S- By Ky6ichi YosHiNAGA

Bull, Kruehg :net. Tceh.

(}1. &- N.S,) ND. e, lg60

On Spaces of Distributions of Exponential Growth

By Ky6ichi YosHiNAGA

ÅqReceived Oet. 31, 1959)

. In a reeent paper [9], J. S. E Silva has developed a theory of ultra-distribu-

!.lla2.igOLr,l.Ii?8,ove,aii!tZ•#xuC,/iC?O;Tlis"ig,i.2hs':Msnlt:is"S,/',rZo"gl,\.tilB,eU{t'8:•,:rt-,-hiD8iFse:••IiZ,ge,År[,

.sP

g,:Pv:ai•ziif,ZOScih?ld,l•ig:sWb,P,:.ioZn:sÅíV:e,/`,iFgag`g',:d".eEOIfgidSel,g,P,IC,e..S,"e8,alLe,2.SP,aZe,d,",,b.Y.

. At.the first glance a distribution of exponential growth proves to be defined Just in the same way as a temperate distribution. Thus we may naturally expect some analogy between ..ST' and Y'. Consequently we are led to define the spaees J7, .9"',,

J7nt corresponding to the spaces LY, Do'c, no At Tespectively.

Most of the spaees and their strong duals are nuelear normal spaees of distri- butions with the property of approximation by truncature and regularization [8]. Besides, some of them have the property (E) [8].

Since the spaee fnf may appear as the spaee of multiplicators of ./`7' into

S- , we should like to say a word in Section 1 about the multiplicator in the sense of Schwartz [8]. We shall prove a proposition whieh will be interesting to Fome extent in relation to Proposition 37 of [8, p. 129]. Section 2 is devoted to i"vestigating the spaees .9', J7', .9"c and Jlc. They are all obtained as the inductive or projective limit spaees of the sequences {eLv(tY)} and {e-V(Y')}

SVhere ty(x) == if1+ ixl L'. After .gfi'Af and .S7'nf are defined in Section 3 we come to

SeCtion 4. There we shall make an argument about the convolution along the PaMe line as in [11]. Among other things we show J7-' * .9i (Yc. By mak- ing use of results in Section 4 Silva's conjecture will be proved in the last Section 5.

The notation and terminology of this paper are essentially those of [1] (local- IY eOnvex spaces) and those of [s], [7] and [8] (distributionsÅr. According to eireUmstances some other terminologies will be used. For example, the spaee Of type (LF) is used in the

'henotionof,,,.itt,dg8."g,raAZded,:g"S,e,gl,gr:,tP.eR•.d,'SgE,E3,'.C,2agg,'X,'a3,1'

[11] or [6]. As for the elements of the Silva space we refer to [10]- 1' Multiplicator. Let Eie`' be a normal spaee of distributions and let,:21!' be anOther spaee of distributions. A distribution s is said to be a mttltiplicator Of 81EF' into er if there exists a continuous linear application [S] of`SV' into

1

ev, ealled mecltiplication defined by S, whese restrietieR on 9 coineides with the u$ual multiplicatigR by S [8:. Sinee 9 (.SZ7"i and t[S] is the usual multi- plieation on S, we see that if S if 8 then [S] is nething but the usual mukipli- eatien bv S.

In his theory of distribution-integrai Schwartz has $hown that if each element of a normal space of distributiens cr is a mult!plieater of Ett`", into 9'ti theR Åqct, TÅr= S[a]TÅqpt)dx for any at c ,fitaf`', Tc `S}EF'. Thus it is not of no interest to state the next proposition which will be used later.

PReFcsi"rioN 1. Let ev be a nermal spgce of di$tTibutions with the paroperty (E). As$iuae that @or(ev and Eg'`(ev, Then foT any eontinmoMs Zinear

form F G , ig"', there exists one and enly ene f E 24 which defines g muttiplication of ev into 9' $eecJ$ tJeat [f],Sii? (91i and

ÅqT, Fl = I [f] T(ptÅrdx, z' E ,s2e`'.

The ficnctien f is determined by f(y) = ÅqFÅqdi), 5(.e -pt)År.

PRgoF. We fir$t remark tbat beeau$e of the property (E), the injection S"-.ev i$ eenti"uous. Letting tT' E ev', we see that

Åqii(SÅr, 5(.e-S•)T(SL)År = g(S)TÅq 9) E (2r'Li)..

feT any uc :27.. Hence

8Åq.S-- jriÅrT(5ÅrÅr ff Åq.fZi)i.i),(9i.) = .Eff7,((9.),,; 9r.).

{Vherefore by definition of the property (E) we observe that 5(.e-s)T(s•) if .sfpt,((@.),; fit/ia" = (s',.i).(,sirdf7.).

Thus by Proposition 36 of [8, p. 129] we ebtain ÅqF(.&), S(."-SÅrTÅqS)År E (D'siÅrr tegether with

(1År ÅqF, TÅr = ÅqF(.e), IS(e -- pt)T( )t)d yÅr =: !ÅqF( .i-År, 6(.e -- y)T( pt) Årdy.

Letti:g ÅqY(ÅíÅr, S(&-y)År =f(y) we see that f(ES', beeause the applieatien Rfi )y.T,S(Åí) a 8' is infinitely Åëontinttously differentiable with respect to M We claim that f is a multiplicater ef ,siie`' intc 9' yielding

[f]T(S)=ÅqF(Åí) , 6(it--SÅrT(SÅrÅr,

Tg prove this St i$ noted that for any u, v E S, we have

ÅqÅqF(.&), 6Åq.e - S)!s(P)År, v(S)År = ÅqF(iÅr, ma(i)v(i)År = Åqffrk), ! 6(fc •-)r)u( yÅrv(y)d:vÅr = I ÅqF(k), S(s -)E)ÅrM(y)v(pt)dor.

X}hereÅíere we ebtain ÅqFÅqÅíÅr, SÅq&-SÅrasÅqP)År =f(S)gÅqS)i

On Spaces ef Distributions of Exponential Growth 3

l'i,/xnio:st3.ik",,S,ot`/7,,".tlL?/f-/i.IBhg/:t:'/$/&:•'gexa:pl"/grl.s-go:liu#Bii2igf`i/ii.tZ,/i"ind,I,/E;ztiii'i.)'i/si'%n-.-i/tl'Zll"tyiin"3;,'

ÅqÅqF(xA), 6(th '- 5År)Ta(S)År, u(S')År = ÅqF(i), Åq6(Åí -- i')Ta(S'), u(S)ÅrÅr = ÅqF(di), u(S')Tut(-Åí')År--,O

ipC"?fi?J':',/{.E'P,hÅq,'i'g"Ri.:'/e`isal'h-'`:'f.ii•:S.X(i',Ps`g,5,(3i,#i•i•'8-m'ZiL"ipTi?•IEa'ti.:hd2Sfi.th,8

ÅqT, Fi ---- SÅqF(.e),6(Åí- --y)TÅqy)Årdy-= I[f]T(y)dy.

This eompletes the proof.

In this connection vve state the following

. PRoposiTioN 2. Let Jil?`' beanormal space of elistributions. Assumethat ff ts a baTrelled. space or a semi-complete space of type (DF). Let S be a m2eltipli- cator of ,filf/ ento 9' 2vith range in 2r'Li: [S](,fiif/ )(S)'L:, Then Sisam2edtipti- eatO T of ew pnto 9'Li ancl conseq2tently the app"cation T.J[S]T(-t-)dx (leLlines a contznttous lmear form on ,Srf.

, PRooF. . This is an immediate consequenee of Theorem 2 of [11], since :2)'Li is a permitted spaee.

2• The spaces .S7", L9i', .9"'c and .SPc. Foreonvenienee'sake let usdenote bY '7(x) the function Vl+ixt[' defined on the Euelidean n-space' R". Vl'e nete that7C.x) +7(r)År7(x+ v) and Dp7 E SZI for any index p, lpl :}i: 1. For eaeh integer

k We giye the (L9)-topology to the veetor space e`7(.Y') = {e`vqp; tp E ,Y} by means :ifoXhe Iinear isomorphism Y ) q)-iFe`vgD E ekv(Y). Then it is seen that the injec-

e`'(L-Y') ) ekV(p--,Fe(k'i)Y(e-V{p) ( e(k+1)v(y)

IS C9ntinuous. Thus we obtain, with respeet to continuous injections, a pro-

JeCtive spectre:

Åq2) LS,P-e'V(.Y)-c-2v(,Y)•e-.,.,

and an inductive speetre:

(3) Ltl,P.e7(LY').e2'(LY'År.•-•. '

In a quite similar manner we may define an inductive spectre:

(2') .S7'.eV(.Y")-e27(, Y")..,.,

and a projeetive speetre:

(3'År L9'-e-VÅqY')-e-2Y(.Y')Åq--•....

!t is not diMcult to see that we may write (eky(Y))' = e'"ky(Y') and (e-"v(Y'))' = e`Y(Y). $inee elyÅqY) and eky(Y') are eentained respectively in e(k"iÅrrÅq.Y) and E("'i)V(Y') as den$e subsets, the dual of the injeetion ely(Y).e{k'iÅr7(Y) may be regarded as the injection ew"Y(Y')-e-(i"iÅrVÅqY') aRd c"nversely.

We n"w denete the pTgjeetive !imits of (2), (3') and the inductive limits of (3År, (2t) as Y, ,.Efi-'irt, .gpu(: and Y"' respeetive}y, Then it helds that J7V' is the strong dual of Yi. This i$ seeR as fellews. Regardless of the topoiogy, the statement is almg$t evident. As for the topology, we first remark that the injeeton of Jf7pt' into the strong dual cf J7 is e"ntiRueus. Thus -`'-"" i$ a Hau$derff spaee. Te preve the cgntinuity of the inverse injeetion we proceed a$ fgl}ews. By definition of the projective limit, Yb is a closed $ubspace cf a dual Silva space [10]:

Lil,P Å~ e--V(YÅr x e-2V(Y) Å~ . =..

Henee.:`x:7 is al$o a dual Silva spaee and the strong dual of ./e is a Silva spaee [10], On the other hand the Hausdorff induetive limit Y' cf the inductive spectre {eky(Y'År} ef Silva spaces is known alsg to be a Silva spaee [10], Thus we may conclude that J7' coineides with the strong dual of Y [le]. This comp}etes the proef. Since the Silva space Y' is the stroRg dual of the dual Silva $paee uS7'" , they are reflexive and .S7pt is the strong dual of .S7pt'.

As for .9if'c and Yc we rnay prove that .Sgc is the $et ef eentin"eus liRear forms onY'c and conveTsely, that is, they aTe duals to eaeh other as vector spaces. The proof Tequires no speeial mention, and will be omitted. It is known that Y'c is a comp}ete spaee of tsrpe (MÅr (--- any elesed beunded subset i$ eempaet) and .7c is a bernel"gical barrelled space.

Further properties of these spaces are shown in the following prepesitien•

PRoposmoN 3. The $paees Y, Y"-', Y'c gnd 0c are agl nuclear mormal 3paees of aistributions with tlte pToperty of approximation by trecncature and regutarization, .S7- and Y' have the propertv (e).

PRoeF. That they are all nuelear is au immediate consequence of TheoreM 9and its Corollary1 of [3, Chap. 2, pp. 4748]. Since they are all ftermal spaces of distributiens the preperty ef appreximatieR by truncatuTe and

regularizatioR gf barrelled spaees Y' , J7V' and 2c may be proved by usMg the remark of [8, p. 8]. As' to the space f'c we mmst verify tha definition Of

the property ef appreximatien by a direet eemputatien. The case of trunCa"

ture beiRg simple enough, only a proof Qf the case of regularization is giVen

here. To show the continuity ef the applieatiofi Y'. ) T.T * p E .siiJ'., it suMeeS

On Spaees of Distributions ef Exponential Growth

5

/l,,iPlli•l.lll'lltiil/5'lliiathgil•ti.•l,sPgIY/i,l,,WICb/%t,`'/Åé,\kf'//,ib/z,,\gla,/shyB,b},Itipig/;,/sk,/tb/kh'so',/ib/r

$/g/l,lgi;,dh,/ts'/lhll'i,g,ill,l.:/ili`li'1mm111pi'/ii,II'lliblil,Bt$b,,,/itdigP,,,B,xbefiiiF'/it/2`1:(,/Sii`i•gt/ii,il/ed•t:t,e,i,

l lfil k, p = .s,u.e l e le7 (# )D pf (x) l .

.W _,e L;I:i9P".Og,e,th.2t,t.2g,d2rEgk:i,x".,D.g'g.?,2o,2e.i2",o."s,o.ps,rs,t?g.b::hL$'L,.-År-'and

Åír,o' //f.'lll131egSgaeCoeSmiltlllte"dasn,Il,Ei7I,1"1,,YII,et,fi,',S.t,?,ta,tte,t,hhe.iO,ifi08"fi:.g.lf,M,M[a,:,Wh,02Sge,

,bo

,zL .illsi'e:,Zf.u,netStoÅíf`1Åíeen"t'hZT"eCt3.0?,t:,}'2f9Z.5t",,S."C6",ihe`i'gi,?,"7,:,eX(ielgÅr'6,i.g

muW it?•p?iO,\lt8Y,rnoftOiflih!i9..dteofilll'!tlOes,Ofprtoh.eeStPha,ce.,`f.PtAf•LettingY",ibethesetof

PRoposmoN 4. jFTor anu distribution f follozving statements are equivatent:

(1) fE .9ir,,;

(2) fc 2S' andfor any index p there exists an integer ic such that e-k'D"f c L';

(3) fE 8' and opDif E L" for any index p and q ( -9'•

g,":,,:zOZF,s),"T,d,hY,g,Ii7it2.g",,is:olti,e,aff,n,ii.L'w",ielIYeo,-e,kSee:',iitgfi.hag:ed':91.=c,i-(ioq,'ria,g/o,gfg,,r'

E R" and a sequence {Uk} gf gpeR spheres Uk with center .r* and radius 2 such

that I ptk l . co, lf(xk) i ;l}: k.e iY(Xk} and UkA Ui = Åë, ft irk i. Let h. e 9 be

h(x)-I5 ig,rI:I-`.S.

Then {gek} where Åqps(pa)=e-ity(Xk}h(x---xh) is seen to be a bounded subset of .9-.

But sinee lf(x',)cpk(x,) l :}!r it, it follews that {fq)k} is ftot beuRded. This is a eeR- tradietien aftd (1År.(2År is preved. (2).(3) i$ cleaT.

Ad (3)"(1): Since efy`p EY for all h and cp c Y-", it is not difficult to see that elyÅqpDs'f ff L"" afid thus E Si9f fer all index p, k and {p E Y-". TheR it results

that `pDJ'fc.S7" for allpand rpEY'. To $ee that the applieation [f]:

V`7pv).fqJGJ7' is continuous, it suMees to shew that the image fB ef any bounded s"bset B(J7- i$ alse a bcuRded sub$et ef J7. Letting

t

M(.as) = ely(XÅr sup j q)(x) l,

Pcfi

we $ee that M(Jx) has the property of the iemma ju$t stated. Thus we "btain a fufictiefi ofjo ff f sueh that M(xÅrSptoÅqx) everywhere. This pTove$

le`rr(X)g(x)D"f(x) I Si ope(x) iDif(x) l

for all g)EB. Therefere {ekYqJDRf; qEB} i$ abounded subset ef Loo fer aRyp and Jit. Ngw it is Ret diMcult to see that {ekygeA {pE B} is a bounded subset of

@ and eonsequently {eqf; g) E B} is a bounded $ubset of .9. This prove$ Åq3).

(1År emd the preof is completed.

Tg give the topo!ogy of Ysf we prove

]PRoposmoty 5. Y,(.9ny; Y-) and Y.(.f7-; Y) indecce on 0"t the same tepclegy ebtai?zed by the fomily {llfll., .} of semi-ncrms aefined as

ilfllp..=: sup Iq,(x)Dpf(x) I, rp c .ST.

nc E Rn

PRooF. To prove the first part we need only to show that the topolegy Cf LSpsAi given by Y,(Y; ..S7) is net strenger than that giveR by Y,ÅqY; -9")•

Letting fAf )f.-,O Sn :2f`'.(C9-; J7) and letting B be any compact disk cenew tained in J7'- , we must shew that iif.opllk.,.e uniformly in qcB, But aS PVpt.O iR Y,(Y; J7V) aRd DP-'B ig a cemp3ct di$k, the p!oblem is brought down to the ease p= (e, •--, O), that is,

.S,UP. I e"'CX)fdi(xÅrtp(x) l .O,

9("

Putting M(x) = gu,girp(x)i, we observe that ely{=}M(x) is a bounded funetion fer

every k. Therefore by the lemma cited above there exists a function opo ti f

On Spaees of Distributions of Exponential Growth 7

sueh that MS{:rpo. Then

.S,".P. Ie""(')fa(x){p(x)1.:{:.S,U.ll lei'(='fa(x)q,o(x)i-.O.

il,iiii):-:epiiXo8giy,kohi,li,l:•T•lcr.ik.2r,!gh.Z,/".'8,:"liil,Llah,retf:8,8:'EeS2aiBfe.:-'i2•-W.'.gr:',sefZ-i'5,stitt/i.ki'

r we may obtain another equivalent family {llfli,,.} of semi-norms stated in the proposition. This eompletes the proof.

In the sequel, by the topolegy of .S;PAf we shall mean the one mentioned in

tRtSs:l:OePO[$tEl9nEbhl.lsi':.sMseaennOtbhye:hVe]'enWexPtOipnrtopoofsim.tiuolntiplicatorswemayobtain

PRoposiTioN 6. fC nt is the space of m2tltiplicators of J7'-' into .:[i'-" 7vith the topology indicced by Y.(..9-'; ."S?"),

PRooF. Lettingfbe a multiplicator of ,.S7"' into .S7"-' and letting q, be any element of .S7-, the continuous linear form on J7-': 7'.Åq[f]T, (plÅr is expressed as

Åq[f] T, qpÅr = ! [g] T(.r) da'

where g is a function belonging to E!g' (Proposition 1). Putting T= vr c :ctz, (e' we may obtain Åqq)f, VrÅr = ÅqYtf, `pÅr == Sg(x) xln(x)dx for all q) E .:`i7- and Yr E :2).

Thgn it is not difficult to seefE EIf'. We now provefc YAf in asimilar manner as .in t])e proof of Proposition 4. If there is no i,- satisfying e'kifc L", there

exist, Just as before, a sequence of points {xk} (R" and a sequence of functions {q)k} bounded in uS'-', Letting B={T.,6}, we see that B is a bounded subset

Of •-:`Z7'. Thus fB== {f(x,)T.,S} must be bounded in J7'. But this is a contradiction since ÅqfB, {qk}År is not a bounded set of numbers. Thusfc YAf.

It is now obvious that every element of .sJPAt defines a multiplieator of , 9"

into J7". Owing to the identity ÅqfT, opÅr :ÅqT, fg,År we may observe that

.E2f',(.:7"; .S7-`') and .-cL;tC'- .(,-S7V; .:;7') give the equivalenttopologieson .Si"Af. This eompletes the proof.

It is now a simple matter to see that .:SP -f is a complete space. Hence it is a Closed subspace of .Sf',(.9"; J7). Thus .SSPAt together with .Sf",(../cr; J7') has the property that every bounded subset is relatively ccmpact (=type (AD) [3, Chap. 1, Corollary 1 of Proposition 19, p. 99]. Consequently .Sli',(.SF-"; ,YÅr and .9Vnf are semi-reflexive.

PRoposiTroN 7. A subset B of .9"Af is bounded if and onty if for any index p there exists a positilve integer k s2Lclv that e'kv(')DnfÅqx-) is imifor?nly bou7ided for

xERn,f(B.

PRooF. We need only to show the "only if" part, Suppose the contrary.

We may assume that there exist xk e R" and fk if B such that lPPfk(xk) le-kv(=s")Årk, ixil.co and lxi-.xilÅr4 (k-S=fÅr. Putting

tp(rc) M- :s"" e-ly(sk)k(.x -- xk)1

k.1

where h is the function used in the preof ef PrepesitieR 4, we see that w E J7 and l6År(x"PPfi(xk)lÅrk, a contradiction. This completes the proof.

PRoposirmoN 8. YAf is a n2tclear normal space of distributions with the pro- peptty of ampreximaticn by truncature gnd reg2elarizgtien. lt is of type H"" and thec$ it has the property (6).

PRooF. That YAf is nttelear fellews immediately from the faet that it is a ele$ed subspace "f the nuclear space .Ef'.(.S7-; J7V) [3, Chap. 2, Theorem 9 and its Coroilary 3, pp. 47-48]. As for the property of approximation we need enly check up the defiltition faithfully. That .9'fif is a spaee ef type He may alse be seen as well. Thu$ Ytr?u has the property (E) [8, p. 56], This completes the proof.

In relatieR to the space .9"M the elements ef Y' may be characterized in the following proposition.

PRopesmeN 9. A digtTibntion T belongs te Y' if and enly ff:f any one of the feZlewing ewtitions is ggti$LfZed:

(1) T is expressecl as T=DPf where f is a continuous ftmction satisfying

e-'Y ff Lmu for a g2titabee k;

(2) There exists a k such that {e-kY(")TAT; h E Rn} is a bounded s2Lbset of 9';

(3) T* ct E gMfor anu cx ff 9;

Åq4) T* a * P E YorM for any ct, i3 ( 2.

PRoeF. Ad T ff J7' e (1År: Since T ff Y' means e-lyT c Y' fer an apprepriate k, the statement fol!ows immediately frorn the cerresponding result con' eerning Y' [5, II, p. 95j.

Ad (3) o (4År: We Reed eRly to shew (4) . Åq3). Let F,= {f; e-Lvf ff L""} be the Banach space with the norrn:

llflik= sup l e"' by{x)f(x) l .

xE Na

Then T * a * B E .ES7M implies DpT * ct * B e F= ÅrL;Fk (direct sum). Therefere

we may ebtain DPT*aEF fer any index p and cxE9 [11, CerellarY Of Theorem 1]. This means T* ct g J7M.

Ad (1År-Åq3År: Thig is shown by a simple e$timation of the following integral;

T * at (x) = DPf * a (x) = if(x -u)DPct(u)du .

On Spaces of Distributions of Exponential Growth 9

R,i"'iljfIK'9i`//1.,/ii,IZiii`;i;hii/lit,lm,S,il,ill/',,(/im`Ig,g,II/t'illi/#/le}.Ik,:-t/,Piifio2fk-:•X'ii•iikStaÅqd,.-gd.hiihi•

z';P'ER ipPÅí':/i'.}2':•gi.9s,y,A•,,hl,sleZk]agn,del,?/Vsbi.e,,3sÅía:s,lli,lligi2o%htlza"uC,'Åígt9Zd.,Zg,S,Sg,1,;,Di,fat

i,/pPtil/ill,l/iOSys:c/ix,/:za,!/P•le/gehi•i,!IS.;e2uEi"lktii'iga3,:b'i,Sn.l,[2i/',O/R..g,"l,'l,Rc,i'"li:6k,ibil,,gu2nldl/,q.'les"gb:s:i,{

In a similar manner the elements of ,-9"(: are eharaeterized as fol!ows.

thePfRoOllPoOtvtitTnigONcotttttilFnfltzS.strsZabt2ttg}Oenclll"beeOngstoLS7ij'(zlfandontytLfanyoneof

sat(zlslfyjl.lnOgr aen,vY,kE" LT.lSaiintte Sum År-]D""fi wherefi are contin2toz(s fitnctions

E32j :Fr O.raa(nYf, .i{cgjl"VaC:);Jh:iEiiS5"} zs a bounded subset of .s.2b';

(4) T*a*BE .:7' for any a,ElE9.

ff,/ro/liPllill"i.li';nilk"2,ilieli31'l,11/Al•IgSi•l,,et/i,:iiPlll•,!,lll/8,lq/?i.11'la/liy,V!11Af/lai(nlieq:hii,li.fSlii"e'c,li,t:/i/ulkr

IIfll = sup l eh"f`='f(x) l ,

=e'R-

We may apply Corollary of Theorem 1 [11]. This completes the proof•

PRoposmoN 12. A bounded subset of .9",: is ciearaeterized as a set B of dis--

trthUtions such that for any k B is expresseal as a set {]S.;DPJf.,i} of fini•te stems

where pi, ns are chosen dependently on k and not on a iihirtieutar eiement of B

le K. YOSHINAGA

and f.,i are centin2sezes fec?zctien$ such that {gkyf.,i} ig a beunded secbset of L"".

PRooF. The proof is essentially simiiar to that of Proposition le and i$

omitted.

We ngw begin to determine the strong dual .f;7'Af of YYAf. It is known, regardless of the topology, that (-Åq"if,(Y; Y-))' i$ isomorphic with .9Y(2}Y' Mnder the c"rrespcndenee:

m

(.s2fP.,(..9"; ,.f7'))Y l fl2 o.'-S;-'(p,opT, E .9'ÅqEl).9'i, tp, E Y, T, E Y',

i-#l

where Åqori, uÅr=XÅqu(q7i), TiÅr for any uEY.(Y'; Y) [1, Chap. 4, p. 77]. Se that, since ..SUnf is a subspace ef Y,Åq,9-;,-9-'År, it is net diMcult to see that A'sf

is the set of the distributions of finite sums ÅírpiTi, q)iEY, Ti EY'. As

noted above, YAf is a elosed subspace of tha semi-reflexive space Y.ÅqY; J7'), we may ebserve that YGf is topelggically isemerphic with (S",(Y'i ..ST'))7

(YA!)e in the eanonical way [i, Chap. 4, p. 94, S 3, Ex. 121. But as Y and Y' are both nuclear eemplete spaces, it is knewn that .Ef",(Y""; Y) me:Jff"[i5Y"' [3, Chap. 2, p. 34, Theorem 6]. It i$ also known that (Y.(J7"s .9-'))'=

Åq..S7W(il5J7')' is a nuelear space of type (LEÅr [3, Chap. 2, p. 48, Theererr} 9 and its Cerellary 3; p. 90, Lemma 9]. Then "'nt, as a quetiept $pace ef a nuclear space of type (L17), it is also a nuclear space of type (LF).

REMARK. The author is not $uecessful to prove that Y.n, is barre}led. If

thi$ is true, it resalt$ that 5psftf i$ refiexive since Spuftf is knewR to be semi-re-

flexive. The same may be conÅëluded in ca$e we ean prove that :S7Af is borno- logical, because a complete bornologieal spaee is barrelled.

4. Convollltion, In this seetion a theory of convolution is developed eofi"

eerning the spaces so far considered. We begin with the follewing remark•

ARy quasi-eomplete Rormal $pace gf distributiens wSth the property of aP"

proximation by truneature and regularization is a permitted spaee [6; 11]•

This is seeft by the identities:

ai(T * pk) - T== (ai -1År(T * k- T) + (T * p, --- T) -f- (a'kT- T),

(ctkTÅr * Fk --- T=Åq(cth "-- 1)T) as pk -(ctA - 1ÅrT+ (T * ij, ---- TÅr +ÅqakT -- TÅr,

where {ak} and {pk} are a sequence ef multiplieaters and R sequence of re' gularization$ respeetively. Here we note that {T*p,---T} and {a,T---T} are relative}y eompaet subsets of the spaee in questiott. Thus we may see that Y, Y', A'c aRd ptM are all permitted.

Let or and cStf be $ets of distributions. We write .fM=,SZF"' or `Srf iptSM'

to mean that SY" is the set ef distributieng cempesable with any di$tribUtiOn

On Spaces of Distributiong of Expanential Grovuth 11

of c9iiP [11].

We first prove

THEoREM 1. It holcts that (1) .S7'==,.S7VtD.9it.=År.9't;

(2) Y'c=ÅrY'.ef'.

A,, - ,-9]t,, M

i/i,lilg.,Rlll,-ny'ncii.iils,.S'i,l,`i'i9Ili',L,:S7iiiig,il'iii,i'?,{'kriliiik.,TEWSie,r,.eS'l'iiidlilii:'il'lillli,:nib;,i;all'lltued'i.,',"'le,g":,2'/llli/li2'$','/'ii,'"ill'

immediate consequence of Theorem 7 of [4]. This proves (1). (2) is provea as follows. We first note that ,:l7c(.Sl.if and .EJ5"Af(.fSi"c. Then on aecount of

lllil)o`9.J'}'h',"gl,lihg,ehO,b.13ie,','Ut-=[tZl'2'?.;}S".ESJi)*'"t"hE,ri?7Ai----,"li:,lb,tF,hg,S."9,),'Shrd

b,Y .,M ,.e ^.a ."s 2,2h,.e,de.fi?i.',,,iO.".Ogi,gO",".gi,UtrO."i,'.h,,eT.ef,O,r,eW,.e,Ob."a2'"if.'c,(,(,`S.7'`.v)".*

ct * I9 E .S7- for each a, I9 E :2y. This proves TE ,9V'c, that is, Yc"(Y'c and

we .get ,.9V,=År,.g]r'c. Observing Ac').95orAf')(J7')*==.C7'c we see 2.ir='=ts2'c.

This proves (2). The proof is thus completed.

We now diseuss the continuity property of the convolution as a bilinear applieation. We use as usual "hypocontinuous" to mean "ÅqC, X,)-hypoconti- nuous" where (5 and Es are the sets of all bounded subsets in the respective spaces.

THEoREM 2. It helcls that (1) Y'.Å~ /C7" ) (S, f) .S.fc JC7';

(2) .sgp '. Å~ ef. )• (s, T) --,) s * Tc .e t,, i

(3) Y' xf ) (s, f) .S *fc Y.;

(4) .9V'x .s7i. ) (S, T) .S * Tc -CV,,

and these bilinear applications are all hypoconti:nitozt•s.

PRooF, Ad (1): We need only to show thatekV(S *f) E LN for all in By )neans Of Proposition 11, this may be verified by a direct calculation,

Ad (2): Letting aE 9, we obtain (S *t T) * a'---S* (T* a'). Then Proposition 11 and the statement just proved we get (S*T)*aEJ`i'- for all a'(S}T. Thus

S * T E ,.9)'c.

ThS•l,d,fi[l)illlY,e,ll8.edbO,"EIY,!O.Sp?,OW,,t.hSttathteir,e.ebxyists,p.gn-spu,c,hp,th,iatl,e.-`6(S*f)cY•

12 K. YOSHINAGA

Ad (4): L6tting cr E 9, we obtain (S * T) * at=S * (T * aÅr ff .Sl (: ( Y" Ar by means of T* ct c J`X' and the statement (3). Thus S * T c Jl"'.

As for the second part, (1), Åq2) aRd (4) are proved by virtue of the eharae- terizatigR of beunded $ubsets ef S""' and f'c giveri abeve. We nete th3t to prove the continuity of a linear applicaticR frgm J7"" or .:SP'c into gnother space we may reduce the pTeblem to the case of Banach space defimftg the space e`y(Y') that is, the space {etyT;T==Dif, -7f-2--,, is abounded continuous function on R"}, We omit the details, The seeend part of Åq3) i$ an immediate consequenee ef [11]. This cempletes the proof.

5. Vltra-distributions. In this seetion the under}ying $pace ef the di$tribu- tions to dea} with will be ecnfined tc the reaHine R. We say, as usuai, that a cemplex funetign q)(":) defi"ed on a subset of the complex plafte C is stewgv increasing if Åq11X'1'ny',)l)k isa bc"Rded functign for a proper n;}io. We now begin with Silva's definition of the ultradistribntieR. Let ?Ik be the set of complex f"ftetions op, helemerphic in lgglÅrk and continuous on l8`.-ll}lrk,

sati$fying that (ll-,ep-, i'.ij..)IiÅrk is bouttded en lfiiY;ll;}:k. ?I,. is a Banach $pace with

the nerm;

i{rf?ii,igwaÅqa!:i{';',i!{i'lli.2l-iiiti•

Then it is not diMcult te see that the inductive spectre with respect to injections:

9..{o e 9.{l - 3?{2 - "''

defines a Silva $pace which will be denoted by Q.(.. On the other hand, letting Tc ,.9", we may write T== T" -tif"' where T", T- f Y" aRd the supperts oÅí T", T-" are contained respeetively in [O, ooÅr,(-- co, O]. Clearly sueh a decompe-

nt sition is uniquely determined up te afi arbitrary pelynemial PÅqS)=.tfo'a.6C'}•

Hence it is $een that T", Z'pt c ely(•.Sptft') if TE eiyÅqY'). 7?heft we may ebserve that the mapping =.e2rkST'(kÅr (resp. .-.e2x's`T-(x)) "f the eernplex half plane

EY:År -2k iff (resp• g=Åq-'21i ) into (0'c). i$ holomorphie. Therefere it follows

that the eemplex functiens;

(4)

tp"Åqsc) :!re:tix=T"(scÅrslx,

ip "' (xÅr = I re.za sii sT- Åqx) dx

On Spaees of Distributfens et ExpenEntial Grewth 13

a.re holomerphie and slowly increaslng ill g;År ". and 3r.-Åq--2k-i teepeg-

i3ioVnetYe'rsi!lyiURoWweNONbetianyashfoUwnettiOa2 q'Est4iC" cteteTn]ined by the pair (of, ept)'.

Åqs) T'(t)=iMe-2'`C"+'")'{p+(.+i;aÅrch,,

T-' (-eÅr= iie-"- :'` (" ' `")"g- Åqu - ia )du,

k. where aÅr2. is an arbitrary fixed nurnber. To see thig we that

e2fi(t--S)(g+ie)T+(t"År E (2'L;År,..(.:a27'.).

Then by using the Theerem of Fubini [8, p. I36], xve cbtain

fiTst observe

Ire-ff2ti{"+iat)s(p.(u+ia)d. =Irfp irf:titLt-s)cu+iajT+(tÅrd.

n ..

== l -.e..-""'`(`-"'eJcr-S(1År(t - 2År • T' (t)ds == Irb-Åqt -- tÅrT'(t)`lt=T'Åqi)•

i[.?

tii ie s SSIIIIInethkeaitdS fer T"h On the ether hand, feT any (fj=(4,+, cfi'ri eq, ane `t :;-,- k,

e•-2ti{e.i"}Sg)+Åqa+iaÅr, e'2giCfi-iO}S,p-(li --iaÅr E (l6 ',.:År'.Åq.9"'.År.

Ilihus the integrals gn the right liand sitle c,f (5) exist akEl tlefine the aistTihg-

tionsT", 7"(J7il Utilizing the Cauchy's l7heerem a .glmple cgrnpqtatign

enables ?s to ggnelude that T',T-- are indepenjeAt ef c;)'•ft, anG their filiprJfJrts are eijt.;;ined :n iij, rsÅr.Åq-eD, ij] Te.3peetively. Henee zare Tnay f)bs(irvff tkat ge,Tg.e,z.ikts,g,gft,.e.a,gkgs::•,grstyhc,s.e;zsf',`,r,I,eÅí2,st.etW?t"...52e.,-ffe4fii.,gi?,?;."

}ei,l:",g,i,eh\.Ii,a./tzh,g--.ff.Sie,t.2fd2.,Ci,Ykn.,e".i,i.a,lj.,rZ,E".e.i',f'.fi,e2.llGe'thats-fi'"'isiifieariyi.e,`j7rfigr.

,r' g T- s3#iÅqT)=(f, c k

Wijere pt i.o tli{j tnrsfiset gf tp=Cpt' ,q")r. ?IJ.ll. "l'he invsstafii ;gX ijf:I'S l3 deJter- ITllnegi }.y (.ijr"]. If 7'Eete7C,Y') fgr a.l!y fixg-af k=l, Z, ••:, thett ge'{=-}, ?Ti'{.i} ffr-'-=

he}gmlj?yhie in #.e:7 -==gk

.--

, #g-t.; z, //-- reegie{:Sively fiit6 it fsllijwe• I•be.t

ip`(fi-)---tp'(.-T-3=:Si.fti""7n'Åqstda.-,Se'f'.---itcr-S-i,

as well as .

. T(Åí')=I g,.'" "" rt'tS(of-Åq#) •--- rf)'" (t))du

It is now almo$t evident that the applicatien tSV-i is eeRtinuous and we may ceRelude that 2Ept, SC-i are mutualiy inverse isomorphisms between the Silva spaee .S7" and the Siiva spaÅëe ft. Accoring to Silva [9], we shall say that U i$ the space ef ?tltra-di$tributions.

Since Y' is a dense subset of ,9i, we see that SZr"i(Y') is dense in ll. On the othar hand if S(Y', then by definition ef Ept'i, we may eensider that

$4'biÅqS) is the "eemplex form" of the "real" Fourier transferm Svi(S). Thus any e!ement fEno"f(Y' is seen to be a multip}ieator of E;""(Y'År inte EX-2(Y'). If the multiplicatien defined by f is extensible te a centinugus 1inear applicatien of U into U we may naturally say that we get a mMttipli- cation with respect to the ultra-distributiett. Silva has shewft that sBeh an extension is pessible whenever f is preleRgable tg an entire function slowly inereasing on every horizontai strip. Here we prove, as Silva eonjectured, that the eonverse of this statement is alse true. Fer this purpgse we mu$t clarify the preblem in the following form. In view of Siiva's idea, it seems natural to say that an ultra-distribution e i$ a multiplicater ef tt inte U wheB St(eÅr ff Yir'c. Theft it fel}cws that [e]fl,==$Vpti($4(e) * $I(d,År) i$ defined for any pa ff U and [e] is a continuous linear application of li into tt ÅqTheorem 2)- Sinee the mapping Cy : . e:t`'`SF(S)(di) E (0'cÅr. is helomerphic, it results that e(x) = !:e:-th:tE zr(e)(x)dx

is aR eRtire function. On accountof Proposition 11, (1) it is not diMeult tO see that e(x•)is slowly increasing on every horizontal strip. Cefiversely anY such eÅqv) defiftes a S $uch that :Y(e) E .9or'c. Te preve this it suMces to shoW that :Y(e) ff Y'c. Then the problem may be reduced to prove that for anY k there exisbs anlsuch that

eky(=) I:g.""2Xfx, ae+/YyÅr,), dy c L"".

We now say that this is an immediate consequenee of the Canehy'$ TheeTeM applied to the strip lg:ls{; i . Thus the statement is proved. HenÅëe in CaSe

that the ultra-distributi"B e--f beleRgs te ePi{ i- the sense ef Silva, we MaY eonclude that f defines a multiplieation of n into u if and only if f is prolenga- ble to an entire function $lowly increasing en every herizefttal strip•

By the way we say a word abput the eomposition operator. Let e be aB

On Spaees of Distributions of Exponential Growth 15

ultra-distribution sueh that {f;(e) E .Slnf. Then it is not diffieult to see that

kS•'

-; R(2.//lllisll`C,'is'ln]Ws.h/ICe,tiiehl,8.l':dtlllnsOl,e,z2W2)c/g./'.tliSe.ggo#ugeS'gei,:',i"gYgLlesE:"iS•eP.olf

(e",e ) of such e is eharacterized as follows. Letting ,S;(e)=gc .9Unf, we obtain e' (z) = ! e2X` :"'g (x) dx, e- (...) = - S O-e.2 '`=g(x)dx.

,B uay ,M ,e.1".S,8,f".h8,Åí9i/rg,i.Lndk2g.',"Lo.",;'li,s,".o,:,dllll,cwa."g.sg,e.`,hg,`,,Z'•.g5.a,:•g

pnts. By this we shall mean that there exists an increasing sequenee of

integerp {k,-} such that e', e- are defined and holomorphic on K9Y':2}lki, :IYx-Åq-ki respeetively, and that there exist

ai == lim z e+ (:)= lim z e'" (z),

zHeg :-"e St4ai SiS-kl

aJ+i:.-i.ltLr.},;""(e'(;)-tl,a.'.)===s///.m.-.,;ii",'(e-'(:)-,:,]""...v),f=i,2,•-F.

Conversely now we may prove that an ultra-distribution e with sueh a re- presentative (e', e'År is a composition operator on U. To see this we must show that

: Z (e) (x) == S:e.-2ti'("'`k")e'(u+ ihle.)du - i e..m2"i=("-`k")e-(u -- ik'.)du

belongs to ,.9'iv, that is, for any p e'-knv(s)DpE I(e)(x) E L- where k. is properly ehosen sueh that e' and e- are expressed as

e+(.)-- a-i +...+ 3s + g;.'{-+g..'.S:,2,gz:}rb.,

" -"-

e-(.)- a.i +,,.+ ef + S'."i +g..-,::,),scsc:s{g--kn

- "--

. W!th bounded g'(:),g-(z) respectively. Now, observing e'e(D+ ct)'S= S(') and so

i :e,.', 2tiS(u+ia)(u + i(r)rdu = ( .i..--)'s(r)

fOr any real nurnber ct and r=O, 1, 2,•••, we obtain

DP,SF (e)(x)= (-2Ti)"Iap.i(j e;2tix(""`k")u+lt:k.du-! e-"2tiX("-ik")u-lik.(lu)

+!re..'ca=("'`krt) gi.(Ii[:.ii,`i//"..År du-Ire.k2xix("-ikn)g("'.(U.-ikXig2,År d.l

=( -•- 27ixiÅrp( -2xiap.} +e2#=ipm !:.. e-reti"u"{llS/ liSl}EIII!!l!-Åqk.)+, k ) du

-e--ortxknI11.. e'ww2"i."`g--i(kas.)"-'.'- i:kn) dul,

This eompletes the proof.

References

[ 1 ] N. BoumuAKi, Espaces vectoriels tepelogigues, Actualites Sei. Ind., no. 1189 (1953), no, 1229 (1955År, Parig, Hermenft•

r 2) C. euEvAmEy, trheorv of distribntibns, Lecture at Col"mbia Univ. in i95e-51.

[3] A. GRoTHauDiEcK, Produits tensoriets topatogiques et espaces nuclEtrires, Memoirs of the Amer. Mgth. $ec. Re. 16 Åq1955).

[4] 'Y. HikATA, en anvoltho7ts in thetheeirv ofddistrdvzetions, J. iki. Hireshima Univ. Ser, A, vol. 22 (1958) pp. 89-98.

[ 5 ] L• ScHwARTx, Thgo?xles des alistributGia?us, I, ]I, Actualltes Sei. Ind., no. Ie91 {1950), no. 1122 Åq1951År, Paris, Hermakft.

il61 , Theorie des nouaux, Proceedings of the International Congress of Mathe-

maticians, Cambridige, Massacrhusetts U S. A., vol. 1 (1950) pp. 220-230.

[7] ---•---- , ffgpaces de fenctiens di•ffic'gnSiabtes h valettTg vecterielles, J. ARalyse Math"

ve}. 4 Åqi954-56) pp. 88--i48,

[8] - - - , Thborie des diseributims d valeurs vectorieltes,Ann. Inst, Fourier,Grenobie, veL 7 Åq1957År pp. 1--139.

[ 9 ] J- S• E SiLvA, Lee ftnictions anelytiqites comtnte uttra-tleistrilÅrutions dans le ealeut opes' ntien- nel, Math. Annaien, voi. 136 (lg58) pp. 58-96.

[10] K• YosmNA"A, On aloccuUy anvex space introtl%cetl by J. S. E Silva, J. Sci, Hiroshima !]fti" Ser. A, vel. 2! Åq1957År pp. 89-98.

[12] --- and H. ectATA, On convatutions, J. Hiroshima Univ. Ser. A, vol. 22 (IS58) PP•

15-24.

[12] K• YosKiNA"A, On tke gpace (drs'År a?$cX the Feewdertraircgfermation, Bull. Kyushu IRst• TeCh•

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Kyushu Institute of Technolcgy