Commentationes Mathematicae Universitatis Carolinae
Hana Petzeltová; Pavla Vrbová
Factorization in the algebra of rapidly decreasing functions on Rn
Commentationes Mathematicae Universitatis Carolinae, Vol. 19 (1978), No. 3, 489--499
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COMMENTáflONlS MáfHBiáTIC.Al OTIV11SITATIS O410I«INAl
1 9 , 3 (1978)
FACTOKZATION IN fHl AXCSBUA OF HAPXDI2Í immJkSim
fUNCTIOiS
Hana PETZIIlPOVi and Pavla VIBOVl, Praha
Abstract: A factorization theorem which is an analogy
of factorization theorems in Banach algebras is proved in
the algebra of rapidly decreasing functions on Bi» The result is closely related to investigations of existence of
faetorizaion in Fr^ehet algebras with an approximate unit.
Key wordsi Hapidly decreasing function, approximate
unit, Fre" ehe t aIge bra.
AMSs 46E25
Let IL be n-dimensional luclidean space. As usual, denote by It]
1 «-(t| +...+ t£)
for t » (t 19 ... 9 t n ) %\
and
l
n
|kl = k 1 +•••+ k^. Let us recall that i » (i 1§ ...^li? k •
a (k 19 «.. 9 k n ) by definition if i-^.4 k-^ i^-tkg,..., i n * kL
and 0 = (OjO,...,©). As usual,
CHI;) •••(;;)•
We shall denote by if the subalgebra of C^(IL) consisting of all functions rapidly decreasing at infinity, i.e.
xiC*tyj
tf * 4
sup |t|3l(i^x)(t)|.* w
for all noa-*
t*^
/
* negative integers j and multiindiees k
- 489 -
J
•XiC*(8,):
u
sup lti(DlcxKt)|<: oo
*n
(t 1 * t^1...^11) -
i,k^O
for all nulti-,
. .
/
indices
v
J
with the topology generated \® the system of pseudonorms
lxliv « max lt|J|(rfSc)(t)| •
* tcf^
Concerning the problem of factorization in projective
limits of Banaeh algebras there exists an approximate unit
in the algebra if which may he regarded as the projective
limit of Banaeh algebras Sf.j.. consisting of all functions
from C00 (M. ) for which the norm
nax I 1 u is finite. NaJ1
0&Uk
mely, the system of characteristic functions of D^ n
s
^
e
\ * ^'^k^ &
=
1,2,.,.) in R smoothened by convo-
lution with suitable functions from C^CIL) forms an approximate unit, unfortunately, this unit is unbounded in
each tf.^. It turn? out that the iterative process which
often provides a positive solution in many proofs of factorization theorems (see, for example, [1] — £ 91) fails to
converge here# Nevertheless, it is possible to prove existence of power factorization on bounded subsets of if with
the help of special properties of the algebra if »
1. Preliminaries*
Denote by w the function w(t) * I t I
for 04-teR • The function w is of class C**. Since
3wP
(t) =- p«w p
2
(t). t s for every integer p, it follows
ft
s
by induction that
(1)
(D k w p )(t) «
X
e(lfkfpMtlp~,kMl,t1.
0*l*k
- 490 -
If e s « ( ^ l s , . . . s <fnB) (a = l f 2 f . . . , n ) then (D**6 wP)(t) «
-
S
c ( l , k f p ) ( p - Ikl - t i | ) | t | P - l ^ - » 1 ! - 2 t 1 + e S +
06Uk
+
25 c U ^ p ^ l t l * - 1 * 1 " ' 1 1 t 1- " e . Hence, i f p > 0 ,
e £lUk
S
l e ( l , k + e 8 f p ) l 6 m a x ( 2 p f 3 | k l ) . max l c ( r f k f p ) | . The l a s t
O^rik
inequality can be easily proved! by considering a l l possible
cases- This, together with c(O f O,p) = l f yields
(2)
l e ( l , k , p ) U (max (2p f 3f.tel)) | k |
|(Dkwp)(t)Udk p,kl |t|p",kl
for t + O, k > 0 , p > 0 .
1.1. Lemma. There exist positive constants Cf 0^ (k>Q)
such that, for every sequence (m(p)) n of natural numbers
with m(p + 1) - m(p)> 5 for p = l f 2 f ... and m(l)>3, there
exists a positive function b*C°^(IL) satisfying:
1°
b(t) = 1 for | t U a ( D - 2
b(t) » wp(t) » lt|p for ( t U < m(p) + 2, m(p + 1) - 2>
(3) 2 0 l(ITcb)(t)UCk(m(p) + 2 ) p
(4)
b(t)*C(m(p) - 2 ) P " X
for )t I i <m(p) - 2fm(p) + 2> , p * l f 2 f ...f ki 0
3° b " 1 * if.
Proof.
Let (m(p)) -, be a sequence of natural numbers
satisfying m(p + 1) - m(p)>5 (p c 1,2,...) and m(l)2:3« We
shall construct a function b having the required properties
and we shall show that the corresponding constants 0y,t 0
0Q
do not depend on the choice of (m(p)) -,. Let a be a positive function defined by
- 491 -
(1 for | t U m ( l )
a(t) -W
[ltl p for |t| 6 (m(p)fia4p + 1)>
p « 1,2,...
¥e shall modify the function a so as to obtain a Qw
function. Take a function
f m 0^(1^) such that 0 h <p 6 lf
<£ s 1 in a neighbourhood of zero, supp 9? is equal to the
unit disc L
positive inside D., and TT<$ a 0 on
in B , y
0D-I for all nonnegative multiindices k. Denote by N^ *
(k2 0). Let (p^ be the function defined
» max l ( D ^ ) ( t ) |
p
« \
as follows
.
t}
,<$(t - m(p)
) for t4sO
V '{
1
0
Clearly, the function (f
class Q*° and supp q
for t = 0
is a well defined function of
* -ftsm(p) - 1 6 |t|6m(p) + If . We
shall show that, for every k £ 0 , there exist constants IL
(depending on k only) such that sup I (w y )(t)l 6 1L for
U\
a l l k ^ O and p = 1 , 2 , . . . . Denote f | ( t )
P
» tt
(1 -
- m i p H t l " 1 ) for t t t € < a(p) - l,m(p) + 1> . Since
(D^c* p Ht) - L ( D e » ) ( t ( l - m(p)ltr 1 ))(D e 8 9?)(t) i t
i»l
follows by induction t h a t ( i r y
) ( t ) i s a polynomial of o r -
der t k l + 1 in indeterminates (D^f ) ( t ( l - m ( p ) | t r 1 ) )
(16tj I 6|k|)f
( D 1 i y | ) ( t ) ( 0 6 1 . 4 k , i * 1 , 2 , . . . , n ) . Hence,
i t i s s u f f i c i e n t t o show t h a t the d e r i v a t i v e s of y f
are
bounded by constants which do not depend on p . We have
(D d ^|)(t) «oTiQ - <f±Bm(p}\tr1
+ a ( p ) t i t 8 l t r 1 for j « e s
and
( D ^ y P ) ( t ) * - m ( p ) [ t i ( D j w " 1 ) ( t ) + ^ ( D * ^ w " 1 ) ( t ) ] for Ut-52.
- 492 -
According t o (1) there e x i s t constants t . sueh t h a t ,
for I j l 2 1, I t I e < a ( p ) - 1, m(p) * 1> ,
I(I5^cff)(t)Uejm(p) | t l " l j l 4 2«J#
Now, s e t
i
CC y p »)*<y )Ct)+Cl-ypCt)iaCt) for H i €<mCp)-2,aCp)n-a>
p • 1,8,..•
a(t)
otherwise*
©lis function belongs to C ^ C ^ ) and satisfies !*• Given a It) e < m(p)-2,aCp)+2> , t-frmCp) we have, according t®
CD,C2)
)CDkb)Ct)|A I / ynCx)a(x)Cl^y)(t-x)dx| + 38 ( i / I C ^ C l • yJHtHD^aK^UNL
p
snp
JC
i ki
:lt-xi^H)+
aS ( xi / %
Q&Uk
S
06Uk
( 3 %
X%/
^
*
S
*m
maxC2p,3|k-i|) , k ~ : y .
OJj^k-i
itJUNk#nC-fx;lt-xUli)Ca(p)+lp^
,t,p-|k-l!-UI
+
aCx)A w «xs
|x«i<m(p)-l,mCp)+l>
S
liaxC2p,3|lo-il) ,lc " i, • Itr l k -%m{p)i.
0.4j*k-i
• 2) p irC k CmCp)+2) p ,
where C, are suitable constants which are clearly independent of (ffl(p))^L«i and ^
is the n-dimensional Lebeaguf me-
asure in 1 • The last inequality follows from
(5)
2p£m(p)-2
p » 2,3,.••
To obtain estimate (4) demte tsy M -*-ft| ^ ( t ) ^ l / 2 ? and
%p*H; ypMz
1/2} . Then
.. 493 -
b ( t ) ^ | a ( t ) for tf.M p and so b ( t ^ j
(M(P-2)P"1
for t c l l t U <m(p)-2,m(p)+2>fx ^
Observe that 1^ »
* 4 t | t - m ( p ) t / | t l € M|. fake € , g' p o s i t i v e such t h a t
- H i l t l * * } c M e l t ; It I 6 e'f
+e
#
and S + €,'< 1 . I f l * cf*>e +
then, for each t e l L , the s e t Kp » 4 x * Mpt Ix-t l-c cf f
contains a b a l l K* »-f x c r f x : l x - M ( p ) t l t l " " 1 l < s i for a l l
p * 1,2,...
b ( t ) g fJ
fhen, for t#ML f we have
a M (x)a(x) c p ( t - x ) d x r 2 " 1 inf a ( x ) . inf gKz).
XpTp
xfiKp
I*!**/
• ^ n (^)^C.(M(p)-2)P'* 1
where 0 is a constant depending on €>, cf,n, cp only.
We have to show now that b" % tf , i.e.
sup
max
lt|^l(Bkb""1)(t)|-<«
p ttU < m(p)-2,m(p+l)-2>
If f, f - 1 # 0*K\)
for all j,k> 0.
then I^f"1 » F ]c (f f ... f D k f)/f ,k,+1
wher© P. io a polynomial of order I k I in indeterminates
I f for 0-fei4k and the coefficients of W^ depend on k only.
Fix j and k. First, let m take It I e <m(p)+2,m(p+l)-2>
for 2p£3l-cl. Then b(t) • w ^ t ) » |t| p . According to (1),
(2), and (5) we have, for arbitrary nonnegative multiindex
m£k
K#wp)(t)l £ dmpl,tt,itlp-,lilkdiaitlp
I t follows that
Itl^KlV^HONItl^
(t)Uw-p(lkl+l)(t)^ |t|J|,
( ^
(lBJwP)(t)))Ik»ltrp(Ikl+l)^
* O^j^k
*)tl*k ( max d 4 ) l k , | t | p l k l . | t r p ( , k l *
where M^,!^ are suitable constants.
- 494 -
1}
- M'ltl^
Now assume |t|c < m(p)~2fia(p)*2 ) • We havef according
to 2°, the following estimate
. i, v -1
t.J|(D-Ъ-)(t)|4
.1
J
|tl
plkl
C'(m(p)+2)
-
C , k , + 1 UlpWl'P-1"!-!*1'
^ C ''(l + — - — ) P l k , + J . (m(p)-2)-P + J + n c , + 1
*
m(p)-2
*
A 0^(1+ i)P' k ' + J. ( m ( p ) - 2 ) - P + J + l l c , + 1 .
The last expression is bounded and so the proof is complete.
2
* x * Theorem.
Let K be a bounded subset of tf . Given
£ > 0, s Q natural and/ j Q E 0, k 0 « (kol,. .•fkon)-S 0, there
exists an a in if and a sequence (K.)^.^ of bounded subsets
s s*~x
of *f with the following property; for x c K there exists an
y a « K n (*f x ) ~ with
9
a
•
1° x * asyfl for s * l f 2 f ...
2° lx-yfl|. k * i, for s = l,2f...fso.
"0 0
Moreover, if x^—** 0 in if then the corresponding y g^ tv»JgQ
for each s.
.Proof. The subset K is bounded, so there are a positive function h and nonnegative constants If^^^n aueh that
sup I t H h(t)<M.<iX}
for all j 2 0 and I (Dkx)(t)t # fvh(t)
UMn
J
for all t t ^ , k ? 0 , x i K (see L103f p. 235)• Denote by
Q„ » max qkv and take a sequence ( €n)*n
of positive numP IkUp
P P-i
k
bers. Since D^r is a polynomial in mdeterminates itif
H P 1 , t p , . , , ^ (t+0) it follows from (2) that we can find
a sequence (m(p))^L1 such that
. - 495 -
( i ) I t | j | ( ^ w p ) ( t ) | h(t) £ e p for |t|^m(p)-2 and a l l
I k U p , O ^ r ^ p 2 , j£p(p+l)
( i i ) I t ! ® + g ° P K D V ) ( t ) | h ( t ) ^ e p for a l l p « 1 , 2 , . . . ,
O^ri-s 0 p, OAk£k g t ttl2»a(p)-2
(iii)
a(p+l)-a(p)£ 5, a(l)2:3
It fellows that
(6)
|ttjt(BV)(t)t l(Dix)(t)|^Qpep
for | t t > a ( p ) - 2 t | i | , | k l * p , O ^ r ^ p 2 , j ^ p ( p + l ) t xcK and
(7) tfl • * i ' P l ( l f t ^ ) ( t ) | l ( S i x ) ( t ) U Q k ep
o
fer | t l 2 a ( p ) - 2 t 0 ^ i t k A k o # 0 ^ r ^ s 0 p f x#K and p » l t 2 t . . #
Let us take a n x c K and consider the factorization of
x in the fora x * b**s(b8x) where be is the function eorresponding to (a(p))
2 aeeording to Leaaa 1.1. Ihe function
—1
b
«-»s? -
belongs to if so that b
belongs to if as well. We ha-
a
ve to show that b x is in tf (s * l t 2 t ...) t i.e.
•up
aax
| tl^l(Dkbilx)(t)|<flp for all j s O t
p •(p)^l|tMi(p+l)4
sfcl, kltO. Hairing known that b 8 x * f it follows that
b l ii(^i)" , # Indeed, t}iere exists an approxiaate unit
(e m )? Ml ia V consisting of functions with compact supports,
so that b*x * lia © j ^ x for x e if , a « l t 2 t ... . Since
e^b11 are in if as functions with eoapaet supports, we obtain b*x*(ifxr.
Fix jtk and s. If p^aax(lkl t j t s) we obtain according
to (6) t for It I a <a(p)+2ta(p+l)-2 > f the estimate
t t t ^ K J ^ ^ x ) ) ^ ) ! » |tlkl(Bk{w*Px))(t)| - 496 "-"*
*|tl j 2
(i)|(i>V p )(t)| K M j t t J U ^ t e . , .
Oid-rk
**
p
Now, assume ttl i<m(p)-2 f m(p)+2> . I t i s easy to see t h a t ,
for f c C ^ C L ) , i r f 8 i s a polynomial P. „ of order s i n i n JC,S
2
determinates ff...fIrf. Again, according to (6), |t|p * p »
|(Dk-ix)(t)l --Qj.
This, together with 2° of 1.1 yields
|t|Jl(»kbsx)(t)|*|t|J
3 * (l:Li ) | ( D 1 b a ) ( t ) | I ( I r ^ x H t ) | 0*i*k
k
s S
( J | P i1 s fl(b(t),...,(Dib)(t))I
i t | d J{JD k " i x)(t)| 4*
%%
0*Uk
»
2
2
K
sp
* * (!)
|t|J"p -p|ttp ^ K D ^ x H t ) ! *
17
xi ss Cm(pH2)
(Ui-ttA
»
£(m(p)+2) s p ( m ( p ) - t ) " p
. Qc
p p
6(1+ —1
) S P 26 ( J U , fl
m(p)-2
O i i * k * i ; x*s
g
( 4 ) K. fi A
% a
0.ti.tk S1
*
Q * i where K. _ are s u i t a b p p
x a
*
le c o n s t a n t s . From the l a s t two estimates we obtain
|tlJ|(Dlcb<,x)(t)Ul^
8
. Qpip for
tt|6<m(p)-2fm(|^I)-2>f
p large enough. According t o (7) we obtain, for s - l f 2 f . . # f s
f
a
p = l f 2 f . . . , similar estimates
*L k^
t t t °|(D °b s x)(t)|.lr Mgpf where Mf M^
fl
are suitable constants.
Given an e > 0 , l e t us choose now ( ^n^nsi
80
that
Q g —» 0 and max Me 6 e / 2 . Using these estimates we can deduce the following f a c t s . F i r s t , b s x # t f for x * K , s = l f 2 f . . . f
a l l bsK » Ks are bounded i n tf and | x - b s x l ., » max | t t °
jfl^o t * I ^
#
|(D ° x ) ( t ) - ( D ° b s x ) ( t ) l « max
t t ( J ° i ( D ° x ) ( t ) - ( D °b S x) ftjjj
tttEm(l)-2
--» £
for s«l f 2 f ... f s 0 . Finally, if x ^ — • 0 in {f then,
for K * (JL)?.,, we obtain y n s baxL tends to zero as well.
Indeed, let us fix j, k and s. Given an % > 0, let us find
- 497 -
p Q so that M^ gQpSp tc e
^or p 2 : p 0 . There e x i s t s nQ suth
t h a t , f o r » B n a f i x n l 1J1AJ l 6 ( 2g f ) sup
| ( D i b i ) ( t ) I )**1
•
*
0*:U1 * i ' |tlim(p 0 )
for l * k . I t follows t h a t , for n £ n 0 , we have
ty„l4v
a
JK
s
aax (max
| t | j HD^VHt)!*
|tlfem(p0)
^
max
1tlJl(DkliV)(t)|) 6 e
ltU*(p0)
The proof i s complete.
Bemark.
Since the Fourier transformation i s a c o n t i -
nuous l i n e a r mapping of tf onto i t s e l f and! takes the pointwise multiplication to the convolution f Theorem 2.1 holds
also i f we replace the multiplication by the convolution.
Acknowledgment. Hie authors wish t o express t h e i r grat i t u d e to J . Fuka, ? . Mttller and M. Zajac for t h e i r kind
help during the f i n a l preparation of the pape^#
B e f e r e n c
es
£1] G.E. AH*ANf A.M. SBfCLAIH: Power factorization in Bainaeh algebras with a bounded approximate unitt
Studia Math. 56(1976), 31-38
[2] 1. (HHISTIANS1N: On factorization in Irdchet algebras
(unpubiisfied)
[3J
P.J. COHEN: Factorization in group algebras, Duke Math.
J. 26(1959), 199-205
[4] l.G. CHAW: Factorisation in
Fr^chet algebras, J. Lon-
don Math. Soc. 44(1969), 607-611
[53
P.C. CURTIS, A. FIGA-TALAMNCA: Factorization theorems for Banach algebras, in Function Algebras, ed. F.T. Mrtal, Scott, Foresman end Co.
Chicago 1966
- 498 -
16] J. KlfŽKOVl, P. ?i©0¥Ís A remark o» a faetorizatiom
theorem, Comment. Math. Univ. Carolinae 15
(1974), 611-614
[7] V. PTÍK: Deux théorěmes de factorisation, Comptes fiendus Ac. Sci. Paris 278(1974), 1091-1094
[8]
?. PTÍK; A theorem of the closed graph type, Manuscripta Math. 13(1974), 109-130
[9]
?. PTÍKs Factorization in Banach algebras, Stuiia
Math. 65,2 (in print)
[ 101 L. SGHWAOTZ; Théorie des distributions, Hermann, Pa­
ris, 1966
Matematicky ústav ČSAV
Žitná 25
11567 Praha 1
Československo
(Oblátům 12.5. 1978)
499 -