,eÊ Ê,p Ê^R. ^Let^J.Lbe the standard Gaussian measure on E* whose characteristic function is given by

L*ei(x,{) J.L(dx) = exp (-~leI5) ,

where (.,.) is the canonical bilinear form on E* x E. The canonical C-bilinear form on_(E~n)*x _(E~n) is also denoted by the same symbol

C·). We denote by (L²) == L²(E*,J.L;C) the complex Hilbert space of square integrable functions on E* with norm 11·110. By the Wiener-It6 decomposition theorem, each</>E (L²) admits an expression

(2.2)

00

</>(x) = L(:x0n :,fn), n=O

x E E* , fⁿ E H~n,

and 11</>115 = L::'=on!lfnl5, where H~n is the n-fold symmetric tensor product of the complexification of H and : x0n : denotes the Wick ordering ofx0n .

Let f3 be a given real number with 0~ f3 < 1. For eachp ~ 0, define

00

11</>11~,f3⁼ L(n!)l+f3lfnl~, n=O

where </>is given as in (2.2). Let (E_p)f3 = {</> E (L2) :1I</>llp,f3 < oo} and let (E)f3 be the projective limit of {{E_p)f3 : p ~ O}. Then (E)f3 is a nuclear space and we have a Gel'fand triple:

(2.3)

where (E)~ is the topological dual space of(E)f3. The triple (2.3) is called the Kondratiev-Streit space [18]. Iff3 = 0, then (2.3) is called the Hida-Kubo-Takenaka space and denoted by (E) C (L²) c (E)*.

An element in (E)f3 (and in (E)~) is called a test (and generalized, resp.) white noise functional. We denote by «., .)) the canonical C- bilinear form on (E)~ x (E)f3. For each if> E (E)~, there exists a unique sequence _{Fn}~=o, Fn ^E (E~n):ym such that

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562 Dong Myung Chung, The Su Chung and Un Cig Ji

00

((if?,cP») = L n!(Fn,/n),

n=O

wherecPis given as in (2.2). In this case weusea formal expression for if? E (E)~ :

00

if?(x) = L(:x®n:,Fn),

n=O

xEE*.

For each { EEc, the functioncPf, E(E){3 given by

cPf,(x) == ~(:x®n:, {=~) ⁼^exp ^{((X,{) -} ~({,{»), ^XE ^E*

is called an exponential vector. Note that {cPf, : {EEc} spans a dense subspace of(E){3' The S-transformofif? E (E)~ is a function on Ec defined by

Sif?({) = ((if?,cPf,»), {EEc·

In [18], Kondrative and Streit proved a characterization theorem for elements in(E)~ and (E){3 by analytic properties of the S-transforms.

By using the characterization theorem, for each if?,'1'E (E)~ the Wick product if?⁰ '1' E (E)~ is defined by S(if?⁰ '1') = Sif? . Sw. In facts cPo'l/J E (E){3, cP,'l/J E (E){3 and the Wick product is a continuous multiplicative operation on (E){3 (see e.g., [24]).

LetL:(X,~)denote the space of all continuous linear operators from a locally convex space X into another locally convex space~. Also for notational convenience we writeL:(X) = L:(X, X). For each 3 E L:«E){3,(E)~), the C-valued function§ onEc x Ec defined by

{,1J E Ec

is called the symbol of operator 3. The following theorem is a characterization theorem for symbols of operator in L:«E){3,(E)~) and in L:«E){3) ([24], [27], [28]).

THEOREM 2 .1. AC-valued function e :^EcxEc ---+ C is asymbolof an3 E L:((E){3,(E)~)ifandonlyifesatisfies the following conditions:

(01) Foreach {,{',1J,1J' EEc, the function (z,w) ^I---? e(z{.+{',W1J+1J')

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is an entirefunction on C x C;

(02) There exist constants K ~ 0, C ~ 0 andp ~0 such that

~,'"EEc·

e,,,,EEc·

Moreover, e ^is the symbol of an operator in £((E){3) ifand only if it satisfies (01) and

(02') For any p _~ 0,€ >0, there exist q_~ 0 and K _~ 0 such that

le(~,",)1 ::; K exp€ (,~,;1~⁺ 1171~^{) ,}

For each y E Ecthere exists a unique operator Dy E C((E){3) such that Dy4>t;, = (y,e)4>t;" e^E ^Ec which is called the annihilation operator. The adjoint operator D; E £((E)~) of Dy is called the creation operator. By Theorem 2.1, we can show that for each K E

(Ec®(l+m»)*, there exists a unique operator 3l,m(K) E £((E){3,(E)~) such that

This operator 3l,m(K) is called the integral kernel operator with kernel distributionK. In particular, D..c = 30,2(T) andN =^31,1(T) are called the Gross Laplacian and number operator, respectively, whereT is the trace defined by (T,e ® ",), e,,,, EEc. It is well-known that for each

K E (E~(l+m»)*, 3_l,m(K) ^E£((E){3) if and onlyifK E (E~l)^Q9(E~m)*.

3. Transformation group on white noise functionals

In this section we obtain a two-parameter transformation group 0

on(E){3. Moreover, we shall prove that there exists an element OK- E 0 such that OK-(3₀,m(K)+^N)O;;l =N. For notational convenience, we define a function f3 :Nu{O} ---+ [0,1) by

{ 0, f3(m) = ^{1 _} ^J.-

m'

m= 0,1 m = 2,3,··· .

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LEMMA3.1. Let^K E (E~m)*andaE C. Then there existsa unique operator9K.,a E £((E){3(m»)) such that

€EEc·

Proof Let ^K E (E~m)* and a E C. For any€,^1] EEc, we put

Then the functione satisfies conditions (01) and (02') in Theorem 2.1 with{3 = {3(m). Hence by Theorem 2.1, there exists a unique operator 9K.,a E £((E){3(m») such that

This complete the proof. o

Let C and C* = C - {O} be the additive and multiplicative groups of complex numbers, respectively. Let GL(x) denote the group of all linear homeomorphisms from a locally convex spacex onto itself.

THEOREM 3.2. LetK E (E~m)* be fixed. And let9= {9<iK,b ; a ^E C,bE C*}. Then 9 isa subgroup of GL((E){3(m»).

Proof By Lemma 3.1, we have90K,1<Pf. = <Pf. for any€EEc, and 9a^IK,b' (9aK,b<PE,) ⁼ exp{(a+a'bm){K,€0^m)}<Pb1bf. = 9(a+a^lb^Tn)K,b' b<Pf.' for any a,a' E C and b,b' E C* and € EEc· But {<pf. : € ^E Ec}

spans a dense subspace of (E){3(m) and for any a E C, bE C*, gaK,b is continuous. Hence it follows that for any <PE (E){3(m),

and

Then9(-ab-m)K,b-1 is the inverse of9aK,b in9. o

Let :FK,a E £((E)~(m») be the adjoint operator of9K,a. Then by using similar arguments as in [5], we obtain explicit expressions :FK,aif!

and 9K,a<P for if! E (E)~(m) and <PE (E){3(m).

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Products of white noise funetionals and associated derivations 565 THEOREM 3.3. Let <Jl(x) = L::=o(: ^{x 0n :,}Fn) ^E (E)~(m)' Fn ^E (E~n):ym' Then wehave, for any positive integerrn,

00 [';;;-1

'L ;r,.() _ ,,(. 0n. ,,1 ^n-mkF ^{~ 0k)}

JK-,a'*! ^X ^- L-J'x ., L-J k!a n-mk®K.

n=O k=O

THEOREM 3.4. For <j)(x) = L::=o(:x0n :,fn) E (E){3(m),fn E E~n,

wehave, for any positive integerm,

( ) _ ~(. ^0n. ~ ^(n+mk)! ⁿ 0k~ ⁾

Q""a<j) x - L.J . x .,L.J n!k! a ^K ®mkfn+mk'

n=O k=O

PROPOSITION 3.5. The operator QK-,a isexpressed by Q""a = e(loga)N⁰ e³⁰,1n(K-).

Proof Let <j)(x) = L::=o(:x0n :, fn). Then by Theorem 3.4, it holds that QO""a<j)= e(loga)N<j)and Q""l<j) =e³⁰,1n("')<j). Clearly, QO""a⁰ QK-,l =

Q""a' Hence we complete the proof. 0

PROPOSITION 3.6. For anyab a2 E C, we have

QK-,a(a1S0,m(K)+a2N) = (a1 +a2m )e-^mlog aSO,m(K) +a2N)QK-,a' Proof Note that for any^K E (Elm)*,

[N,SO,m(K)] = -rnSO,m(K).

Therefore, we have

e³⁰,'rn(K-)N = (N+mSO,m(K»e30,1n("') and

e(1oga)N30,m(K) =e-TnlogaSo,m (K)e(10ga)N •

Hence by Proposition3.5, we obtain that

Q""a(a1S0,m(K)+a2N) = (a1 +a2m )e-^mlogaSO,m(K) +a2N)QK-,a'

Thus we complete the proof. 0

COROLLARY 3.7. Let K E'(Elm)* and let QK- = Q_..!..",l ' Then we

' r n '

have

Proof The proof is straightforward from Proposition 3.6. 0

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566 Dong Myung Chung, Tae Su Chung and Un CigJi

4. Products on white noise functionals

In this section, we introduce a new class of continuous multiplicative products^0K on (E)j3 indexed by^K, E (E~m)*.

Let 1£ be a linear homeomorphism on (E) j3. Then we can define a continuous multiplicative operation011. on (E)j3 associated with 1£ by

</>,'If; E (E)j3, where°is the Wick product.

PROPOSITION 4.1. Let3 E £«E)j3). Then 3 is a derivation with respect to the multiplication011., Le., 3(</>011.'l/J) = 3</>^011.'If;+</>011.='l/J,

</>,'If; E (E)j3 ifand onlyif1£31£-1 is a derivation with respect to -the Wick product.

Proof The proof is clear from the definition of011.. o

From now on we only consider the case 1£ = 9^I'. for a fixed kernel

K, E _(E~m)* with integer m _~ 2, where gK is given in Corollary 3.7.

We put 0K = 0g",. That is,0K is a continuous multiplicative operation on (E)j3(m) defined by

(4.1) </>,'If; E (E)j3(m).

~,,,.,EEc,

(4.2)

PROPOSITION 4.2. Let ^K, E (El^m)*. Then there exists a unique operatorTK E £«E)j3(m), (E)~(m» such that the following hold:

</>t.^0K</>." = T,;(~,"")</>t.+.", and

Proof. By using the equation (4.1) we can easily see that

~,,,.,EEc.

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Note that the function811:(e,^1]) = exp{~^(K,(e+^1])0^m ^- e^0m ^- ^1]0^m^{) }}

satisfies (01) and (02) in Theorem 2.1. So, there exists a unique operator TII: E £«E){3(m),(E)~(m)) such that (4.2) holds. Now for any

e,^{1], ( E} ^Ec we observe that

((<Pc;011:<Pf" r/J'J1» = ((TII:r/Jc., <PE.» «<pc.+f" <P'J1»

=e(E.,'J1) «(T: <PE., <Pc.» «<pc., <P'J1»

=e(f,,'J1) «(T:<pf,0 r/J'J1' <Pc.).

Since 811:is symmetric, we haveT: = TII:, and hence (4.3) holds for all

<P=<Pc., ( E Ec· Thus the proof follows from the fact that {r/Jc. ; ( E Ec}

spans a dense linear subspace of(E){3(m). 0

EXAMPLE 4.3. Let K E (E~2)* and'K,E £(Ec, El;) be given by

Then we have

Hence we obtain that TII: = r«'K,+'K,*)/2), where reA) is the second quantization operator of A E £(Ec, Ec). In this case, the equation (4.3) becomes

<PE (E)I3(Tn) ,e,^1]^EEc·

Moreover, ifK E (E~2);YTn' then we have

In particular, 0'Y_r becomes the ,-product 0"1 studied in [21. Further- more, 00 is the Wick product and Or is the Wiener product (see [2], [24]).

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5. First order differential operators

In [2], Chung and Chung discussed thefirst order i-differential op- erator:=: E £((E» with coefficient q> E El:. ® (E), where :=: is given by

:=:= l ^q>(t)^0"(^Ot^dt

as a formal integral expression. We now introduce a first order differential operator associated with the OK-product.

PROPOSITION 5.1. Let ^K, E (E~m)* and let q> E Ec^® ^(E){3(m)'

Then there exists a unique:=: E .c((E){3(m» such that

§(~,1J)= (((q>,~)^OK<Pe,4>TJ))' where (q>,~) E (E){3(m) isgiven by

(( (q>,~),4») = ((q>,~®4»),

~,1J EEc,

4>E (E){3(m).

Proof. The proof is immediately from that the function

~,1JE Ec

satisfies (01) and (02') in Theorem2.1 withf3 =f3(m). o

DEFINITION 5.2. Let ^K, E (E~m)* and q> ^E Ec^® ^(E){3(m)' ^The

operator:=: defined in Proposition5.1 is called afirst order K,-differential operatorwith coefficient q> and denoted by

:=:=

L

^q>(t)^OK.^Ot^dt

asa formal integral expression.

THEOREM 5.3. Let ^K, ^E (E~m)* andq> E Ec^® (E){3(m)' Then for :=: E£((E){3(m» the following statementsare equivalent:

(i) :=: is a first order K,-differential operator with coefficient q>.

(ii) For any~E Ec and n ~ 0, we have

:=:((:-l2m:,~0n) = n(:-0(n-l):,~0(n-l»)OK. (q>,~).

(iii) Forany~ EEc andn ~ 0, :=:((-,~)OKn)= n(.,~)OK(n-l)oK(q>,~).

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Proof (i)::::} (ii) Since 3 is a first order ~-differentialoperator with coefficient ~, the symbol of 3 is given by

§(~,¹⁷⁾ =«3</>t" </>.,,)) =^«(~,~) ÔK</>t" </>.,,)) , ~,17 ÊÊe.

Hence we obtain that for any ~ E Ee, 3</>t, = _(~,~)OK </>t,. Therefore for any ~E Ee, </> E (E) and ^Z E C, we have

f ~«3(^(: ^{.®n :,}~®n)), ^</»)zn

n=on.

= ~ ⁽ⁿ ~^1)!«(~,~)^{OK (:} ^.®(n-l) :,~®(n-l»),</»)zn.

Thus the proof follows.

(ii) ::::} (i) The proof is obvious.

(ii) ^{9 (iii) Note that for any ~ EEe, g;l«.,~))=^(-,e). Hence by the definition ofÔK, we obtain that for anyeÊ Êe

f ~(.,~)ol<n ⁼ g;;l</>t, =e-!:.(K,t,@m}</>t,.

n=On.

Therefore (ii) implies that for any~ ~ Ee

f ~3«-,~tl<n) ⁼ e-!:.(K,t,@m) f ~3«:-®n:,~Q?)n))

n=On. n=On.

=

t

~(.,~)ol<n^OK (~,~).

n=On.

Hence (ii) implies that for any €E Ee and zE C

f ~3«-,~}Ol<n)zn ⁼ f ~(-,~tl<n^OK (~,€)zn+l.

n=On. n=On.

Similarly (iii) implies that for any ~E Ee and z E C

f ~3«:-®n:,~®n))zn ⁼f ~(:-®n:,~®n) ^OK (~,~)zn+1.

n=On. n=On.

Thus we complete the proof 0

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THEOREM 5.4. Let K, E (E~m)* and 3 E £((E),8(m»). Then 3 is a derivation with respect to01<;. ifand only if3 is a first order K,-differential operator with some coefficientq, EEl:.® (E),8(m).

Proof Let 3 be a first order K,-differential operator with coefficient q, E El:. ® (E),8(m). Then we note that 91<;.{-'f,) = 9;1(.,f,) = (.,f,) for all f, EEc· Then the operator 3' = Qx:3Q;1 is a first order Wick differential operator with coefficient 4>', where q,' E El:. ®(E),8(m) is given by (4)',f,) =QI<;.(q"f,) for eachf, EEc. Infact, we have

3'((: .em :,f,0 n») = Q1<;.3Q;1 ((., f,tⁿ⁾

=Qx:2((.,f,)o,.n)

= 9x:(n(.,f,)o,.(n-1)Ox: (q"f,»)

= n(: .0(n-1) :,f,0(n-1» 0 (<I>',f,).

So, by Theorem 4.5 in [1], 3' is a derivation with respect to0 and hence by Proposition 4.1, 3 isa derivation with respect to OX. _

Conversely, let 3 be a derivation with respect toOX. Define a map4> : Ec ^--? (E){3(m) by ~(f,)⁼ ^{3( (.,}f,», f, EEc· Then ~ ^E £(Ec, (E),8(m»).

Hence there exists a unique q, E El:.®(E){3(m) such that f,EEC.

Since 3 is a derivation with respect to 01<;., for any f, E Ec and n _~ 0 we have

Thus by Proposition 5.3, 3 is a first order K,-differential operator with

coefficient 4>. 0

EXAMPLE 5.5. For each y E El:. the differential operator Dy is a first order K,-differential operator with coefficient y ®l.

EXAMPLE 5.6. Let K,E (E~m)*. Then by Corollary 3.7 and Propo- sition 4.1,30,m(K,)+N is a derivation with respect toOx:. Moreover, by Theorem 5.4, 30,m(K,)+N is a first order K,-differential operator with coefficientq,o, where (4)o,f,) = (30,m(K,)+N)((.,f,» = (.,f,). In particular, _I!:i.G+N is the first orderIT-differential operator with coefficient

<Po (see [2]).

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6. Applications

In this section, we shall discuss the eigenvalue problem, Cauchy problem and Poisson type equation associated with 30,m(K)+N, K E

(E~m)*.

We now consider the eigenvalue problem associated with 3_0,m(K.)+N, Le., we consider

(6.1)

where'If; E (E);3(m) and ,\E Care unknown.

By using the fact that _{3 0,m(K.)} +N = Q;1 NQK" we easily have the following proposition:

PROPOSITION 6.l.

(i) ,\isan eigenvalue o130,m (K.)+^{N if}andonly if,\ isaneigenvalue olN.

(ii) <P is an eigenfunction of_{3 0,m(K.)}+N if and onlyif QK.<P is an eigenfunction ofN.

(iii) Theset ofall eigenvalues of30,m(K.) +^N is {O, 1,2"" }.

Now, we consider the following Cauchy problem:

(6.2) Uo = 4>E (E)(3(m)'

Note that Ut = QO,e-t is the one-parameter subgroup ofGL((E)(3(m)) with infinitesimal generator -N (see [5], [17], [24]). Hence we can easily check that Ut = Q;1Qo,e-tQK. is the one-parameter subgroup of GL((E)(3(m)) with infinitesimal generator _-(30,m(K.) +N). Thus we have the following theorem.

THEOREM 6.2. Let 4> E (E)(3(m). Then Ut = Q;1Qo,e-tQK.4> E (E)(3(m) is a unique solution of theequation (6.2).

Finally, we consider the following Poisson type equation:

(6.3)

where^</>E (E)(3(m) and ,\ > o.

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The>..-potential (>.. >0)of test functional</>E (E){3(m) is defined by H>.</> = 100e->..tQo,e-t</>dt,

where the integral is a white noise integral (see(17], (24]). For the case

>..= 0, define the normalized potentialof</>E (E){3(m) by

G</>=100^QO,e-t(</> - E(if» )dt,

whereE(if» is the expectation of</>.

THEOREM 6.3 [17]. Let </>E (E)j3(m)' Then we have NG</>= </> - E(</» and (N+>"I)H>..</>= </>.

THEOREM 6.4. Let K E (E~'Tn)* and </> E (E)j3('Tn)' Then u Q;;lH>..QK,</> E (E)I3('Tn) isa solution ofthe equation (6.3).

Proof Let </> E (E)I3('Tn)' Then by Theorem 6.3, v = H>..QK,</> is a solution of the equation(N +>..I)v =QK,</>. Hence we obtain that.

(Q;;l(N+>..I)QK,)Q;;lv = </>.

Thus by Corollary 3.7, we have

(30,m(K) +^N +>"I)Q;;lv= </>, That is,u =Q;;l H>..QK,</>satisfies the equation (6.3).

THEOREM 6.5. Let</>E (E)j3(m)' Then wehave (30,m(K) +N)Q;;lGQK,</> = </> - E(QK,</»' Proof Let </>E (E)I3(m)' Then by Theorem 6.3, we have

NGQK,<I> = QK,</> - E(QK,</».

Hence we have

Thus by Corollary 3.7, we complete the proof.

o

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Dong Myung Chung

Department of Mathematics Sogang University

Seoul 121-742, Korea Tae Su Chung

Department of Mathematics Meijo University

Nagoya 468, Japan Un Cig Ji

Global Analysis Research Center Department of Mathematics Seoul National University Seoul 151-742, Korea