Multiplier extension and samplin theorem on hardy spaces

Publicacions Matemàtiques, Vol 38 (1994), 441-454 . MULTIPLIER ExTENSION AND SAMPLIN G THEOREM ON HARDY SPACE S QIYU SU N Abstract Extension by integer translates of compactly supported functio n for multiplier spaces on periodic Hardy spaces to multiplier space s on Hardy spaces is given . Shannon sampling theorem is extende d to Hardy spaces . 1 . Introduction and statement of result s The purpose of thispaper is to establish a natural extension fro m multiplier spaces M(p) on periodic Hardy spaces HP (T) to multiplie r spaces M( p) on Hardy spaces H P (R) by integer translates of a functio n ~ and to extend Shannon sampling theorem to Hardy spaces . It is th e continuation of [13] on stability of integer translates of a function bu t with different interest . In [13], the following stability problem of intege r transiates of 0 ( 1 )  c-l llf IIHP(Z) E f(n)O(x — n ) nEZ Cllf 11HP(Z ) HP (R ) was considered which arises in the interpolation of sequences by function s and plays an important role in multiresolution analysis, where o C p < oo, f = {f( n )}Ez is a tempered sequence, H P (Z) and HP (R) denote s Hardy spaces on Z and R respectively, and Z is the set of integers . A natural replacement of the norm in (1) when p = oo is the nor m as multiplier operator on H P (T) and HP (R) respectively, which is a n This project is partially supported by Postdoctoral Fellowship Foundation of Chin a and Natural Sciences Foundation of China . 442  Q . SU N original inspiration to consider multiplier extension here . To this end , we introduce some notations . ~ Let ~ be a smooth function such that supp ~ c {4ó Ç lxl Ç A} , O(x) > Co on { ó < l x l < 1} and EmEZ .P (2 m x ) = 1 for x O, wher e ¿)(x ) Co is a positive constant and the Fourier transform is defined by _ f e 2 ' i ' y+ P(y)dy . Denote (D,(x) = 2 m c l9(2 m x) for m EZ . Now we defin e Hardy spaces H P (R) b y H P (R) = f E S'(R) ; I_ (E Om * fi2 ) -m .E Z L p (R ) and define Hardy spaces H P (T) (c .f . [3D b y HP ( T ) = .Í(x) = E fke2irzks E S t( R ) . kEZ E P .* f 1 2 m> ~ 2 L P (T ) where we denote the space of tempered distributions by S'(R), the nor m of p - integrable functions on R and T by ii • 1ILP(R) and by Il • l , respectively and T = R/Z denotes the torus . For a measurable functio n m on R, we say that m is a multiplier on H P (R) i f (2)  II F IIHPR) Ç c mll f 1111P(R ) „  . . holds for any Schwartz function f, where F = m f . We denote th e infinum Cm in (2) by IImIIMP . For a sequence m = {ñ(n)}, we sa y that m is a multiplier on H P (T) i f (3)  IGII H P T ) ~ c rn II9II HP(T ) k 27rikac E S ' (R) 5 kE Z llf 111IP(T)  ifol + E f k e 2nik x 2 m C~k 1 C2 7 ' 41 i p di MULTIPLIER ExTENSIQN AND SAMPLING THEOREM  44 3 holds for every trigonometric polynomial g(x) = E kez g k e 27rik ' , wher e G(x) _ E kezrn(k)g k e 2 " rikx . Also we denote IIrnI1]() the infinum C, ñ in (3) . The classical result of de Leeuw [7] on multiplier said the restriction t o the integer lattice of a continuous multiplier on L P (R) is a multiplier o n L P (T ) when 1 C p < oo . In 1992, Liu [8] extended the above conclusio n to Hardy spaces . The multiplier extension was considered by Jodeit [6] , Berkson and Gillespie [2] . Let be a continuous function with compac t support . Denote the space of sequences by S and the linear span o f integer translates of by S(0) = {> mEZ C(n)O(x — n) ; {C(ri)} E S} . Define a natural map O' from S to S(0) b y 0 * ' :  {C(n)} C(n)O(x — n) E S( q5) . nE Z W e say that the integer translates of are globally linearly independen t if O*' is one - to -one . Denote the restriction of O ' on M(p) by 1 . Berkso n and Gillespie [2] proved that 1 maps M(p)to M(p) boundedly unde r the hypotheses 1 C p < oa and ~ = x [_1,1] * Ao, where x [_1,11 is th e characteristic function of , * denotes the convolution operato r and Ao is a bounded variation function supported in [-, 1] . In thi s paper, we will prov e Theorern 1 . Let o < p < oa and ~ have compact support . I f f I~~ x} Imin( 1 fp) dx C +oo, then 1 maps M(p) to M(p) boundedly . We improve Berkson and Gillespie's result since under their hypothese s i Ç C(1 + I4 —2 and f I0 ^ (x)i dx < oa . Applying to Bochner-Ries z summation operator Bs, we reproved that B b maps HP (Rn ) to LI P (R n ) when b>p — 2 and o C p < 1 [11], [12] when we let  _ To consider the inverse of Theorem 1, we introduce the paraproduc t P h and show that P h maps HP (T) to H P (R) . Theorem 2 . Let ~ be a continuous function with compact support . I f O' : S—> S(0) is one -to -one, then 1 has bounded inverse 1 —1 : M(p) n S~~} —+ Ú(p) . 444  e? . SU N In the proof of Theorem 1 and Theorem 2, Lemma 2 plays an important role . If we assume Lemma 2 is true, or m(x) = > flEZ C(n)O(x—n) E L°° implies {C(n)} E l°°, then it suffices to assume 1 is one- to -on e in Theorem 2 . In particular Theorem 2 can be written as that 1 ha s bounded inverse 1 —1 : M(p) M(p) provided 1 has bounde d inverse 1' : M(2) f1 S(0) ~ Ñ(2) and is a continuous function wit h compact support, where 4 < p < oo . The continuity condition oi l can be dropped in one spatial dimension since for any distribution o n R such that is one -to -one there exists a univariate spline B k suc h that ~ = B k * ~ is continuous and ~ *' is one -to -one . But I do no t know how to construct this modifier B k in high spatial dimensions . B y Fourier transform characterization of global linear independence in [9] , the box spline and Daubechie s ' scaling function satisfy the condition o n ~ in Theorem 2 . Shannon sampling theorem [10] plays an important role in signa l analysis . It says a function with its Fourier transform supported i n [—1 + E, — E] for some 0 < E < 2 has its L P ( R ) norm comparabl e to its 1 P (Z) norm of its restriction to integer lattices Z where 1 < p < oo . In 1990, R . Torres [14] extended the aboye conclusion t o Besov spaces . Let {m}m>~ be a family of sequences such that ~ 7 z ~  —  ~ is smooth, supp 4) m (1) C{2 - m - 2 < 111 ç 2_ m }, O~ Co o n T m , T m , where ■ :Dm lIl = > IIfl E z 4)m (n)e 2 ' znl , To = { $ Ç IS I Ç 1} and Tm = {2m S I~I S 12 — 1 . Define (c .f . [13] or [14] } H °( Z ) = S {/( n )}nEZ ;  (I rn* f(n)I 2 ) a ) P <+o0 1 , nE Z nEZ m ~ U where f = {f(n)} is a tempered sequence and ~,,,,,* f (72) = >.IkEZ 4),,, ( n — k) f(k) . , . Theorem 3 . Let 0 < p < +oo . If f E S' (R) with supp f C [—1 + e, — e] for some 0 < E < 2 , then the inequalit y C—1 11{f( n )}IixP(z) Ç ~~f~~xP(x) ~ c íí{f( n )}IIxP(z ) holds for some constant C dependent of e and p only . MULTIPLIER EXTENSIaN AND SAMPLING THEOREM  44 5 For simplicity in the exposition we restrict ourselves to one spatia l dimension, all results can be extended to high spatial dimensions . Th e results of Theorem 1 and 3 can be extended to spaces of Triebel-Lizorki n type trivially . The big letter C will denote different constant at differen t occurance . 2 . Sorne lemma s To prove our theorems, we will use the following fundamental lemmas . Lemma 1 . c .f . [15] . Let f E S ' (R) have its Fourier transform contained in a compact set . Therefor e (4)  * f( x )l  cm (1fi r )( x ) holds for every Schwartz function '?P and o < r < 1, where M denote s Hardy-Littlewood maximal operator and the constant C depends on th e seminorm of r and the radius R for which f is supported in the bal l with radius R and center zero . Proof of Lemma 1 : Without loss of generality we assume x = O , M(I f 19(0) C +oo and supp f c H, 4 ~ by dilation invariance . Writ e (s)  f (x) = E f(n)y)(x — n) , nE Z for some Schwartz function cp such that supp cp c H, 1] . Hence ~ f (0)IT < E ne z 1 f(n)I r (1 +  To prove (4), we first prov e >InEZ 1 f(n) i r (1 + Ini) —3 < + oo . Recall that f E S'(R) and supp f C -4, . Therefore f(n)~ < C(1 + 1 n1) N for some constants C and N . On the other hand we hav e I~  1 T E f(m)ir(1 +~m—nl)-(N+3 ) mE Z by (5) and if( n )l r ( l + lni)- 3 raf Ç2 k + x 446  Q . SU N by summing over Il Ç 2 k and integrating over 161 C ~o < 1 , where So is chosen later and C N is independent of 6o . Denot e Ak = E2k- x <InIC_ 2k 1+ InIY3• Therefore E 3 . - A 3 < 2Cs 0 + 2C N 160 I r Ak+ 1 . Conversely if we assurne E i= l ° A j ~ 4Cs o for some ko , s then E j k —° + 1 s  I A j ~ 1 + 4CN so I r C 60 , which contradicts Ek .° 1 IAI ç k 0 + s C E j=l (1 + IC2 sN provided 6o is chosen small enough . Thi s proved EThZ I .f( n ) i r (1 + I< +ao . Furthermore by (5) we hav e I U+ á ) I `+ C l á l' E I  n — m D - a mE Z for some constant C independent of f and by integrating over 161 Ç 6 o for some sufficiently small 6o > o we ge t ~ 3 <C J ~+ IxD3dx < CM(' i~~)( 0 ) . n  R Therefore Lemma 1 is proved . ■ Lemma 2 . Let the integer transiates of the continuous function b e globally linearly independent . If m(x) = E nez C(n)O(x — n) E M(p) , then {C(n)} E  and l< Cilmii M ( p ) • Proof of Lemma 2 : First we prov e ( 6 )  li m ilL—(R)  Climilm ( p ) . Obviously (6) is true when 1 < p < oo since Marcinkiewicz real interpolation, 1 m MM(2) = IIm~~L~~R} and II m IIM(P) = iimIIm(p,) wher e p' = --~-- ' Hence the matter reduces to proving (6) for o C p Ç 1 . Fo r p— 1 f E H P (R), we have the atomic decomposition f(x) = Er_ o Akak (x ) with C—1 II f MHP(R) ç(Er o IÀ k 1» ç C f ~ f IIHP(R), where a k are (p,2,$ ) atoms and s ~ p ~ — 1. We call that a is an (p, 2, s) atom if there exists a n interval 1 such that supp a c 1, 11 a M L 2 ( R) < 1 1 1 1- t , and f x"a(x) dx = o for o < cx < s . It is easy to show Ex} is continuous, Iá(x)i ç 1-- 1 and I< CIx l s+l I I i P s+2— ~ . Hence I< Cixi~ 1 — 1 CIII P  an d I ç Er o I a kl lá k (x)I ç C(E°° o  Denote f t (x) = t~ p f () . Therefore f t (x) =  a k ( a k ) t (x) and C 1 (o I a k I) p k MULTIPLIER EXTENSION AND SAMPLING THEOREM  44 7 C ( E k o IAki P ) P . Recall that m E M(p) for 0 < p < 1 . Hence I C II .ft II HP ( R )  c ll .f II HP(R ) and Im(x)t ( l P ) f(tx)I < _ c II .fIIx p ( R ) I x I P- i for all t > O . Therefore l ( A ) < C ll f Iixp (R) for every x  0 when we let t = ixl —1 and (6) is proved for 0 < p < 1 by choosin g f E H P (R) such that f 1 on the unit sphere . Second we prov e ( 7 )  II{C(n)}iii . . .(z ) By [l, Theorem 1 .3], there exists a local algebraic dual {A n } of {O(x — n)}, which says An O (x — k) = S n k and there exists a bounded set K such that A te, f = 0 when f E S( 0 ) and supp f fl (K + n) = O, where w e define the Kronekker symbol 6nk by S nn = 1 and 5 nk = 0 when n k . Recall that ~ is continuous . Hence there exist finite points x i E K an d weights C( x 2 ) such that A nf = E i C( x 2 ) f ( x i + n) for every f E S(0) . This shows IC(n)I = Anmi < for every n E Z . Therefore (7 ) holds and Lemma 2 is proved by combining (6) and (7) . ■ Lemma 3 . ([4] or [5, Theorem A .1]) . Let 1 < p < ~-oo and 1< q < -loo . Therefore the following Fefferman-Stein vector-valued maxima l inequality IkEZ  L P (R) kE Z Cc LP(R ) holdswhere M denotes the Hardy-Littlewood maximal operator on R a s usual . Let h and rj be two Schwartz functions such that supp ~ c {I xj ç 20 } and (x) = 1 on {IxI Ç 40 } . Let be as in the definition of H p (R) . For f E H P (T) we introduce a new type of paraproduct operator P h defined b y (8)  P hf ( x ) - E (7h-ri * h ) ( x )( 1 'm * f) ( x ) , m> o where 77, n (x) = 2 m r] ( 2 m x ) . 448 Q . SU N Lemma 4 . Let P h be defined by (8) and h be a Schwartz function . Then P h maps H P (T) to H P (R) , ll P hfiiHPR)  C II .fIIHP ( T) • Proof : Observe that  * h (x)l ç C N (1+ I far every N> 0 an d some CN independent of m . Also observe that supp(eri, n *h) N) , . 1 z * f))" C {2m Ç lxi ç 2ó 2m } and  * f (x + k) = ~ m * f(x) for all k E Z . Therefor e ii P hfiiP( R ) < c L l y kEZ" " G C II .fIIHp ( T ) (E 1(97m * h )( s )( T m * f)( s ) 1 2 1 <CE dx ((1 + 1k~}--max(4,1)4~jm * f ( x )1 2 and Lemma 4 is proved . ■ 3 . Proof of theorem s Proof of Theorem 1 : Let {C(n)} be a multiplier on HP (T) an d sup p q5 c HM, 4 M ~ fo r som e M > 1 . Denote m(x ) = E C(n)O( x — n) . Le t ■ : I) k be as in th e definitio n o f H P (R) . Writ e f = f o + f 1 + f 2 , wher e f o = Ik<2 M 1 * f and f i = >k> M 1 d› 2k+ z * f fo r i = 1, 2, wher e M 1 is a positive integer such that 2 M1 > 20M . Observe tha t m(x)( c l , k * fr(x) =  E  C(1)O(x — 1)( ■ b k * f) A (x ) 1l 1<2 2M i -F M for k < 2N~ 1 . Write O'(x) = InEZ O' (Z) 7/)(x - -  for som e ~ E S(R ) wit h sup p z j ~ c {lx I < $ } , wher e O' denotes invers e Fourie r trans - 44 9 MULTIPLIER EXTENSION AND SAMPLING THEORE M form . Therefor e ( E 1(m(1 .k*f)A)'(x)1 2 L k<2M 1 k C 21V1 1 ~ ~ dx ~ .c  E  l c ( l ) I p I11<2 2M 1 +M min(p,1 ) max(p,1 ) o V fz 2 p 2 kC21Vf 1 dx ~ ClI{C(1)}111 . 3(z) f ( E (M('b k R rc<zM l 2 * f) r (x)) 1  dx II f M(p)  H P (R) 5 where 6 C r C min (p,1) . The first inequality follows Hólder inequalit y and >nEZ la n l Ç (>nz lfor 4 C p Ç 1, the second inequalit y follows from Lemma 1 an d sup E I+ n)I min(p,1) C kd x (see [13, Lemma 6]}, and the third inequality follows fro m Lemma 2 and Lemma 3 . For k write 0 (x )(1 . k * f) A (x + n) = EiEZ C k,n (I)e 2 ' /lVl ñ(x'M) for some Schwartz function ~ wit h supp ~ c {fxI Ç D .. Therefore we ge t (m(k * f }n)V `x) =E  E  C(n)C k, n(1)e -27ri'77(Mx _ 1 ) iEZ 2k —2 ~InlC2 k since 2m1 3 20M an d C'k n (1) e - - 27rin x J 2k -25InlC2k  , f 0(Y) (E(f  j+ )e_ 2 ) 7rinac e-27Tidy dy nEZ (f*k)(x+n)(x+n _ y1 E C i l M~