Bull. Kyushu Inst, Tech.
(M. & N. S.) NTo, l1, 1964.
ON LIOUVIU ,E,S DM;ERE]!-eTIATION By
Ky6ichi YosHnsTAGA
(Rcceived Dec, 15, 1963)
By Liouville's differentiation we mean here the differen.tiat.ion of.non- integral order. In the theory of distributions sueh a differentiation is studied as the eonvolution of Yn:= 17t.,) Pf.(xM'i)xÅro carried out on the space (e;)[8, II, -p. 30]. But if one hopes to get a satisfactory theory, eertain improve.ments
must be made on the spaee to aet on. In a recent paper [5] P.I. Lizorkin has preposed a theory of Liouville's differentiation based on a certain spaee of gen- eralized functions. Such a spaee, named M' by Lizorkin, is the strong dual of the spaee O composed of all qEY such that
IXg•op (s'1, ••+, -xj, •--, .xn)dxi -= O
for any i--'1, ••-, n, p=O, 1, 2, +-+• .
It will be one of the purposes of the present paper to make a further m- vestigation of these spaces. Seetion 1 is concerned with preliminaries. We .shall define the spaees Åë, O' and their respective Fourier transforms !Zr, V', It.will be seen that O', T)P"' are Silva spaces and O, !F are dual Silva spaces [9]. We also nete that LP(e'. In Seetion 2, after giving a characterization of UT due to Lizorkin, we define the multiplieator on V7', T' and the operator of convolution on e, to'. Then, in Seetion 3, we may define Liou'ville's differentiation as an operator of eonvolution of particular type:
ddiNf= ( rXil:Åí) cE}••-o i(Ei"i,i) )*f, fEÅë',
Where -r=(.xi, ••+, xn), Z= (li, •-•, R.). [Phe subsequent sections will be intended for an applieation of this differentiation. In Seetion 4 the concept of the multi- plier of type (p, p) due to L. H6rmander [3] is studied. The most part of this seetion wil! be devoted to a proof of Mikhlin's theorem which states a suMcient condition for a given funetion to be a multiplier [7]. The proof exposed there will be slightly different from that given by Mikhlin [7] and Lizorkin [5]• In Seetion 5 we shall examine a spaee .sie,g,...), a generalization of the spaee 3e'(m,s) of H6rmander [4, p. 51]. It will be shovvn that such a space may be identified
z
F
with the spaee defined, by means of Liouville's differentiation, in a similar manner as given originally by S. L. Sobolev. Further investigation of such a spaee will be given elsewhere.
Most of the notations and the terminologies in this paper are essentially those of N. Bourbaki [1] (locally convex spaces) and those of L. Schwartz [8]
(distributions). .
1. Prelimiiiaries. Let R" be the real Euclidean n-spaee of the points x=
(.ri, --•, x.), Let Åë be the set of functions q7 E sp such that S-X,P'ip (Xi, •••, xi, --+, x.)a.r,• iO •
for any i--1, •+•, n, p= O, 1, 2, •••. Then Åë is a elosed subspace of the dual Silva (=Sehwartz (F)) space y". Hence it is also a dual Silva space [9] and one may easily verify that
-0
t,e = di, Åë= o, a.j Åë(Åë, e*y (o
for any h== (hi, •+-, h.), i--- 1, •-+, n. If if E @A Åë, then it holds th at IP(rc) g(x) dx
=:O for any polynomial P(x) and consequently we get opiiO. This shows us that
@AÅë= {O}. Sinee e isa closed subspaee of Se, we may infer that the strong dual Åë' of di is a Silva spaee isomorphic with the spaee ,gp70e, where ÅëO is the polar of pt taken in y". Letting IIi be the set of polynomials of the variable xi and letting se;• be the set of temperate distributions in the (xi, •-•, .vi..i, .rj.i, d••,
xn)-space, one may observe that diO is a closed subspace of Y' spanned by UiopSO{,
+•• , j7nCEbSeA. To see this it is enough to note that Åë is the polar in s" of the
subset oÅí ,spi:
{O"a, (Xi)( i) '''X O"ai.-, (xi-i) (8)xe'i (El) 6a,•+, (xi+i)Q'''(8) 6a. (x") ; i -- 1, ' t', n,
pi--O, 1, 2, •••, and (ai, ••-, a.) is any n-tuple of real numbers},
where 6.j(x'i) is the Dirac rneasure at .ri---ai in the spaee of the real number .vi.
We now define some symbols about the Fourier transformation, For any q7E,s7 we write
ip (y) : j e'2rt ir' op (x) dx, ip (x) = S e2"`x)'q (y) dy,
and fdr any TE .9"', S? and T are given respectively by
A
ÅqT, opÅr == Åq T, ipÅr, ÅqT, qÅr = Åq T, di År,
satisfied by any ip E Y.
On Liouville's Differentiation 3
Putting 4T==di, we see that ZIT is a dual Silva spaee elosed in .9' and we get ep-=Åë-H.-oN, di,.ft=lt, IP'=yr,
e,tmp==4r, yjT(T, T•.gp(vr,
+for any h==(hi,•••,h.),i---1, --•, n. Since opEaT means ij/ EÅë, it is not diffieult to see that ?Ir is the set of funetions ip EY such that
oOyP?. op(yb ••+, y,-i, O, 7j+i, ••-, cr.) i!i O
for any ii--1, d••, n,p=O, 1, 2, -••. The strong dual Vr' of Ur isa Silva space iso-
morphie with the spaee .9"/IVO where IYO is the polar of ?P taken in Y". We now define the Fourier transformation between Åë' and Yr' in the following way.
Letting fE di', g E lr', one may determine f, .jF c SV' and g, g'" E di' by means of the next identities:
A---
Åqf, vtÅr == Åqf, eÅr, Åqf, opÅr= Åqf, opÅr, Åqg, qÅr .. Åqg, ipÅr, Åqg-, qÅr=Åqg, diÅr,
satisfied respeetively by any Åë E !Zr and by any ep E di. In like manner we also define the symmetry "v" respective!y in e' and !Zr' by virtue of
vv
Åqf, qÅr = Åqf, diÅr, Åqg, ÅëÅr = Åqg, eÅr•
Then we get tl'=di'=diN' :tP', Åë'=2r'=/fi'==di' and it is not diffieu!t to see that
Ve=diO=diO. Therefore we may conelude that qO is 'a closed subspaee of SP'
spanned by nicgÅr,sp{, --•, n.(g):JeA, where rti is the set of polynomials :.]ci,6(')(yi) of the Dirae measure o"(yj) at yj ----O in the space of the real number ri. In case of n=1, it is a simple matter to see that YrO=l7' (=the set of polynomials Ek,O(')) and hence diO==IT (==the set of polynomials 2c.x'). Therefore in this case we may conclude that LPAOO == {O}, lsgpsg+ co. This leads us in the general case of n=1, 2, ••• to the next
PRop osr rioN 1. LP AÅëO = {O}.
PRooF. Let feLPAÅëO, ,and we must show that f=-O. Put
fi (-Xi) = I f(-ri, -as2, - • •, xn) q2 (x2) • • - ipn (xn) dx2 • - -d-Vn
where goi(xi) is any funetion of Åë in the one dimensional xi-space, i---2, ••+,n-
Then one may conclude that fi G LPAeO in the .xi-spaee, and thus fi=O. Taking
Åëi(x:) E e and putting
f2 (-X2) = If(Xi , X2, X3, ' + ', Xn) Åëi (xi) q3 (x3) - - • can (.rn) dxi d.x3 • + - dxn
we get f2 c LPAdie and so we have f2=O. Proceeding in this way we may verify that the funetion
i fk (Xk) = !f(Xl, •••, Xk-1, Xk, Xk+1, •••, Xn)Åë1(pt1)•--Åëk-1(Xk-1)qk+1 (Xk+1) '''can(-rn)dxl'''d-xk-1dxk+1'''dxn
belongs to LPAee and henee fk-=O, for any Åëi(xi)Ee,i---1, --+,k--1. [l7herefore one may eonclude that
If(Xi, •+•, Xn) 9r i (•x'i)- -•On (xn) axi••Laxn = O
for any Åëi(.asi)E9,i=1, -•-,n, and consequently we get fi!O as desired. This completes the proof.
Owing to this proposition vve may infer that the application LP)f-Årf+
ÅëO E sp'/eO=Åë' will be a continuous injection and henee one may well write LP(Åë' and ZP E ]e', where """ may be regarded as taken in SP' as well asdin !V'.
Z Multiplicator a nd operator of convolution. Let ly l =(yl + • • - + y;, )l and put
lif( J') = max ((1 + l yI 2) l`, t vl, L , ''', ] rl. I ),
Mk (y) = (M(y))k,
fQr any )r==(yi, -••, y.) ER",k=O, 1, 2, •••. Then it holds that 1 == M,(y) sg da (y) sg M2(y) s{; -•-,
1
and Mk(y).+oo(k.+oe), where we assume o =+ee, coO==1. Let IFk be the
set of k-times eontinuously differentiable function ip(y) such that llÅëllk = sup Mk(y)1 D'ip(y)1Åq+oo,
ypezs,
where we write D"= ay{,O,i .lioy#. with p=(pi, •-•,p.), [pl=:pi+-+-+pn- Then Vfr
beeomes a Banaeh space wlth the norm llÅëIIk, and owing to the theorem of Ascoli
it is seen that the eontinuous injeetion llTk.i D ep-Åë E ITk will be compact. There-
fore we obtain, with respeet to compaet injection, a projective spectre
On Lieuvillc's Differentiatien ' 5
To-UTI-Yz-•••, co
whieh defines as its projective limit a dual Silva space ,A.,qk [9]. Let us prove the next
oe
PRoposmoN2. !ir=A!Vk.
h"O
PRooF. It' is elear that ATk(V with the eontinuous injection. Henee we k=O
di needonlyto prove ,I(A!Zrk [9]. To this end let eEilr. Then by means of k=O
Taylor's expansion we may write ak
D"Q (y) = yS• ay,k. DPQ (yi, •••, yi-i, eyi, yi+i, ••-, yn), OÅq eÅq 1,
.1
for any DP,k=O, 1, 2, -•• and i---1, ••-, n. This shows us that the fUnCtiOn Lrjtk
DPÅë(y) is bounded in R". Therefore bearing ip G sP in mind we may concluqe that Mk()t)PPÅë( v) isa bounded function in Rn, that is ÅëE?)Uk for any Ic. This completes the proof.
We here remark that for any ÅëE]P', DP and k=O, 1, 2, -t.•, it holds that MA(y)D"Åë(y).O (M(y).+oo), because
M,(y)lDpip(y)1= AAI,k.:1-7---)) M,.,(y)lDpÅë(y)]s: MC(y).
To define the multiplicator on iV it seems convenient to prove the following PRoposrTioN 3. Let Tbe a dist7'ibution, TE@'. Thetre vt.epT cleLtines a con- tinmous linear application of V into gV if and oney if the following two conditio7zs
are satisLfieal.
i) T= T(y) is an inJfinttely di.fferentiable functionfor y==(yi, --•,yn), yi'••+'yn ajo.
ii) .FTor any DP there exist an integer lcp:;}rO and a positive number Cp stteh tltat ID'T(7)1E{iC)Jlfk.(y) for any y, yi••••'yn=tO•
PRooF. The "if" part is easily seen as follows. Let ip E Yr and put Åë(y)T()t) ==O fO!' y, yi'•••'yn=O• Then ÅëT beeomes an infinitely differentiable funetion defined on R". By using Leibniz' formula it is not diMcult to see that
Mk(y) 1D"Åë(y)T(y)1Sn(k(y) Åri] cp, lDaT(y)D"'qyt (y)1 o'--eSp
Sg Mk (y) :E] cp, C, Mk, (y) 1D "MqÅë(y)1 '
Osesp
S{: c,tO..,Mlk+k,( v') lD"eg)( r) 1 Kc' 1[gb1it,
where cpe == '(ipiL i-/!) ! g! =(Pg',)-r•(Pgll), O s{:g E{gp means Os{ggi s{;pi, i--- 1, ---, n and l is
an arbitrary integer such that l2}tk+lv, for all g, Of{;gs{;p. This proves the "if"
part.
Let us now prove the `Conly if" part. Since the first eondition is elear, we .
need only to show the seeond, and with no loss of generalities we may assume
p = (O, t•-, O). Suppose that there exists a sequence {)r(k)}, y(k) =(ylk), ..., )rl,k)) G R", y(i"'•••••ptS,k'#O such that
l T(y(k)) l 2}i: kMk (7("')
for A:---O, 1, 2, +--, and it wi11 be shown that we should arrive at a contradietion,
Let h(Jr)Ge be
h(cr) = ii for ]yi :{g -li- ,
kO for lyl21,
and put
sbk(v)= M,(1),{k)) h(s;?/Lr()tim'ySk'), ••+, ";k) ()r.-yfik'))•
We first remark that ipkL E @ and that for any DP one may write (1) DPgtk(y) = M,(l y,,,) y2,i kP
)ip DPIt(r?k) (y,-ytk'), ••t, pt;,,,()t.-)tS,k')),
where )(k)P= vik)Pi•.++-itfikÅrPn. Therefore if any one of the inequalities I Jri-)tS-k'I }i: i lySt")l, i=1, ••t,n, is satisfied, in particular if yi•---•y.=O, it holds that
DPÅëk(7)=O. This proves dikEI?T, h=:O, 1, 2, •+•. We next show that the set {Åëk}
is a bounded subset of Vr. To this end take any integer l2}lO and we shall prove that the set of numbers {IIÅëklli}k-o,i,2,," is bounded. To give an estimate for c+)p KMtpi(y(k)) and next th at, as Mi(7)D'ipk(y), lpl:{gl, we first remark that
I'
noted above, it is enough to consider only those y which satisfy l7i"-orS•k']Åq S [yS•h'], i•e• S lyStk'1Åq.yiÅq : lyS•k'l,i----•1,+••,n-
For any such ay we may observe that
On LiouvilTe's Diffcrentiatien 7
(2) M( v) :i{gl max ((1 + 2 lv(k)I2)l, IJ,?k) r , ly?,k)t )S; 2M( )'(")),
and thus in virtue of (1) and (2) we get `
1 21Pt
M,(y)1DPgfk(y)1S{;1lf,(7) M,(p.ck)) lyÅqk)Pi 'C S(2M(y(")))`(M(y(k)))-k2iPiMtpi(y("))•c = 2iPt+t (M(yCfi)))t+ iPi-kc s: 22i (M(7(k)))2t •-kc,
where c is a constant satisfying ID"hl:{;c for any p, [plKl, It new follows that
il,(:i.I.).i,elÅëg.(l'a,iS.bt"."e,e,d8,/j,f9',M.XY,`:,I:.eG.a:,1•,Y"ia',9},1;,Z',b'E:•,l'ie2`l4Åëf,2i(,i,li2'i
:;}llc shows us that {ytk}tT is not bounded in !T. This beeomes a contradiction and the proof is complete.
It is noted that if in Proposition 3 TE IZ7'", then it holds T( v)=O for x 1•i•
•--• y. IFO, Thus T satisfies the conditions i), ii) and we obtain ÅëT=O for any
gb E yr.
Let gE]g' and let TE :;P' be a representative of g in ,9":g==T+!ZT". If T satisfies the conditions i) and ii) of Proposition 3, then for any gb E !Zr, gbTE Ur is
determined independently of a particular ehoice of a representative T of g.
Putting gÅë=ÅëT and we obtain a continuous Iinear application of pt into Yr. We call such a g E gVr' a multiplicator on Ur,
Let go E SF' be a multiplicator on YTi Then by means of the follovving identi- ty one may associate with any g E IP' a continuous linear form on gF, whieh will be denoted by gog:
Åqgog, 9t År == Åqg, goÅëÅr for any ip E ilt.
It is not difficult to see that the applieation IZr' )g.gog E IP" will be continuous and we shall say that go defines a multiplicator on V'. It is noted that if in addition g is also a multiplicator, onemay obtain gog=ggo and gog is a multipli- cator as well.
We now define an operator of eonvolution on Åë. Let fcÅë' and let f be a multiplieator on Yr. Then j7ipEep' for any caEÅë and it is seen that jFe= diS' for any representative S E SP' of f. Therefore we get (.fip)A=S*ca, where S*ca should be understood as the usual convelution between SE sp' and eeY [10]. This allows us to write f*g= (-iFe)A=S*q and we say in sueh a ease thatf is an opera- tor of convolution on O. It is noted that the applieation Åë) op.f*ip ee will be continuous, and if fis an operator of convolution then so is also f.
For any operator of eonyolution fe on Åë and for any fe e' one may define a
eontinuous.lipear form on e, which will be denoted by fo*f, by means of the following identity,
Åqfo *f, opÅr = Åqf, fo* opÅr for any qE Åë.
It rhay elearly be observed that the applie4tion Åë' ).f.fo*fE.di' will be eontinu- ous. Thus we shall say that fo defines an operator of convolution on Åë'. In case that both fo andf are operators of eonvolution, one may easily verify that fo*f is also an operator of convolution and we get fo*f= f*fo.
3. Liouville's uaerentiation. Basing on the preceding diseussions we now define Liouville's differentiation in the following way. We will first restrict
ourselves to the case n=1. Let Z be an' y complex number and let us define the functions:
,i--(xx Ig; ;kg•, ,i-Ig,i. Igsc2.}zg]
(y+,o)k=I:,x.,lyl, lgi ;ÅqÅrOol (y--.io)x=gi,.,I7i. igi ;ÅqÅrOo]
With these functions one may assoeiate pseudo-functions in Y' and elements in e' or tP' represented by these pseudo-funetions. In order to simplify the notation we shall make use of the saTrie funetional symbol to denote the pseudo- functien and the corresponding element in Åë' or iV'. Such a convention will not eause any centextual confusion.
rt is known that [2, pp. 83--84]
(pt + iO)X =yti + eirtXyxt,
(ot - iO)N =: yllL + e""X yi, (Z ]t - 1,,-2, ••-)
and
(v+ io)'dm .. y-'m -. iZ (.( ,- -l i)Mii 6(m-i),
(v-io)-m .., v-m+ i!i'iiiiliii)M!-' 6cm'o, (ni=i, 2, ••-).
We also notice that
(( y + io)N) '(x) = (2 r, )- xe iS'4X flll) ,
(( y- io)A)'(x) == (2n)'xe' i'i'Sti
rX' (-"llÅ}'`ll) ,
on Liouviltc's Diffc"rcntiation 9
for any complexZ[2, p. 2!9]. Here and henceforth for h=O, li2, ••• we shall takethesymb.ols 71' f Ii5--!-t) and rX(E';i) tostandfor(-1)fta(")-ando"(k)respeetive-
ly [2, p. 80]. Let us rewrite these identities in the following forms: - -X-1
x-
((-- 2r,i(y + iO))X)-(nt) = r(-Z) ' - -X-1
((27r,i(y- iO))N)'(x)= rX('- a) • '
Then it folloiks that fpti(--'l!'l) and r"'(-'I'i) wiil be operators of convolution in
e', beeause (--2r,i(y+iO))X and (2ni(y-iO))K are clearly seen to be mu!tiplicators on IP'. Therefore we may define the continuous linear alpplications:
-X--1
- -A-1
'Åë' il f. rX("-- a) *f E Åë', e 9 q.. rX("- z) *q E Åë
whieh we shall write ad .l,)f and ddiX, qrespeetively. Then by definition we get Åq i.lxf, ipÅr=Åqf, at..AN ipÅr for any fE Åë', gGO and in particular for z=k=o, 1, 2, ••• this becomes Åq d\.i f, qÅr =(-1)"Åqf, d4.k, opÅr. We here remark that
Åq d!It.f, C7År is an analytic funetion ofafor any fixed fEO' and gpEÅë. There- fore by means of the identity (2ni(y-iO))"(2r,i( v-iO))-" --- (2r,i(J,-ie))X"" holding for MAÅrO, suptÅrO, one may observe that
dK di- d•N+"
d.vN dx" ' d.rN'P
for any Z, Ii. Furthermore dgi,f is an operator of convglution wheneverfis so and it holds that
d4.XK (f*g) -( Åíi,K f) *g= f* ('daSK- g)
cl" .Liouvitle's cli.ffleo'e7ztiatien of for any gEÅë'. We will say the operator d.x
order Z. ' ' ,
-'' It is now immediate to defihe Liouville's differentiation for the case nÅr1•
Letting sr=(vi, •--, vr.)ER" and letting Rt(Zi, ,.., a.) be any n-tuple of eomplex t
10 iK. YOSHINAGA
numbers, we write
(2ni (y - iO))" = (27z i (yi - iO))"i @- • -X (2ni (or. - iO))Xn.
-
Then it holds that
((2r•i(y-io))x)'(x)-= flil'ii) (g).•,(s} )ll'(";"f.')
fOr .r==(x!, •--, ptn), and we shall write for any fEÅë' da."kf== o2.il;IH.5:"x..f==( rXii'i,') (2)'''(D 71\(tttX"ii) )'f•
Furthermore we get Åq ddx", f, geÅr=Åqf, di", caÅr for any gEO, where we write dd'.K, op = o.O'i;lliii':. op=( TX(Tl'i,') x-•+(s) 7X(-"i"i',I) )*q•
Further properties of Liouville's differentiation for nÅr1 may be easily observed from the ease n=1.
4. Multiplier. A continuous linear applieation A from LP to Lq, lsgp, g:{:+ oo, is said to be translation invariant if t,,A==Ath, h E.R". It is known that
any sueh applieation may be expressed as Aip=T*op for any op E J;n, where TE Y' is uniquely determined. f will be cal!ed a tnTultiplier of tmpe (p, g) [3].
PRoposiTroN 4. Let f be the multiplier of tiJpe (p, p) associated 2vith a t?'a7vs- lation invariant applicatik)n A. Then we itctve ff eZP and Af==(tf)v for any fE LP if any of tlie follo2ving conditions holdls.
i) 1 s{;ps{g 2 and t is a boacnded fecnction.
ii) ISp E{; + oe and t is a multipeieator on iP'.
In case of ii), if in particular t E 0M, we get Af== T*f fo7' any fG LP.
PRooF. Letting fe LP, 47k E SP and letting opk.f (h-)- + oo) in LP, we observe Aqk.•Af in LP.
i) If ls{gps{g2, then it is well-known that ipk-fEL", l; + -i;,- ==1 and
11ipk--fllLp's{ilIipk--fIlp--)bO (k.+oo). Therefore it holds tbat l1e(ip.k--f)1lLp.'-:,tO .iÅí
i is a bounded function, and hence tipk-,if in Y". Thus Aqk=(Tipk)'.(Tf)' in sp'. This proves Af=(tf)- if i) is,satisfied.
A AA
ii) In case that f is a multiplieator on 7F', we get TÅëk--)hTf in Y', because ipk.f in Y"'. Therefore (Tiph)-.(th' in e'. This proves Af= (ff)"' as desired.
+ If in particular fEe}f it is knowti that (Th-=T*f. This eompletes the proof
On Liouville's Differentiation 11
of ii).
A suMcient eondition for a function to be a multiplier of type (p, p), 1ÅqpÅq + oo, will now be given by the next theorem [7], [5].
THEoRE]f 1. Let F(or) be af2Lnction cleLfineal on R" ancl let a?il'.i'iX".k. F(y) be
continuousfunctio71sfoT Jt=(yi, •-•, y"), yi•••••vn=\O, satfisf?Jing
*) yfi••••'yE" o?i•l;l'l"a;'k,, F(y)1sM
for any lcj---O or 1,i=1, --+, n, whe7'e M is a positive co7zstant. Then F is a multiplier of type (p, p) for any p, 1ÅqpÅq + oe. FurtivaTmore it holcls tlzttt li P-'*op ll Lp s{; C•MIIqll Lp for any ca E {s;P,
where C is a positive constant aepending onty on p and n, so that tJtere exists a uniquely determined traozslation invariant continteous linear applicatdon A: LP ) f.AfE L", llAfllLp s{;C•MiifllLp, such tluzt Agp=li'*op for any q, E y.
The proof of this theorem given by S. G. Mikhlin [7] and P. I. Lizorkin [5]
is based essentially on the next-Marcinkiewicz' theorem [6]. Let Q., aÅrO, be the n-dimensional open cube {pt; .r=(.ri, ••-, .x.), -aÅqxiÅqa, i=1, ••-, n} and let us
denote for any function f(m), 7n == (n}i, -+•, m,t), .
dif(m)=f(nti, ---, nti--i, mi+ [3,II1 , mi+i) •+t, mn)
-f(Mr, •••, Mj.tl, Mi, nti+b •••, Mn), (Mi =IE O).
Then Marcinkiewiez' theorem says
Let ?L(nb) be af2enction of m=(mi, •••, rn.), 7nid-O, Å}1, Å}2i ---,i---1, ---, n, ana let A( be a pesitive consta7zt stccjt that for any integer afi, i--1, -+-, n,
2thj:+!1 2ctik'!1
**) IE] •-• Z ldi,•••di,Z(nb)lE{;M
lmit1-.2ai1 tmJ-kl-2aik
is satisjied, where ii, ••• ik are any k(ic=O, 1, •••, i}) diffeTent cltoice of 1, ••-, n, and nbi'--2dij"i or -2"i"iforiSFii, .•.,ih. Then given tjte 17Tozzrier se?'ies IIIi]c(m)e`Mr of a ftenetionf E LP(Åq2.), 1ÅqpÅq + oo : c(nz) == ! ,.f(x)e'-""'dx, it holds tltat Iil]a(m)c(m).
e"MX is also the }?ourier series of a f2e7zction gELP(Q.), a7ta we obtain llgllLp !{1 C.MllfllLp, 2vhere C is a constant clepending oney on p and n.
Being based on this theorem the proof of Theorem 1 will now be reproduc-
ed here in a slightly different manner from that of Mikhlin and Lizorkin•
PRooF of TEiEoREM 1. a) We first show that the theorem may be reduced to the case where F(y) has the compact support. Let h E e,
h(y'=[: i:i iil2il '
andassume oillil;'iX".,. h(v) is{Ici Åío.r kj=O or, 1,i--,1, •d•,n. Putting ht(Jt)=
h(-7-), l=1, 2, •••, one may infer that hiF has the compact support. By observ- ing
sO i,Tht(,)F(y)- } ,O,'l (4-}i--)•F(7)+it,(y) ,;, F(y)
and ht(y)=O for lyi1Årl, we get yi o?, hi(y)F(y)is{g2ciM, Hence,quite similar!y, it is not diMcult to obtain
yEt•--yfin oeii;l'.'5 "k.. ht(or)F(y) s;2"i""'knciMsl;2"ciM.
Therefore hiF satisfies the condition *) with M replaced by 2"ciM. By assump- tion that the theorem holds true for any F with the eompact support, we may
observe that gi=(htF•ip)"' == (hi.l7)-*g e L" with llgtllLp f{;C•2"ciMilql]Lp for any ip e ,9'
and henee that the sequence {gt} is bounded in LP. Furthermore given Åë E SP, it follows immediately that the sequence {Åqgi, ÅëÅr} will be convergent as l.+ oe and this implies that {Åqgi, f'År} will also be eonvergent for any f' E Li", }+-i:T,
==1 [1, Chap. III, p. 23]. Aceording to the theorem of Banach-Steinhaus [1, Chap. III, p. 27] we may find geLP such that. Åqg,, f'År.Åqg, f'År (l.+ eo) for any f' EL?' and hence IIgllLp:!gC•2"ciMIIqllLp. On the other hand it may be im- mediately seen that gi.F\' ip in @' and therefore we get g=i7*op E LP. The state- ment is thus verified.
' b) We next prove the theorem for Fwith the compaet support (Q.. Sup- pose op E@ with support (Qb and let us write ip(y)=!Q.p(x)e'2"'XYax with bÅqc.
Then we get
' e7(x) :(i)" :il]4b(-.-g:ti-)eiEcmr,
where the right hand series will be unifotm!y convergent o'n Qb. Putting Z(m)=
On Liouville's Differentiatien 13,
.di,.,.dik,t(nt) = !Åq."i"]iil,Mi'iiMJ'it)r2C..I(.M,iJlel' ,Mikt'MikiÅrt2C a),i,?;oy,ik F(ly)dlJtj,•••cl l'jk,
where yi----mi--- Å}2"j"i for i='Fi' i, +•-, ik. ' Hence we have
1 ai, +•• dikz (m) 1 s; ( iid'/ Mi'ii , •-- Illt'k l) ( i )k = i .i, li.i. .,i, 1 •
Consequently it follows that
.Oii+!1 2ptjk+!1 2CSil`;' ILI 2tuik+ 1.1 i
,,,Ihll.ll2ai, '''i.iA} .2aik 1aii ''' djkJZ (77i)l :{g Att [mh;.Il2ai,''' imi k;.l2a,-k lnzi,•••mi-kI
:EIIg 2oj,4.{2ptjk 20ii-••2`jJt•2"=2kMs{g2nAit.
- This shows us that a(m) satisfies the eondition **) with M replaced by 2"M. Thus letting g=(Fe)- =fi*op and letting
.#
g,(rx) = ( 21c )" :;;z(m)ip( 2Mc )etiMX
we get g, EL"(Qb)' with
!,, lge (-x) 1'd-x Sg C'MPI,, lq(x)1'dpt, d
and finally for any xE Qb we may find g (X) - gc (-") = I ,,F(or) Åqb (y) e2"X,dot
- -2a.ct.=mi-.z.c'F(-llli-)ip( :c )e2"i2nt=( 2ic )"-o, J :1.H.,n
as c-,+ee. It now follows by Fatou's lemma that
Ie, 1g(nc) l'{i-x sigC"ittt"I,, 1 {p (x) lP(itx,
and letting b-++ oo, one may obtain llgHLpsgCMIIgollLp for any go Ee. Then, sinee e is dense in LP, a standard procedure enables us to extend the applieation op-Årg uniquely to the eontinuous linear application LP )f.AfGLP so as to get il.tdfllLp s{gC•MllfilLp and Aop==i)*q for any qEsp as desired. This completes the proof.
5. An application. Let p, m,s be real numbers such that 1ÅqpÅq+ee,
m:;}le, s:}E:O and let abe any integer Os{;ctsgn. For any y=(yi, -•-,yn) we write y'=(yi, -••, r.) (62)1). By Y=ifcP.-,.,.) we shall denote the set of a!1 fG sp' sueh
that
tn sA ...
' (1+lyl 2) L2- (1+1 y'1 2)T f( y) EL'.
Then :2e' is a Banach space with the norm:
M SA
111flll = il((1 + IJt l2)r2'(1 + Iy' I 2)L2-f( y'))"llLp.
In fact, letting {fk} be any Cauchy sequence in .s2f' and putting gk== ((1+ l y12)Mt2.
(1+ ly'l2)"t2fk(y))-', we observe that {gk} is a Cauchy sequence in L" and thus gk.g (k. + oe ), g (f LP. Putting f= {?/(1+ [ yi 2)M'2 (1+ [ F' 1 2)S'2, we see fE LP, beeause 11(1+ IylZ)Mi2(1+ ly'12)Si2 is a multiplier of type (p, p) contained in ojt [Theorem 1, Proposition 4]. Henee we get fG .g' and fk.f in .sY. This proves
the statement. "
We next denote by .i/l=.t(l(g,.,,) the set of all funetions fELP sueh that for any e==1, -t•, 6, i'--a+1, •••,n
C)m+s Om
Om+s
o.T+s fe LP, o,,,ny fE LP, o,tl,v o.,s fE LP•
Let us show that .lt is a Banaeh spaee with the norm:
mfm'=UfllLp+t9,l]aO.M""+S,fllLp+,.tt.,lloO.Mm,.fllLp+,.tti.,t9,[loO../l"oS...,sfHLp•
- If {fk} be any Cauchy sequence in .It, then there exist f, f,,fi, ft,ELP such that in LP
Om+s Om
6}m+s
fk'f, oxy+sfk'ft, 0xTfk'fi' 0tny,Oxifk'fiL
as k-)F+oo, c=1, --•, if, j---a+1, •••, n. For any qEÅë we may write
om+s i;m+s
Åq oxp+s fk, opÅr= Åqfk, otpt, ÅÄs qÅr, and letting k.+eo we obtain
Om+s
Åqf. qÅrc=Åqf, o.f,n+s 9År'
This proves f, = aO af' .,' .S . f. Quite similarly we may infer fj--- E9.Mf. f, Dz=a.O.,M.o'-;.ff•
Therefore one may obtain fk--Årf (k--+ee) in .ld, and it follows that ./l is a
On Liouville's Diffcrentiatien ' 15
Banach space.
The purpose ef the present seetion is to prove the next theorem.
THEoRE"f2. .S!f'=.Ill Their7zormsareequiOalent.
PRooF. We begin the argument by provihg .,(t (.S?. LettingfE .i(d one may
observe for c=1, •••,6 r
(2n,i( v, - io))m+sf( v) = ( c/lllMi.', f)" c z,.
Then we get 1 r,1""'f(or)=Fi()t) ((2r,i(y,-iO))M'Sf()t) where 1 pt,1m+s
Fi ( P') = (2.i( y, -io))m+s '
Owing to Theorem 1 and Proposition 3 it is possible to see that Fi(y) is a multi- plier of type (p, p) and a multiplucator on V'. Hence by Proposition 4 it follows
that ly,IM'Sf(y)eZP and the application r
(1) L') oO .M t,liS. f.(1 y,IM'Sf(y))-ELP
is continuous, e=1, +--, o. In like manner we may conclude that 1),ilMb•,iSf(y) E ZP, lrj1Mf(y) E Z' and that the applieations
(2) LP ) o.IZ,iM i..i f. (l vi 1m[ y,[sf (1•))- GLp,
Om (3) L"e a..,T f.(1 rjlMf(y))-'EL"
are eontinuous for i=a+1, •--,.n, e=1, •-•, o. On the other hand let us write for e, c'==1, •-•, cr, l v,1M1y,,1Sf( v)=F2( v) (1+I v,lM'S+l v,t1M'S)f( r) where
- 1or,1'nly•,.ls
-F2 ( }t) - 1 + ly, [m+s+ I v,.Im+S '
- Then by means of the inequality l v,1M1 y,,1':ES siM +, [ pr,1M"S+ .,i\, ls'ttlM"', it iS not difficult to see from Theorem 1 that F2(y) is a multiplier of type (p, p)- Moreover Proposition 3 tells us that F2(y) is a multiplicator on Sl7'. Therefore by using Proposition 4 it is possible to conclude that lr,IMIy"ISf(y) E ZP and
that the applieation ,
(4) LP ) ((1 + 1 y,1m+s +• ly,,1m+.s) f-( y))-.(ly, 1ml sr,,1SjP (Jt))' E LP
is continuous for e, c'=1, •-•, 6. Quite similarly, by writing lsr,i`f(7)=F3()')(1+
ly, iM+S)f( v), ly, 1 MJfi( v) == F,( y) (1 + ] y, 1m+s)f"( y),
F, (y) = 1+lkl `1 .., , F, ( v) = 1 +l ;1 }1 ,Ml .., ,
we may conclude i v,lSf"( y) E Z", I y,lMf( v) EZ', and the applieations (5) LP p ((1 + l v,IM'`)f( r))' -)- (i)t,lSf-( y))' EL',
(6) LP) ((1 + ly,IM+s)f(Jr))-•.(1 v,1"f (y))'" ELP
are eontinuous for t=1, •••, 6. It is now a simple matter to see from (1)-(6)
r that (1+ ll]IJtiiM) (1+Sl v,l`)jfi(y) EZP and that the applieation
.i=1 `=1
e ''
-/t' p f--,• ((i + tl., l iti I"') (i + ,7, 1 v, l`) jf(y))- e L'
m is continuous. Letting (1 + 1y1Z)M'2(1+ l)t'12)S'2Ji'( y) = Fs()") (1 +,1.E..;, 1 ri1M) (1 + e 2 I7, 1 S)f(pt) where
s=1
(1+ 1 v1 2)Mi2(1+ l yt 1 2)si2
F,(y): E or ,
(1+ :E] 1 yj1M) (1+21 y,lS)
)' =1 ,=1
we may observe from Proposition 3 that Fs(y) is a rnultiplicator on Y'. Further-
more, sinee n'" Mt2 1y 1 M sl;l . ,1 vi l M ESn Iy 1 M, cr-Si2 i y' l S:{gt9, ly, 1 'sg6 i y' l S, it is not difilt-
cult to verify from Theorem 1 that FsÅqy) is a mu!tiplier of type (p, p). There- fore Proposition 4 enables us to see that (1+ l pt 12)Mi2(1 + 1 v' i2)Si2f( v) E ZP, that is
fEY, and /l)f.fE.s? isacontinuous injeetion.
To preve :2e (.t/l it is enough to remark that for any c= 1, •-•,c, i--6+1, +•t,n
the functions .
'
1 (2zi(y,-io))m+s
(1+ ly12)mt2(t+ 1 y,,12)s12 , (1+1 I, 12)ml2a+] yti2)st2 , (2ni( yj --- iO))M (2zi(yi-iO))M(27ri(y,-iO))S (1+ l yt12)Mi2(1+l v' 12)Si2 ' a+ 1rl2)mi2(1+ l v' l2)siZ
are all multiplierS of type (p, pÅr ah`d multiplieators on gP'. Then for any fE .S?
we get .f ( Ir) E Z', (2zi( v, -- iO)) ": "f( v) E Z', (2ni( yi -- iO)) nt E ZP and (2 T, i( yi -- iO))M•
(2ni(y,--iO))'f(y) E Z', and therefore fE LP, oO,':.,'.S. fE L' aa.:pt fE L', o,O,.,M. o'i, fE LP
as desired, This eompletes the proof.
' on Liouvme's Differentiation 17
ReÅíerences
[ 1 ] r,iT. Bourbaki, Espaces veeteriels topologiqucs, Act. Sci. Ind,, no, 1189 (1953), ne. 129-9 (1955), Paris,
Hermann.
[2] I. M. Gelfand and G. E. Silov, Generalized functions, r, 1959, Moskva (In Russian).
[3] L H6rmander, Estimates for trans]atien invariant operatoTs in LP spaces, Acta Math,, vol. 104 (1960) pp. 93-140.
[4] ., Linear partial diffTerential eperators, 1963, Berlim.G6ttingen.Heidelbcrg, Springer.
[s] P.L Lizorkin, Generalized Liouvillc's difrercntiation and functional spacc Ltp(E.), Imbedding theorems, Mat. Sbornik, vol. 60 (!963) pp. 325-353.
[6] J. Marcinkiewicz, Sur 1es multiplicateurs des series de Fourier, Studia Math., vol, 8 (1939) pp. 78-
9L i [7] S. G. Mikhlin, On multiplicators ofFourier integrals, Doklady Akad. Nauk SSSR, vol. 109 (1956) pp, 701-703.
[8] L, Schwartz, Theericdcs distributions, I, II, Act. ScL Ind,, no. 1245 (1957), no. 1122 (1951), Paris, Hermann.
[9) K. Yoshinaga, On a Iecally convex space introduced byJ. S. ESilva,J. Sci. Hiroshima Univ. Ser.
A, voL 21 (1957) pp. 89-98.
[10] andH. Ogata, On convelutions,J. Hiroshima Univ, Scr. A, vol. 22 (1958) pp. 15T-24.
Department of Mathematics Kyushu Institute of Teehnotogy
'
