VALUES OF VECTOR-VALUED DISTRIBUTIONS By

VALUES OF VECTOR-VALUED DISTRIBUTIONS

By

Ky6ichi YosmNAGA

(Received Nov. 30, 1966)

.

Values and limits of distributions have first been investigated by S. Lojasie- wicz [6]. Lately, motivated by the study of the smoothness about the semi- group distribution, the author of the present paper has considered the value and the limit of a vector-valued distribution of a single variable [14]. It seems probable that such notions will have a variety of applications in connection with the local structure of distributions. An attempt to define the multiplicative produet of distributions has been made along this line elsewhere [15]. In such a case one must handle the value of a distribution of several variables, and com- pared with the case of a single variable one must be faced with some peculiar technica' 1 difficulties at the starting point of the argument. Yet in the work of Lojasiewicz [7] concerning the fixation of variables in a distribution one may find an effective means to overcome such obstructions. The purpose of the pre- sent article is to make a basic study of the value and the limit of a vector-valued distribution of several variables. Our main result is Theorem 1 which states a local structure of a distribution having the value or the limit at a given point.

This is a generalization of a corresponding theorem of Lojasiewicz [7] to our vector-valued case and will play a fundamental r61e in the subsequence of the present paper.

Section 1 is devoted to the preliminary remarks. Various known theorems requisite for our succeeding arguments are stated. Among them some (Pro- position A and B) are already proved in [11] and [14], and the other (Theorem A) is the theorem of K6nig-Lojasiewicz [7] extended to the case of a vector- valued distribution. In Section 2 we shall introduce an operation of prolonga- tion L2,i defined for any indexp==(pi, •••,p.), pi:}}il (i= 1, •••, n), of differentia- tion relative to x :(xi, •••, x.)ER" and for any interval I== IiÅ~•••Å~I.(R". A funct.i.'on f' on I taking values in a locally convex space E is extended to a func- tion f'----Ll,i7 by setting outside I the polynomial of degree pi----1 in xi(i=1, ...,

n) obtained by an interpolation formula of Lagrange. We close this section by

giving estimates of the growth of the function affected by this operation. We

define the value and the limit of a vector-valued distribution in Section 3. This

will be done in accordance with Lojasiewicz [6], [7] within the framework of

,

Schwartz's theory of distributions. We prove our fundamental structure theo- rem (Theorem 1) asserting that a distribution, vector-valued in a quasi-complete locally convex space E with a fundamental sequence of bounded subsets and having the value or the limit eEE at the given point x=xoER", may be ex- pressed as

T= DPfi, F'(x) = (X -p- "! O)P e+ Ix-xo1iPib(x --xo)

on a given bounded open set, wherep=(pi, •••,p.) and 6(x) E 80(E), 6(x).O (x-ÅrO).

The proof given is rather complicated and may be seen to be erected on the basic idea of Lojasiewicz modified in the case of a locally convex space. In our case where distributions of several variables are in question one might be puz- zled how to take the limit at issue in defining the value of the distribution. In

Section 4 one may find some types of limiting processes concerned with such a problem. Among others we shall show that the notions of the value and the limit we have defined above has an invariant character under the change of local coordinates of class CeO. We shall also make mention of the boundary value of a distribution. In the last Section 5 it will be proved that in case E is a locally

convex space of a certain more restricted but familiar type (Silva space [13]) any E-valued distribution T admitting limT(xo+ZÅí+ or) may be considered on

)v-,O '

y-o

some neighbourhood of xo to be a scalarly locally integrable E-valued function.

The notations and the terminologies in this paper are essentially those of N. Bourbaki [1], [2] (locally convex spaces), [3] (integrations) and those of L.

Schwartz [8], [9], [10], [11], [12] (distributions). We also assume elements of A. Grothendieck [4].

1. P:treliminaries.

Let R" be the real Euclidean n-space and let N" be the product set ofn factors of the set N of the nonnegative integers. For any p=(pi, •••,p.),

g==(gi, •••, gn) E IV" and x==(xi, •••, xn) e R" we write Ip1 ==pi+•••+pn,p! =pi!•• •pn!, (E) = (El) ' ' ' Åë:), DP =" o.{9 l. l5.2. , xP == x{i • • • x2" and we aiso denote p ;;?: g to mean

pi;;}l gi, i---1, •••, n. Aceording to circumstances we use Dk instead of DP for p==(k, •••, k). An interval in Rn is a cartesian product of n factors of bounded

or unbounded intervals on the real line R.

Let E be a locally convex Hausdorff space (for short, LCS). A sequence of

bounded subsets of E is said to be fundamental if any bounded subset of E is

contained in a member of this sequence. The continuous functions defined on

a subset S2 of R" with its values in Eform a vector space 9E(9). It is an LCS

with the topology of uniform convergence on every compact subset of 2. If 9 is an open subset, the space {9E(S2) coincides with the space esl(E) of Schwartz [8, p. 93], [11, p. 49].

In order to develope our theory suMciently the following proposition will play a fundamental r61e.

PRoposmoN A. Let E be a quasi-complete LCS with a funclamental sequence of bounded szebsets and let 2 be a bozendeel open siebset of R". If a subset H of gt(E) satis;fZes that {Åqi, ÅëÅr ; T E H} is a boundeel szebset of E for any Åë E s7 with its szeppoTt supp Åë( S2, then it hotcls that

i) His an equiboiended szebset of Y(e2-; E),

ii) theTe exists an m e IV such that H is an eq2eiboundea szebset of Y,(ez; E)

== (9SM).(E),

iii) the topologies of s7fo(E) and of (:2,3M)g(E) coincide on H.

PaTticielaTly,

iv) any aistTib2etion T E 9'(E) is locally of .finite order, i.e., for any bounded

- open subset 2(R" theTe exists an m E Ar such that T E (93M)g(E), and fuTthermoTe foT any compact subset K of R", T i$ a bouncled linear applieation of 0K into E.

This is a slight modification of a proposition due originally to Schwartz [11] and the proof will be found in [14]. The next is also given by Schwartz [11, Proposition 24] and we shall state it without the proof thus

PRoposmoN B. LetEbe an LCS and let S2 be an open subset of R". Then, given mE2V, there exists upEY((93M)ni esS) foT each pcIV", lpl-Åqm+n+1, in szech a way that

i) any T E (g3m).(E) is expTessea as T == Z DPfp, fp=(upX I)T 6 dp2(E) lplSm+n+1

where J is the ulenticat mapptng E-ÅrE,

.

ii) if a smbset H((92M)2(E) satisifies the condition that V T(B) is a bounded

TEH

szthset of Efor each bozended s2Lb$et B of 9Z, then foT any compact K(2 it holels that Vfp(K) is a bounded szebset of EfoT each p, lp1.Åqm+n+1.

TEH

For the later purpose we prove the following corollary known very well for scalar-valued distributions.

CoRoLLARy. Let E be an LCS, m E IV and let S? be an open szebset of R". Then any (listribution T E (93M)g(E) of s2Lpl)ort {xo}, xo E S2, is expTessecl as

T(Åí)== = (DP6)(x-xo)Qep, epEE.

IplSm+n+1

PRooF. For convenience' sake we assume xo=0. Take any ÅëE9g and we

write

Åë(X) = t,r.llli.l.., 'ilgie- DPÅë(O)+ Åë(x),

where ÅëE es and (DPÅë)(O)=O for eachpeN", Ip1-Åq rn+n+1. Then we get

ÅqT' ÅëÅr =ipisllll+l].+i ÅqT(Åíp)i ÅíPÅr DPÅë(O)+ Åq T, ÅëÅr•

.

For any e'cE' the scalar distribution ÅqT, e'År is of support {O} and of order

.- - .

-Åq m+ n+1 because T(&)= = D"f(Åí). Thus by a wellknown theorem [9, p.

IPISm+n+1

-ÅÄ -

93, Th6orbme XXVIII], ÅqÅqT, ÅëÅr, e'År==ÅqÅqT, e'År, ÅëÅr--O which proves - ÅqT, ÅëÅr=O. This completes the proof.

The following is a generalization of a theorem of Lojasiewicz [7] to the vector-valued case.

THEoREM A. Let Io=]ai, bi[Å~ ••• Å~ ]an, bn[ be a bo2Lndeel open inteTval in R", Jbe anotheT opuen inteTval in R" containing Io, p==(pi, •••, p.) E N" and let E be a given qzeasi-eomplete LCS. The totality of the nonvoid subsets a of cro== {i; piÅr-1}

is denoteel by A. .EoT any cr E A we wTite Qa= {g; g== (gi, •••, gn) E IV", O-Åq gi -Åqpi-1

foTiea and gi=OfoTica}. Then for eveTy aEA and foT every gcQ. we may

dejine a fzenetion a.,(xi,, •••, xi,c.))c9i. where a={ii, •••, ile(a)}, iiÅq•••Åqik(a), and Ia ==]ai,, bi,[Å~ ''' Å~]ai,(.), bi,(.)[ So as to holcl

i) if TE s)}(E) and DPT = O, then T = 2 IE] T.,,•Åíq, wheTe T.,, = aEA PEOa

. ÅqT, aa,qÅrxi,.-,xi,(.) and Åq , Årxi,, "., xi,(.) means the inner pToeluct with Tespect to xil) •••) )Cik(a)e

ii) iff- E es9(E) and D'fS=O, then 7(x)= 2 PS)i a"i,,(xi, •••, xi-i, xi+i, •••, xn)x;•,

iEao p=O

.- Where di,v==.ÅrAl](t)qEo2.,4,..,Åqf, aa,qÅrxii,-•,xi,{.) and .=GA(i) iS the s2Lm only foT szeeh T2Lnning af E A that i is the least index: i =ii, and censeqzLently a"i,, E (5'9(E).

PRooF. The proof of the statement i) may be given, mutatis mutandis, nearly in the same way as given by Lojasiewicz [7] and thus will be omitted.

As for the statement ii) we need only remark that since E is quasi-complete, fS( s9(E) may be considered as 7 c :o}(E) [11, Corollaire 1 of Proposition 21] and

hence the assertion i) is applicable. This completes the proof.

2. Theoperationofprolongation.

Before giving the definition of the value and the limit of the vector-valued

distribution, some mention must be made of the prolongation concerning the

vector-valued functions.

-

Let Ibe a closed interval of Ri and let f(t) be a given function on I with

values in a quasi-complete LCS E. According to Lojasiewicz [7] we now define

-- the operation L9=L9,i : f-Årf, 1-Åqp E 7V as follows.

(i) In casepl}})2 take toÅqtiÅq•••Åqtp-i in I where tj are arbitrary except that to and tp-i must be the end points of Iwhenever such comes about. Then define

fN(t'=[till/li'7(,,)eg:,-.t,:iO,i`,ii2

v=O (ii) In casep=1 we define -

f(t) for tc l,

-

f(`'==l,2iZe..O,{.,7.2`.gh,e.,C.O,pe.M,l".e.",d,,P,Olng,9g.i,a,ndforteL

Then tr=L?fri is a function defined on Ri and one may easily verify that L9 c .S?({lfE(I); 80(E)), and if f(t)=aotP-'+ait"m2+•••+ap-i, ai E E(i=O, 1, •••,p-1), on Ithen f(t)i=aotP-'+aitP-2+•••+ap-i on -Ri also. Furthermore letting C be the convex circled envelope of the set Ir+4(tlp-,;tE II in E we may find a positive number K=K(p, I) such that f'(t) e K(1+ 1t1P-')C for all t c Ri. Using this L9,i we next define the operation L2==L2,i==L.Pi,,i,•••L.Pn.,i. fOr anY p==(pi, •••, p.) E Arn, pi År-1 (i=1, •••, n), and for any closed interval I= IiÅ~ •••Å~ I.(R". Then

we may infer that

10. L2,iE.!2e'(`ZfE(I);esO(E)), -

20. for any fEVE(I) such that DPf==O in 9f(E), it holds that DPf==O in g'(E), 30. givenp and Iwe may findapositive number K=K(p, I) in such away

-+ - that for any fE %'E(l) it holds that f(x) E KArp(x)C for all x E R" .where IVp(x)

==

ini(1+1xilPt'i) and Cis the convex circled envelope of the set IfNp((X.));xE II in E.

-

Except 20 it is not diflicult to prove and 20 is seen as follows. Since D'f==O on Iand piÅr-1, i=1, •••, n, it follows by Theorem A that f(x) =: E] di,,(xi, •••, t==1v\t

p= o o xi-i, xi+i, •••, xn)x;• on Zwith a'i,, 6 80io(E) and therefore

- pj-1 n pi-1

L21,i,f== = a"ivx;• +Åril IZ L21•,i,di,,•x;••

v=O t==1V=:O iiEj

. Hence DPL.Plaf==O for every 7'=1, •••, n and thus on R"

6 K. YosHIN.A,GA

, DPf == DPLe• l,T,•••L2:,z.i= O•

Letting now Px== {x; lxiIÅqZ, i--d1, •••, n} for ZÅrO, we state the following vector version of the lemmas of Lojasiewicz [7].

LEMMA 1. Let E be a quasi-complete LCS and let OÅqptÅq-S- , p= (pi, • • •, pn) EN", pill}lrl (i--1, •••, n), be given. Then theTe exists a positive constant K=K(p) in

szech a way that for any g-'(x) E 9E(P-x-P,,) satisfying DPg'=O on Px--15. we may constTuct a g(x) c {gE(R"-P.) so that

i) Dpg =o on Rn-p"., ii) g(x) == g(x) on A- P.,

iii) g(x)EK•IVp({li--)Cfor every xER"--P. where C is the convex ciTclea

envelope of the s2ebset {g'(x); x E P-N-P.} of E.

PRooF. Owing to the similarity we may assume Z= 1 and thus with no loss 1 of generalities we may take pt== 2 . Then putting P(le)={x; lxk+ilS;1, •••,

lxnlKl} (k=O, 1, •••, n-1), and P(") =R", a successive application of the opera-

tion L.P':. (i---1, •••, n)

10. from Ix ; -1 -Åq xi -Åq - -i-Ii- , 1 ti+i1 .S; 1, •••, l xn l -Åq 1) to Ipc ; ui fE{l: '-- -li- ,

lxi+ilK1, •••, lx.Isgll, and

20. from Ix ; -l}- .s{: xi .s{: 1, l xi+i l -Åq 1, •••, I xn 1 SI II to Ix ; t .Åq xi, l xi+i1

Sl, •••, Ixnls;;ll will allow us to obtain a prolongation from P(i-')-Pit2 to P(')-Pit2 for i=1, •-•, n. The function g(x) thus finally attained from g(x) will meet the requirements. This completes the proof.

LEMMA 2. Let Ebe a qzLasi-complete LCS and let p =(pi, •••,p.) E Ar", pill}itl (i=1, •••, n), be given. Then theTe exists a positive eonstant K=K(p) in such a way thatfoT any g'(x)E{gE(A) satisfying DPg---O on Px zve may con$tTuct a g(x) c {eE(R") so that

i) DPg=o on Rn,

ii) g(x) == g(x) on p,,

iii) g(x)EK•IVp(-i[-)Cfor every xeR" wheTe C is the eonvex eiTcled en-

velope of the s2ebset {g'(x); x E P-N} of E.

PRooF. Nearly similar to the preceding.

3. The limlt and the value of a vector-valued distribution.

Let E be an LCS, S? be an open subset of R" and let xo E 32.

-

DEFiNiTioN 1. A distribution TE9fo-{.,}(E) is said to have the limit limT(Åí) =eEEat x=xo if T(xo+ZÅí)-Åre (Z--ÅrO+) in gfen"{.,}(E), i.e., ÅqT(xo+ZÅí), x-Xe

Åë(Åí)År = ÅqT(N), zl. Åë( jt :ii 'CO)År--ÅreSÅë( v)(l t (Z-ÅrO+) for every g5 E 9Rn.{o}.

DEFiNmoN 2. A distribution T E efo(E) is said to have the value T(xo) == eEE at x=xo ifi(xo+Zfi)---Åre (Z--ÅrO+) in okn(E), i.e., ÅqT(xo+ZÅí), Åë(Åí)År== ÅqT(te), zl. Åë( hi iXO)År -ÅreSÅë(x)ax (z-Åro+) for every ÅëE gRn.

REMARK 1. Since suppÅë(ÅíIIXO)==xo+Z suppÅë(a) it follows that for any .

ÅëE9Rn-{o} or Åëe9Rn, ÅqT(xo+ZÅí), Åë(di)År may be defined for any sufliciently small Z in the respective cases. Furthermore vkTe may find that the limit and the value thus defined above have the local character, i.e., if one of two distri- butions eoinciding on a neighbourhood of xo has the limit (resp. the value) at x =xo, then the other has also the same limit (resp. the same value) at x==xo.

REMARK 2. In the previous article [14] the present author has studied the limit of a distribution on R' in a slightly different manner. One says that T has the (unilateral) limit on the right lim i(&) (resp. Iimit on the left lim T(Åí))

x-xo+O x.xo-O

at x == xo if there exists a distribution S' E 9t(E) satisfying

--

ÅqS(&), Åë(&)År= lim ÅqT(xo+ZÅí), Åë(Åí)År x-o+

for anyÅëEe with suppÅë([O, +oo[(resp. with suppÅë(]-oo, O]). Then it is . seen that S(Åí) is a uniquely determined constant distribution on ]O, + oo[(resp.

on]-oo,O[), S'(Åí)=eE E, written as lim T(&) =e (resp. as lim i(di)= e). When

x-xo+O x-xo-O lim i(di)= lim T-(te)=e, T- is said to have the limit at x=xo and we write x-,xo+O x-xo-O

lim T- (hi)== e. According to an observation of Proposition 5 given below one may

x-Xo easily verify that both of these definitions are identical.

REMARK 3. In case T'e9fo(E) one may define, besides the value T-(xo), the limit lim T' (Åí) as well. Clearly the existence of the latter is a consequence of x-,Xo

the existence of the former, but the converse is not always true as is seen easily

.

by the example T==6([De, e E E.

The next proposition based on the original of Lojasiewicz [7] will be of

fundamental importance in our theory.

PRoposmoN 1. Let E be a quasi-compZete LCS with afundamental seq2Lence of boundeel s2ebsets and let {Z,} be a given sequenee of positive numbers such that

Z,+1

År0ÅrO, v=1, 2, •••, wheTe e isa.fixeel eonstant. Letfurther Z,-ÅrO (v-År oo), 1År

Zv

oÅrO be given and let S2 be an open s2ebset of R" containing P-.- {O}. If T-'E9fo(E)

- satisfies T(Z,Åí)-ÅrO (v---Åroo) in 9fenH{,}(E), then it holds that i) theTe exist p E Ar" and i7 E 80(E) so that

---

T-=D"i3onP.-{O} ana LF.(IX,l,--ÅrO (x-ÅrO) in E,

ii) itfollows that T'(ZÅí)-ÅrO (Z-ÅrO) in 9kn-{,}(E), i.e., limT-(ZÅí)==O and fur-

x-O

- theTmoTe foT any subset B of 9Rn-{o} bo2endeel in 9Rn we obtain Åq T(ZÅí), Åë(Åí)År -ÅrO (Z.O) unifoTmly in ÅëEB.

'-iF If in partie2Llar O E S2 ancl T(Z,Åí)-ÅrO (v-Åroo) in s7fen(E), then

iii) the expression T'=DPfi in i) is vahcl on P. a7zd it holcls that T-(ZÅí)-ÅrO

- (Z-ÅrO) in e'(E), i.e., T(O)=O.

PRooF. Proof of i). Owing to the similarity we may assume 1ÅroÅrO so as to satisfy Pi- {O} (2 and then we put P=Pi. Taking a subsequence we may also assume zi==1 and 1ÅreÅr-i-i-'-'Åre2 (v==1, 2, ...). Letting 6, -Zl2LÅr6Åro, be

v

sufficiently small and putting 9o=Pi.s-Pin e2,2-s we suppose 2o(9. Then by Proposition A and B applied to 9=9o and H={T-(Z.Åí)} we may find an mE Ar and functions lp, E 8Z,(E),pE .N", lp1"Åqm+n+1, v== 1, 2, •••, so as to obtain

--) -

T(1,Åí)= E] DPfp,(Åí), y==1,2,•••, IPISm+n+1

and 7p,-O (v-oo) in es2,(E) for each p, lpl-Åqm+n+1. Letting hc 9, O-Åqh(x)

bÅq1,

h( ,,) .,. Il fOr x E JP- Pe2i2,

for xcPe2i2-sl2V(R"TPI+si2), to

and putting

. a) g,,(.)-{3(x)fp"(x' igi:aSg;

we observe g"p, E 80(E), g-p.-ÅrO (v-Åroo) in esO(E). Denoting ep.(x) =ZLP[g',,( fi-),

-- we may construct a function Fp,E80(E) from Gp. first by taking rn+n+1-pi

times repeated indefinite integrals from O with respect to each xi, i=1, •••, n, and then by multiplying thus obtained by the scalar function h (-jli7). Then it

p

is not difficult to see F- p,-ÅrO (v--Årcx)) in 80(E) for each p,lp1hÅqm+n+1. Let us write fa, == 2 fip.(v =1, 2, •••) and let us observe in Px.-Px,e2t2 that for v =1,

lplgm+n+1

2, ...

DM'n'ifiv(hi)=ip1,,lll.l]..,DPUpv(Åí)== ip1s;Il]...i(DPgpv)( zX", )

== fpis;F+]..,(DPi7ipv)( zX", ) ., jil(te).

.

Let Cp, be the convex circled closed envelope of the set {fp.(x);xEPi+s,2 -Pe2t2msi2} in E. Then since E is quasi-complete this envelope Cp. is easily seen to beacompact subset of E [2, p. 9]. We here note that Cp.-{O} (v-Åroo) for each given p, i.e., any neighbourhood of O in E contains almost all Cp,, v=!, 2, •••.

On account of the definition (1) of g-•p, it holds that g'p.(R")( Cp, and thus G,.(Rn)

. (ZLPiCp. from which it turns out to get Fp,(I-'x,)(ZSM""'i)"Cp.. Therefore it follows that

(2) i,(i5N,)( = Z(,m+n+i)nCp,=Z(.m+n+i)nc.,

lpl {;m+n+1

where C,= 2E] Cp, is a convex circled compact subset of E and it holds that lpTSm+n+1

-År- C.-År{O} (v--Åroo). Putting ij,=F,+i-F. we may observe on the open subset (Px.-jPN,e2t2)A(Px,+,-I5N,+;e2t2) =Px,+i--Px,o2i2 that

- - -) --

Dm+n-1ij, = Dm M+IF,+1-DM-M+IF, == T- T== O.

Because of Z" ^{2e2 Åq Z"} i' we may now use Lemma 1 and consequently we find

ijN E gE(R"-A,ei2) so as to satisfy the condition that ij,==ij, on I-'N,.-Px,e2i2 and Dm+n+iij', =O on R" -- A,e2i2. 0n the other hand for any x E i5x,.,-Px.e2i2 one getS by (2) that

g',( u) =: .iil,+1( )c)- F,(x) E Z(,m++1n I 1)nC.+1+ Z(,m+n 'F 1)nC, ( Z(,m ln'lm1)n(c.+1 + c,),

and thus owing again to the same lemma there exists a positive constant K so that ij.(x)EKZSm+n+i)n(C.+C,.i)i"" =,(1+ {.i, M+") for any xEP-P-'x,e2t2. This implies

(3) ij,(oc)EL.ze(zev+in+Isulm+n)n(c,+c..1)

10 K, YosHrNAGA

for every xEP-P- x,ezi2, where L=K/e2"(M'"). Let us now define C,, v =1, 2, ..., as follows.

-- -

Gi=:i7i on P-I]'e2i2,

--

G,.,=IGF.l++iij, .O.npPXL+lbll,I,fti+ie2i2',21.

. To see that suchaG,.i may well be defined it is enough to note that for any x C (PN,+i - A.+ie2i2) = Px,+i - A.e2i2 one obtains e,(x) + ij,(x) = fi,(x) + ij,(x)

== F.+i(x). Therefore we see that G,, v =1, 2, •••, is a continuous function defined on P-Px.e2i2 with values in E, and it is not difficult to verify by induction on v that

(s) Dm+n+la,=T- on P--P,.e2i2.

Letting v2}irl, oE Ai and observing G,+.-j, =ij-,+.-i+•••+ij, valid on P-A.e2i2 we may obtain by (3) that

- -+

G,+.(x)-G.(x) c L.ZY+..i(ZT++." + l x l M+n)"(C,+.-i+ C,+.) + +L.Zy+.-2(Ze++.".i+ l x 1 M+n)"(C,+.-2+ C,+.-i)+ •••

+ L•Ze(Zy.+in + l x l m+n)n(c, + c..,), for everyx E P-A.e2i2 and thus it follows that

-+ -

G,+.(x)-G,(x) E L.(Zy.+ln + I x 1 M+n)nZe [en(a-1)(C,..-1+ C,+.)+

+ en(a-2)( C. ..m2+ C, +.-i) + • • • + e"(C.+i + C,+2) + (C, + C, +i)]

for every x E P-jPx.e2/2 where we use Z,+kÅqekZ,. Let Ube a given convex circled neighbourhood of O in Eand let yu ENbe chosen so that C,(U for each v2}iivu.

Then for any such v and for any u E P-P- x,e2i2 it holds that

-- (6) G,+.(x)-G.(x) c 2L(Zy.' ,n + 1x i m+n)nze( u+ en u+ ...+en(cr-i) u)

1

( 2L 1-e. (Zer'i" + 1 x l m"")"ze u.

Therefore owing to the quasi-completeness of E one may conclude that the - sequence fG,} .is convergent uniformly on each open se.t P-A,,e2i2, pt==1, 2, •••, to a function F E 9E(P- {O}), and furthermore from (6) it follows that

"i7(`v)-Ov( u) E 2L 1le.(ZY."in+ 1 v lM'n)"zeu

for each v;;}lvu and for any x c P-A,e2i2. We also find from (5) that

Dm+n+ii-- T'-' on P-- {O}•

-- On the other hand one knows from (2) and (4) that G,(x) =F,(x) E ZSm'""'"C. for each v2}:vu and for any x E Px.--A,e2i2, which proves that

i7(x) E I12-Le.(ze.',"+ I u 1m'n)"ze+zsm+n+i)nl u

for the samevand x. From this one observes -

1 ic Iliml(+X.)+i). c Ii2.Le.((lpx+i)m+n+i)"( ivl )" +( lZil )(m+"+i)n}u

e2z

and therefore since lxl;;}it 2" for xEPx.--Px.e2i2 we may conclude ' .

1. IF,Åíln)+i)n 6 I12-Le.(( 2eZ,"z",')M'"+1)"( e2, )"+( e2, )(M'"'i)") u,

and finally because of OÅq Z i;' Åqe, we obtain

-

lx iFtÅílil.!). E Ii2.Le.((-ll-)M'"+i)"( e22 )"+( e22 )(m+n+i)n} u

for each v;;}lvu and for any x EA.--A,e2t2. This proves F-(x)

.o (x-Åro) lx1(m+n+i)n

in E and thus by setting fi(O)=O it follows that fi E es$(E). Then multiplying F-' by a suitable function of 9 we get a new F' E 80(E) satisfying the statement i).

Proof of ii). It suMces to prove the second assertion. As just proved we may write T'---DPfi on P.-{O} where F-(x)=IxliPi6(x), bE80(E), and 6(x)-År O (x-ÅrO). For any Åë E B we find by a straightforward calculation that

Åq T-(ZÅí), di(dr)År=:(-1)iPiz,pilz1nÅqF-(Åí), (DPÅë)( ft )År

.. ( -- l: l )iPi S 1 . vp: 6(zx)DPÅë(x)ax

which proves Åq T' (ZÅí), Åë(hi)År -ÅrO (Z--ÅrO) uniformly in Åë E B, because B(eRn.{o} is a bounded subset of eRn. This completes the proof of ii).

Proof of iii). Employing just obtained in i) we write T:o=D-.P,i7E96.(E).

Then it is not diMcult to see To(O)==O. Since the support of T---- To is {O} we

12 K. YosHrNAGA

may apply Corollary of Proposition B and thus we see

-ÅÄ -

T-- To= X eq([il)Dq6•

lalsk

For any Åë E 9 and for any sufficiently large v let us observe

--År -

Åq T(Z,Åí)- To(Z,Åí), Åë(Åí)År=: IZ ÅqDq6(Z,&), Åë(Åí)Åre,

lqlgk

= ,;ll,2.,(z'.nl+)i:ii Åq6(Åí), (DaÅë)(-i})Åre,

= ,iili, i.,( fn, 1+ )i iq i DqÅë(o)e,.

The first member of this identity tends toO as v-Åroo for any ÅëE:2 and there- fore we must have e, =O for all g, Igl s{gk, i.e., T-= T-o on P.. This proves iii) and the proof is completed.

REMARK 4. Given g E N" the indexp E Ar" in the proposition may be chosen in such a way that g-Åqp.

THEoREM 1. Let Ebe a quasi-complete LCS with afunclamental sequence of

bou7zded s2th$ets. Let 2 be an open szebset of R" and let tu, di (9, be another boundecl open smbset containing a given puoint xo. Then it holds that

i) for any iE s7b-{.,}(E), a necessaTy and suLfiicient condition foT lim i(Åí)

x-Xo

- .+ =e is that theTe exist ap E 7V" and an FE (80(E) in szeeh a way that T is expTesseel as

T= DPi on tu -- {xo} and i(x) == (X "p X! O)Pe+ 1x --- xol iPi 6(x-xo)

where b E 80(E), 6(x).O (x-ÅrO),

ii) for any T'E s7fo(E), a neeessaTy an(l s2LX7icient eondition for T'(xo)==e is that the expuTession T-'=DPi of i) is valut on tu.

PRooF. The sufficiency of both statements is clear. To prove the necessity

we may assume xo=Oande==O. Take bounded open subsets 9i, 2o such that 2)ni)9i)9o)9o)to, and let P. be tu)P.. Then by Proposition 1 we may

write T'=Dqfii on P.-{O} for some ge7V" and for some F-icesO(E) with it(x)

=lxli4ib(x). LetS be the restriction of T-DqFi on S2i-{O}. Then S=O on .

P. --- {O} and by Proposition A and B, S may be expressed as

.-- .- S= = D'f,

lr L()m+n+1

for some m E Nand f-', E ifOg,.{o}(E). Letting h E 9, Os:h(x)S:1, h(.).={6 ffO.r, .X,Epoj.T,e2(aR'3;--s2,),

one observes on 2i--- {O} that

hS- ---- i,ts;lll+l .+ ihD'1' == ,,,.;ll.] .., ,Z., (- 1)iSi (Z)D'-S((DSh)lr).

Since (DSh)1,==O on {P.i3v(R"-2o)}A(2i:{O})==(P.,3-{O})V(9i-2o), setting O on P.i3v(R"-2i) one may prolong (DSh)f to a function g',, E 80(E). We then construct a function e,, E 80(E) from g-",, by taking m+n+1-ri+si times re- peated indefinite integrals from O with respect to xi for each i---:1, •••, n. Lett-

ing

e=iris]iF.;.., ,IZ.. ('- 1)iSi(g)j,,,

.

we observe G=O on P.i3 and

Dm+n+ii ,== hs- =h( T-'- DqA) on 2i- {O},

---

=T-DqFi on tu-{O}.

Thus T--=Dm+n+ie+Dqfii on tu-{O}. Since U, F-i E esO(E) and since C=O on P.i3 and fii(x)== l x 1 iqib(x) on P., a proper repeated indefinite integration from O will

prove the necessity of the statement i). To show that ii) is necessary, let uts employ F- in i) just proved and write T-'o==DPfiE96(E). Then we may prove T-•= T-o on to just in the same way as given in the last part of the proof (Proof of iii)) of Proposition 1. This completes the proof.

' CoRoLLARy. Let E be a q2Lasi-complete LCS with a fundamental $equence

of bounded subsets. Let 2 and to be open szebsets of R" such that 9) di and suppose tu is bo2Lnded. Let xo be a given point containea in to. Then it holels that

i) if iE 9font{.,}(E) and lim T'(hr)=e, then theTe exists a uniqieely determineal

x-Xo

T-"o E S)6(E) such that T-o(xo)=e and T"---= T-'o on tu -- {xo},

ii) if T-'E9fo(E) and lim TO(Åí)=e, then there exists a 2eniquely deteTmined x-ÅÄXo

io E l76(E) s2ech that T-o(xo)=e and TF' is expTessea as

- --

T= To+ 2 DP6(x-xo)Åq8)ep lplSk

on to foT some kENand ep E E.

PRooF. The statements are almost evident from Theorem 1. The proof is

14 K. YosHiNAGA

briefly outlined as follows. T-' o is given as T' o.=DPi and its unicity may be seen just in the same manner as in the last part of the proof (Proof of iii)) of Pro-

position 1. This completes the proof.

4. Miscellaneouslinits.

We here consider various types of limits of vector-valued distributions and their consequences.

PRoposmoN 2. Let Ebe a quasi-complete LCS with afundamental sequeveee of bounded smbsets and let {Z,} be a given sequence of positive numbeTs such that Z..O (v.oo), 1År Z i"' ÅreÅrO, v=1, 2, •••, wheTe e is a.fixecl eonstant. Let 2be an

p open smbset of R" containing the point xo and let T-E9fo(E). Sztppose A=(ZiD is

an nÅ~n real matrix with detA 7!=O and put AT(A)==J/,lj-l,Z?•i• Suppose fuTtheT

.-e,

t.hat T(xo+ZÅí).-e E E(v-• oo) in s2'(E) and take any boundeel subset B of R". Then zt follows that T(xo+AÅí+IV(A)or).e in 9'(E) 2LnifoTmly with Tespect to yEB when 7V(A)--ÅrO in such a way that IV(A)"==0(detA).

PRooF. We assume, as we may, xo==O and e=O. It is already known by

Proposition 1, iii), that i(ZÅí)--ÅrO (Z--ÅrO) in 9'(E). For any ÅëE9 we write

Åq T(AÅí+N(A)y, Åë(Åí)År = Åq T'(x), Åë(A"i(Åí '--' Ar(A)y)År 1d,l

tAl

-

= Åq T(AT(A)Åí), ÅëA,y(Åí)År,

where Åën,,(x)== 1(\gfÅr"{ Åë(N (A)A-'(x--- pt)). Let ptii be the (i, i) element of A-i.

Then pti7•=:deA iih where Aji is the cofactor of Zji in detA and so 1pti,•lsg;

ind,-t lA) i IV(A)"-'. Therefore 1 IV(A)pt. 1 K lfllgf)A" l (n - 1) ! f{g C(n - 1) !, c being the

gonstant presupposed by Ar(A)"=0(deSA). Hence {ipA,,; Ar(A)"=O(detA), yE B}

is a bounded subset of s7 and thus Åq T(IV(A)Åí), ÅëA,,(Åí)År -ÅrO (Ar(A)-ÅrO) uniformly with respect to such ÅëA,,. This completes the proof.

REMARK 5. IV(Ami) == )/ ,} "= , A?. j/ l det A l and so if .IV(A)" -Åq- C1 detA l , then

Ar(A-i) ggVn2AI(A)2(nmi' (n -- 1) !)2/ I detA1 = 2V(A)"-in!/1 detAl

2!=--!1 1

Sn!,C " ldetAl'ii.

This shows us that Ar(A)"==0(detA) if and only if Ar(Api) =O(detA'i). We also note that as a norm of A, AT(A) is equivalent to the norm 11All=suplAxi.

Ixl=1

We now consider the behaviour of the limit under the change of local coordinates.

PRoposmoN 3. Let E be a quasi-comptete LCS with a fundamental sequence of boundea subsets. Let 9 and 2' be open smbsets of R" and let x'=A(x) be a Coo homeomoTphism of 9 onto 9'. Denote the Jaeobian of A(x) by JA(x) and assume JA(x) 7C=O everywheTe. Given xocS2 we wTite x6 =A(xo). Then it holas that

--

i) if T-'E s2)fatH{.;}(E) and lim T(Åí')=e, then lim T(A(Åí))==e,

tt

x"xo x.xo

ii) if T E gfo t(E) and T(x6) == e, then T(A(xo)) == e.

PRooF. We assume, as we may, xo==nc6==O, e==O, and we only prove i), because ii) is very much similarly done. Letting ÅëE 0Rn-{o} and putting C==

suppÅë(R"-{O}, we choose 6ÅrO so that A(ZC)(2' for any Z, OÅq1ZiÅq6. Then for such Z it holds that

(7) Åq T(A(zÅí)), Åë(te)År -= Åq T(z&r), Åë( lz A-i(zÅí'))IJA-i(zÅí')lÅr.

Setting gbN(x')=:Åë( 1 A"(Zx'))l,JA-i(Zx')l we first prove that the set {gbN; OÅq lZl

Åq6}(eRn-{o} is a bounded subset of 9Rn, To this end let us put C'=

,ÅqY,,Åq, 1 A(ZC) and let us show that C' is a bounded subset of Rn. In terms of coordinates, A is expressed by a system of functions

xl•=ai(x) =ai(xi) •••) xn)) i==17 •••7 n.

Then for any vEC and for any Z, OÅq lRl Åq6, it follows that l ai(zx)=te.,ai.,(ezx)xh oÅqeÅqi,

i

andhence C'(P. for someaÅrO. This proves that suppqN= z A(ZC),OÅqiZlÅq6,

are all contained in a fixed compact set l']'.. That {lDPÅëx(x')l; OÅq 1Z1 Åq6, x' E C'}

is bounded is a consequence of a straightforward calculation, and so we find that {Åëx;OÅqlZiÅq6} isabounded subset of e. Owing to Proposition 1, ii) we may conclude from (7) that Åq T(A(ZÅí)), Åë(Åí)År--ÅrO (Z.O). This proves i) and the proof is completed.

REMARK 6. The value and the limit of the distribution are invariant under

the change of local coordinates of class COe and so one may speak of such notions

on a Coo real manifold.

When a distribution Thas the value at x= xo we have seen that the struc- ture of Tis fairly restrictive in a neighbourhood of xo (Proposition 1), and so a much stronger character of convergence to T(xo) will be expected. In order to obtain an answer to such a question, we denote Px(xo)={x; lxi-xoilÅqZ, i---1, •••, n} for each ZÅrO. Then one may prove

PRoposiTioN 4. Let E be a quasi-complete LCS with a fundamental sequenee of bounded subsets. Let 2 be an open subset of R" and assume xoE9. If

Tc 9fo(E) has the value T(xo)=e, then there exists ap E N" so that foT any family {zx} of xxE9p,(.,), given for su;fiiciently small ZÅrO and satisfying jxx(x)ax .1 (Z-ÅrO+) and D"xx(x)=O( z.1.ipi), it holds that Åq T, xxÅr--Åre (Z--ÅrO+).

PRooF. We assume, as often have done, xo==O, e==O. On account of Theo-

rem 1, taking p E Ar", fi E es O(E), fi(x) = 1 x I iPi b(x) we write T=r- DPfi in a given

open bounded neighbourhood tu, di(2, of O in R". Then we get

Åq ITi, xxÅr=(-1)iPi Åq.fi, DPxxÅr ==(---- 1)iPiÅq 1Åí1 iPi6(Åí), DPxx(Åí)År.

Letting Cx be the convex circled closed envelope of 6(A) we observe that Cx--År {O}

(Z-ÅrO+) and hence it follows that

Åq 1Åí i iPib(Åí), D'xx(te)År = j., lx 1 iPib(x)DPXx(x)ax E (V-E-JRL)iPi z.9tpi (2Z)"Cx =niPi/22"C.C,,

where c is a given constant Ixx(x)l:s{g z.9,p, . This completes the proof.

The next proposition will be the first step towards the study of boundary values of distributions. Let o be an open subset on the surface of the unit sphere IZ= {x; 1xl=1} and for any 6, OÅq6is{g+cx), let us denote r(d; 6)=:Ix;

OÅqlxIÅq6, l:l cC)• xo+r(O; 6) is a cone of length 6 with vertical angle o at the vertex xo.

PRoposmoN 5. Let E be a quasi-compeete LCS with a f2endamental sequence of boundea smbsets and let {Z.} be a given sequence of puositive numbers such that Zv+1

Z,.O (v--Åroo), 1År

z ÅreÅrO, v==1, 2, •••, where e is a fixed constant. If v

TE er(.,s)(E) has lim T(Z.Åí)=O in s7i(.,..)(E), then for any open s2Lbset oo of 2,

p--oo

troÅqo, and for any positive n?LmbeT 6o, 6oÅq6, theTe exist p E IV" and F-' E (S'O(E) so that

-

T=DPfi on r(oo;6o) ancl IF.(F),,.O(x-ÅrO) in E•

-.+ Consequently it holds that T(Zx)-ÅrO (Z--ÅrO+) in 9)$(.,..)(E) and furthermoTe foT any $ubset B(er(.,,..) bounclea in 9 we obtain Åq T(ZÅí), Åë(Åí)År-ÅrO (Z-ÅrO+) uni- foTmly in ÅëEB.

PRooF. Take 6i, 6År6iÅr6o and choose an open subset di of =, o)ai)oi)ao.

Let cr(u) be such an infinitely difEerentiable function defined on :E] that

.(.)-It io,i:[g:l

and let further B(r) be another infinitely differentiable function defined on the positive real numbers with values

B(r)=-I6 ig:O,,Åq.'.fi6o'

Setting

s-(Åí) .. Ia(l lg 1)B(lÅí 1) T(&) for v( r(d; 6),

10 for xer(o;6),

we see S" E efen-{o}(E) and S-' ---- T on T(do; 6o). Given Åë E eRn-{o} we write Åqs'(z.te), Åë(ft)År = Åq iTi(ZvÅí), cr(I2 1)B(Zp l ft l )Åë(Åí)År

and observing that B(Z,1x1)==1 for each sufliciently large v independently of xcsuppÅë, we may see for suchv

af(1 i; i )B(Zp 1 5b 1 )Åë(6b)=af(l `.M 1)Åë( 2) E 9r(a,..)

and thus ÅqS-' (Z.Åí), Åë(th)År.O (v-År oo) may be concluded for any Åë E eRn-{o}. There-

fore by Theorem 1 it follows that there exist pEIV", F'E80(E) and S-=DPF' on . _{r(d;6), IF} ^.( ff,)pt.O(x--ÅrO). The last statement may be seen just in the same

way as in the proof of ii) of Proposition 1. This completes the proof.

CoRoLLARy. Let E be a quasi-complete LCS with a fundamental seqecence of

bounded s2ebsets and let 2 be an open subset of R". Szeppose that xo is a given bozendary point of 3? such that having the veTtex at xo theTe exists at least one cone contained in 2. Let r(oo; 6o) be given ancl asszeme that x'=A(x) is a continu- ous homeomoTphism of 9v{xo} onto T(oo; 6o)v{O} which is at the same time a CeO homeomorphism of 2 onto T(do; 6o) with nowhere vanishing Jaeobian JA(x).

If TE e$(.,,s,)(E) has lim T(ZÅí')=e 6 E in 9$,.,,..)(E), then T(A(fu)) E :2fo(E) has

x-o+

atso ,1.i,{p. i(A(xo+ke))==e in 9)1(.,,..)(E), wheTe oi is an open subset of = elejinea by

Oi =V{O;=)d is open an(Z 2)xo+T(o; 6) for some 6ÅrO}.

PRooF. The proof may be carried out just in the same way as that of Pro- position 3 and will be omitted.

In giving the definition of the value of a distribution T, the distributional

limit of i(xo+ZÅí) was supposed to be existent and moreover to be a constant distribution, lim T(xo+me)=e E E. According to Proposition 2 it is not difficult

x-o+

to see that if Tc gfo(E) has a constant distributional limit lim T(xo+ZÅí)==e in

Xeo+

s2'(E), then it follows that lim T(xo+Z.Åí+ pt,) = e for any sequence of numbers V-oo

{Z,},O=7!Z.-ÅrO (v.oo) and for any sequence of points {pt.} such that(-i:'tl is a bounded subset of R". The next proposition will give a converse of this state- ment.

PRoposmoN 6. Let Ebe a quasi-complete LCS and let 9 be an open s2ebset

- .-

of R". SzLppose xo e S2, TE S2fo(E), S E 9r(E) and suppose a sequence of numbeTs

-- {Z.}, O 7! Z.--ÅrO (v.oo), be given. If lim T(xo+Z.hi+ or,) =S(Åí) in 9'(E) foT any

p-}oo

sequence {or.} of points of R" such that or,4O, v==1, 2, •••, ana (-ill'?] isaboundea

v

siebset of R", then S'(Åí) is a constant elistTib2Ltion S"(te)==e EE and it holds that lim(xo+ZÅí) ==e in g'(E), i.e., T(xo)=e.

x-o

PRooF. We may assume xo = O. Observation of the identity bl(Zp(fu + or)+ ptv)= T(ZvÅí+(Zvor+ ptv))

.- - shows us that S(di+ cy)==S(fo) for any yER" which proves that for any e'EE the scalar-valued distribution ÅqS' (Åí), e'År is a constant distribution: ÅqS(a), e'År

=e(e'). Since e(e') becomes a continuous linear form on E2 it follows that e E (E2)'==E, i.e., S-(Åí)=e. We next show lim di(ZÅí)==e and to this end we assume

N-,O

e=:O. LetÅë(s7 be given and we write ,

T(Z.di), Åë(Åí)År = Åq T(Z,fc)- T(Z,te+ pt,), Åë(Åí)År + Åq T(Z,Åí+.y,), Åë(fu)År == Åq T(Z,Åí+ y.), Åë(it+ zcr" )-Åë(th)År+Åq Ti(Z.Åí+ pt.), Åë(iv)År,

v

where {or,} is any sequence of points such that or. =7kO and I-zZ:) is a bounded subset of R". The first term of the last member converges to O as v-År oo because IÅë(te+ zor" )-Åë(Åí)l is a bounded subset of 9. Therefore Åqdi(Z.Åí), Åë(i)År-,,O (v-År oo)

p

is obtained and the proof is completed.

5. Furtherpropertiesandapplications.

In order to obtain a further information about the structure of a distribu- tion in a neighbourhood of a point where the distribution has its value, we prove the following lemma owed originally to Lojasiewicz [6].

LEMMA 3. Let Ebe a qzLasi-complete LCS and let TE s2fo(E) wheTe S? is an open s2ebset of R". Szepupose a seq2Lence of n2Lmbers {R.}, 04Z,-ÅrO (v-Åroo), a f2Lnetion Åë E :0, SÅë(x)dx=1 a7zcl anotheT open szLbset co of R" are given so that ib+

Z. suppÅë( S2 foT each v=1, 2, • • •. Letf2LTther C be the convex ciTclecl closeel envelope of {Åq T(Åí), Åë,(Åí-or)År; v==1, 2, •••, ptEdi} wheTe Åë,(ac)= lÅíl.Åë( f, )• Then it

holas that

Åq T, aÅr cV2(jlcr(x)lax)C

foT eveTy a E s7., and if in paTticielaT a is Teal-val2Leel this may be Tecl2Lee(Z to

Åq T, aÅr e(S1cr(x)1ax)C.

N PRooF. We divide the proof into three steps.

i) In case a is nonnegative, we may assume Sa(x)ax = 1. Since cr E s2.

there existsa yoEIV so that Åë,*aEs2År. for every v2})yo. Letting z(x) be the characteristic function of suppa we may observe for every v that the convex

-. circled closed envelope K, of the set {z(or)Åq T(Åí), Åë,(di -- or)År; orE to} is a bounded

complete and even compact subset of C. Thus for each v, vÅrinvo, we get by [3,

20 K. YosHINAGA

p. 14, Proposition 8] that

Åq T,Åë,*aÅr=SÅq T(Åí), Åë.(Åí- or)Åra( y)a)

c (Sa( or)aor)K. ( c.

On the other hand, because of Åë,*af-Årcr (v-Åroo) in g. it holds that Åq T, Åë,*evÅr.

Åq T, aÅr (v.oo) and therefore Åq T, crÅr EC as desired.

ii) In caseaisa real-valued function we proceed as follows. To begin with, takesÅrO so that the e-neighbourhood of suppcu is contained in tu. Then, according to a lemma of [15], a is decomposed as follows: a =a'-ev" where cr', a" are nonnegative functions of e with supports contained in the e-neighbour- hood of suppa, hence cr', a" E 9. and furthermore OmÅqa'-cr'=cr"-a--Åqs, a" = sup(a, O) and a-=:sup(-a, O). Therefore one obtains

Åq T, aÅr == Åq Z cr'År-Åq T, cr"År E (ja'(x)ax+Sa"(x)ax)C

( (j(a'(x)+ e) ax + j(a -(x) + e) ax)C( (S l cr(x) 1 ax + 2ES dx)C,

where A, ==(suppcr')v(suppa"). Letting s-ÅrO+ one gets ÅqT, aÅrE(j1a(x)l dx)C as desired.

iii) In case a is a complex-valued function we write a=cri+ia2 where ai, a2 are real-valued functions of s2.. Then

Åq T, aÅr =Åq 7I, criÅr+iÅq T, a2År E(S(lcri(x)1+1a2(x)1)ax)C ( V2(I 1 ev(x) 1 ax)c.

This completes the proof.

Let E be a given LCS. According to A. Grothendieck [4], a sequence {x.}

in E is said to converge in the Mackey sense to O if there exists a sequence {Z,}, Z,-ÅrO (v.cxo), of positive numbers such that the sequence I-Xz:-K-l is a bounded v

subset of E. Eis said to satisfy the Mackey condetion if every sequence con- vergent to O converges also to O in the Mackey sense. For any bounded closed convex circled subset A of E let us denote by EA the normed space generated by A with the norm llxllA = inf lZl, x E EA. Eis said to satisfy the stTict Mackey

xEXA

condition if for every bounded subset A of E, there exists a bounded closed

convex circled subset B)A such that the topology on A induced by E is identi- cal with the topology on A induced by EB. E is said to be qzeasi-noTmable if for any equicontinuous subset A of E' there exists a neighbourhood V of O in E such that the topology induced on A by the strong topology of E' is identical with the topology on A induced by Ee..

We now prove the following

THEoREM 2. Let Fbe a metTizable quasi-normable LCS with a co2Lntable clense szebset and let E be the strong cluat Ft of F. Szeppose that 9 is an open s2Lbset of R", xoE9, and assume that a sequence of nzembeTs {Z.}, O=ÅrFZ,.O (v.oo) is given. ThenfoT any TE9fo(E) thefollowing statements i), ii) aTe equivalent:

.

i) theTe exists lim T(xo+Z,Åí+ or) in e'(E), ;::

.

ii) on some open neighbourhooel to of xo, T is ulentijied 2vith a scalaTly locaZly integTable funetion 7(x) taking valiees in EpTovielecl with the weak topology

--

c(E, F), and strongly continuous at x==xo. in this case T has the value f(xo) at x =Xo.

PRooF. To begin with let us observe the following circumstances. By our assumption that F is metrizable and quasi-normable, it follows that its strong

dual E is a complete LCS with a fundamental sequence of bounded subsets and satisfies the strict Mackey and hence Mackey condition as well [4]. Moreover any metrizable LCS is bornological and thus quasi-tonne16 ( =infratonne16 [2, p. 13, Ex. 12)]) [4], i.e., every bounded subset of its strong dual is equicontinu- ous. Therefore the bounded subsets of E is identical with the equicontinuous subsets of F'. Letting xo==O we now proceed as follows.

Ad i)--Årii). By virtue of Proposition 6 we first note that lim T(Z,Åí+ y) is a y:rr

constant distribution which may be assumed to be O. Take any fixed Åëce, SÅë(x)ax==1, and set Åë.(x)=lzl.Åë( zX, ). Then we get

--

(8) Åq T(Åí), Åë.(Åí--- Åre)År :Åq T(Z,te+ Årc), Åë(k)År.O (v---Åroo, y`-ÅrO).

Let us take eÅrO so small that {x; lx1Ke}+Z,suppÅë (2 for any v, -!L f{gvE N,

E -

V,,s={Åq T(fu), Åë.(Åí- or)År; vSI pt l yl Åq6}

for any pair (v, 6), OÅq6Åqs, -l-Kv. The filter di on E generated by these {v,,s}

has a countable base and by (8) one finds that di is convergent toO in E. Accord-

ing to Grothendieck [4] it then follows that there exists at least one of {V,,s},

say V,,s, bounded in E. Letting C == C.,s be the convex circled weakly (==o(E, F)) closed envelope of this V,,s and setting to==tus={x; lxIÅq6} one obtains by Lemma 3 that

(9) ÅqZ aÅr cV2(Slaf(x)l ax)C

. for any a c 9,,. By the fact that C is a bounded subset this proves TE Y(9g; E) and therefore Tis a vector-valued measure on to with values in E. Since C is a convex circled weakly closed equicontinuous subset of F' ==E and since F has a countable dense subset, Theorem of Dunford-Pettis [3, p. 46, Corollaire 3 of Th6orbme 1] tells us that T has a density on tu, i.e., there exists a function f(x)=fs(x) defined on to=:tos with values in Eequipped with the weak topology c(E, F), scalarly locally integrable and satisfying

Åq T, aÅr =Scr(x)1(sc)dx w

for any cr E 9.. At the same time, owing to the very same theorem, the expres-

-- -- -+

sion (9) shows us that f(to)(V2C. Clearly it holds that fs= fs,=Ton each tus,, OÅq6'Åq6, and on account of (8) we get C.,,st-År {O} (v'-), oo, 6t--ÅrO). Thus by chang-

--

ing values if necessary and putting f(O) =O we may conclude that f is continu- ous at x=O.

Ad ii).i). We assume 7(O)=O. Take any aEs7 and choose a positive number s so that Z,suppa+{x; Ixl s{{;E}(tu for every v;}}i!. Let further As, e

OÅq6Åqe, be the convex circled weakly (=:o(E, F)) closed envelope of the set If(Z,x+ or); v;}) -3}, I orl Åq6, x E suppa]. Then again by Grothendieck [4] cited

above we find that some As is a bounded subset of E, because by assumption of

.

the continuity off at x :O it follows that As--År{O} (6.0). Consequently there exists a 6ÅrO such that each Ast, OÅq6'Åq6, is an equicontinuous and hence weakly compact subset of E. For any such Ast we may infer that

j7(Z,A; + cy)a(x)dx c Astj 1 cr(x)l ax,

for every y, y2}i:-ill, , and or, l orI Åq6' [3, p. 12, Corollaire of Proposition 5]. Thus

ÅqT(z.Åí+or), a(a)År-O (v--Åroo, y.O), i.e., limT(z,te+y)=O=7(O). This com- ;:x

p}etes the proof.

In a previous article [13] the present author has considered the Silva space.

An LCS E is called a Silva space if it is an inductive limit of an increasing se- quence of normed spaces {Ek} such that each injection Ele.Ele.i is compact. Such a space is characterized, among others, as a Montel space of type (DF) satisfy- ing the strict Mackey condition. Hence a Silva space is a reflexive space and its strong dual is a Schwartz (F) spaee. Thus we may state the following CoRoLLARy. Let E be a Silva space and let S2 be an open siebset of R", xo E S2.

Then for any TE 9k(E) the following statements i), ii) aTe equivalent:

i) There exists lim T(xo+Zte+ or) in gi(E), }:8

ii) on some open neighbourhooa tu of xo, T is identified with a sealarly locaZZy integrablefzLnctionfwith values in E, continzeozes at x==xo. In this case Thas the value f(xo) at x=xo.

PRooF. Setting F==Et one knows that F is a Schwartz (F) space and hence metrizable and quasi-normable [13]. Furthermore since F is a Montel (F) space, byatheorem of Dieudonn6 [5, p. 373] it has a countable dense subset. This completes the proof.

In order to study the fixation and the limit of Lojasiewicz [7] the present

theory may be applied as follows. Let 9 be an open subset of the space RMÅ~Rn of points (x, pt) and let it be supposed that a point xoERM and an open subset tu(R" are given so that {xo}Å~to(2. Given Tc9fo it is said that one may .fix x=xo on co in T or T has a seetion on (D at x== uo if there exists an S(5År)E96 such that lim T(xo+ZÅí, S)== S(S) in 9fem... This means that

N-O+

(10) Åq T(xo+me, 5År), x(fi, 5År)År = Åq T(N, 5År), -Jil.r x( " z

x -- xo, o) År

--År Åq S(s), Sx(x,'s) ax År (z -År o + ),

for any xEeRm... S(0)== T(xo, 5År) is called the section of T on co at x==`vo.

Given Tc9b-({.,}.Rn) we say that Thas a limit as x-Årxo on tu if there exists an S(0) E ED6 such that limT(xo+Z&, SÅr) =S(5År) in 9(Rm-{,}).., i.e. (10) is true for x-"o+

any x E :2År(Rm-{o}).. and S(S)=lim T(Åí, 0) is said to be the limit of T on to at x=xo.

x-Xo Given two distributions E efo(resp. E 9fo-({.,}.Rn)) if they are identical on cDiÅ~ co

(resp. on (tui-({xo} Å~ R")) Å~ tu), toi being an open neighbourhood of xo, and if one

of them has the section on tu at x==xo (resp. the limit on tu at x=xo), then the

other has also the same section (resp. the same limit), i.e., the fixation and the

limit have the local character. When T has the sections both on tu and on tu'

at x==xo, Thas also the section on tovtu' at x==xo. Hence there exists a maxi-

mal open subset to, {xo} Å~ to(2, of R" on which Thas the section. Consequently

24 K. YosHINAGA

the inquiry about the fixation may be reduced to the case where tu is a bounded open subset of R" such that {xo} Å~ to(2. Precisely speaking, take an arbitrary h E e2 which is identically equal to 1 on a neighbourhood of {xo} Å~ to contained in S?. Then the problem will turn out to consider hTEYE(sDRm; 9)k)==9fem(9k) where K is a suitably chosen compact subset of R" containing di and 9k is an LCS composed of the distributions in 9fen with supports contained in K and provided with the topology induced by s7fe,t. Then it is not difficult to see that 9k is a closed subspace of the Silva space 8fert and hence a Silva space as well.

We obtain therefore hT E g2fem(ek)( g7fem(8fen) and all the consequences thus far obtained about the value will be applicable. As for the limit the situation will

be quite similar.

References

[1] N. BouRBAKr, EsPaces vectoriels toPotogigues, ChaP. I, II, Actualit6s Sci. Ind., no. 1189 (1953), Paris,

Hermann.

[2] , Espaces vectoriels tePologieues, ChaP. III, IV, V, Actualites Sci. Ind., no. 1229 (1955), Paris,

Hermann.

[3] , Jntegration, ChaP. 6, Actualites Sci. Ind., no. 1281 (1959), Paris, Hermann,

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[5] G, KOTHE, ToPologische lineare Rdume I, Springer, Berlin.G6ttingen.Heidelberg, 1960.

[6] S. LoJAsiEwrcz, Sar la valeur et la limite d'une distribution en unPoint, Studia Math., vol. 16 (1957) 1-36.

[7] , Sar lajxation des variabtes dans une distribution, Studia Math., vol. 17 (1958) 1-64.

[8] L. ScHwARTz, EsPaces de fonctions dtll71grentiables d valeurs vectorielles,J, Analyse Math., vol. 4 (1954-

56) 88-148.

[9] , The'orie des distributions I, Actualit6s Sci. Ind., no. 1245 (1957), Paris, Hermann.

[10] , The'orie des distribution II, Actualit6s Sci. Ind., no. 1122 (1959), Paris, Hermann.

[11] , The'orie des distributions a' valeurs vectorielles, ChaP. I, Ann, Inst. Fourier Grenoble, vol. 7 (1957) 1-139.

[12] , The'orie des distributions a' valeurs vectorielles, ChaP, U, Ann. Inst. Fourier Grenoble, vol. 8 (1958) 1-209.

[13] K. YosHiNAGA, On a locall2 convex space introduced b" J.S.E Silva, J. Sci. Hiroshima Univ. Ser. A, vol. 21 (1957) 89-98.

[14] , Values of vector-valued distributions and smoothness of semi-grouP distributions, Bull, Kyushu Inst, Tech. Math. Nat. Sci., no. 12 (1965) 1-27.

[15] , A note on the value and the multiPlicative Product of distributions, Bull. Kyushu Inst. Tech.

Math. Nat. Sci., no. 13 (1966) 5-13.

Department of Mathematics

Ky2Lsh2e institute of Technology