Ultradistributions and quantum fields: Fourier-Laplace transforms and boundary values of analytic functions

REPORTS ON MATHEMATICAL

Vol. 16 (1979)

ULTRADI[STRIBUTIONS TRANSFORMS

AND QUANTUM

AND BOUNDARY

VALUES

PHYSICS

FIELDS:

No. 2

FOURJER-LAPLACE

OF ANALYTIC

FUNCTIONS

F. CONSTANTINESCU*and W. THALHEIMER+ Institut

fur Angewandte

Mathematik

der Johann Wolfgang Goethe-Universitat, F.R.G.

Frankfurt

am Main,

(Received October 4, 1977) (Revised February IS, 1979) Dedicated to the memory of W. Tbalheimer

The distribution theory is the mathematical framework of the axiomatic quantum field theory of A. S. Wightman. The axioms are satisfied in the case of the free field and in some non-trivial models studied in the constructive quantum field theory introduced by J. Glimm and A. JatIe (two and three dimensions). No non-trivial example in four dimensions is known. The vacuum expectation values in the Wightman theory can have at most polynomial increase in momentum space. A. Jaffe has extended the axioms in order to allow non-polynomial increase in momentum space. In this paper we discuss the ultradistribution framework which is the most general framework for JatIe fields (strictly localizable fields). The ultradistributions have been introduced by A. Beurling, G. Bjiirck and independently by C. Roumieu. Ultradistribution theory is a natural generalization of the distribution theory. We study the Fourier-Laplace transform of ultradistributions, extenting results of A. Jaffe [8,9] in several directions. A BochnerSchwartz theorem for ultradistributions is also shown to be valid. We expect ultradistribution theory to play a role in constructive quantum field theory.

1. Introduction

It is well known that distribution theory is the mathematical framework of the axiomatic quantum field theory [21]. The axiomatic quantum field theory, introduced by A. S. Wightman in 1956, is given by a collection of postulates which are called “axioms” by physicists. From these axioms one can deduce physical properties such as PCT, spin and statistics, crossing, etc. They are satisfied in the case of the free field and in some nontrivial models studies in the constructive quantum field theory [ll], [8] (two and three dimensions). For a four dimensional space-time, no non-trivial example has been found so far. * Supported in part by the Deutsche + Died in August 1976.

Forschungsgemeinschaft

F. CONSTANTMESCU AND W. THALHEIMER

168

One of the consequence of the traditional Wightman axioms (formulated in the frame of Schwartz tempered distributions) is the fact that the vacuum expectation values (which are called sometimes Wightman “functions”) can have at most polynomial increase in momentum space. A. Jaffe [8] has extended the axioms in order to allow non-polynomial increase in momentum space. Exponential models in three and four dimensions are simple examples of fields with stronger than polynomial increase [8], [IO], [l]. In his work A. Jaffe has introduced some spaces of test functions and the corresponding duals which are similar to S-type spaces of Gelfand and Shilov. Both S-type spaces of Gelfand and Shilov and the Jaffe spaces are particular cases of spaces of ultradistributions. Ultradistribution spaces have been introduced in full generality by C. Roumieu [17] and by A. Beurling [2] and G. Bjorck [3]. A very nice unified presentation of ultradistributions has been given by H. Komatsu [12]. Following A. Jaffe we will call strictly localizablefields those fields which can be accomodated in the ultradistribution framework (see 0 2). A further extension of the axiomatic framework appeared in [S] (the localizable fields). Recently Wightman axioms have been formulated in the frame of hyperfunctions [14]. In this paper we discuss the ultradistribution framework which is the most general framework for the strictly localizable fields of A. Jaffe. We study Fourier-Laplace transforms of ultradistributions as boundary values of analytic functions. Results of this type have been already obtained by A. Jaffe in [8], [9]. We extend the results in [8], [9] in several directions. In particular, the theorems in this paper apply to general ultradistribution spaces characterized by an indicatrix function o defined in 0 2. Our notations are standard. For A, B c R" the notation A c B means that the closure 2 is contained in the interior of B. Some results of this paper have been already used in [6]. Q 2. Ultradistributions In this chapter we introduce the Beurling-Bjiirck ultradistributions and some other spaces of generalized functions which are intimately connected to ultradistributions. The spaces of generalized functions which we discuss are characterized by means of an indicatrix function defined as follows DEFINITION2.1. Let %B be the set of all real valued functions o defined and continuous in the interval [O, + co) and having the following properties: (a) w(t) is an increasing continuous concave function on [0, + co) and w(O) = 0,

(y) there is a real constant a and a positive constant b such that for all positive t o(t)

2 a+blog(l+f).

ULTRADISTRIBUTIONS AND QUANTUM FIELDS

169

From (cl) it follows that w(t) is a subadditive function on [0, cc): < w(t>+o(s).

o(t+s)

The condition (3) is called sometimes the Carleman criterium. We define now our test function spaces. DEFINITION2.2. Let o E!B~. We denote E L1(R”),-%m(R”) such that P&,(8

by

= ,“P” ~~“x’V%(~)l

is finite for all multiindices a and all non-negative the seminorms P=,~. DEFINITION2.3. Let CJJE fl. such that $ E 6Pm(R”)and

the set of all functions

JY~ >

9

(2.1)

L. The topology on JZ~ is defined by

We denote by +?a the set of all functions q E L’(R”)

%.l(V) = sup {t+(““IDar$(E)I] EGR"

(2.2)

are finite for all multiindices a and all non-negative 1. The topology on ga is given by the semmomrs 3d,,1. In Definition 2.3 $ denotes the Fourier transform of 9~.It is defined as

DEFINITION2.4. Let o E m. We denote by Ym the set of all functions 9~E AtinVO. The topology on Pa is given by the seminorms pa,2 and n,,, . Remark: The spaces JY_, Va, 9, are locally convex spaces. They are also countably normal spaces in the sense of Gelfand and Shilov [7]. For the particular case o(t) = t”‘, a > 1, the spaces JZa, %mand P’, can be related to the spaces of type S defined in [7]. In particular, one has JZjr,+ = limproj SOL**. A-0 It is easy to see that we get equivalent topologies on J#~, VZO and Ym by using following seminorms:

(2.3)

(2.4) We

have the following evident statements PROPOSITION2.5.

with the Schwartz

Let w(t) = log (1 + t). Then the spaces Au, %Zm andY,

space 9

of strongly

decreasing functions.

ail coincide

170

F. CONSTANTMESCU

LEMMA2.6. The Fourier transform a continuous automorphism of 9,.

AND W. THALHEIMER

is a continuous

isomorphism

of AS

Now we introduce the space of test functions SBa which corresponds in distribution theory.

and V* and

to the space $I?

DJZFINITION 2.7. Let K be a compact set in R".We denote by G@_(K)the set of all functions e, E L’(R”) with support in K such that the seminorms (2.5)

are finite for all f. s 0. It is easy to see that the elements of ~8, (K) are in fact C” functions with support in K, i.e. go(K) c g(K). DEFINITION2.8. Let {K,,) be a family of compact sets in

R" such that 6 5 = R” r-1

and & is contained in the interior of Ky+I for all v. We define as usual gU(R”) = limind g&J.

The properties of .9JR”) are studied in [2]. In particular, one shows that (for w E 102) .GB~ is not trivial, i.e. there are non-zero Cm functions with compact support in [email protected] remark that for o(t) = log(1 + t) the space B,,, coincides with the Schwartz space 9 of C* functions with compact support. The following results should be also mentioned LEMMA2.9. Let w E %Jl and let K be a compact set in R”. The family norms on 9, is equivalent to the family of seminorms

(2.5) of semi-

IlldIll = ~~~{l@(ol~“E’~).

(2.6)

The proof of Lemma 2.9 is based on a generalization of the Paley-Wiener theorem and may be found in [3], Theorem 1.4.1. The elements of the dual space 9: of S@m are called ultradistributions. The duals %?L and 91, are spaces of ultradistributions. The space 4; is a distribution space. 0 3. Preliminary results

We give some properties of the test function space AU, GF?~ and Y, which are in part contained in [3]. The proofs are standard and therefore left to the reader. LEMMA3.1. Let w E %I. The spaces A%?~and ‘ip, are topological wise multiplication and convolution. LEMMA3.2. Let w E %Q. Then gW is dense in A_,

LEMMA3.3.

Let w E !BI; ‘p E A!&XJ G

%‘,,,and 9,.

and u’ E A!:(%?~). = +ii.

algebras under point-

Then q~

E .#i(UJ

and (3.1)

ULTRADISTRIBUTIONS

LEMMA3.4.

AND

Let o E 92. The translation a

and the multiplication

PROPOSITION 3.5.

QUANTUM

operator

171

t., a E R”:

= 9Xx-a)

by exp(ixa) are continuous operators

Let co E ‘%?, v E AJS?,,,) G(E)

FIELDS

on Mu and %ZO.

and u E A?~(%‘~). Then we have

= (@*II>(~) = u(x)(p(x)e’““).

(3.2)

For a proof see for instance [5], Satz 10.5. LEMMA3.6. The topology of 9, is$ner we can look at A?: as a subspace of 9:).

than the topology

on 9:

induced by .NW (i.e.

6 4. A basic lemma and its simple consequences

Let us introduce for the beginning some notations. A set r c R” is called a cone in R” (with vertex in the origin) if from 9 E r follows izq E r for all 1 > 0. Let S,_ 1 be the unit sphere in R”. We denote by Prr the set FnS,,_, . Let now rbe an open convex cone in R” and let q be in I! For an integer m large enough one can find m different vectors qj, j = 1, . . . , m in I’ such that (a) 7 lies in the interior of the convex hull of the vectors vi, i.e. T =

2

m s?jjs

0 <

Aj

<

1,

c

a] = 1.

(4.0

j=l

j=l

(b) The set ql-q, . . . . q,,,-q spans the space R”. Let r’ be an open cone contained in r such that Prlr’ c Prr. We suppose that 17E r and choose the vectors qj EI” such that besides (a) and (b) the following condition is also satisfied:

(cl

-

= 1;i;,’ Ii,%I ___

cos(x, y)

(4.2)

for all x on the boundary of I” and all y on the boundary of I’. Geometrically condition (c) means that the angles between 7 and qj are smaller than the smallest angle between the two cones r and r’. We introduce the function

a@,99 ?$I=

exp(-4 m

zI

(4.3)

exP(-X%)*

We prove the following lemma which is an extension of results obtained by Schwartz Dgl, Vladimirov [22] and Gaffe [9]. LEMMA4.1. Let T be an open convex cone in Rn andP be an open cone contained in r such that Prr’ c PrI’. For each q E P one canJindan integer m and vectors qj, j = 1, . . . , m in I’ such that we have for the derivatives Dia(x, q, qj): \DEa(x, 7, ~j)l < C,e-dlxt ‘11’.

(4.4)

F. CONSTANTINESCU

172

AND W. THALHEIMER

Here d > 0 is a geometrical constant which does not depend on x, q and u and C, are polynomials in rj. Proof The first part of the proof follows Jaffe [9] (see also [5]). Let q* be an arbitrary vector in r’. We can find, as mentioned above, vectors qf, j = 1,2, . . . , m, such that (77) satisfy the conditions (a)-(c) with $ EI”. From (4.1) we have m

c4(11*--7ljf)x=

(4.5)

0

i=l

for all x E R".Because the vectors v* - qf span R” and 5 > 0, there are positive as well as negative terms (Q*- @)x in the sum (4.5). Let us denote (4.6) h,(x) is the cosinus of the angle between T*-r$ and x. We know that for each x E R" at least one hj(x),j = 1, . . . . m, is positive. We claim that for all x E R" with 1x1= 1 there is a positive constant C such that P(X) = max {hi(x)) 2 C. 1
(4.7)

This is a consequence of the fact that ,u(x) is a continuous function which attains its infimum on the compact set x E R",1x1 = 1.The above remark then implies that C > 0. Now observe that h](x) is invariant under scaling so that (4.7) is valid for all x E R" with a constant C > 0 independent of x. From (4.6) and (4.7) we get min ((VT--7*)x} = 1x1mm {-hj(x)l~~-~*l} 1 Sjsm

ldj
< -CIxI,r$&{lrl:-71*I~.

(4.8)

Let now 7 be an arbitrary vector in I”. We can imagine that 7 was obtained from 1;1*by rotation and scaling. Let us apply the same rotation and scaling to 77. We get a system (7, Q} which satisfies the conditions (a) and (b). The condition (c) on {q*, T$} was necessary in order to assure that all qj are in r. Without loss of generality we can take 7: = aq* (a # 1) and min I$-7,1*1 = IT:-71 = la-11 Irl*l.

lsl=sm

(4.9)

By applying the rotation and scaling we get min IQ--rl

lSj
= la-11 1171.

(4.10)

The relation (4.8) remains also valid after rotation and scaling. From (4.10) and (4.8) for the system (9, v$} we get finally

ULTRADISTRIBUTIONS

AND

QUANTUM

FIELDS

173

where d = CJa- 1I and therefore U(X, 7,

qj) <

(4.11)

e-d’xll$l.

The derivatives of a(x, 7, qj) with respect to x equal a(x, 7, r/i>multiplied by polynomials in 11.This completes the proof of the fundamental lemma. We now prove the following corollary to Lemma 4.1. FQOP~SITION 4.2. Let 7 E I’ and let qj, j = 1, . . . , m, be such that the conditions (a) and (b) are satisfied. Then we have a(x, q, qj) E .M@for all co E %.R

Proof: In the proof of Lemma 4.1 we derived (4.8) without using condition follows that SUP (&(‘x’)lDoLIZ(X, XERl

17, r]i)l> <,SJll

(c). It

{@(lXWae-d’lxi)

with a constant d’ > 0 which depends on q and ~7~.Taking into account (2.1) we have only to prove that for all A the function Ao(lxl) -d’lxl, x E R” is bounded from above. This is a simple consequence of the conditions (a)-($ defining w in m. Now we introduce some new definitions. DEFINITION

4.3.

Let I’ be an open convex cone in R”. We define

JcJm = {uE

Kv9 =

rY(R") : e-x’k.+)

(24tz9’(P):

Kxr> = {u E 9k(R”): Y’(r)

=

E

A;,

Vrj E r],

e+%(x)

E %A, Vy E r},

e-“%(x)

E 9’&, Vy E r},

%L*(l+,x,)(l?.

We have LEMMA

4.4.

fit

o Em. Then &:(I’) = Y’(r).

Proof: We have trivially .&;(I’) I> YQ. We prove now that Y’Q Let u E A’LQ. Then u E 9 and we have (see (4.3)) e-XQu(x) = +)c(x, Using Proposition

7,

~j)(,~evxq')

=

ah,

c &Q.

(4.12)

fl, r]i) $WemsqJ

4.2 we can choose the vectors T,rjsuch that

a(~, 7,~)

E Am.

From

the hypothesis it follows that fu(x)e-Xqj E AL. In (4.12) we multiply a distribution from i=i AA by a function from Aa( We claim that the result belongs to Y’(R”). Indeed let ~1~9, YE&~. Then (4.13) The sum in (4.13) is finite because each of its terms is finite. It follows that the mapping v t+ 9~ from 9’ to .air, is linear and continuous for w E Aa. For u E Ai( q~E 9,

F. CONSTANTINESCU AND W. THALHEIMER

174 p e Am we can define

(4.13) This shows that uw can be interpreted

as a tempered distribution.

5 5. Fourier-Laplace transform We prove first two theorems which show that the elements in &kQ, I’ being an open convex cone in R",can be represented as boundary values of analytic functions in some tubular domains. The third theorem studies the singularities of the analytic functions near the real axis and the behaviour at infinity. We follow closely methods used in the axiomatic quantum field theory by A. S. Wightmanand R. Streater [21] and A. Gaffe [8], [9]. THEOREM5.1, Let I’be an open convex cone in R” and let u E &,(I’). Laplace transform of u:

9(u)(c,

7) = S[e-X’ru(x)l(B

Then the Fourier-

= F(E+iq)

(5.1)

has the following properties (i) F(E+ iv) is an analytic function in Y,

= R”+ilY

(ii) Let K be a compact set inl7 Then there is a polynomial PR, which depends only on K, so that for all 11E K we have

(5.2) Conversely, a function in Y,, with the properties (i) and (ii) is the Fourier-Laplace transform of a distribution u E &i(r). Proof The proof followsimmediately distributions u E Y’(P) (see[21], [5]).

from Lemma 4.4 and the classical results for

Remark: Theorem 5.1 gives no information about the behaviour of the function F at infinity and near the real axis. In the next theorem we study the convergence l/y _fZ’(u)(E+iv). THEoREM 6.2.

Let I’ be an open convex cone ii R”, u E $B’(R”) and u E &~(I’).

The

limit ~ ;yEr,-Wu)(6+ir) -+, exists for each open convex cone r’ c 7 in the sense of the weak topology of %?kif and only ifu E &L(R”). Proof The proof of this theorem is standard [29] and we give only the idea. Suppose u E LY and e-%(x) E A& for all q E r and 11= 0. One proves first that lim e-%(x) ll+o,?IEP

= u(x)

(5.3)

in the sense of Ai. The first part of the theorem follows by applying the Fourier transform on the both sides of (5.3). Conversely, let us suppose that for all y e %‘, and all

ULTRADlSTRIBUTIONS

AND

QUANTUM

FIELDS

175

q E r’ the following limit exists lim9(U)(f+iq)(ry(5)) r1’0

= f? 9(e-Xqu)(~)(yl(~)) *

= lime-“qu(x)(9(y)(x)) r1’0

(5.4)

Because the Fourier transform is a surjective mapping from VU to Aa, it follows that the limit ~~e+u(x) exists for all q~E A,. of the limit

(V(X))

This implies, because of the weak completeness of A!;, the existence lime-xQ(x) t1+0

= u’

in AL. Let now x E 9. Then we have $

e-XQ(x) (x(x)) = u’(x) = ~mou(x)(e-““x(x)) = u(x). -t

(5.5)

But 9 is dense in AD and therfore we get u’ = u E Am. We study now the singularities of the analytic functions near the real axis. THEOREM5.3. Let co E %I, I’ be an open convex cone in R” and F be an analytic function in Y,, = R” +iI! Suppose for each compact set K c I’ there is a polynomial PR such that for all 7 E K IF(E+irl)l < P,(f). (5.6) Let r’ be an open cone in r such that Prr’ c PC. The analytic function F(E+ iq) has a boun&ry value in %‘&in the weak topology of $9;for 17+ 0, 7 E r’ if and only iffr each R > 0 there is a polynomial P and a constant a > 0 such that for all q E I”, 171c R IFG+i$l

< P(&W?l)

(5.7)

where A,(t) = t i @m(X)e-x’dx.

(5.8)

Proof: From Theorem 5.1 it follows that F(l+ iq) is the Fourier-Laplace of a distribution u E A!#‘), i.e. F(E+i$

= &‘(u)(t+iq) = p(a(x,

7,

~j)~e-Xq'O)(E)

j=l

transform

(5.9)

where 7, qj E I’. From (3.2) follows that A F(C+ &I) = >I e- *“Ju(x)(a(x, g, qj)@“).

(5.10)

j=l

Suppose that F(5‘+iq) has a limit in %?Lfor 7 + 0, 9 EF.

From Theorem 5.2 it follows

176

F. CONSTANTINESCU

AND W. THALHEIMER

that u E AL(R”). This implies that the set of distributions in Ai:

where qj is chosen as in Lemma 4.1 and R > 0 is arbitrary, is pointwise bounded in Ai. Indeed if once the set {vi} satisfying the conditions (a) to (c) for 7 given was chosen, then T,I remain in a definite geometric relation to 17and for q -) 0 in r’ we have rli + 0 (see the proof of Lemma 4.1). It follows that there are constants C, k, 1 such that (5.11) for all Tq E ‘u. We have to find a bound for (5.1 l), i.e. for

From (4.4) we get lo”(a(x, 7, ~&?“~)l < P(E)e-dlqllx’ where P is a polynomial (which depends on r’ and R) and da positive constant. We concentrate on the bound (5.12) sup G+ (IxOe-+il 1x1). XER"

Let to 2 0 be the point on which the function ti(*)-dlolt attains its maximum for t 2 0. We get Q) co (5.13) sup {&“(‘)e-dlql’) = dll;ll@(‘o) i eTdl”ltdt < djql 1 &ufo-dlqlf dt f>O

to

because co(t) is increasing on [0, co). On the other hand dlql

j &‘)-dWdt to

< 171 j

eIb3\*k”“dt

0

and therefore because of the property (a): sup,+N’-dl~l’ < I91i e~(‘)-itllt dt t>O 0 where u > 0. This is the first part of the theorem. Conversely, let

We have

ULTRADISTRIBUTIONS

AND QUANTUM

177

FIELDS

and

00

00

=

t2] e-Ytdyi

e2ao(x)-xrdx = A2.(t).

We have then

(5.14)

IF(5+N)l G +(P’(0+&.(lrl))

On can find a positive constant y and an multiindex 01such that for all q E P, 171 < R +P’(E)

G SUP{YlD”(Q(x, 11, ~jb+“‘)l}+~ XER"

We find a similar estimate for A,&]).

= WC~.~(~X, 7, Tj)@‘)+y.

(5.15)

Indeed, using the subadditivity of cc)we get (5.16)

Let us consider now a vector x E R"such that n - 1 components vanish: x(‘) = . . . = x(“) = 0 and the first component x(l) is nonnegative. We have x(l)qo) e-3 = Ia(x, q,qj) = m & e-“‘l’q~’ T, eexQ and therefore e-X’l’W< mU(X, ?7,

(5.17)

qj)

where w = rrnjym(7 u)- qf’)>. From Lemma 4.1 follows that w > 0. From (5.16) and =S (15.7) we get (5.18) ~2.(lql) < ~;;g{e40a(ix%(~, r, qJeixdl). Combining (5.14), (5.17) and (5.18) we get

l~(~+~$l =

ICe-xqJu(x)(u(x, j=l

r], qj)eixb)l < CkA(a(x,

7,

where u E &I’:,, 7 E .P and the constants C, u, 2 do not depend on 7. Following Jaffe [9] we show now that the set {u(x, q, q,)Pb:

t E R”)

qjk’“‘)

(5.19)

F. CONSTANTINESCU

178

AND W. THALHEIMJZR

is a total set in .&JR”). If this is not the case there is a nontrivial distribution v E J%:(P) such that ~(5.20) 0 = n(x)(a(x, 7, ~j)f+"')= ~(~(x>h, 73 Il,>)= O* Because a(x, 7, qj) has no zeros, the multiplication by a is an automorphism of ga. It follows that for all y E ga we have v(y) = 0 which implies v = 0, which is a contradiction. Now from (5.19) follows that 5 e- xqj U(X)can be uniquely extended to an element j=l

of A!:(P)

such that 12 e-X~Ju(x)(~)l < CP~,A(P~) j=l

for all v E AU and all q E r’, 1~1< R. We choose the set in a given geometric relation to q E I” as in Lemma 4.1. Because e -xq~ is differentiable in 11 (see for instance [18], p. 301) and beacause multiplication by x is continuous, we can find constants A, 1, k such that

This implies (here rj’), 7;‘) are vectors in R”)

<

2 I#‘-#)IA c ~a&‘)

It

follows that 2 e-Xqju converges for 17--f 0 in Jih(R”).

(5.21)

/cxl
j=l

This limit is clearly u such that

j=l u E

.ML(R”).

We now apply Theorem 5.2 and complete the proof.

EXAMPLES

(i) Let o(t) = log(1 +t). We get the well-known 1~1-“ singularity near for the Fourier-Laplace transform of distributions in ~~~~~~~~~~~ = Y’. (ii) Let o(t) = tlla, tl > 1. Then co E ‘illl. ~I,,(lqj) produces singularities ‘/(a-1) (see also [5]). The result of this second example is (with ew(lllrl) modifications) applicable also to the case of S-type spaces Si, a > 1 of Shilov [7] (see remark in $2). For other related results see also [20].

the real axis of the form some minor Gelfand and

ULTRADISTRIBUTIONS

AND QUANTUM

FIELDS

179

0 6. Further results

The ultradistribution theory is the adequate general mathematical framework for studying strictly localizable fields in the sense of A. Jaffe [8]. In this paper we have studied, following closely the axiomatic approach [21], [8], [9], the Fourier-Laplace transform of ultradistributions. We remark that recently a Bochner type theorem for ultradistributions in $@Awas also proved [23]. It says that a positive-definite ultradistribution is the Fourier transform of a positive measure with “exponential increase”, i.e. with an increase like em(lxl)where o is the indicatrix function. More precisely, we have [23]: Let co E !lJl . Each positive dejinite T E 59; is the Fourier transform THEOREM 6.1. of a positive measure which has o-exponential growth. Conversely, the Fourier transform of a measure with o-exponential growth is a positive definite ultradistribution. We say that a measure p on R” is of co-exponential growth if there exists a positive constant iz such that

(6.1) Finally, we remark that the results of this paper concerning the Fourier-Laplace transforms of elements in .&‘h can be reformulated for the case of the space 9;. We give here a typical result, leaving the proof to the reader. THEOREM 6.2. Let cc)E ‘$I and let r be an open convex cone in R” and F an analytic function in Y,, = R”+ iI’. Suppose for each compact set K c F there are constants C, and & > 0 such that for all 17E K IF(t+iq)l < CKea@(lci).

Let P be an open convex cone in T such that Pr r’ e Pr r. The analytic function F($+ ir]) has a boundary value in 9’: (in the weak topology of 9’:) for 11+ 0, q E P if and only if for each R > 0 there are constants C, 1, a > 0 such that for all 17E r’, 171-X R IF(5+i$l

< Cd”(rei)&(lql)

where A,(t)

(6.2)

= tydw(x)e-X’dx. 0

An analogous result can be stated for Fourier-Laplace in %L. The bound (6.2) has to be replaced by

transform of ultradistributions

IF(S+iq)l < C&(~cI)IqI-” where C, ;Z and a are positive constants and q ~rl, reader.

(6.3) 1~1< R. We leave the details to the

REFERENCES [I] Albeverio S. and R. H8egh-Krohn: The exponential interaction in R”, Preprint Bielefeld, September 1978. [2] Beurling, A.: Quasi-anaIyticity and general distributions, Lectures 4 and 5, A.M.S. Summer Institute, Stanford 1961.

180

F. CONSTANTINESCU

AND W. THALHEIMER

[3] BjSrck, G.: Arkivfdr Mutematik 6 (1966), 351. [4] Bros, J., Epstein, H. and V. Glaser: Comm. Math. Phys. 6 (1967), 77. [5] Constantinescu, F.: Nuovo Cimento, I 1 (1969), 849; J. Math. Phys. (1971); Distributionen und ihre Anwendung in der Physik, Teubner, Stuttgart 1974 and Distributions and their Applications to Physics, Pcrgamon Press, 1980. [6] Constantinescu, F. and W. Thalheimer: Comm. Math. Phys. 38 (1974), 299. [7] Gelfand, I. M. and G. IS. Shilov: Generalized Functions, Vol. II, Academic Press, New York 1964 [S] Jaffe, A.: Phys. Rev. 158 (1967), 1454. [9] -: High-Energy Behavior of Strictly Localizable Fields, Mathematical Tools. (unpublished). [lo] -: Ann. Phys. 32 (1965), 127. [ll] Jaffe, A. and J. Glimm, T. Spencer: On the particle structure of the weakly coupled Pi model and. other applications of high temperature expansions, I, ZZ, Lecture notes in Physics, Vol. 25, SpringerVerlag, Berlin 1973. [12] Komatsu, H.: J. Fat. Sci. Univ. Tokyo, Sec. Z A 20 (1973), 15. [13] Lions, J.: Journal d’dnalyse Mathematique 2 (1952-53), 369. [14] Nagamachi, S. and N. Mugibayashi: Comm. Math. Phys. 46 (1976), 119; 49 (1976), 257. [15] Osterwalder, K. and R. Schrader: Comm. Muth. Phys. 31 (1973), 83; 42 (1975), 281. [16] Rieckers, A.: Dissertation (unpublished), Miinchen 1970. [17] Roumieu, C.: Ann. Sci. Ecole Norm. Sup. 77 (1960), 41. [18] Schwartz, L.: Theorie des distributions, Hermann. Paris 1966. [19] Simon, B.: The P(a))z Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, N.J. 1974. [20] Soiovev, M. A.: Teoret. i Matem. Fizika 15 (1973), 3. [21] Streater, R. and A. S. Wightman: PCT, Spin and Statistics, and AN That, Benjamin, New York 1964, [22] Vladirnirov, V. S.: Methods of the Theory of Functions of Several Complex Variables, MIT Press, Cambridge, Mass. 1966. and Generalized functions in mathematical physics (in Russian), Nauka. Moscow 1976. [23] Weingartner, 0. and P. Constantinescu: Math. Zeitschrift 147 (1976), 175.