Bull. Kyushu Inst. Tcch.(M, Se N. S.År No, 9, 1962.

ON SPACES OF FkNITE PARTS OF HO"IOGEATEOUS FUNCTIO!"S

By Ky6iehi YosHiNAGA (Received Nov. 31, I96!)

Let P be a Iinear partial differential operator on R" with smooth eoeMeients

and of homogeneous order rn. Then P--FAM where A=-211t-V-d and Fisa

singular integral operator [3]. Hence partieular studies of such operators Fhave been made in their conneetion with the operator A [3], [4], [5]. Thekernel of the operator F is given by F(x, x--g): F(g)--- ! F(x, x-})rp(g)clg where

F(x, ij) is a distribution expressed as

F(x, e)==c(x)6(g)+vtp. f(x, g)•

Here c(.x) and f(x, e) are m times boundedly continuously differentiable func-tions with respect to nc, S(g) is the Dirac measuTe relative to g and v.p. denotes ethe Cauchy's principal value in e. The existenee of vlp. f(x, g) is assured by

the assumption that f(.z•, g) is in gER"- {O} a function, infinitely continuouslydifferentiable, homogeneous of degree -n and with spherical mean O. The spaceof sueh distributions is denoted by g3'M(S) [11]. If c(c) and f(rc, g) are aetually

independent of x, then F(rp) is easily computed by taking Fourier transforms.Distributions of sueh a partieular type constitute a nuclear spaee :EL] of type (F).Then we may observe that {IM(K-")=:;IYM Ci5 'LSt , where g3'M is the space of ,n times

rteontinuously differentiable funetions on R" with bounded derivatives. Stimulated by these investigations, the author of the present paper hasexamined a generalization of the space of kernels of such singular integra!operators. Let E be any quasi-complete locally eonvex Hausdorff space, and letZx(E) be the set of functions defined on R"-{O} with values in E, infinitelycontinuously differentiable and homogeneous of degree x, where N is any com-plex number. In ease that E is the complex number field, we write E]A in placeOf t=k(E). Then E'iA is a nuclear space of type (F) and we may infer that:!]K(E)=År]x 6E for any E. Furthermore if in particular E is a space of type (F)we obtain IXk(E)='.}EIN (l]i E. Hence in such a ease various properties of elements

rtof S=N(E) may be derived from those of elements of X and E. Section 1 is devoted to the study of ];!.]A and of Pf. 2]A, the spaee of finite

1

parts of functions in 2-]x. We shall prove by means of the Fourier transforma-tion that Pf,tL2:A is isomorphie with Pf.';t -(..,), if )L =7C h and ]y 7! -(n+h) for any

integerh[l}iO, As for the case x=h, Pf.2],, is transformed into t;J-(.+h)(Pf•Z-(n.h)• We sha}1 investigate the structure of z]!(..h) and show that it isisomorphic with År-]h. Thus in particular we obtain NL.]I.=t=. In Section 2 weshallconsider the space År.:J',(E). We first prove that X-',(E)=:'Årt:A8E• Then de-fining the finite part of the vector valued function in t-'x(E), and denoting the

set eonsisting of them by Pf. X',(E) we may prove that Pf.N-vL. 'A(E) is mappedonto Pf. 2]..(.+A)(E) by the Fourier transformation provided that Nl/t and N=i:e=-

-(n+h) for any integer hl;2 O. The argument for the singular cases Nt ==h orX==-(n+h) is quite similar to that in Section 1. Applying the preceding re-sults to cases N=Oandx=-n we shall study singular integral operators inSection 3. There we shall make use of the fact that SllM(S;])== NLj!. 'Q's)N @M. Then

rtvarious problems concerning the continuity of singular integral operators intheir conneetion with the operator A rnay be reduced to the next type: what

eou}d be said about the continuity of the operator T,((p)=s* (b Z:T.li) -b(s *g.",t"',-.),

where sE 2]l. and bE S3'M are preserjbed elements. We shall give an answer byProposition 7, so to speak a lemma of Friedrichs' type if we may use such aneipression [6]. Thus the rest of this seetien rnay be seen as a mere applica-tion of Propesition 7.

Notations and terminologies of this paper are essentially those of [1] (loeal-ly convex spaees) and those of [12], [15] (distributions).

1. Spaces tL]A and Pf. t;n. Let R" be the real Euclidean n-space and let Nbe any eomplex number. A distribution TE LY" is called ho7nogeneous of degreeh ifÅqT, q)(r'-ix)År==r"'"ÅqT, gÅr foy any q)E.EL,P and for anyrÅrO. In case that 1'=f

is a locally integrable function, the said condition is redueed to that for any

rÅrO it holds that f(rx)=r" f(.x) almost everywhere on R". Thus a continuousfunetion on R" is homogeneous of degree O if and only if it is a constant, and aninfinitely eontinuously differentiable function on R'; might be homogeneous ofdegree N only if x is a non-negative integer. According to [14] we write

(f'`(X)=m,,iJ;T, Si.,i-, q)(ixl-x')d.v'

for any continuous funetion fp, where (c. is the surface area of the unit spherelx'i==1. (plt is called the spherical meanof rp. Then it isnotdiffieultto see that

q,.rp# isacontinuous linear applieation of 9 into :2r [14]. Therefore its ad-joint, denoted by the same symbol "ig", may be defined by

ÅqTlj, g,År=ÅqT, (piÅr for TE9', tpE2.

On Spaces of Finite Parts ef Hemogeneous Functions 3

Since rpk may be expressed as the mean of g) with respect to the retation groupon R", it is not diffieult to see that IA'tr--TAtr for any TEY'. Here and in the fol-

lowing sections of this paper we often make use of ir to denote the Fouriertransferm ,va`;(T)(Jr) = ! T(.x•)e'2rt+ `Åqrt-vÅr clx.

Let K-'i, be the set of funetions f(-x') defined on R"-{O}, infinitely continu-

ously differentiable and homogeneous of degree x. It is a space of type (F)provided with the subspace topology of 8(R"- {O}). For any integer hl2 0, wewrite

';t 9,={f; fE År.;h and (D"f)lt=:O for any p, lp1 ==h}

and

2.l9(n+h)={f; fE 2.].(n+hÅr and (xPf)"=O for any p, lpl =h},

where p--(pi, •-t, p.), lp1 =pi +•--+p., and

alpl DPf=== exfi... e...p.n f, xP=xei-••xp.".

For any fE Zx the .tinite part Pf.foff is defined in the following way [7, pp.367-389]. Letting q)E9 and putting

" (')== !,.,, . , f(x') ev (rx') d.r'

we write

ÅqPf.f, q,År-1,i-u,} II[I"r""N-' it.(r)dr+tS"g t) iliCl(2).:.Ij E"'""'l,

where Jt is any non-negative integer such that --1ÅqSnx+n+Jt. This expressionmay have no meaning if x is a negative integer of the form x= -- (n+Z). Insueh a case --,-,tF,L'x-"-llil-i must be replaeed by log Lc. Thus for sirnplicity's sake let

us agree to think of i-O as logsif such is the ease. Here we remark that

it,(')(e) = ,tli}) ,(p(P)(O) !,.,., f(.v)a•P cl;v.

As an immediate eonsequence of the definition we obtain

PRoposiTioN 1. Let VrE8(R'") be homogeneous. Then it helds that vrPf.f=Pf.ilnffor any fE X-',t.

PRooF. We first note that the degree lt of the homogeneity of ilr is a non-

negative integer. Letting opE:i}r, we write

ÅqvnPf•f, opÅr= ÅqPf•f, ff,yrÅr == 1,i-m., IIrr"'A'-i u(r) dr +te-i zt-(,Y212-(.-Ol i) e""A"' l

and

ÅqPf•ilrf, q)År=ll'-m, II:r"'"'"di v(r)dr+te6 t-! (,, ll(:'(l9}, .7) E""'""`l ,

where

"(') == j,.,., f(X) Yr (rX) op (rx) d-x =rh !,.,., f(x) ilr (x) (p (rx) dx= r" v(r) .

Then sinee we have

O for OSIsgh-1, ( U{')(O)=iTaJl.i,)fT- v(idi:)(O) for h sgl,

it is not diMcult to see that ÅqvrPf.f, q,År =ÅqPf.vrf, rpÅr. This completes the proof.

PRoposmoN 2. it holcls that (Pf.f)4=Pf.f" for amJfEi 12]A.

PRooF. Letting g)E:2) and putting

q)4(•r)=-EII.- !,.,, .., rp(lx1x') dx'=h(Ix1 ), f= S,.,., f(x) cix,

we obtain f"(x)= 1 lxl"f, and thus

e)n I,.,-, f(X) ff" ('x) dx --h(r)f== I,.,=, f' (x) q,(rx) ax.

Therefore it yields that

Åq(Pf.f)", rpÅr==ÅqPf.f, q)"År=ÅqPf.fh, q,År

for any vJE2. This completes the proof. It is not difficult to see that Pf,f is a distribution E.Y" and in case thatxs! -(n+h), h=O, 1, 2,•••, Pf.f is homogeneous of degree x. Although Pf.f=f forfE t;A if Sl)LÅr -n, yet in general the set of distributions Pf.f, fE! !2],, should be

denoted by Pf. :E]K. The topology of Pf.2K is given by that of =A. For the caseX= -(n+h.), h=O, 1, 2,-i•, we obtain

On Spaces of Finitc Parts ofHomogcneous Functions 5

PRoposmo:N 3. LetfEÅr.].(.+h). Then Pf.fis homogeneous of degree -(n+h)if and oney iffE Z]9cn+h)•

PRooF. Letting g)E:2tr and letting

U (r) --- !i.,-, f(x) q) (rx) dx

we obtain ÅqPf•f, op (t-i x)År=l,ith II:r-h" u(t-'r)dr +tL: l`ii-"('l',(2)l)E-h" + t'h iUil,h'(O) log el

=t'h l3'tla IIr-,,r-h'iu(r)dr +2,ii i:,?21'SO, ).--i)- (t-iE)'-'t"-E- !LiSl'!l',1,IO-/ 1og (t'-iE)l

+t-h .ut71 Si O)- log t

=t'-"I Åqpf.f, (pÅr + `lt(-hzl(!O-)i log tl

Thus Pf.f is homogeneous of degree --(n+h) if and only if u(h)(O)=O for any

{pE 2r• This means !,.i.,f(x)xP dx=O for any p, 1p1 ---Jt, that is (fxP)'==Ofor any

p, lpl=lt. This eompletes the proof. We now prove the next lemma for later use.

LEMrfA 1. Let TELS,P' be a fttnction on R"-{O}, in,tinitely continzcously dif-ferentiable and homogeneous of degree x, Then its Fourier transform T is also afunction on R"- {O}, infinitely continuously di.fferentiable and homogeneems of

degree -(n+X).

PRooF, For any {p E :2F we obtain

Åqlr, q,(r-' )t)År=ÅqIT', r" q")(rx)År=r'" ÅqT, (7,År-

Therefore T is homogeneous of degree -(n+N). To see that T is infinite}y eon-tinuously differentiable on R"- {O} it is enough to show that dhT is a eontinuousfunetion on R" -- {O} for any k [12, II, p, 47]. For this purpose let lt E9 be

(1 for lxlSl, li(X)=io for lxl22,

and let l be any integer :;}l -l- (2k+1+n+S", ). Then by the theorem of Paley-

Wiener we get ,v".' (di 1 .x l 2" JiT) E es.

On the other hand we get a'1x[2k(1-h) TELi, since there exists a positive con-

stant c sueh that

1 `t' e .x' 1 2k (1 -h) T(.r) 1 gc ixl d2"2k'StX for 1x [ ÅrH 2.

Thus teR='(d'Lx[2k(1-h)T) are continuous function. Therefere j`"k'(d'1.x12" T.) and

hence 1ptl2tak T are continuous functions on R". This completes the proof.

We now prove the following theorem.

TiiEoRE:T 1. Let )L be a complex nz"mber $iLch that N4h ancl )t iE --(m+h) for

any integer hÅrHO. Then .:"k2' is a topological isomorphism of Pf.ZK onto Pf•IZ.(n+x)+

PRooF. It is enough to eonsider the case E,JlxÅr -n, because if Sl)Ls{g --n thenthe problem is reduced to the ease S{(-(n+N))i)O concerning the inverse :iv-i.We first prove that fE!Pf.=.(..N} for anyf(!Årl]k It is knevvn by Lemma 1that f is identieal with an element gE tj.(..k} on R" -- {O}. The.refore f and Pf.gcan differ only at the origin of R';. Hence f-Pf,g--ÅrlcpDPS and thus wehavef-:`R"' (Pf.g)=S--"cp(--2nt7)b. The left side of this identity is a homoge-neous distribution of degree N. Therefore cp =O exeeptp with lpl=N, and thelatter is excluded by the assumption xSli. Hence we get f=Pf.g as desired.Quite similarly we may prove that for any gE2.(.+x) there exists an fEXxsueh that f-----Pf,g. Thus we see that ,;`R' is a linear isomorphism of År]ft onto

Pf.NL]-(..a). To see that this is topo!ogical we need only to prove that it is con-tinuous because '.Sl-'K and Pf.'it d(..A) are both spaces of type (F). The closed

graph theorem [9, Chap. I, p. 16] enables us to reduee the problem to show thatif fk.O in Eli and fk.Pf.g in Pf.tL:-(..u, then g=O. This is shown as fol-lows. Sinee it holds that fk.O and fk.Pf.g in LS7P', we may obtain for any

qEY ÅqPf.g, q)År=lim Åqfk, qJÅr=1im Åqfk, q")År---O, k-• op k- oethat is, Pf.g=O. This eompletes the proof. For the case of singular values of X: X=lz or X=-(n+lt), we first prove the

next theorem.

Ti-TEoREM 2. Let h be anzl non-negat`ive integer. Then t`" is a topologieal iso-

morphism of 2L]2 onto Pf.=O-(..J,).

PRooF. We begin the proof by showing fEi Pf.20-(..fi} for anyfEZ2. Itis known by Lemma 1 that f==gES--i.(..ft} on R"-{O}. Then since f and Pf.gcan differ only at the origin of R", we may write f-Pf.g= ]X cp DP 6, and thus

Åqf, q,(r-iy)År-ÅqPf.g, a,(r'-i y)År--= )'z]" cp ÅqD' 6, q,(r-'y)År

On Spaces of Finitc Parts of Homogeneous Fu=ctions 7

for any qpE:i). Using an identity obtained in the proef of Proposition 3 we

get

Åqf, opÅr-Åqpf.g, q)År- "(h ;,IO) log r=::E] cp(-1)iPi rh-L" rp('}(o)

for anygE9,whereu(r)=I,.,.,g(x)op(rx)dx. It now follows first that cp=O

for any p, lpl 7E h, and next that u(h)(O)=O for any rpE.si). This means

I,.,.,g(x)xP d-x =O fOr anY p, Ip1 ---h- Consequently we getgEZO-(..h) and f=

Pf•g+i ,1:.;, cp DP 5• Henee we may write

f=-e-i (Pf.g) t,:zii i,cp(-2ri)h xP,

and thus for any p, lpi=h, we see

DP f= ."=" ((27ri)h yP Pf.g) +cp(-2rti)h p!,

where p ! --pi !- - • p. !. Therefore it follows that

(DP f)i=(2 7ti)" .:a,='" f' (Pf.(yP g)') + cp ( -- 2;T, i)h p !,

since b, is commutative with -a'-i and Pf. is commutative with )fP and.ta,. Thus

we may conclude that cp=O, because (DPf)i=O and (yPg)"=O, This proves/7EÅr:,if5,il}.i-gC./7ite"jZe,p,t,,ea,`tC:,"TV,g/,,Ss?gi,ggeifill:O[ili',V'i?,igEeh,n:o:u,f/i,gfte'siiWs,:},kht?iD,'2-t/L

onto Pf.Nie-(..h). That !`r is topological may be seen just in the same way as in

the proof of Theorem l. This completes the proof. . In order to cievelop furtheT arguments eoncerning the spaces =h and:g'.'k:getfia"6Gge,Zhi3.".I?.ki:",iedZf,;,h,eiS?lxll/•,rY.)"..gp",O.`da`]kO,"S,,-,.Iir,P,tblh,:e,t,h,2S,2tdqf-

stributions of the form K"' cp DP 8. Then ilh, ll.-Åq..i,) may be considered as com-

IPI-hplex Euelidean ( :"! -`(}li-:--ii/!-! -spaees, and ll.-(..h) is the Fourier transform of nht

Here we should like to show that the set of distributions Pf,t;O.-(.+h) and-ll-(..,,)have no element in common except O. In fact let Pf.f=it--, i,cp DP S forfE L'O.(.+h)•

Then for any epeE) such that op(.v)=O on a neighbourhood of the origin IxiÅqS,

we obtain

I,.,., f('V) q' (X) d'V =' ÅqPf'f, wÅr=Åq, tf, i`p D" 6, opÅr=O•

Henee it follows that f(x)=O for l.xlÅrS, and sinee 6ÅrO is arbitrary it yieldsthat Pf.f=O as desired. Therefore we may eonsider that Pf•t;O.c.+h)+i7-Åq.+h)is a topological direct sum whieh we shall denote by El'L(..i,). We now state

the next

Ti-iEoREif 3. :"R:' is a topotogica,l i,somorphism of t;fi onto ';t !(n.h)•

PReoF. For anyfE 2],, we ebtain f=gE't p(..,,) on R"-{O}. Then just inthe same manner as in the proof of Theorem 2 we rnay conclude that gE 2]9(.+h)and f-Pf+g=ijll/i?icp DP 6, that is fE2]!(..h)• Let conversely T==Pf•g-ITpli.;, cpDP8

with g E ZO- (..h). Then we get

,)t:`:. -i(T)=.ea-i (Pf.g)+ E] c,(-2"i.x)'E Z2 + llh ( 'St h ,

1pl-I:

that is TE,te,e(X2'h). Therefore we have proved that ,;`ZR'(E]h)==tA'i.(n.h). By means

of Theorem 2 it is not diffieult to see that tcr," "' is a continuous application of

E]!(n+h) onto t;h. Hence -er is a topological isomorphism of X--'h onto X;:-(..h).

This eompletes the proof.

CoRoLLARy. It hol(ls that =h=S--TX+]7h in the sense of tepotogical (lirect

sum. PRooF. It is suMeient to remark that Pf.Ze.(..h)+rr-(..h) is a topologieal

directsum. Thiseompletestheproof.

2• Spaees S-SIK(E) and Pf.E]x(E). Let Ebe any quasi-complete loeallyeonvex Hausdorff spaee. A vector valued distribution T on R" with values in E:TE 2T'(E), is ealled homogeneous of degree x if the scalar valued distributionÅqT, 'e'År is homogeneous of degree N for any 1'EE'. Let Z;x(E) be the set ofvector valued functions f defined on R"-{O} with values in E, infinitely con-tinuously difEerentiable and homogeneous of degree N. It is a locally convexHausdorff space provided with the subspace topology of 8(R"- {O} ; E) [13]. We

first show that

PRoposiTJoNr 4. ZA (E)= El.]A SE.

PRooF. Sinee S-J'K is a closed subspace of e(R"- {O}), it is a eomplete Montelspace. Therefore its strong dual S.,'.]'x eoincides with ('lt S), and henee we have

-t,sE=29,('=t KiE). Thus the statement is proved if we show that N";',(E)=.9,(År-]S; E). Let i be the injection of )IE]A into ff(R"- {O}. Then its adjoint 'i is

a continuous linear application of 8'(R"-{O}) onto År[]K. We first prove thatany .7El' (Åí'x; E) may be considered as an element of År]k(E). For this purpose

we put

On Spaccs of Finite Parts of Homegeneeus Functions 9

f(x) =f(ti (s(.))), x E ,Rn - {o},

where 6(.) =T.S and 7(•) on the right side means the image in E by the linear

application f. Since it holds that

Åq'i S(rx), fÅr =f(Tx)=rKf(.r)=rk Åq'i S(r), fÅr

-- .- --for any f(! S-:K and rÅrO, we have 'i 6(,.)=rXti S(.). Thus .f'(rx) ==rX f(x), that is, f(sc)is a homogeneous function of degree )L defined on R"-{O} with values in E. Toprove that 7(x) is infinitely differentiable it is enough to see

.7(ti (Dp s(.)))=(- 1)[pt .7'(ti De s(.))=(- 1)iP[ D.p f-(ti s(.))=(- 1)hpi D.p f(.r)

for any p. Since the mapping R"-{O})x.DP6(.)E8'(R'L- {O}) is continuous,the derivatives of 7(x) are all continuous. Thus we get fEX.,n:-th(E). Next weprove that any 7ES-nA(E) may be eonsidered as an element of .E2e(tr'R; E). Tothis end let S be any element of 8'(R" -- {O}) and define a linear form Åqf, SÅr on

E' by the following identity:

ÅqÅq1, sÅr, e'År-ÅqiÅqCr, -e'År, sÅr for 'e-'(E E',

Then it is known that Åq7, SÅrEE beeause E is quasi-complete [13]. Let nowTE '-5;"'x and observe that there exists an SEe'(R"- {O}) such that T=ti S. Put-

ting

f(T)- Åqj7, SÅr E E

we must show that ÅqL7, SÅr=O for any SEs'(R"-{O}) with 'i.S==O. In fact for

any E'EE it holds that

ÅqÅqr, sÅr, E'År-Åqi Åq7, E'År, sÅr-ÅqÅqf, vtÅr, ti sÅr =-o.

\8",C,e,vr8,h,a."2.11,li.År,:,O,ls7d,e.'lre.d6,Te,eg,eioge,{8T,L.is,.uni.'u.ei,y,,d,eke,r,I:ie,eg,•

1ltRjLHg;";,i{Rg,}1,?,"nld,i.gehzCg'?,2,e,si,gv"sbls:pagchs/rg:n,3,'Åíh,Illg.kWhsa,:c"flZn,itF.iXEÅé,Pgli9illgalr,gT?yhiicftfgO!5i

g:;iig,8Pag,i(,T,)--..ffkeS?.:?.,cgg`S\",oe,s,ae.d.`,h:g,{,fg,t'i6N.•E&,,W,8,T8.e,",O,Y'

have also the same topology, we consider the fundamental systems of neighour-

hoods of the origin in respective spaces. Let

v= pT(c, m, n) == {f; fE s.,'.",, 1 DP f(c) 1 sg l, for any p, lp 1 sg nz}

where Cis any compact subset of R"-{O} and m2IO, nÅrO are any integers.

10 K. YOSHINAGAThen by definition {V(C, m, n)} is a fundamental system of neighbourheods ofthe origin in E]A. Taking any eonvex circled neighbourhood U of the origin in

E and putting

-- -i ll7-= IJ7(U, V)== {f; fE .SZ)(]2]1; E), f(U")( V},we obtain a fundamental system {n7(U, V)} of neighbourhoods of the origin in

:2eE (2K; E). By writing

Jv- {.r; .r E! .{e, (2ls E), 1 Dp jX(uo) (c) l s{ ", for any p, lp l s; nt}

={1; jX E '-}:-; 'h (E), D' .7(c)(--1- u for any p, ip1.Åqm},

we infer that Il7 is a neighbourhood of the erigin in K-:'A(E). Furthermore this

identity shows us that any neighbourhood of the origin in ZA(E) contains amember of {JI7(U, U)} because the last side of the identity written above formsa fundamental system of neighbourhoods of the origin in NLI]x(E). This com-

We now define for any jX E2A(E) a veetor valued distribution Pf.f E! :i)'(E)

in the following way:

ÅqPf•.7, q,År =lilp, II:r""X'-i il(r)clr+t-.k", -z-i/-(?,C'i(OÅr,)+l) E'i"k"t]

for any qE g, where k is any non-negative integer such that ERx+n+kÅr -- 1 and

r'(pt)== !i.i-, 7(x)rp(rx)d-T• As in Seetion 1, this expression may have no meaning

when x is a negative integer of the form x=-(n+l), and in such a case log smust be written in plaee of -.-tr, "NA':l . The set of distributions Pf.7, rEZk(E),

is denoted by Pf.S--"x(E) provided with the topology of '-S'x(E). Then by meansof the following sequenee ef isomorphisms:

Pf.'=t ,(E).'..S:-.",(E)-:".-],EE-,-(Pf.'LSt .)eE== .s2f7,(E:; Pf. 2:'l,),

it holds that ÅqPf.f, 1'År--Pf.Åqf, -e'ÅrEPf.X.,:-.-'K for any 'e'EE'. Keeping this m

mind we may write Pf,'-Y'A(E)==(Pf.=x)EE. In case of N=h or N=-(n+lt)where le is any integer 2;E)O we have defined ttK' in Section 1. In just the same

way we put t;2 (E)= {.Ts .7 E ".S,-:lh(E) and (D" .1)h --O for any p, ip1 -- h}

and 20"cn.h) (E)={.7; 7E E]-{.+h) (E) and (.x'bi=O for any p, lp1 =h},

On Spaces of Finite Parts of Homegcncous Functions 11

where ip, denotes as before the spherieal mean of di: dit(-r)= -,l/-.'!,.,i.,ip(IXI•T')dX"

We now prove the next

PRoposmoN5. År.]9(E)==t29EE, '-'t O-(..h)(E)=':tSe.(..h)EE.

PRooF. It is enough to prove X-'?,(E)=:Y,CÅrt:i:'; E) and --t O..c.+h) (E)=!Z'e(2O..'cn+hÅr•; E)•

Ad År:)2(E)=2",(N;?'; E): Letibe the injection ÅrE]Y,--,8(R"-{O}) and maHkeuse of the notations in the proof of Proposition 4. We first prove that any fEY(5--"?'; E) may be considered as an element of År]Y, (E). To this end an E-valued

funetion f(x) is defined by the next identity:

ji (x)= .7 ('i (6Åqr))), .r E Rn- {O} •

Then quite sirnilarly as before we infer that .TE';t h(E). To prove (DP .7)"=O,

we first observe that D" r(r) --(-- 1)i"i 7('f DP S(.)År and next that for any fE X-'? we

obtain

!i.,-, Åq'1' D" 6(x), fÅr dX=('1)i"i!,.,., ÅqScx), D"fÅr dx=(-1)'Pi I,.,t, D'f(+r)d•x=e•

Then it follows that !,.,.ii DP 6Åq.) (lx=O and hence (Dh 7)t=O as desired. We next

shew that any rE År]Y (E) may be eonsidered as an element of .Y'(Åril':'; E). Theproof is sketched as follows. For any SEes'(R"-{O}) we define a linear form

on E' by the identity:

ÅqÅq.7,SÅr, E'År-ÅqjÅq7, 'e"År, sÅr for E'EE'.

Then it may be seen that Åq7, SÅrEE. Hence putting for any 1'Et;2'

-- f(T) =Åqf, SÅr EEwhere T=:ti S with SE es'(R"- {O}), we obtain a linear app!ication ef X-'2' int.o 4'.

By observing that `i is a topological homomorphism of e`(4"- {O}) onto N;?' it isnot difficult to see the eontinuity of .7(T). This proves fE.if (Årl9';E) and we

say that ÅrSP (E) and .g',(K-"Y'; E) are the same vector spaee. Finally wg mayprove that they have also the same topology just in the same way as in the

proof of Proposition 4 and details are omitted. . Ad År.ig(n+h)(E)=.S?E(t;"-'(.+h); E): Let i' be the injection X--'O-{..,,,.(sP(R"-{O}).

Then by putting Ji (x)= ji (`i'(6(.)), xE R" - {O}

for any .7E.sZ'(ZO-',..h)i E), we must prove that the function f belongs to

12 K, YOSHINAGA5LJ9(.+h)(E)- Except that (xP7)-=O, the proef goes along the same line as inthe previous ease. That (ptP .jF)t==O is seen as follows. For any fEIZ`1(n+i,) it

ho}ds that

i,.: ., Åq'1' XP S{i), fÅr dX "= i 1.t -, ÅqXP 6(x)] fÅr `l 'V -- !,., ", 'XP f( -V) dX == O

Therefore I:.i., `1 xP S(.) ax=O and henee (xP 7)#=O. This completes the proof.

If the set of distributions Pf.7, 7E 20d(..fi, (E), is denoted by Pf.'LSt ".T(.+h)(E),

then we may write Pf.Åri]2(..h}(E)=(Pf.';t'C-'{..h))sE. Let ITh(E) be the set ofhomogeneous polynomials of degree h with coefficients in E:ipS`F..,xP ep, it, E E, and

let ll.(..i,)(E) be the set of linear eombinations of the formEp?I.ll,, DP S(g) Ep• Then

llh(E)=ITh eE and LT-{.+h) (E)=17-(..h) eE. Hence in the sense of the topologieal

direct sum we obtain S-5Jlh(E)=';t :(E)+llh(E). Let us denote the topologicaldireet sum Pf.Z9(..h)(E)ÅÄll.(.+h)(E) by K-'l(.+h)(E). Then we obtain S.L" ,'!(..h)(E)

=Y!(n+h) EE. Since 2]a, '-K'n'2, =O-{..h) are all defined to be elosed subspaces of

the complete nuelear spaee 8(R"-{O}), we may infer that they are nuelearcomplete spaces of type (F) [9, Chap. 2, p. 47], and thus they have also the pro-perty of approximation [9, Chap. 1, p. 165], Therefore if E is a space of type(F) we may write that S..-.'x(E)= ÅrEl;x(il5 1Ir, E]h(E)==hQE,r•- [15]. Thus, for ex-

rt tample, any element fe t=A(E) is expressed as

-. eef=S"' atkfk -e'k, k-1

where X-op" IctklÅqoo,fk.e in E]x, 'ek.O in E [9, Chap, 1, p. 51]. The same holds

k-1also for Pf.N;x(E), tL[]1(n+,,) (E),••`•

It is now a simple matter to define the Fourier transformation on Pf.5LSK(E).Let Ibe the identical application on E., Then the application ,:diR=el is called the

Fourier transformation. It is defined on Pf.=A(E), 5;i-(.+h)(E),--i, becausePf-X-'x(E)=(Pf.';t N) EE, )E]!(n+h)(E)=':tS!(n+h) EE, •••. By virtue of the preceding

results we obtain

PRoposfTfoN 6. The l7oacTier transformation dejines the topological isomor-

phisms between 1) Pf.5nn(E) and Pf.X..(n+h) (E) (N is any compeex number other than O, 1,

2,•-•, -n, -(Jt+1), -(rz+2),•-•), 2) Z: (E) and Pf.';t O-(n+h) (E) (h=O, 1, 2, •-•),

3) Zh (E) and =i.(n+h) (E) (h=O, 1, 2, -•-).

PRooF. Evident.

on spaccs of Finirc Parts of Homogcncous Functions 13

3. Singularintegral operators. In the theory of singular integral opera--tors studied by A.P. Calder6n and A. Zygmund particular r61es are p]ayed bythe spaces X-'o and '2t i... [3], [4], [5], [11]. Therefore in the rest ef this paper

we shall restrict our attention to ')tz;o, ';t i-. and investigate the same subject

matter as the one diseussed in [3], [5] in view of an applieation of our pre-

ceding results. Let HM, m=O, 1, 2,•••, be the spaee of distributionsf on R" sueh that DPfE L2

for any p, [plE{;lm. It is known that IIM is a Hilbert space provided with the

scalar product antl the norm

(f, oa)m= '-t (D'f, D'g)L2, llfll.=(f, f)k

lplsmrespectively. In the first plaee let us consider the convolution of .EE";t t-. by

hEL2. Since sE .E27Z, and hE :2Tl,, we may eonclude that Y'-convolution s(S)lt=s*h is defined [10]. Then by virtue of the exehange formula we get (s*h)-=

n case that hEHM, we . Therefores*hEL2 and Ils*hIIL: s{Ig1 S!l.1]hIIL2. ISm'

aiAY[iinOf]er that ,.hEHpt and n,*n".s{g 11s!!.Elh"., an immediate consequence of

the faet that DP(s*h) =s*DPh for any p. We now remark that for any s Ei Pf.'Årt]'O.. it holds that

S'h(::)=1,ith !,,i.., 1'(Xi-Y)S(Y)dY

in the sense of convergence in L2. It is already known that the right hand side of this identity really exists in the sense of convergence in L2 as well as ot con-

iZr,gXeS,eClii,M,O,S.t,f",g',Zw,?.er,e,[3,]f.,'P,eJeSgig,g\e,,s.tgge,g},e."8,}1g.pealLy,2v,idgn.Y•

a, b, OÅqaÅqbs{; eo

( s(.x) for aSI.a:1Åqb, Sa-b(X)=i o otherwise•

We first eonsider the ease bÅqoo. Then the Fourier transform of ,sa,e may be

xvritten as

Sa•b(g)=" !,.,i.,s(x')1(Åq-r', g'År; a, b, p) dxt

where g== pg', pl}iO, lg' 1 =1 and we put

I(ct)=I(a; ., b, p)==Ii.IP,: e'-ita7tt-:i'l'r dt.

since r(ct)= -1- (e-2ifipb"-e'2"ip"o), we obtain

a

I(ct)=

le':""'"9`!-B e-Z?if--l""a'.-. dB for ctÅrO,II( - 1) + ! "- , e'2'i" b"A -ts e'2"i""S dB for at Åq o.

Therefore we get

u(a)ls{: ! -2 iOg ct for ctÅro,

Åq c--2 Iog :al for atÅqO

where

C"2,..s.u,p,., ,Iil:: -EE'!nt-t dt!.

We now pass te the case b== oo. By Lebesgue's convergence theorem we mayinfer that

1,i-M.. Sa,b(g)=!,.,,., s(x') l(Åqx', g'År; a, co, p) dx'

for any g, and hence it holds that

Sa,co(e)=!t.,i-, s(x') I(Åqx', 8'År; a, co, p) dx',

because S.,b-S.,tu (b. oo) in L2. Consequently it holds that for any OÅqaÅqb soo

I Sa.b(g) 1 s{g I,..,., 1s(x') I (c -2 log 1 Åq.x', gÅr {) , l..rÅr! ,,

where ci is a eonstant independent of a, b, se. Thus for anyhEE L2 we get s.,bfiE L2 with li s.,b nli L: sg ci ll i, ll L2. Again by Leb esgue's convergence theorem it

holds that s.,b E.s.,. ii in L2, beeause sa,b(g)E(g). S.,. (ij) i, (ij) almost everywhere.

By the inverse Fourier transforrnatien we get sa,b*h.sa,..*h in L2. 0n theother hand it is clear that for almost everyx we get

S"'b ' h("-) '-- !..1,].b h(X "" )')S(Y) d)' "-"!..1,,l h(`pt -Y)S()f) d)' (b'co)'

Hence we obtain

Sa• eo ' h ('r) = !. r- ,,, h (x -y) s (.J') dy (I! LZ.

JWe now eonsider the case a-O. For any fixed pÅr)O we observe that

On Spaccs of Finite Parts of Homogeneous Functions 15

li(a)1 f{:

2i: 1sin Tpf(sb-")191 di3s2t, p(b-a) for ctÅrO,

2 g2=p" 1 sint 1 dt +2 [O Sin 7, tP$ib '-- a) Bi dlg sg 67,rp (b - a) for at Åq O•

J2tpa t J-1 tJ lTherefore it holds that l(a; a, b, p)--FO (b-a.e) uniformly in a, 1atI E{g1, and in

op confined in a bounded set of positive numbers. Hence for any h.EL' we see g)li(g).•O(b-a.O). Thus by the inverse Fourier transformation we maySa,b(assert that there exists a funetion gE L2 such that h*s,,..--pg (E.O) in L2. Henee

for any qp E :2r it holds that

Åqg, q'År=1,i-M, ! op (X)d"' !,,,., h('XH)')S(Y) dpt

==lim g (p*ri(y)s(y)dy=Åqs, q)*iLÅr=Åqs*Ii, tpÅr,

e-•o Jt)•tzE

that iS s*h=g =lg' -m, !,,,.,h(x-y)-s(pt)dy, as desired. Here we remark that for

any sE K-'O.. it holds that Pf.s==v.p. s, Cauchofs principal value of s. By the way

we note that the Y'-convolution is defined for any si, s2EÅíln and we have si*s2

ES`i"6,b,e,Za8,:e,S.igSz,E,`{}."&e",2.`Bb*,i•tl•G.-=g'Sidi2,:t"O'

LE"frma 2. LetfE';t'.. and v. p.fbe deLtinea. TjtenfE'2t O...

PRooF. Letting h(t) E 9(Ri), h(e) 2}I O and

h(,)=[ ll lo,rr 11ill lili121

we put {p(x) =h(i-x•l)forxER". Then

Ii=ize f('T) `P(")dX `i,., hS') d'S,.,., f('x)dx'

This shows us thatg f(x)dpt= O, because if it is not the case we get

JLxl-1 lim i f(.x) q) (x) dx--, Å} oe

g--O JIxleei3`,/AegtlZ,?ti2.\.t-.'Rhl,XPX2:,e.e:tSBes'ghf,iesi,:,gMe,Io'X9iitt.•.t6sPg',go;fili:i=nu,o,,u?},\=dg6ff,ere?:i?gi:

Banach space (mÅq oo) or a space of type (l7) (m== ee), with the usual topology;

uniform eonvergenee on R" of derivatives of all orders up to m. Next pro-position is a Iemma of Friedrichs' type, if we may use such an expression [6].

PRoposmoN 7. Let bE S3rM, m2 2, sE 2]!. anel let for 1'=1,-+-,n,

T,(f)-=s*(b-,a-.if-',)-b(s*-iiO[.tl(?)•

Then Tj is a continuous linear application of Ht into IIi, O-ÅqISm-2,

pRooF. We first remark that Tj(f) is meaningful beeauses, aO.llli E2'L2,

and hence the convolutions are defined. Furthermore it is not diHieult to seethat Ti(f) is a eontinuous linear application of Hi into g'. Therefore to prove

the statement it is enough to see that there exists a constantcsueh thatllTi(op)llis{:cllrplli for any wES. Since we may write

Db Tj (q)) -.E. ]-, (e) Is * (DP '- r b • -QeDi ,g-'-) - (Dp -rb) (s * ag.C ,iZi )l ,

the problem is redueed te show that there exists a constantc such that ]iTj(q))llL2

s{gcll(pllL2. This is proved as follows. We assume i=1 and sE:2]g.. Then byan easy computation we first infer that Ti((p) (x)=lim I,(x) everywhere on R", e --- owhere

(1) Ie(x)=I,,,., (b (xdy) "-'b(nt)) go (xHpt)s(y) opi(y) dpt

-Ii,ize b"(X-Y) ep (X-Y)S(Y) dy +I,,,., (b (X-Y)'b(X))rp (X-Y)Syi(7) `IY

== lSi'(x) + I;2' (x) + 1E3'(x).

Here ni(pt) is the first component of the outer normal unit veetor on the spherely1=e. Sinee bEg3r2 it is not diffieult to prove 1lim ISi'(x)1 sgci@(.r)1 and thus e"o

(2) illim I:"IIL2 s{Icill{pliL2, e-,owhere ci isaconstant depending on b.j,i--1,•••, n, and s. We next dominateIS2) as follows:

(3) "1im lE2'1iL2 S; ll(bxiq,)'sllL2 .-, iiSllee llbxi (pl1L2 S{l;c2Ilq)IjL2,

e-.owhere the onstant c2 depends on b.i and s. To estimate I53}(x) we write

(4) I:3'(X) == Ii.vi2i + SiÅri.i2e (b (X -Y)-b(X))9(X -7) Syi(Y) dy --"-J(X) +Ji(X)'

On Spaces of Finitc Parts of Homogeneeus Functions 17

Since l (b(x' -y) -b(.T)) cp (.r - y) I . {g 2 11 b ll .. I `p (.ft -)t) l and l s,,(y) 1 .s: c3 1 :y 1 'C""' for

17 1 :}}i 1 with c3 = ,?,1 l"Il I s)+, (pt) I , we infer th at

(s) 11JliL2 S: c4 1i cp ll L2,c4 being a eonstant depending merely on b and s,,,. Finally, Ji(x) is dominated

in the following way. Taylor's expansion shows us that

n b (x' -y) - b (x) = --- = yk brk(x) +b(x, y), k-1where lb(x, y)1:{:csly12 and cs is the maximum of lib.,.i[I.., i.i---1,+--,n. There-

fore

(6) Ji(-T)=-t-" .', bxk(x) I,.L,,., v) (x -- Y)yk s.vi()') dy + !,.i,,.,qp(-x '-' v)b(x,)')tsn(y)a}t•

The absolute value of the s.econd term is dominated by c6 I ,.il.g(•z'-)')1 1.y1 -"'idy

where c6 is aeonstant depending on cs as well as on iE.ui.p, tss,(y)1 and not on 8-

Therefore its L2-norm does not exceed c7 [Iq) I[L2, c7 being a eonstant depending only

on c6. As for the first term we remark that pths,,,(y)ES-"O-. because v.p.yk ss,()r)

does exist. Consequently to dominate the L2-norm of the first term by aconstant multiple of llrpllL2, it is suffieient to con$ider the integral of the type:

I,.,,i., q'(X'-)')S(pt) dy, Where e)EY and sEÅrL;9.. Then, using the notation in.

troduced above we see that it is nothing but the convolution op * se,i. Hence weget II(/i) * s.•,illL2Sgcs!Iq)ilL2, where cs=s,u.p, els,,ill.. Thus it follows from (6) that

(7) 11 .Xi Il L2 .:EI{: cg il q)11 L2,

cg being a constant independent of E. By means of (1), (2), (3), (4), (5) and (7)we may conclude that IITi(q))IIL2s{;cll(pllL2 where the constant c depends only on

s and b. This eompletes the proof.

REifARK 1• To estimate ls,,i(ij)1 we may put J(ct)=!i.:, e'"at-eP' de instead

of I(a). Then using IJ(at)1s{ic-log 1atI we obtain the same result [3].

RE"fARK 2. Further observations show us that when s and b vary withinbounded subsets of :Eg. and g]M respeetively, the corresponding c will alsp be

C.O,":liZe,d,i,",a.,b,O.",",d,e2,s.e,F,s.f,?•flthtGge,,n,u.[II.e,rsi.go,,lg,a3,d'R,zu,cl},e..c,zse.v,.arious

As an application of the preceding proposition we state

18 K. YOSHINAGA PRoposiTioN 8. Let bE@M, nt:}l2, sE E]9. and let

T(f) -- s * (bf) -b (s *f).

Then T is a continuous linear application of H' into H'"i, Os!:tsg nz --3.

PRooF. By means of the Fourier transformation it is not diflieult to seethat the application

(fo• fi:•''• fn)'f="fo " 8f.-', "'''" ul ft

is a eontinuous homomorphism of Ili'ix ... Å~Ht'i((n+1) faetors) onto Hi, HenceT(f) == T(fo)+ Ti(fi)+ -•• + T.(f.) is a continuous application of U' into Hi'i. This

completes the proof.

RElifARK 3. By virtue of Remark 2, it is noted that if {T.} is a family ofT's defined by s. and b., eonfined in bounded subsets of ]2)9. and SilM respee-tively, then {T.} is an equieontinuous subset of y(H'; H'+').

We now define the operator A by the identity:

xlf== cn (f' Pf' ,nl+i)' dn (Af' h, nl-i)

r(Lt-)Where cn=, .:2 t-.(-4,)=(n-"1)d.. It is seen that A is a continuous linear

operator of Ht into Ht-i, l=1, 2,+-•, beeause v.p. k, E:Åí9.. Sinee it is known rthat 4y`"'(Pf• ,.1.,)= ,1. r[12, II, p. 113], we obtain for any (pELspP

Arp=,s`R"""'((P t r).

Therefore it is not difficult to see that A2=-IC.lr:i d and henee we may write

A=-21., v-d.

To define the operator F corresponding to FE 'lt L.(g3fM) we first remark that12]1.(@M) may be considered as the space of m times eontinuously differentiablefunetions F(x) defined on R" with values in t;it. and with bounded deTivativesef all orders up to m, Thus for any xER" and hEH"' we may define F(.v)*h.

We now prove

PRoposmoN 9. For any FE Ei]!.(fiii'M), there exists a uniquely determined con-

on Spaces of Finite parts of Homogeneous Functions 19

tinuoacs linear operator F: h-•F(h) on liM such that for any (pEY

F(rp) (x)=(F(x) * rp) (x).

PReoF. It is known that Zl.(EEIi'M)= Y3'Mal5=L., and that both 9il"' and :;:]!.

tare spaees of type (F). Therefore F may be expressed as

F=Si] ak bh sk+bS k.1where :{] lctklÅqoe,b(! g3r", bk.O in {IM and sk.O in 12]2. [9, Chap. I, p. 51].

k.1Here the right hand sum is taken in 2L.(gilM), and thus it is not diMcult to seethat for any op E .Y we obtain

(F(x) * g,) (x)=:S] ctk bk(x) sk * q)(x) +b(x) {p(x)

h-1everywhere xE R". On the other hand if we put

F(h) = ISI.] ct, b,(s, * h) + bh

k.1for anyhEHM, we get F(h)EHM. To see that F is continuous it is enough to

remark that

Ilb,(s, * h)Il., Ki 'S-: llDP (b, (s, * h))il,2

lplsm sg,,Z,.. ,:EL ]i.", (i;) llD'-' bkll ..Iisk1lmllD' hIiL2KckIVi I•lm

wherel ck is a constant such that ck-O(k.eo). Since F(q) (x)= (F(x)rop) (x) for any opE.Y, it is clear that such an F is uniquely determined, This com-

We now introduee some operators related to F. To-simplify the wording vve call 17EN--"o(giirM) the symboe of the operater F. Then fi is also thg .symbol of

i"x,2,Åq\i,k'//.Ofi•mg",YL&,-i9,f16,e,l,,/,O,S.[k`i}l,e;.ft/p,,gg,e;"t,?I"i',F?9'io:,pie;i,get,okOg,gza'lsihV,:,ii•ih,,ek,:.IKmt"6egii/iicell,

[5].

A(FBRF02P2SF"ir}02N:Ond'f'ftLIF::'ifF-,':Er;'Ei'ij'cin"ti'h"ubfs(El•"•aF•j'bSf;hfSr-•1'•f2\l

O:E{l l sl:Tn-2, ]n :;2 2.

PRooF. It is clear that eaeh of the operators in question belongs to

20 K, YOSHINAGA.Y'(I-I'; 9'). Thus to prove the statements it is enough to show that their re-striction on 9 are continuous from H' into H". To this end let the symbols ofthe operators F, Fi and F2 be expanded in the following forms:

fi(x, g)=].2:"] aj bj (x-) sAi (g) +c(.v),

J-: -iPh(X, e)==].ilE] atki bki(.r) Ski (g)+ck(x), th --- 1, 2,

1-1

eo ewhere i., l ctj 1 Åq oo, JE, ]- , lakj l Åq co, c, ck E S?I M, bi, bhi .O (i- oo) in EllM and si, shj-- .O (i.oo) in V-g.. Then the statements are proved as foliows.

Ad FA-AF: Lettingq)E9,wewrite

nt (FA-AF) (tp)=it-, ctj{bi (sj * Arp)-A(bi(sj * q}))} +cAgD-Ac{p

='dn Ij2.op, ai (bi (Si ' rnl-i * drp)'-- ,nl.i ' d(bj(s)L ' q))+c(,nl-i* dq)- rnl.-i ' d(cop)l-

Thus the problem is reduced to prove the equicontinuity of

P(9) =b( ,nl-i ' dg)- ,nl-i 'd(bg)

with g---s * rp or g=op, when b and s are confined in a bounded subset of g3rM and

a bounded subset of 20-. respectively, Writing

p(op)-,-",(b(ea.,. ,.i.t*8.g-,)-- aa., ,.i-a* oa., (bg)l

-t-."i,Ib(-,aG.I-,7.i=,*-,-O.g-,)-,?.,-,.th-*b,a.E.-,a.,,,.i.,',a.b,.gl

Oland observing axj ,.-, =r-(n--1) v.p• ,.X.i, E 2]9., we may dominate the lastterm as follows:

eb ae.,. ,nli',aÅí. gl,s{gciI a,kg' (S*op''

k axf (P,i

Proposition 7 to the first two terms we get

sg ci ll {pllt,

where the constant ci depends only on s and b. On the other hand, applying

lb(•o-O., ,.i.i *-aeg)--o-a•-.7. ,.i., *b-:al.ErI,,Sciilgi!isgc2Ilcplli,

On Spaccs of Finiie Parts of Homogcneous Functions 21

where the constant c2 depends only on s and b. Thus we see 1!P(q,)iii-/,,c3ilrp!rrt

and the constant c3 may be ehosen independently ofs andb confined in a bound-ed subset of År-;2. and a bounded subset of gil" respeetively. This completes the

proof.

Ad F'zi - AF': Letting {p E 2, we get

(F * zt - AF *) (q,) = x"O"i eei {s-j * (bi Afp) - A (gi * (bj rp))} + oA(p - ne g,.

j.1Hence it is enough to prove the equieontinuity of

Q((p) --s * (bztrp)-A(s * b{T}),

when b and s are Testrieted in a bounded subset of S{IM and a bounded subset ofÅíO.. respectively. To this end, we write

Q(cp)-dn tl.] Is*b( ea.., -?ii-.--, * -aDI/Z-,. )- oO., -ifii;r' -b?/i;.- (s*brp)l

=dns'j".,Ib (mal.lj ",rnl;i-"-II']t,-",. )- ali.IJ ,nl-i '(b Pollllj )--oa,,i -,,rl-T ' -to-3'llll.- `pl'

Then paying attention to aa .i ,.1-, E20-., and utilizing Preposition 7, it is not

difficult to see IIQ(cp)iiisgc411(7)lli, where c.t is a constant independent ef b and s

restricted in a bounded subset of glY" and a bounded subset of Årt.]O.. respeetively.

This completes the proof. Ad (F'-Fe)A; For any r/) E 9, we get

m (F * --- F#) zi (gp) = t; esi {gi * (bi zt op) - bi (SJ- * Aa,)} • j-1Therefore, as repeatedly done, the problem is reduced to show that the applica-

tion

{p.s * (bAqp)-b(s * ztrp)

is equicontinuous when b and s are confined in fixed bounded subsets of g3rM andÅr-]gn respectively. This again turns into the same problem concerning the ap-

plieation

R((p)=s*b( oa.,,, -,ili.--, '-?di,Il-. )n-b(s' -aR.erini-i "-2.".'i)

=s* Ib (-oP.-,- ?-i!-T * --eo-ti,-) -- o?',, -,ii;-i- *b-gi,", l

22 K. YOSI-IINAGA +(-s * -ba.xi- ,.i-,)*b -ao-x`7:;'. -b((s* a?,.,, -,.i-i)* -g.q:'iJ)•

Then observing s* -ea ..j -,,1.-, E X-'O-. and using Proposition 7, it is not difficult to

conclude that IIR(tp)Il,:!{;csiirpl!, where the constant cs is independent of b and s

confined in prescribed bounded subsets of Y3'M and 2]9. respectively. This com-pletes the proof.

REtsfARK 4. Decomposing the applieation gp.s* (bAq))-b(s * ricp) into q,.Aq).s * (bArp)-b(s * Arp) and applying Proposition 8, we may conclude the sameresult whenlÅrnl. But to overcome thecase l==O along the sameline it seernsdesirable to eonsider the spaee H" for negative nz. Ad A(F'---F#): This is aconsequence of the following identity:

n (F*-F#) --- (AF* -F* Z) + (F*-F#) A+ (F# A- AF #).

RF."tARK 5. It is not diffieult to see by Proposition 8 that F"-F' is a eon-tinueus operator of IIt into H"i for O:fglsSm-3. Henee by composing A onthe left of F'--F', we obtain the desired result for OS:ls{;nb-3. But it seemsto the author that the ease t==nt-2 requires a partieular argument. Ad (FieF2-Fi liT2) A: An expansion of (FioF2-Fi F2) A(rp) shows us thatthe problem becomes to prove that the application

t7) .bi {b2 (si * (s2 * Aq)))-si * b2 (s2 * Ag))}

is equieontinuous when bi, b2 describe a bounded subset of Y3iM and si, .s2 describe

a bounded subset of t;]g., This turns out to the same problem under the similarcircumstanees relative to the applieation of the form

(p --Årb(st*s2 * gxq.)i )-sl*b(s:* -Oexrp,. )

-b (s, * s, * 8--kt3. )-s, i: s,• *(b gt.i)+s, * Is,* (b -:tde-l--) --b (s,* gt.,.-)l.

Thus the same argument as in the case (F'-F"•)A shows us the desited eon-clusion. Thiscompletestheproof. Ad Fi F: A-iU7i F2: Quite similarly as before, an expansion of (Fi F2 A-dl;'i F2) (g) reduees the problem to show the equicontinuity of the application

tp.bi(si * b2 (s2 * Atcp))-A{bi(,ri * b2(s2 * tp))}

when bi, b2 and si, s2 run over prescribed bounded subsets of @'" and ZO-. re-speetive]y. Then it is not difficult to see that this is a direet eonsequence of the

On Spaccs of Finite Parts of Hornogcneous Funetions 23

same statement concerning the applieation

q,-,bi(si*b2(s2*-..-)O-.].- }.1:-f* -8- .qi);))-:oTaE.i]-. Li.1--.-r* ta-..i {bt(si*b2(s2*{p))}

--bi(si rk b2 (rt-'`li" ',,}1;i 's'- ' -PelC'i ))

- tQ..:;-' ,nl-i 'bi (si' rdD.r, b2(s"-'(p))-me?i7. ,ii *;Oolb,tr, (si'b2(s2'r/'))'

As to the last term everything is obvious. Letting g= s2 * g, the first two terms

may be written as

bi (tsi * b2 ( a'O'.llr Li.1-ri" -s.O-.,g,', ))- -eE) ).v ,,il-I-i' * bi (si* b2 -iaii.llJ.-)

- iO.,- ,nl--i 'bi (si'(8ft.2i g)).

Again there needs no mention of the last term, svhile the first two become

bi (si*b2( oa.;L r,i-, * .d-exgi. ))-bi (si* a9.i Li.!.r*bz -Sdxg-i )

+bi( ea.i ,.i.-i*si*b2 -eQ.g-i-)---eaJ.i -,.i-, *bt (si*b2 -igl..g-j )

=biIsi '(b2 ( ,aii -,n'-.i ' -iil.g;)"bLO.,-- ,ni- i- 'b2 8./,-)l

+bi(e'O., ,nl-i'(si'b2 aO.g, ))- a?., -,.ill-i`'b; (sr'b2 DaiJ, ).

Then applying Proposition 7 it is not diMeult to get the assertion as desired.

This completes the proof. Ad A(Fi e F2-Fi F2); An immediate consequence of the expression

ri (Fi " F2-Fi F2)={A(F! o F2)-- (Fi v F2)A} +(Fi b F2-Fi F2)ri+(Fi F2 A-dFs F2)-

This completes the proof.

Referenees

[ 1 1 N• Bourbaki, EsPaces vecteriels topoto.girfues, Actualites Sci. Incl., no. 1189 (l953), no. 1229 (1955), Paris,

Hermann.[2] A. P. Calder6n and A. Zygrnund, On singtttar integrats, Amer, J. ef Math, vol, 78 (19S6) pp, 289--

309.[31 ------ ---- - -- - , Singtttar integrai operators and dslOTerential eauations, Amer, J. of Math. vol 79 (19S7)

24 K, YOSHIN AGA pp, 901-921.[4] A, P. Calcler6n, Unietteness in the Cauch), prebtem for partiat dt;6Terentiat equations, Amer. J. of Math.,

vol, 80 (1958) pp. I6-36,

[5] , lntegrates singutares )sas apticaciones a eeuacienes diferenciates hi/)erbeticas, Cursos y Scrninarios dc Matcmatica, Fasciculo 3 (r960), Universidad de Buenos Aircs,[6] K• O. Friedrichs, On tite dVerentiabllit), of tlte seltaions of tinear cttiptia drLbllerentiat eeuations, Comm.

Pure Appl. Math, voL 6 (1953) pp, 299-325,[7] L M. Gelfand and G. E. Silov, Cenerati;edfunctions, I, Meskva, 1959 (In Russian).[8] A. Grothendieck, Sttr les espaces (F) er (DF), Summa BrasiL Math. vol. 3, fasc. 6(19S4) pp. 57-122,

[9] --- -----, Produits tensen-ets topetogiques ct esptttes nttctE'aires, Memoirs of the Amcr. Math.

Soc. no, 16 (1955).:10] Y. Hirata and H. Ogata, On t/te etchange formula fer distributions,J. Sci, Hiroshima Vniv. Ser. A,

vol. 22 (1958) pp. I47•-152.[11] B. Malgrange, Unicite du tJrebtane de Cauchy d'apri}s A. P, Catder6n,, Seminairc Bourbaki, Expose I78,

SC'cTetariat mathernatique, Paris, 1959.[12] L, Schwartz, ThCjerie des distn'butiens, l, ll, Actualites Sci, Ind. no. 1245(1957), no. 1122 (1951), Paris,

Hermann.[13) -, EsPaces de fonctions dtVerentiabtes di vatettrs vectorieltes,J, AnaLyse Math.vol,4(1954 -56) pp, 88-148.[14] , ZLes ePEvateurs invariantsPar rotatien, t'epie'rateur d, Seminaire SchwaTtz (Equations aux derivees partielles, Expose 7), Secretarlat mathC"matiquc, Paris, 195S.

[15] , 7Tmoerie des distributions d valeurs vecteriettes, Ann. Inst. Fourier, Grcnoble vol. 7 (l957) pp. 1-1cl1.[16] K, Yoshinaga, On a tocallJ• convex space introduced b), J. S. E Silva.J. Sci. Hiroshima Univ. Scr, A, vol.

21 (1957) pp. 89--9S.

Department of MathematicsKyushu Institute of Technology