VNỤ JOURNẠ OF SCIENCẸ Mat... tlu' symbol .V... TAF CHI KHOA HỌC DHQGHN, KHTN, t.
Trang 1VNỤ JOURNẠ OF SCIENCẸ (Mat Scị, t x v - 1999
M U L T I P L I E R S F O R G E N E R A L I Z E D
e n t i r e D I R I C H L E T s e q u e n c e s p a c e s
T r i n h D a o C h i e n
Girl L hi Ed u c Ht i o i i i ì i ì d Tvri i i i i ng De pHi ' t I i i e nt
Ị I N T R O D U C T I O N
Given a ipquencc (ÀA-) with Ậ G c , 0 < |Aạ| t +00 and p > 0 , consider tho gpneraliz'i
entire Dirichbt series
oc
k=\
where coefficents an' fomplpx nunibers and Ep[z) is tho Mittag-Lefflei funct ion:
( r being the G a m m a function)
, f r o r ( i + i)
In [5] W( piovfd that if the spiios ( 1.1) convorges absolutely for all 2 G c
log In
l i n i s u p - ^ f — = - 00
a n d conversev, if t H coofficient.s of th e s.n ios ( 1 1) satisfv coiiditioii ( 1.2 ) and if
lôA‘
i i i n s u p — < -1-00
|Aa-flien the serũs ( 1.1 couvvv^cs absolut(‘ly for all z e c
N e x t , i n t h e C<SP ÍÀ;.) t h ' r r , n l i t i o , i (I :i) i „ car,., o f [2],
w<-t h e f o l l o wi n g s o q uo uí ' s pace
4 = {q : ca- satisfies (1.2)} = {(r^.): lini s u p | a | I" = 0}
A ^oc
D e n o t e d b y tli( Kothe d u a l of Ạ ịẹ
CXj
= {("a ) : ^ |r/, < + 0 0 for all (f>) e A}.
Ả-=1
we proved th it A ‘^ = c = A Honco C" = A whc-io
c = |(i/Ạ) : lim S lip < + 00Ị
F u r t h e n i o i p , each c = (c*.) e Ạ wo dpfined
i /IA a T
A = sup Ct
*■>1
Trang 2Iu [5], 1)V u s i n g t h e saiiu' i n r t h o d as iu [2], \V(' Ị)rovO(l tliat Ả is a C()iu])l('t(' s('pariHlt)l('.
noii-iiormablc n i o t i i z a h l r space, whoK' th(' iiiPtiic is j;iv('u t),v
( ỉ I > ) = II" - ^’11.4 ; " = ("/ ) e
I n t l i i s n o t e VV(' c o i i t i i n H ' t o s t u d y m u l t i p l i i ' i s t x ' t w i ' c n t h e s o s p a c e s a n d o t l i M s e q u i c n c i '
spaces oil spaces A ami c.
w v K'call th a t for two S('qu(nic(’ spaces A' an d y tlu' symbol (.V Y ) -li'iiotrs the
s('(ỊU('ncp s p a c e o f m u l t i p l i e r s f ro m A' t o i P.R [1]).
(A' Y ) = (a"A- ) e 'i' foi all ( a ) € A'}
It is o bvious tliat if
A'l c A\> and V'l c Yo- tlieii (X->- i ' l ) c (A'l, V'i) (1.4)
Also it is c lea r t h a t , t h e K o t h c d u a l of a soqiiencp s p a c e is, in fact, t h e soqtLK'iK'c spaco of m u l t i p l i e i s fr om t h i s s p a ce t o /, i.e (-4, h) = A ' \ A q u ps t i o n arises: w l i a t aibout
multipliois from A an d c to I,, (0 < p < + o o ) and vifc-veiHH'.' This is the su b je c t i:)f the
p m s e n f n o t e
1 would like t o expif' ss my d e e p g r a t i t u d e to P r o f XRuyt-ii Van M a n a n d I ) i L.f' Flai Khoi for helpful stiggostioiis ill t h e p r e p a r a t i o n of t h i s p a p o l ,
II MULTIPLIERS FOR GENERALIZED E NT IRE rJIRICHLET S E Q U E N C E s p - \ c ? ; s
Fi r s t \V(‘ not(' tli(' followiui> li'sult
L e m m a 2.1 U e hrive
A d i , c l o o C CẢ) < p < + 00.
Wo provo th e following l(>ninia.s
L e m m a 2.2 W'e lìỉ ìvc
a) C ) c C
l > ) C c A ) ,
c) c c (C„ C).
Proof:
a) Lf't (i/A.) e {A C) Sr.pposo th a t (»A.) ệ c T h e n for a ib itr a r y M > 0 a n d foi a
s cquriicp (f,,), 0 < e„ i 0, tlK'io exist s an iiicrea.sing scquencf' (A-„) of positive
s uch t h a t
> M - £ n V/; > 1
We defiue t h e s e qu e nc e (c^.) as follows
nk 1/2 i f n - 1 , 2 , ,
Ck =
Trang 3M u l t ọ l i e r s f o r Generalizec E n t i r e 3
1 / 2
<
li o n Wf* liaví'
-oc
S' ( a - ) G A H o w e v o r w e lavo
;1I11
n —oc
l i i n s u p ( A / - f - > o o as M + O Ũ .
n—oc
T his iiplies th a t (Cf, m.) ệ c w iid i leads to a rontradictioii
T i p implications b) and c) aiP obvious □
N i w WP c a n p r o v e t h o f o l l c w i i i g r e s u l t
T h e o e n i 2 1 We have
( ^ 0 = (/,„ C) = (/^ , C) = (C, C) = ( ^ , = ( ^ , Ì,,) = ( A /oo)
PTOof.Vvom Loiiiina 2.1, Lenur.a 2.2 ami (1.4), it follows th a t
C’ c ( A c {A I,) c ( A loc) c {A, C) c c,
= c.
an d
C’ c {C, c c (/oc C) c (/;„ C) c ( A C) c c
T f tli(*oiPin is proved c
we prove the
followiiiJ-L e r n n i 2.3 We ỈIỈÌVC
a)'/,, .4) c .4,
hj C, U ) c A
c ) A c i C, A )
Proof:
a )-irst, we Iiotp th a t (q.) E a ÌỈ and only if { (ị) G A (with any ap p ropriato choicp
of th e ow n ) Furthoi more, we can check th a t the soquencp (Aa-) satisfies C ondition (1.3)
if and aily if tlioir exists a > 0 such that
CXD
*-=1
Nw let (;/*.) e {Ip, A ) Suppose th a t (i^.) Ệ A , which means t h a t {uị) ị A Then there eists M > 0 such t h a t for a sequence (£„) ị 0 , there exists an increasing sequence
(A:„) of)ositivo num bers such th at
Trang 4T ỉ-i nh D a o C h i i e n
Ai > M - £„ , V7Í > 1
This implies th a t
kn
{£„ - A/)|Aa„ , Víí > 1
Dofinp a sequoiicp (r^.) as follows:
exp
Ck =
0 ,
, if k = k'n, ÌÌ — L 2,
ot herwiso,
where 7 < i\I — a an d a > 0 is defiiKHl by (2.1) T h en , we have
'( 7 - i n ) aa -„ r]
<
due to (2.1) which shows t h a t (ca-) € Ip However,
lini sup
k—*oc
log
A a = lira sup
log f'A-„ »A'„ p
Aa-„ p
= l i m s u p (7 - e„) = 7 > - o o ,
n—*oo
= 0 < + 0 0
duo to (2.2) and (1.3) Hence (ct) G c. Howrvor
sup | c A i/A-| = sup l a - , , i/j.,, I = sup h„ = 4-00
Hence {Ckĩỉk) ị loc- a contradiction.
c) T h e im plication A c (C, A ) is obvious □
We can prove th e following
( ( 2 2 )
which moans th a t ịịct,-ÌIk-)’’) Ệ Ả or {ckiik-) ị Ả. T h is is a contradiction Hf“iìC(’ {Ij,, ^ 4 ) c
b) Let (iu.) G (C, /oo)- Assume th a t (?/.*.) ị A , th en t h n c exists au increasing s v q n fv m v
(Ả'„) of positive' Iiuiiihois suc’li tliat
lilll |»A.„ I " = + 00
n —► OC'
Consider a soquonce (aO as follows:
f A - „ / | ỉ / a - J , if />■ A-;,, ÌÌ = 1 , 2 , , ,
C ị ,- — <
Then we have
Trang 5T l e o r o m 2 2 W’v liiiv(>
(C 1 ^ ) - {C I„) = (C (/^ ^ ) = 4.
P tio /: Fioiii l c i i i m a 2.1 l.ciiiiiia 2,:i a n d ( 1, 1) it foll ows t h a t
^ c {C A) c {1^ A) c A ) c A
R f i n a i k riKHjH'in 2.1 a n d 2.2 for i h ( ‘ (Ji(linai \' Di l iclil('ĩ s('i i('s OÍ 0 1 H' a n d sf'\'(Mal ('(Jiiiplrx
varal)l(' s wTvr ỊM‘0V(‘(1 in a n d [4ị.
R EFE R E N C E S
[1 J M A i i i l c r s o u cV A L S h i e l d s C'cx'fficient I i i u l l i p l i c i s o f B l o c h f u n c t i o n s , Tnnis Aiiier Math Soc. 2 2 4 ( 1 9 7 6 ) 2 5 5 -2 0 5.
2 Lc I lai K ho i i l o l o i i H H p h i c Dilicli lot s e r i e s iiis(>v(‘i a l v a i i a h l c Math Scnriil 7 7 ( 1 9 9 5 ) 85-11)7.
3 Lc Hai k h o i M u l t i p l i e r s for I^iiichlf't s c r i e s in tli(' coiiiiili'x |)laii(' S o u t h - E a s t A s i a n
Mat h Bull ( Ĩ(J aỊ)Ị)(‘a i ' ).
;4 \ j ' Hai l \ h o i C o e f fi c ie n t m u l t i p l i ( ' i s f(ji SOUK' classc.s o f Di ricl il ct s c r i e s ill s('C(nal Í'()U1Ị)1(‘X \ ar i aỉ ) l ( ' s A c ỉ a Md i ỉ i \' ỉ( t ì i af i i ỉ ( ‘ti ( í o apỊ)(' ai ).
5 l n n l D a o ( ’l u c n S c q u c u c c S])acc ot Co f ' ff ic ' in i th o f o('iicializ(-(l (-nt ii c D i i i c li l ct S('IÌ('S
\ 'N Ư Journal o f S c i n u T X a t Sci., I X I V \ o K 1 9 9 8 ) 8-15.
TAF CHI KHOA HỌC DHQGHN, KHTN, t x v , n‘’ l - 1999
M IA X r i ” CVA KHOXC; C;iAX 1)A\'
i ) iH ! ( 'iií j: r XCỈUYKX s r v íU).\c:
l Y i n h Đ à o C h ìố ii
S ờ (Ỉiỉía (lục viì Đ à o ĨỈÌO (Ỉiỉì ĩ^ỉỉi
V (/] l.ai klioiip, ^ian (là\- A \'à V, khoii^ gian ílã>' cua các Iilian tir t ừ .V \'ào y \ kv
là ( A ) ) (lìrtrc x;íc (lịnh ỉilur sau ( A '.)') := {(///.): G V(V‘A-) G A'} Xót khuip ” ia:i (lãv A cái' liọ s u c ù a cli uỗi Di r ic l il ct s u v r ộ n g (lạiio Ỹ2 í r o i i o (ló
E ậ ị là h à i n M i r t a o - Lofflor Q u a i n o t ả klioiift nịaii A ' ' đ ố i n g ẫ u K ỏ t h e c ù a A t a t h ấ y ră:ií { A l / } ~ t r o i i o (tó /i = {(///^.); |//^.| < o c } M ộ t c á u h òi đ ặ t r a : k ố t q u ả sõ
n h ư th ế nao (lối vái các khono »,ian (lãy n i a các Iihán từ fừ A A ‘' vào rá c khong «ian
qupi t h u ọ ' khác, c h ẳ i i - han /,,(0 < p < o o ) , / o o v à Iigirợc lại? B à i b á o Iiày S(' đ ồ cập
đ ế n c á c noi (luiiíi íló.