Research Article Littlewood-Paley Operators on Morrey Spaces with Variable Exponent

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1 Hindawi Publising opoaion e Scienific Wold Jounal, Aicle ID 79067, 0 pages p://dx.doi.og/0.55/204/79067 Reseac Aicle Lilewood-Paley Opeaos on Moey Spaces wi Vaiable Exponen Suangping Tao and Lijuan Wang ollege of Maemaics and Saisics Science, Nowes Nomal Univesiy, Lanzou , ina oespondence sould be addessed o Suangping Tao; aosp@nwnu.edu.cn and Lijuan Wang; wang22004x@63.com Received 23 June 204; Acceped 2 July 204; Publised 7 Augus 204 Academic Edio: Józef Banas opyig 204 S. Tao and L. Wang. Tis is an open access aicle disibued unde e eaive ommons Aibuion License, wic pemis unesiced use, disibuion, and epoducion in any medium, povided e oiginal wok is popely cied. By applying e veco-valued inequaliies fo e Lilewood-Paley opeaos and ei commuaos on Lebesgue spaces wi vaiable exponen, e boundedness of e Lilewood-Paley opeaos, including e Lusin aea inegals, e Lilewood-Paley gfuncions and g μ-funcions, and ei commuaos geneaed by BMO funcions, is obained on e Moey spaces wi vaiable exponen.. Inoducion and Main Resuls Le ψ L (R n and saisfy e following: (i R n ψ(xdx=0; (ii ψ(x ( + x n ε ; (iii ψ(x + y ψ(x y γ ( + x n γ ε, x 2 y, wee, ε, γ ae all posiive consans. Denoe ψ (x = n ψ(x/ wi > 0 and x R n.givenafuncionf L loc (Rn,eLusinaeainegaloff is defined by S ψ,a (f (x =( ψ f(y 2 /2 dy d Γ a (x n+, ( wee Γ a (x denoeeusualconeofapeueone Γ a (x ={(,y R n+ + : y x <a, a 0. (2 As a=, we denoe S ψ,a (f as S ψ (f. Now le us un o e inoducion of e oe wo Lilewood-Paley opeaos. I is well known a e Lilewood-Paley opeaos include also e Lilewood-Paley g-funcions and e Lilewood-Paley g μ-funcions besides e Lusin aea inegals. Te Lilewood-Paley g-funcions, wic can be viewed as a zeo-apeue vesion of S ψ,and g μ-funcions, wic can be viewed as an infinie-apeue vesion of S ψ, ae, especively, defined by g μ (f (x =( R n+ + g (f (x = ( ψ f(y 2 d /2 0, (3 ( + x y /2 μn ψ f(y 2 dy d n+, μ>0. (4 If we ake ψ o be e Poisson kenel, en e funcions defined above ae e classical Lilewood-Paley opeaos. Leing b L loc (Rn, m,ecoespondingm-ode commuaos of Lilewood-Paley opeaos above geneaed by a funcion b ae defined by [b m,g ψ ](f(x =( 0 [b (x b (y] m 2 /2 d ψ (x yf(ydy R n, (5

2 2 Te Scienific Wold Jounal [b m,s ψ,a ](f(x =( Γ a (x [b (x b(z] m 2 /2 dy d ψ (y z f (z dz R n n+, (6 [b m,g μ ](f(x =( R n+ + ( + x y μn [b (x b(z] m 2 /2 dy d ψ (y z f (z dz R n n+, (7 wee μ>0. Te Lilewood-Paley opeaos ae a class of impoan inegal opeaos. Due o e fac a ey play vey impoan oles in amonic analysis, PDE, and e oe fields (see [ 3], people paymucmoeaenionoisclass of opeaos. In 995, Lu and Yang invesigaed e beavio of Lilewood-Paley opeaos in e space BMO p (R n in [4].In2005,ZangandLiupovedecommuao[b, g ψ ] is bounded on L p (ω in [5].In2009,XueandDinggavee weiged esimae fo Lilewood-Paley opeaos and ei commuaos (see [6]. Tee ae some oe esuls abou Lilewood-Paley opeaos in [7 9]and so fo. On e oe and, Lebesgue spaces wi vaiable exponen L p( (R n become one class of impoan eseac subjec in analysis filed due o e fundamenal pape [0]by Kováčik and Rákosník. In e pas weny yeas, e eoy of ese spaces as made pogess apidly, and e sudy of wic as many applicaions in fluid dynamics, elasiciy, calculus of vaiaions, and diffeenial equaions wi nonsandad gow condiions (see [ 5]. In [6], uz-uibe e al. saed a e exapolaion eoem leads e boundedness of some classical opeaos including e commuao on L p( (R n. Kalovic and Lene also independenly obained e boundedness of e singula inegals commuao on Lebesgue spaces wi vaiable exponen in [7]. In 2009 and 200, Izuki consideed e boundedness of veco-valued sublinea opeaos and facional inegals on Hez-Moey spaces wi vaiable exponen in [8, 9], especively. In 203, Ho in [20] inoduced a class of Moey spaces wi vaiable exponen M p(,u and sudied e boundedness of e facional inegal opeaos on M p(,u. Inspied by e esuls menioned peviously, in is pape we will conside e veco-valued inequaliies of e Lilewood-Paley opeaos and ei m-ode commuaos on Moey spaces wi vaiable exponen. Befoe saing ou main esuls, we need o ecall some elevan definiions and noaions. Le E be a Lebesgue measuable se in R n wi measue E > 0. Definiion (see [0]. Le p( : E [, be a measuable funcion. Te Lebesgue space wi vaiable exponen L p( (E is defined by L p( (E ={fis measuable: ( f (x p(x dx< E η Te space L p( loc (E is defined by L p( loc (E ={fis measuable: f L p( (K fo some consan η>0. (8 fo all compac subses K E. (9 Te Lebesgue space L p( (E is a Banac space wi e nom defined by f L p( (E = inf {η>0: E ( f (x p(x dx. (0 η Remak 2. ( Noe a if e funcion p(x = p 0 is a consan funcion, en L p( (R n equals L p 0 (R n. Tis implies a e Lebesgue spaces wi vaiable exponen genealize e usual Lebesgue spaces. And ey ave many popeies in common wi e usual Lebesgue spaces. (2 Denoe p := ess inf{p(x : x E, p + := ess sup{p(x : x E.TenP(E consiss of all p( saisfying p >and p + <. (3 Te Hady-Lilewood maximal opeao M is defined by M(f(x = sup Q f(y dy. ( Q Q x Denoe B(E o be e se of all funcions p( P(E saisfying e condiion a M is bounded on L p( (E. (4 Le p( B(R n.denoeκ p( = sup{q > : p( /q B(R n and e p( is e conjugae exponen of κ p ( (see [20]. Definiion 3 (see [20]. Le p(x L (R n, <p(x<. If ee exiss a consan >0suc a, fo any x R n and >0, Lebesgue measuable funcion u(x, : R n (0, (0, saisfying χ B(x,L p( (R n j=0 χ u(x,2 j+ u (x,, (2 B(x,2 j+ L p( (R n en one says u is a Moey weig funcion fo L p( (R n.one denoes e class of Moey weig funcions by W p(.

3 Te Scienific Wold Jounal 3 Definiion 4 (see [20]. Le p(x B(R n, u(x, W p(. Ten e Moey spaces wi vaiable exponen M p(,u (R n ae defined by M p(,u (R n = {f is measuable: f M p(,u (R n <, (3 wee f M p(,u (R n = sup x R n,r>0 u (x, R fχ B(x,RL p( (R n. (4 Remak 5. ( If u(x,,enm p(,u (R n is e Lebesgue spaces wi vaiable exponen L p( (R n. (2 Noice a if p(x p,<p<,isaconsan funcion, en fomula (2 canbeewienasaninegal in fom. To be pecise, fomula (2 canbeewienine following fom (see [20]: u (x, (x, d u n/p+, >0, n/p x Rn. (5 Le 0 < α < n. By e e condiions of Moey weig funcions menioned in [2] u p (x, d (x, n α/p+ up, >0, n α/p x Rn (6 and Hölde s inequaliy, via simple calculaion, we ave u (x, d n/p+ u p /p (x, { d n α/p+ u (x,. n/p { /p d αp + (7 Fom is, i follows a if p(x p, < p <,isa consan funcion, en condiion (2isweakeancondiion (6. Tus, e class of e Moey spaces inoduced in Definiion 4 is moe wide an a saisfying condiion (.8 in [2]. Moe sudies of common Moey spaces can be seen in [22, 23]andsofo. (3 If u(x, = B(x, /p(x /q(x, p(x q(x, en e space menioned in Definiion 4 is e Moey space wi vaiable exponen inoduced in [24]. And wen < s < κ p (x, /p(x /q(x < /s, iiseasyosee u(x, saisfying condiion (2. Ta is because i follows fom p(x B(R n, <s<κ p (x,a(see[20] B(R n, < <, en ee exiss a consan > 0 independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,efollowinginequaliy olds: S ψ,a (f /Mp(,u(Rn f. /Mp(,u(Rn (9 Teoem 7. Suppose a g ψ is defined by (3. Tenunde e same condiion as e one in Teoem 6, ee exiss a consan > 0 independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,e following inequaliy olds: g ψ (f /Mp(,u(Rn f. /Mp(,u(Rn (20 Teoem 8. Suppose a g μ is defined by (4 and μ>3+ 2(ε + γ/n, 0 < γ < ε. Ten unde e same condiion as e one in Teoem 6, ee exiss a consan > 0 independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,efollowinginequaliy olds: g μ (f /Mp(,u(Rn f. /Mp(,u(Rn (2 Fo commuaos [b m,s ψ,a ], [b m,g ψ ],and[b m,g μ ],we ave e following esuls. Teoem 9. Supposeafuncionψ L (R n saisfies (i (iii and [b m,s ψ,a ] is defined by (5.Leb BMO(R n, p( B(R n, m, <<.If,foanyx R n and 0 >0, funcion u saisfies χ B(x, 0 L p( (R n j=0 χ j m u(x,2 j+ 0 u(x, 0, (22 B(x,2 j+ 0 L p( (R n en ee exiss a consan > 0 independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,efollowinginequaliyolds: χ B(x,L p( (R n χ B(x,2 j+ L p( (R n 2 jn(/s, x R n, > 0, j N. (8 Fo Lilewood-Paley opeaos S ψ,a,g ψ,andg μ,inis pape, we ave e following esuls. [bm,s ψ,a ](f /Mp(,u(Rn f. /Mp(,u(Rn (23 Teoem 6. Suppose a funcion ψ L (R n saisfies (i (iii and S ψ,a is defined by (. Ifu W p(, p( Teoem 0. Suppose a [b m,g ψ ] is defined by (6. Ten unde e same condiion as e one in Teoem 9,eeexiss

4 4 Te Scienific Wold Jounal aconsan>0independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,e following inequaliy olds: [bm,g ψ ](f /Mp(,u(Rn f. /Mp(,u(Rn (24 Teoem. Suppose a [b m,g μ ] is defined by (7, μ > 3+2(ε+γ/n,and0 < γ < ε.tenundeesame condiion as e one in Teoem 9, ee exiss a consan >0 independen of f suc a, fo any funcion sequences {f = wi f / Mp(,u (R n <,efollowinginequaliy olds: [bm,g μ ](f /Mp(,u(Rn f. /Mp(,u(Rn (25 Remak 2. ( I is easy o see a condiion (22 in Teoem 9 is songe an condiion (2 in Definiion3. Teefoe, if a funcion u saisfies condiion (22, en u W p(. (2 Te funcion u wic saisfies (22 exiss. In fac, if we ake τ:0 τ</e p( suc a funcion u saisfies u (x, 2 u (x, 2nτ, x R n, > 0, (26 en u W p(.taisbecause,foanyτ</e p(, ee exiss s<κ p ( suc a τ< /s< /κ p ( =/e p(.hence, i follows fom (8a χ B(x,L p( (R n j=0 χ j m u(x,2 j+ B(x,2 j+ L p( (R n j=0 j m 2 jn( /s 2 jnτ u (x, u(x,. (27 We end is secion by inoducing some convenional noaions wic will be used lae. Tougou is pape, given a funcion f, wedenoeemeanvalueoff on E by f E =: (/ E E f(xdx. p ( means e conjugae exponen of p( ;namely,/p(x + /p (x = olds. always means a posiive consan independen of e main paamees and may cange fom one occuence o anoe. 2. Peliminay Lemmas In is secion, we inoduce some conclusions wic will be used in e poofs of ou main esuls. Lemma 3 (see [0] (genealizedhölde s inequaliy. Le p(, p (, p 2 ( P(R n. ( Fo any f L p( (R n, g L p ( (R n, R n f (x g (x dx p f L p( (R n g L p ( (R n, (28 wee p =+/p /p +. (2 Fo any f L p( (R n, g L p2( (R n,wen/p(x = /p (x + /p 2 (x,oneas f (x g (x L p( (R n p,p 2 f L p ( (R n g L p 2 ( (R n, (29 wee p,p 2 =(+/p /p + /p. Lemma 4 (see [7]. If p( B(R n, en ee exis consans δ,δ 2, > 0,suca,foallballsB R n and all measuable subses S B, χ BL p( (R n χ SL p( (R n χ SL p ( (R n χ BL p ( (R n χ SL p( (R n χ BL p( (R n B S, ( S B δ, ( S B δ2. Remak 5. Fom fomula (2, i follows a (30 χ B(x,L p( (R n χ u(x,2 j+ u (x,, B(x,2 j+ L p( (R n j N. (3 Tus, by Lemma 4,weave, j N, u(x,2 j u (x, χ B(x,2 j L p( (R n χ B(x,2j 2 nj. B (x, B(x,L p( (R n (32 Lemma 6 (see [8]. If p( B(R n, en ee exiss consan >0,suca,foallballsB R n, B χ BL p( (R n χ. BL p ( (R n (33 Lemma 7 (see [25]. Le b BMO(R n ; m is a posiive inege. Tee exis consans >0,suca,foanyk, j Z wi k>j, ( b m sup B (/ χ B L p( (R n (b b B m χ B L p( (R n b m ; (2 (b b Bj m χ Bk (k L p( (R n jm b m χ B k. L p( (R n Lemma 8 (see [26]. Le ψ L (R n saisfy (i (iii. If p( B(R n, <q<, en fo all bounded compacly suppo funcions f j suc a {f j j= L p( (l q,ais, ( j ( f j q /q L p( (R n <, e following veco-valued inequaliies old:

5 Te Scienific Wold Jounal 5 ( ( j S ψ,a (f j q /q L p( (R n ( j f j q /q L p( (R n, (2 ( j g ψ (f j q /q L p( (R n ( j f j q /q L p( (R n, (3 ( j g μ (f j q /q L p( (R n ( j f j q /q L p( (R n, wee μ > 2, 0 < γ < min{(μ 2n/2, ε. Lemma 9 (see [26]. Le b BMO,ψ L (R n saisfy (i (iii, m N \{0.Ifp( B(R n, < q <, enfoall bounded compacly suppo funcions f j suc a {f j j= L p( (l q,ais, ( j ( f j q /q L <,efollowing p( (R n veco-valued inequaliies old: ( ( j [b m,s ψ,a ](f j q /q L p( (R n ( j f j q /q L p( (R n, Tus, u(x 0, 0 S ψ,a (f / χ B(x0, 0 u(x 0, 0 S ψ,a (f 0 + u(x 0, 0 S ψ,a (f j D +D 2. j= L p( (R n / χ B(x0, 0 L p( (R n / χ B(x0, 0 L p( (R n (36 (2 ( j [b m,g ψ ](f j q /q L p( (R n ( j f j q /q L p( (R n, Fo e em D,noiceasuppf 0 Lemma 8 and (32, i is easy o see a B(x 0,2 0 ;using (3 ( j [b m,g μ ](f j q /q L p( (R n ( j f j q /q L p( (R n, wee μ>2, 0<γ<min{(μ 2n/2, ε. 3. Poofs of Main Resuls Nex, le us sow e poofs of Teoems 6,especively. Poof of Teoem 6. Le {f l M p(,u (R n ;foanyx 0 R n, 0 >0, denoe f (x f 0 (x + f j (x, (34 j= D u(x 0, 0 u(x 0,2 0 u(x 0, 0 f 2 n sup y R n,r>0 f f0 u(x 0,2 0 / u(y,r / /Lp( (Rn χ B(x0,2 0 L p( (R n χ B(y,R L p( (R n (37 wee f 0 =f χ B(x0,2 0, f j =f χ B(x0,2 j+ 0 \B(x 0,2 j 0, j N \ {0. Noing a, in ode o pove Teoem 6,iisenougo sow a e following inequaliy olds: u(x 0, 0 S ψ,a (f / χ B(x0, 0 f. /Mp(,u(Rn L p( (R n (35 f. /Mp(,u(Rn We now un o esimae D 2.Todois,weneedoconside S ψ,a fis. Wiou loss of genealiy, we may assume a a.le Γ ={y R n : y x <a, y 2j+ 0, Γ ={y R n : y x <a, y>2j+ 0. (38

6 6 Te Scienific Wold Jounal Ten, by Minkowski s inequaliy, we ave S ψ,a (f j (x =( Γ a (x n ψ( y z f j 2 /2 R n (z dz dy d n+ R n fj (z ( Γ a (x 3n ψ(y z 2 /2 dy d dz fj R n (z ( 3n 0 Γ ψ(y z 2 /2 dy d dz + fj R n (z ( 3n 0 Γ ψ(y z 2 /2 dy d dz. (39 Obseve a if x B(x 0, 0, z B(x 0,2 j+ 0 \ B(x 0,2 j 0, j,en + y z x y a + y z x y + y z a z x a Teefoe, i follows fom condiion (ii a 3n 0 Γ ψ(y z 2 dy d 3n ( + 0 Γ 2 j+ 0 0 Γ 2ε n z 2a. y z 2(n+ε dy d ( + y z dy d 2(n+ε (40 On e oe and, we denoe 3n 0 Γ ψ(y z 2 dy d 3n 0 Γ ψ(y z ψ( y dy d 2 + 3n 0 Γ ψ(y 2 dy d I+II. (42 Noe a if y>2 j+ 0,en > x y /a ( x y /a > (2 j+ 0 0 /a 2 j 0 /a. Tus, by condiion (iii, we ge I 3n ( z 2γ 2 j 0 /a Γ ( + y 2(n+γ+ε dy d (2 j 0 2γ 3n 2γ dy d 2 j 0 /a x y <a (2 j 0 2γ a n 2n 2γ dy d 2 j 0 /a a 3n+2γ (2 j 0 2n. (43 And by condiion (ii, simila o e esimae of I,weobain II 3n ( z 2γ 2 j 0 /a Γ ( + y 2(n+ε dy d 2 j 0 /a Γ a 3n (2 j 0 2n. 2ε n ( + y dy d 2(n+ε Hence, fom e esimaes above, i follows a (44 S ψ,a (f j (x a3n/2+γ+ε (2 j 0 n fj L (R n. ( j+ 0 Γ a2(n+ε 2ε n ( + y z dy d 2(n+ε 2 j+ z 0 2ε n dy d 2(n+ε 0 x y <a + a2n z 2n 3n dy d 2 j+ 0 x y <a (4 Tus, D 2 u(x 0, 0 j= a3n/2+γ+ε (2 j 0 n fj L (R n / χ B(x0, 0 L p( (R n 2 j+ z 0 a n 2ε dy d 2(n+ε 0 a2(n+ε + a n 2n dy d 2 j+ 0 a 3n+2ε (2 j 0 2n. a 3n/2+γ+ε fj (2 j u(x 0, 0 0 n χ B(x 0, 0 L p( (R n j= /L(Rn. (46

7 Te Scienific Wold Jounal 7 Teefoe, applying e genealized Hölde s inequaliy, Lemma 6,and(2, we ave D 2 a 3n/2+γ+ε (2 j u(x 0, 0 0 n χ B(x 0, 0 L p( (R n fj j= /Lp( (Rn χ B(x 0,2 j+ 0 L p ( (R n (2 j u(x 0, 0 0 n B(x 0,2 j+ 0 j= χ B(x 0, 0 L p( (R n χ B(x 0,2 j+ 0 L p( (R n fj /Lp( (Rn χ B(x 0, 0 L p( (R n u(x 0, 0 j= χ u(x 0,2 j+ 0 B(x 0,2 j+ 0 L p( (R n u(x 0,2 j+ 0 f sup y R n,r>0 / u(y,r f f. /Mp(,u(Rn χ B(x0,2 j+ 0 / Adding up e esimaes of D,D 2,weobain u(x 0, 0 S ψ,a (f / χ B(x0, 0 f. /Mp(,u(Rn Tis complees e poof of Teoem 6. L p( (R n χ B(y,R L p( (R n L p( (R n Now le us pove Teoems 7 and 8 in bief. (47 (48 Poof of Teoem 7. Fo g ψ, simila o e esimae of S ψ,a,via asimplecalculaion,wegea(see[26] if x B(x 0, 0, supp f j B(x 0,2 j+ 0 \B(x 0,2 j 0, j,en g ψ (f j (x (2j 0 n fj L (R n. (49 Hence, simila o e poof of Teoem 6, ifollowsfom inequaliy (2 in Lemma 8 a Poof of Teoem 8. Fo g μ, by e definiions of S ψ,a and g μ, we ave g μ (f (x =( 0 R n ( ( 0 x y < + l= + x y ( 0 2 l x y <2 l /2 μn ψ f(y 2 dy d n+ ( + x y ( + x y /2 μn ψ f(y 2 dy d n+ μn /2 ψ f(y 2 dy d n+ S ψ,a (f j (x + ( + 2 l μn/2 S ψ,2 l a (fj (x. l= (5 Accoding o e esimae of S ψ,a in e poof of Teoem 6, we know a if x B(x 0, 0,suppf j B(x 0,2 j+ 0 \ B(x 0,2 j 0, j,en S ψ,a (f j (x a3n/2+ε+γ (2 j 0 n fj L (R n. (52 Tus, as μ> 3 + 2(ε + γ/n,weobain g μ (fj (x a3n/2+ε+γ ( + l= (2 j 0 n fj L (R n 2 (3n/2+ε+γ μn/2 a 3n/2+ε+γ (2 j 0 n fj L (R n. (53 Hence, also simila o e poof of Teoem 6, andfom inequaliy (3 in Lemma 8,ifollowsa g μ (f /Mp(,u(Rn Tis finises e poof of Teoem 8. f. /Mp(,u(Rn (54 Poof of Teoem 9. Le b BMO, {f l M p(,u (R n. Fo any x 0 R n, 0 >0, denoe g ψ (f /Mp(,u(Rn Tis accomplises e poof of Teoem 7. f. /Mp(,u(Rn (50 wee f 0 N \{0. f (x f 0 (x + f j (x, (55 j= =f χ B(x0,2 0 and f j =f χ B(x0,2 j+ 0 \B(x 0,2 j 0, j

8 8 Te Scienific Wold Jounal To finis e poof of Teoem 9, we only need o pove Tus, u(x 0, 0 [bm,s ψ,a ](f Tus, f. /Mp(,u(Rn / χ B(x0, 0 L p( (R n (56 E 2 u(x 0, 0 j= a 3n/2+γ+ε (2 j 0 n (b ( bm f j / L (R n u(x 0, 0 [bm,s ψ,a ](f / χ B(x0, 0 u(x 0, 0 [bm,s ψ,a ](f 0 + u(x 0, 0 [bm,s ψ,a ](f j E +E 2. j= L p( (R n / χ B(x0, 0 L p( (R n / χ B(x0, 0 L p( (R n (57 Fo e em E,noiceasuppf 0 B(x 0,2 0 ;by Lemma 9 and inequaliy (32, we ave E u(x 0, 0 u(x 0,2 0 u(x 0, 0 2 n sup y R n,r>0 f0 /Lp( (Rn u(x 0,2 0 f u(y,r f f. /Mp(,u(Rn / / χ B(x0,2 0 χ B(y,R L p( (R n L p( (R n (58 Now we un o esimae E 2. Accoding o e esimae of S ψ,a in e poof of Teoem 6, weseeaifx B(x 0, 0, z B(x 0,2 j+ 0 \B(x 0,2 j 0, j,en Teefoe, S ψ,a (f j (x a3n/2+ε+γ (2 j 0 n fj L (R n. (59 χ B(x0, 0 (2 j u(x 0, 0 0 n j= L p( (R n (b ( bm f j /L(Rn χ B(x0, 0 L p( (R n (2 j u(x 0, 0 0 n (b b B(x 0, 0 m χ B(x0, 0 L p( (R n fj j= /L(Rn + (2 j u(x 0, 0 0 n χ B(x 0, 0 L p( (R n j= (b b B(x 0, 0 m f j /L(Rn Using Hölde sinequaliyandlemma7,we ge E 2 (2 j u(x 0, 0 0 n b m χ B(x 0, 0 L p( (R n j= fj /Lp( (Rn. χ B(x 0,2 j+ 0 L p ( (R n + (2 j u(x 0, 0 0 n χ B(x 0, 0 L p( (R n j= (6 [bm,s ψ,a ](f j (x = S ψ,a [(b (x b m f] (x a 3n/2+ε+γ (2 j 0 n (b ( bm f j L (R n. (60 fj /Lp( (Rn (b b B(x 0, 0 m χ B(x0,2 j+ 0 L p ( (R n

9 Te Scienific Wold Jounal 9 (2 j u(x 0, 0 0 n [j m +] b m j= χ B(x 0, 0 L p( (R n fj L p( (R n χ B(x 0,2 j+ 0 L p ( (R n. And en, i follows fom Lemma 6 and (22a E 2 b m (2 j u(x 0, 0 0 n j m B(x 0,2 j+ 0 j= (62 χ B(x 0, 0 L p( (R n χ u(x 0,2 j+ 0 u(x B(x 0,2 j+ 0 L p( (R n 0,2 j+ 0 fj /Lp( (Rn b m j m χ B(x 0, 0 L p( (R n u(x 0, 0 j= χ B(x 0,2 j+ 0 L p( (R n u(x 0,2 j+ 0 sup y R n,r>0 u(y,r f f. /Mp(,u(Rn / χ B(y,R L p( (R n Hence, Adding up e esuls of E,E 2,weave u(x 0, 0 [bm,s ψ,a ](f f. /Mp(,u(Rn / χ B(x0, 0 Te poof of Teoem 9 is accomplised. L p( (R n (63 (64 Poof of Teoem 0. Fo [b m,g ψ ], accoding o e esimae of g ψ in e poof of Teoem 7, weseeaifx B(x 0, 0, supp f j B(x 0,2 j+ 0 \B(x 0,2 j 0, j,en Tus, g ψ (f j (x (2j 0 n fj L (R n. (65 [b m,g ψ ]= g ψ [(b (x b m f] (x (2 j 0 n (b ( bm f j L (R n. (66 Hence,similaoepoofofTeoem 9,andfominequaliy (2 in Lemma 9,ifollowsa g ψ (f /Mp(,u(Rn f. /Mp(,u(Rn Te poof of Teoem 0 is compleed. (67 Poof of Teoem. Fo [b m,g μ ], accoding o e esimae of [b m,g μ ] in e poof of Teoem 8,wegeaifx B(x 0, 0, supp f j B(x 0,2 j+ 0 \B(x 0,2 j 0, j,en Tus, g μ (fj (x a3n/2+ε+γ (2 j 0 n fj L (R n. (68 [b m,g μ ]= g μ [(b (x bm f] (x a 3n/2+ε+γ (2 j 0 n (b ( bm f j L (R n. (69 Hence, also simila o e poof of Teoem 9,ifollowsfom inequaliy (3 in Lemma 9 a [bm,g μ ](f /Mp(,u(Rn f. /Mp(,u(Rn Te poof of Teoem is accomplised. onflic of Ineess Te auos declae a ey ave no conflic of ineess. Acknowledgmen (70 Suangping Tao is suppoed by e Naional Naual Foundaion of ina (Gan nos and Refeences [] E. M. Sein, Te developmen of squae funcions in e wok of A. Zygmund, Bullein of e Ameican Maemaical Sociey, vol. 7, no. 2, pp , 982. [2]. Kenig, Hamonic Analysis Tecniques fo Second Ode Ellipic Bounday Value Poblems BMS 83, Ameican Maemaical Sociey, 994. [3]S.A.ang,J.M.Wilson,andT.H.Wolff, Someweiged nom inequaliies concening e Scödinge opeaos, ommenaii Maemaici Helveici,vol.60,no.2,pp ,985. [4] S. Lu and D. Yang, Te cenal BMO spaces and Lilewood- Paley opeaos, Appoximaion Teoy and is Applicaions,vol., no. 3, pp , 995.

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