Triebel Lizorkin space estimates for multilinear operators of sublinear operators

Transcription

1 Proc. Indian Acad. Sci. Mah. Sci. Vol. 3, No. 4, November 2003, pp Prined in India Triebel Lizorkin space esimaes for mulilinear operaors of sublinear operaors LIU LANZHE Deparmen of Applied Mahemaics, Hunan Universiy, hangsha 40082, People s Republic of hina lanzheliu@263.ne MS received 23 March 2003 Absrac. In his paper, we obain he coninuiy for some mulilinear operaors relaed o cerain non-convoluion operaors on he Triebel Lizorkin space. The operaors include Lilewood Paley operaor and Marcinkiewicz operaor. Keywords. Mulilinear operaors; Triebel Lizorkin space; Lipschiz space; Lilewood Paley operaor; Marcinkiewicz operaor.. Inroducion Le T be he singular inegral operaor, a well-known resul of oifman, Rochberg and Weiss [6] which saes ha he commuaor [b, T ] = Tbf btf where b BMO is bounded on L p <p<. hanillo [] proves a similar resul when T is replaced by he fracional inegral operaor. In [9,], hese resuls on he Triebel Lizorkin spaces and he case b Lipβ where Lipβ is he homogeneous Lipschiz space are obained. The main purpose of his paper is o sudy he coninuiy for some mulilinear operaors relaed o cerain convoluion operaors on he Triebel Lizorkin spaces. In fac, we shall obain he coninuiy on he Triebel Lizorkin spaces for he mulilinear operaors only under cerain condiions on he size of he operaors. As applicaions, we prove he coninuiy of he mulilinear operaors relaed o he Lilewood Paley operaor and Marcinkiewicz operaor on he Triebel Lizorkin spaces. 2. Noaions and resuls In he sequel, will denoe a cube of wih sides parallel o he axes, and for a cube, le f = fxdx and f # x = sup x fy f dy. For r< and 0 δ, le M δ,r f x = sup x δr/n fy r dy /r. We denoe M δ,r f = M r f if δ = 0, which is he Hardy Lilewood maximal funcion when r = in his case, we denoe M f = Mf, see [2,3]. For β>0 and p>, 379

2 380 Liu Lanzhe le Ḟp β, be he homogeneous Triebel Lizorkin space, he Lipschiz space β is he space of funcions f such ha f β = sup x,h h 0 [β] h fx / h β <, where k h denoes he kh difference operaor []. We are going o consider he mulilinear operaor as follows: Le m be a posiive ineger and A a funcion on. We denoe DEFINITION R m A; x,y = Ax α m α! Dα Ayx y α. Define Fx,y, on [0,. Then we denoe F f x = Fx,y,fydy and F A R m A; x,y f x = x y m Fx,y,fydy. Le H be he Hilber space H ={h: h < }. For each fixed x, we view F f x and F A f x as a mapping from [0, o H. Then, he mulilinear operaors relaed o F is defined by T A fx= F A f x ; we also define ha Tf x = F fx. In paricular, we consider he following wo sublinear operaors. Fix λ>. DEFINITION2 Le ε>0 and ψ be a fixed funcion which saisfies he following properies: ψx x n, 2 ψx y ψx y ε x nε when 2 y < x. The mulilinear Lilewood Paley operaor is defined by g A λ f x = [ ] nλ /2 F A f x, y 2 dyd n, where F A f x, y = R m A; x,z x z m fzψ y zdz

3 Triebel Lizorkin space esimaes 38 and ψ x = n ψx/ for >0. We denoe F f y = f ψ y. We also define nλ /2 g λ f x = F f y 2 dyd n, which is he Lilewood Paley operaor [3]. Le H be he Hilber space H ={h: h = h 2 dyd/ n /2 < }. Then for each fixed x, F A f x, y may be viewed as a mapping from 0, o H, and i is clear ha gλ A f x = nλ/2 F A f x, y, g λ f x = nλ/2 F f y. DEFINITION3 Le 0 <γ and be a homogeneous of degree zero on such ha S n x dσx = 0. Assume ha Lip γ S n, ha is here exiss a consan M>0such ha for any x,y S n, x y M x y γ. We denoe Ɣx ={y, : x y <} and he characerisic funcion of Ɣx by χ Ɣx. The mulilinear Marcinkiewicz inegral operaor is defined by where µ A λ f x = [ F A f x, y = y z We denoe ha F f y = We also define ha µ λ f x = y z ] nλ /2 F A f x 2 dyd n3, y z R m A; x,z y z n x z m fzdz. y z fzdz. y z n nλ /2 F f y 2 dyd n3, which is he Marcinkiewicz inegral operaor [4]. Le H be he Hilber space H ={h: h = h 2 dyd/ n3 /2 < }. Then for each fixed x, F A f x, y may be viewed as a mapping from 0, o H, and i is clear ha µ A λ f x = nλ/2 F A f x, y, µ λ f x = nλ/2 F f y.

4 382 Liu Lanzhe I is clear ha Definiions 2 and 3 are he paricular examples of Definiion. Noe ha when m = 0,T A is jus he commuaor of F and A, and when m>0, i is he nonrivial generalizaions of he commuaors. I is well-known ha mulilinear operaors are of grea ineres in harmonic analysis and have been widely sudied by many auhors see [2 5,7,8]. The main purpose of his paper is o sudy he coninuiy for he mulilinear operaors on he Triebel Lizorkin spaces. We shall prove he following heorems in 3. Theorem. Le gλ A be he mulilinear Lilewood Paley operaor as in Definiion 2. If 0<β</2and D α A β for α =m. Then a gλ A maps Lp coninuously ono Ḟp β, for <p< ; b gλ A maps Lp coninuously ono L q for <p<n/βand /p /q = β/n. Theorem 2. Le µ A λ be he mulilinear Marcinkiewicz operaor as in Definiion 3. If0< β</2and D α A β for α =m. Then a µ A λ maps Lp coninuously ono Ḟp β, for <p< ; b µ A λ maps Lp coninuously ono L q for <p<n/βand /p /q = β/n. 3. Main heorem and proof We firs prove a general heorem. Main Theorem. Le 0 <β<and D α A β for α =m. Suppose F,T and T A are he same as in Definiion, if T is bounded on L p for <r< and T saisfies he following size condiion: F A f x F A f x 0 D α A β β/n Mf x for any cube wih supp f 2 c and x. Then a T A maps L p o Ḟp β, for <p< ; b T A maps L p o L q for <p<n/βand /q = /p β/n. To prove he heorem, we need he following lemmas. Lemma []. For0<β<,<p<, wehave f Ḟ β, p sup β/n fx f dx sup inf c β/n fx c dx L p L p.

5 Triebel Lizorkin space esimaes 383 Lemma2 []. For0<β<,p, wehave f β sup β/n fx f dx sup β/n fx f p dx /p. Lemma3 [,2]. Supposeha r<p<n/δ and/q =/p δ/n.then M δ,r f L q f L p. Lemma4 [5]. Le Abe afuncion on and D α A L q for α =mand some q>n. Then R m A; x,y x y m /q D α Az dz q, x, y x,y where x, y is he cube cenered a x and having side lengh 5 n x y. Proof of Main Theorem. a Fix a cube = x 0,l and x. Le = 5 n and Ãx = Ax α! Dα A xα, hen R m A; x,y = R m Ã;x,y and D α Ã = D α A D α A for α =m. We wrie, for f = fχ and f 2 = fχ \, F A R m Ã; x,y f x = x y m Fx,y,fydy R m Ã;x,y = x y m Fx,y,f 2 ydy R m Ã; x,y x y m Fx,y,f ydy Rn Fx,y,x y α α! x y m D α Ãyf ydy, hen T A f x T Ãf 2 x 0 = F A f x FÃ f 2 x 0 F R m Ã; x, x m f x x α α! F x mdα Ãf x Thus, FÃ f 2 x F Ã f 2 x 0 =Ix IIx IIIx. β/n β/n T A f x T Ãf x 0 dx := I II III. Ixdx β/n IIxdx β/n IIIxdx

6 384 Liu Lanzhe Now, le us esimae I, II and III, respecively. Firs, for x and y, using Lemmas 2 and 4, we ge R m Ã; x,y x y m sup D α Ax D α A x x y m β/n D α A β. Thus, by Holder s inequaliy and he L r boundedness of T for <r<p, we obain I D α A β Tf x dx D α A β Tf L r /r D α A β f L r /r D α A β M r f x. Secondly, for <r<q, using he following inequaliy []: D α A D α A fχ L r /rβ/n D α A β M r f x and similar o he proof of I, we gain II β/n T D α A D α A f χ L r /r β/n /r D α A β M r f x. For III, using he size condiion of T A,wehave III D α A β Mf x. D α A D α A f χ L r We now pu hese esimaes ogeher, and aking he supremum over all such ha x, and using he L p boundedness of M r for r<p, we obain T A f Ḟ β, D α A β f L p. p This complees he proof of a. b By same argumen as in he proof of a, we have T A f x T Ãf 2 x 0 dx D α A β M β,r f M β, f.

7 Triebel Lizorkin space esimaes 385 Thus, T A f # D α A β M β,r f M β, f. Using Lemma 3, we gain T A f L q T A f # L q D α A β M β,r f L q M β, f L q f L p. This complees he proof of b and he Main Theorem. To prove Theorems and 2, we need he following lemma. Lemma 5. Le <p<,0<β<and D α A β for α =m. Then g A λ and µa λ are all bounded on L p. Proof. For gλ A, by Minkowski inequaliy and he condiion of ψ,wehave gλ A f x fz R m A; x,z x z m nλ /2 ψ y z 2 dyd n dz fz R m A; x,z x z m nλ 2n y z / 2n2 0 0 nλ /2 dyd n dz fz R m A; x,z x z m [ n x y Noing ha n nλ dy y z 2n2 M x z 2n2 x z 2n2 and 0 d x z 2n2 = x z 2n, nλ /2 dy d] y z 2n2 dz.

8 386 Liu Lanzhe we obain gλ A f x fz R m A; x,z d /2 x z m 0 x z 2n2 dz fz R m A; x,z = x z mn dz. For µ λ, noe ha x z 2, y z x z x z 3 when x y, y z, and x z 2 k 2 k2, y z x z 2 k3 when x y 2 k, y z,wehave µ A λ f x [ nλ ] y z Rm A; x,z fz 2 /2 y z n x z m χ Ɣzy, dyd n3 dz R m A; x,z fz x z m [ 0 x y R m A; x,z fz x z m [ 0 k=0 2 k < x y 2 k R m A; x,z fz x z m/2 R m A; x,z fz x z m/2 ] nλ /2 χ Ɣz y, dyd x z 3 2n 2 n3 dz ] nλ /2 χ Ɣz y, n 3 dyd x z 2 k3 2n 2 dz [ x z /2 ] d /2 x z 3 2n dz [ ] /2 2 knλ 2 k n n 2 k d k=0 2 2 k x z x z 2 k3 2n dz R m A; x,z fz x z mn dz R m A; x,z fz x z mn R m A; x,z = x z mn fz dz. Thus, he lemma follows from [2]. [ ] /2 dz 2 kn λ k=0

9 Triebel Lizorkin space esimaes 387 Now we can prove Theorems and 2. Since g λ and µ λ are all bounded on L p for <p< by Lemma 5, i suffices o verify ha gλ A and µa λ saisfy he size condiion in he Main Theorem. For gλ A, we wrie F A Rm Ã; x,z f x, y = x z m ψ y zf zdz Rm Ã; x,z = x z m ψ y zf 2 zdz Rm Ã; x,z x z m ψ y zf zdz x z α ψ y z α! x z m D α Ãzf zdz, hen β/n gλ A f x gãλ f 2x 0 dx nλ/2 = β/n F A f x, y nλ/2 F Ã f x 0,y dx nλ/2 R m Ã; x, β/n F x m f y dx nλ/2 β/n α! x α F x mdα Ãf y dx β/n nλ/2 F Ã f 2 x, y nλ/2 F Ã f 2 x 0,y dx :=III III. For I and II, similar o he proof of Lemma 5 and he Main Theorem, we ge I D α A β g λ f x dx D α A β g λ f L r /r

10 388 Liu Lanzhe D α A β f L r /r D α A β M r f x and II β/n β/n /r g λ D α A D α A f χ L r /r D α A β M r f x. D α A D α A f χ L r For III, we wrie = nλ/2 F Ã f 2 x, y nλ/2 [ x 0 y ] x z m x 0 z m nλ/2 F Ã f 2 x 0,y R m Ã; x,zψ y zf 2 zdz nλ/2 ψ y zf 2 z x 0 z m [R m Ã; x,z R m Ã;x 0,z]dz [ nλ/2 ] nλ/2 x 0 y R mã; x 0,zψ y zf 2 z x 0 z m dz [ nλ/2 x z α α! x z m ] nλ/2 x 0 z α x 0 y x 0 z m D α Ãzψ y zf 2 zdz := III III 2 III 3 III 4. Noe ha x z x 0 z for x and z \. By he condiion of ψ and similar o he proof of Lemma 5, we obain β/n III dx β/n \ x x 0 x 0 z mn R mã;x,z fz dz dx

11 Triebel Lizorkin space esimaes 389 For III 2, by he formula [5] and Lemma 4, we ge D α A β D α A β D α A β k=0 k= 2k \2k x x 0 fz dz x 0 z n 2 k 2 k fz dz 2 k 2 k Mf x k= D α A β Mf x. R m Ã; x,z R m Ã;x 0,z= η <m R m Ã; x,z R m Ã;x 0,z D α A β β/n x x 0 x 0 z m. η! R m η D η Ã; x,x 0 x z η Thus β/n III 2 dx β/n D α A β R m Ã;x,z R m Ã;x 0,z x 0 z mn fz dzdx x x 0 fz dz x 0 y n \ k=0 2k \2k D α A β β/n Mf x. For III 3, by he inequaliy: a /2 b /2 a b /2 for a b>0, we gain β/n III 3 dx β/n [ nλ/2 x x 0 /2 ψ y z R m Ã; x 0,z f 2 z x z m nλ/2 ] 2 dyd n /2 dzdx

12 390 Liu Lanzhe Rn f 2 z R m Ã; x 0,z x x 0 /2 β/n x z m d /2 0 x z 2n2 dzdx D α x x 0 /2 A β x 0 z n/2 f 2z dz D α A β Mf x. For III 4, by Lemma 4, we know ha D α Az D α A Dα A β x 0 z β. Thus, similar o he proof of III and III 3, we obain β/n III 4 dx x x0 β/n x 0 z n x x 0 /2 x 0 z n/2 f 2 z D α Ãz dzdx D α A β 2 kβ 2 kβ /2 Mf x k= D α A β Mf x. Thus, III D α A β Mf x. For µ A λ, similarly, we have β/n µ A λ f x µãλ f 2x 0 dx nλ/2 = β/n F A f x, y nλ/2 F à f 2 x 0,y dx nλ/2 R m Ã; x, β/n F x m f y dx nλ/2 x α β/n F α! x m Dα Ãf y dx

13 Triebel Lizorkin space esimaes 39 nλ/2 β/n F Ã f 2 x, y nλ/2 F Ã f 2 x 0,y dx :=III III and I II β/n D α A β µ λ f x dx D α A β µ λ f L r /r D α A β f L r /r D α A β M r f x, β/n /r µ λ D α A D α A f χ L r /r D α A β M r f x. D α A D α A f χ L r For III, we wrie = y z nλ/2 F Ã f 2 x, y nλ/2 [ x 0 y x z m x 0 z m y zr mã; x,zf 2 z y z n dz nλ/2 y zf 2 z y z y z n x 0 z m [R m Ã; x,z R m Ã;x 0,z]dz [ nλ/2 ] nλ/2 x 0 y y z y zr mã; x 0,zf 2 z y z n x 0 z m dz nλ/2 F Ã f 2 x 0,y ]

14 392 Liu Lanzhe [ nλ/2 x z α α! y z x z m ] nλ/2 x 0 z α y zd α Ãzf 2 z x 0 y x 0 z m y z n dz := III III 2 III 3 III 4. By he condiion of and similar o he proof of Lemma 5, we ge β/n III dx x x 0 β/n \ x 0 z mn R mã;x,z fz dz dx D α x x 0 A β fz dz k=0 2k \2k x 0 z n D α A β Mf x, β/n β/n β/n III 2 dx D α A β R m Ã;x,z R m Ã;x 0,z x 0 z mn fz dzdx x x 0 fz dz x 0 y n \ k=0 2k \2k D α A β β/n Mf x, III 3 dx β/n R mã; x 0,z x 0 z m ] 2 dyd n3 [ nλ/2 x x 0 /2 χ Ɣz y, f 2 z nλ/2 y z n /2 dzdx Rn R m Ã;x 0,z f 2 z x x 0 /2 β/n x 0 z mn/2 dzdx D α A β Mf x

15 Triebel Lizorkin space esimaes 393 and Thus, β/n β/n III 4 dx x x0 x 0 z n x x 0 /2 x 0 z n/2 f 2 z D α Ãz dzdx D α A β 2 kβ 2 kβ /2 Mf x k= D α A β Mf x. III D α A β Mf x. These yield he desired resuls. References [] hanillo S, A noe on commuaors, Indiana Univ. Mah. J [2] hen W G, Besov esimaes for a class of mulilinear singular inegrals, Aca Mah. Sinica [3] ohen J, A sharp esimae for a mulilinear singular inegral on, Indiana Univ. Mah. J [4] ohen J and Gosselin J, On mulilinear singular inegral operaors on, Sudia Mah [5] ohen J and Gosselin J, A BMO esimae for mulilinear singular inegral operaors, Illinois J. Mah [6] oifman R, Rochberg R and Weiss G, Facorizaion heorems for Hardy spaces in several variables, Ann. Mah [7] Ding Y, A noe on mulilinear fracional inegrals wih rough kernel, Adv. Mah. hina [8] Ding Y and Lu S Z, Weighed boundedness for a class rough mulilinear operaors, Aca Mah. Sinica [9] Janson S, Mean oscillaion and commuaors of singular inegral operaors, Ark. Mah [0] Liu Lanzhe, oninuiy for commuaors of Lilewood Paley operaor on cerain Hardy spaces, J. Korean Mah. Soc [] Paluszynski M, haracerizaion of he Besov spaces via he commuaor operaor of oifman, Rochberg and Weiss, Indiana Univ. Mah. J [2] Sein E M, Harmonic analysis: Real variable mehods, orhogonaliy and oscillaory inegrals Princeon, NJ: Princeon Univ. Press 993 [3] Torchnisky A, The real variable mehods in harmonic analysis, Pure Appl. Mah. New York: Academic Press 986 vol. 23 [4] Torchnisky A and Wang S, A noe on he Marcinkiewicz inegral, olloq. Mah. 60/