LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITLLEWOOD-PALEY OPERATOR. Zhang Mingjun and Liu Lanzhe

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1 3 Kragujevac J. Mah. 27 (2005) LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITLLEWOOD-PALEY OPERATOR Zhang Mingjun and Liu Lanzhe Deparen of Maheaics Hunan Universiy Changsha, 40082, P.R.of China (e-ails: zhangingjun2004@sina.co, lanzheliu@263.ne ) (Received March, 2005) Absrac. In his paper, we will sudy he coninuiy of ulilinear couaor generaed by Lilewood-Paley operaor and he of funcion b, wich belongs o Lipschiz space, in he Triebel-Lizorkin, Hardy and Herz-Hardy space.. INTRODUCTION Le T be a Calderón-Zygund operaor. Coifan, Rochberg and Weiss proved [4] ha he couaor [b, T ](f)(x) = b(x)t (f)(x) T (bf)(x) is bounded on L p (R n ) for < p < and b BMO. Chanillo proved a siiliar resul [2] when T is replaced by he fracional operaors. Janson and Paluszynski sudy hese resul [7, 5] for he Triebel-Lizorkin spaces and when b Lip β, where Lip β is he hoogeneous Lipschiz space. The ain purpose of his paper is o discuss he boundedness of ulilinear couaor generaed by Lilewood-Paley operaor and coninuiy of b Lip β in he Triebel-Lizorkin, Hardy and Herz-Hardy space.

2 32 2. PRELIMINARIES AND DEFINITIONS M(f) denoes he Hardy-Lilewood axial funcion of f and M p (f) = (M(f p )) /p for 0 < p <. denoes a cube of R n wih side parallel o he axes. Le f = f(x)dx and f # (x) = sup x f(y) f dy. Mark Hardy spaces by H p (R n ). I is well known ha H p (R n )(0 < p ) has he aoic decoposiion [, 6, 7]. For β > 0 and p >, le F p β, be he hoogeneous Tribel-Lizorkin space. The Lipschiz space Lip β (R n ) is he space of funcions f such ha f Lipβ = sup x,y R n x y f(x) f(y) x y β <. Lea. [5] For 0 < β < and < p <, f F p β, sup f(x) f + β dx n L p sup inf f(x) c dx c + β n L p. Lea 2. [5] For 0 < β < and p, f Lipβ sup sup f(x) f + β dx n ( /p f(x) f β dx) p. n Lea 3. [2] For r < and β > 0, le M β,r (f)(x) = sup x βr n /r f(y) r dy if we suppose ha r < p < β/n, and / = /p β/n, hen M β,r (f) L C f L p..

3 33 Lea 4. [5] If 2 hen f f 2 C f Λβ 2 β/n. Lea 5. [0] Le 0 < β, < p < n/β, / = /p β/n and b Lip β (R n ). Then gψ b is bounded fro L p (R n ) o L (R n ). Definiion. Le 0 < p, <, α R, B k = {x R n, x 2 k }, A k = B k \B k and χ k = χ Ak for k Z. ) The hoogeneous Herz space is defined by K α,p = {f L Loc(R n \{0}), f K α,p < }, where f K α,p = k= 2 kαp fχ k p L 2) The nonhoogeneous Herz space is defined by /p. K α,p (R n ) = {f L Loc(R n ), f K α, (R n ) < }, where f K α,p (R n ) = [ ] /p 2 kαp fχ k p L + fχ B 0 p L. k= Definiion 2. Le α R and 0 < p, <. () The hoogeneous Herz ype Hardy space is defined by H α,p K (R n ) = {f S (R n ) : G(f) K α,p (R n )}, and f H K α,p = G(f) K α,p ;

4 34 (2) The nonhoogeneous Herz ype Hardy space is defined by HK α,p (R n ) = {f S (R n ) : G(f) K α,p (R n )}, and f HK α,p = G(f) K α,p where G(f) is he grand axial funcion of f. The Herz ype Hardy spaces have he characerizaion of he aoic decoposiion. Definiion 3. Le α R and < <. A funcion a(x) on R n is called a cenral (α, )-ao (or a cenral (a, )-ao of resric ype), if ) Suppa B(0, r) for soe r > 0 (or for soe r ), 2) a L B(0, r) α/n, 3) R n a(x)xη dx = 0 for η [α n( /)]. Lea 6. [6, 4] Le 0 < p <, < < and α n( /). A eperae disribuion f belongs o H K α,p (R n )(or HK α,p (R n )) if and only if here exis cenral (α, )-aos(or cenral (α, )-aos of resric ype) a j suppored on = B(0, 2 j ) and consans λ j, j λ j p < such ha f = j= λ j a j (or f = j=0 λ j a j )in he S (R n ) sense, and f H K α,p (or f HK α,p) j /p λ j p. Definiion 4. Le ε > 0 and ψ be a fixed funcion which saisfies he following properies: ) Rn ψ(x)dx = 0, 2) ψ(x) C( + x ) (n+), 3) ψ(x + y) ψ(x) C y ε ( + x ) (n++ε) when 2 y < x. Le be a posiive ineger and b j ( j ) be locally inegrable funcions and b = (b,, b ). The ulilinear couaor of Lilewood-Paley operaor is defined

5 35 by where g b ψ (f)(x) = F b (f)(x) = R n ( 0 F b (x) 2 d ) /2, (b j (x) b j (y))ψ (x y)f(y)dy, j= and ψ (x) = n ψ(x/) for > 0. Le F (f) = ψ f. Define Lilewood-Paley g funcion by [7] g ψ (f)(x) = ( 0 F (f)(x) 2 d ) /2. Le H be he space, H = {h : h = ( 0 h() 2 d/) /2 < }. For each fixed x R n, F (f)(x) ay be viewed as a apping fro [0, + ) o H, and i is clear ha g ψ (f)(x) = F (f)(x) and g b ψ (f)(x) = F b (f)(x). Noe ha when b = = b, g b ψ is jus he order couaor. I is well known ha couaors are of grea ineres in haronic analysis and have been widely sudied by any auhors ([]-[4][7]-[0][2][5]). Our ain purpose is o esablish he boundedness of he ulilinear couaor on Triebel-Lizorkin, Hardy and Herz- Hardy space. Le be a posiive ineger, j, b Lipβ = j= b j Lipβ and C j be he faily of all finie subses σ = {σ(),..., σ(j)} of {,..., } of j differen eleens. For σ C j, le σ c = {,..., } \ σ. For b = (b,..., b ) and σ = {σ(),..., σ(j)} C j, le b σ = (b σ(),..., b σ(j) ), b σ = b σ() b σ(j) and b σ Lipβ = b σ() Lipβ b σ(j) Lipβ. 3. THEOREMS AND PROOFS Theore. Le 0 < β < in(, ε/), < p <, b = (b,..., b ) where b j Lip β (R n ) for j and g b ψ be he ulilinear couaor of Lilewood- Paley operaor. Then

6 36 (a) g b ψ is bounded fro L p (R n ) o F, p (R n ). (b) g b ψ is bounded fro L p (R n ) o L (R n ) for /p / = /n and /p > /n. Proof. (a) Fix a cube = (x 0, l) and x, when = [0]. Now consider he case when 2. Le b = ((b ),..., (b ) ), where (b j ) = b j(y)dy and j. If f = f + f 2 where f = fχ 2 and f 2 = fχ R n \2, hen F b (f)(x) = (b (x) b (y)) (b (x) b (y))ψ (x y)f(y)dy R n = (b (x) (b ) ) (b (x) (b ) )F (f)(x) + ( ) F ((b (b ) ) (b (b ) )f)(x) + j= σ Cj ( ) j (b(x) b ) σ R n (b(y) b ) σ cψ (x y)f(y)dy = (b (x) (b ) ) (b (x) (b ) )F (f)(x) + ( ) F ((b (b ) ) (b (b ) )f )(x) + ( ) F ((b (b ) ) (b (b ) )f 2 )(x) + j= σ Cj ( ) j (b(x) b ) σ F ((b b ) σ cf)(x). Therefore g b ψ (f)(x) g ψ (((b ) b ) ((b ) b )f 2 )(x 0 ) F b (f)(x) F (((b ) b ) ((b ) b )f 2 )(x 0 ) (b (x) (b ) ) (b (x) (b ) )F (f)(x) + (b(x) b ) σ F ((b b ) σ cf)(x) j= σ Cj + F ((b (b ) ) (b (b ) )f )(x) + F ((b (b ) ) (b (b ) )f 2 )(x) F ((b (b ) ) (b (b ) )f 2 )(x 0 ) = I (x) + I 2 (x) + I 3 (x) + I 4 (x).

7 37 Thus + n + n + g b ψ (f)(x) g ψ ((b ) b ) ((b ) b )f 2 )(x 0 ) dx I (x)dx + I 4 (x)dx + n = I + II + III + IV. By using Lea 2, we have for I, I + n + n I 2 (x)dx + + n I 3 (x)dx sup b (x) (b ) b (x) (b ) g ψ (f)(x) dx x C b Lipβ n + n C b Lipβ M(g ψ (f))(x). g ψ (f)(x) dx Fix r, such ha < r < p. Le µ, µ be he inegers such ha µ + µ =, 0 µ < and 0 < µ. By using Hölder s ineualiy, he boundedness of g ψ on L r and Lea 2, we ge II j= σ Cj C C C j= σ Cj j= σ Cj j= σ Cj + n + n + n + n C b Lipβ M r (f)(x); ( ( ( b(x) b ) σ g ψ (( b b ) σ cf)(x) dx ( b(x) ) /r ( b ) σ r dx g ψ (( b ) /r b ) σ cf)(x) r dx ( b(x) ) /r ( b ) σ r dx ( b(x) ) /r b ) σ cf(x) r dx r ( ) b σ Lipβ µβ n bσ c Lipβ µ β /r n f(x) r dx By Hölder s ineualiy, we have for III III = + n g ψ ((b (b ) ) (b (b ) )f )(x) dx

8 38 C C C + n + n + n /r g ψ ( (b j (b j ) )f )(x) dx /r R n j= /r /r (b j (x) (b j ) )f(x) r dx 2 j= /r ( ) /r b Lipβ n f(x) r dx C b Lipβ M r (f)(x); 2 Since x 0 y x y for y (2) c, using by Lea 4 and he condiion of ψ, we have for IV, F ((b (b ) ) (b (b ) )f 2 )(x) F ((b (b ) ) (b (b ) )f 2 )(x 0 ) ( ψ (x y) ψ (x 0 y) f(y) b j (y) (b j ) dy) 2 d /2 0 (2) c j= x x C 0 ε 2 /2 0 ( + x 0 y ) f(y) b n++ε j (y) (b j ) dy d j= C x 0 x ε x 0 y (n+ε) f(y) b j (y) (b j ) dy (2) c j= C x 0 x ε x 0 y (n+ε) f(y) b j (y) (b j ) dy k= 2 k+ \2 k j= C 2 kε 2 k+ f(y) ( b j (y) (b j ) 2 + (b k+ j ) 2 k+ (b j ) )dy k= 2 k+ j= C 2 kε 2 k+ n b Lipβ M(f) k= C b Lipβ n M(f) 2 ( ε)k k= C b Lipβ n M(f). The following holds IV C b Lipβ M(f). Puing hese esiaes ogeher, aking he supreu over all such ha x

9 39 and by using Lea, we obain This coplee he proof of (a). g b ψ (f)(x) F C b p, Lipβ f L p. (b) Like in he proof of (a), we have g b ψ (f)(x) g ψ (((b ) b ) ((b ) b )f 2 )(x 0 ) dx I (x)dx + I 2 (x)dx + I 3 (x)dx + I 4 (x)dx C b Lipβ (M, (g ψ (f)) + M,r (f) + M,r (f) + M, (f)). Thus (g b ψ (f)) # C b Lipβ (M, (g ψ (f)) + M,r (f) + M, (f)). By using Lea 3 and he boundedness of g ψ, we have g b ψ (f) L C (g b ψ (f)) # L C b Lipβ ( M, (g ψ (f)) L + M,r (f) L + M, (f) L ) C f L p. This coplees he proof of (b). Theore 2. Le 0 < β, ax(n/(n + ), n/(n + ε)) < p, / = /p /n and b = (b,..., b ) where b j Lip β (R n ) for j. Then g b ψ is bounded fro H p (R n ) o L (R n ). Proof. I is enough o show ha here exiss a consan C > 0 such ha for every H p -ao a, g b ψ (a) L C. Le a be a H p -ao suppored on a cube = (x 0, r), a L /p and R n a(x)xγ dx = 0 for γ [n(/p )]. When = see [0]. Now consider he case 2. ( / ( g b ψ (a)(x) L g b ψ (a)(x) dx) + x x 0 2r = I + II. x x 0 >2r g b ψ (a)(x)dx ) /

10 40 Choose < p < /β and fro L p (R n ) o L (R n )(see Lea 5), we ge for I such ha / = /p β/n. By he boundednss of g b ψ I C g b ψ (a) L r n( ) C a L r n( ) C. Le τ, τ N such ha τ + τ =, and τ 0. We ge for II F b (a)(x) (b (x) b (x 0 )) (b (x) b (x 0 )) + j= σ Cj (b(x) b(x 0 )) σ c B B (ψ (x y) ψ (x x 0 ))a(y)dy (b(y) b(x 0 )) σ ψ (x y)a(y)dy C b Lipβ x x 0 ψ (x y) ψ (x x 0 ) a(y) dy B + C b Lipβ x x 0 τβ y x 0 τ β ψ (x y) a(y) dy τ+τ = C b Lipβ x x 0 ( + x x 0 ) n++ε + C b Lipβ τ+τ = C b Lipβ ( + x x 0 ) + C b Lipβ ( + x x 0 ) Thus g b ψ (a)(x) C ( b Lipβ 0 + C ( b Lipβ 0 so B B x 0 y ε a(y) dy x x 0 τβ ( + x x 0 ) n+ r+ε+n( p ) n++ε r+n( p ). n+ ( + x x 0 ) n++ε ( + x x 0 ) n+ C b Lipβ x x 0 n r +n( p ), ) 2 d ) 2 d B /2 /2 ( II C b Lipβ r +n( p ) x x 0 n dx x x 0 >2r C b Lipβ. This coplees he proof of Theore 2. y x 0 τ β a(y) dy r +ε+n( p ) r +n( p ) ) /

11 4 Theore 3. Le 0 < β, 0 < p <, <, 2 <, / / 2 = /n, n( / ) α < n( / ) + and b = (b,..., b ) where b j Lip β (R n ) for j. Then g b ψ is bounded fro H Proof. Le f H α,p K (R n ) o K α,p 2. α,p K (R n ) and f = j= λ j a j, suppa j = B(0, 2 j ), a j be a cenral (α, ) ao, and j= λ j p < (Lea 6). When =, we have g b ψ α,p C p K 2 k 2 2 kαp k= j= 2 kαp + C k= = I + II 2. j=k By he boundedness of g b ψ on (L, L 2 ), we have for II 2 II 2 C b p p Lip β 2 kαp λ j a j L C b p Lip β k= j=k 2 kαp k= j=k p λ j p 2 jα p λ j g b ψ (a j)χ k L 2 p λ j g b ψ (a j)χ k L 2 k= j=k λ C b p j p 2 (k j)αp, 0 < p Lip β k= 2 ( kαp j=k λ j p 2 jαp/2) ( j=k 2 ) p/p jαp /2, < p < j= λ j p j+ C b p k= 2(k j)αp, 0 < p Lip β ( ( ) k= j=k λ j p 2 (k j)αp) p/p 2 j=k 2 p 2 (k j)α, < p < C b p Lip β j= λ j p. For I, we have F b (a j )(x) (b (x) b (0)) (ψ (x y) ψ (x))a j (y)dy + ψ (b (y) b (0))a j (y)dy [ x β y ε C b Lipβ ( + x ) a j(y) dy n++ε y + Bj β ] ( + x y ) a j(y) dy n+

12 42 Thus [ x β C b Lipβ y ε a ( + x ) n++ε j (y) dy ] + y ε a ( + x ) n+ j (y) dy [ x β ) α) C b Lipβ 2j(ε+n( ( + x ) n++ε ] ) α) + 2j(β+n(, ( + x ) n+ g b ψ (a j)(x) C b Lipβ Fro ha, we have so ( + 0 ( 0 ( + x ) n+ ( + x ) n++ε ) 2 d /2 ) 2 /2 2 j(β+n( [ C b Lipβ x (n+ε) x β 2 j(ε+n( ) α) ] x n 2 j(β+n( ) α) C b Lipβ x n 2 j(β+n( ) α). g b ψ (a j)χ k L 2 C b Lipβ 2 j(β+n( ) α) I C b p Lip β C b p Lip β ( x β 2 j(ε+n( ) α) ) α) C b Lipβ 2 j(β+n( ) α) 2 kn( 2 ) ) /2 B k x n 2 dx C b Lipβ 2 [j(β+n( ) α) k(β+n( ))], 2 kαp k= j= k= k 2 ( k= 2 kαp k 2 C b p Lip β λ j 2 [j(β+n( ) α) k(β+n( p ))] j= λ j p 2 (j k)(β+n( ) α)p, 0 < p j= λ j p 2 p 2 [j(β+n( ( k 2 j= 2 p 2 [j(β+n( ) p/p ) α) k(β+n( ))] ) ) α) k(β+n( ))], < p < j= λ j p k=j+2 2 (j k)(β+n( ) α)p, 0 < p j= λ j p k=j+2 2 p 2 [(j k)(β+n( ) α)], < p <

13 43 Then C b p Lip β j= When 2, we have λ j p. g b ψ (f) K α,p C b Lipβ ( 2 g b ψ (f) α,p C p K 2 j= k 2 2 kαp k= j= 2 kαp + C = I + II. k= λ j p ) /p C f H K α,p. j=k By he boundedness of g b ψ on (L, L 2 ), we have for II II C b p Lip β k= C b p Lip β k= 2 kαp ( 2 kαp ( j=k j=k λ j a j L ) p λ j 2 jα ) p p λ j g b ψ (a j )χ k L 2 p λ j g b ψ (a j )χ k L 2 C b p Lip β { k= j=k λ j p 2 (k j)αp, 0 < p k= 2 kαp ( j=k λ j p 2 jαp/2 )( j=k 2 jαp /2 ) p/p, < p < C b p Lip β For I, we have j= λ j p. F b (a j ))(x) (b (x) b (0)) (b (x) b (0)) (ψ (x y) ψ (x))a j (y)dy + (b(x) b(0)) σ c (b(y) b(0)) σ ψ (x y)a j (y)dy j= σ Cj C b Lipβ x ψ (x y) ψ (x) a j (y) dy + C b Lipβ τ+τ = C b Lipβ x ( + x ) n++ε + C b Lipβ τ+τ = x τβ y τ β ψ (x y) a j (y) dy y ε a j (y) dy x τβ ( + x ) n+ y τ β a j (y) dy

14 44 Therefore g b ψ (a j )(x) = C b Lipβ x ( + x ) ( + C b Lipβ τ+τ = F b (a j )(x) 2 d ) /2 ) α) 2j(ε+n( n++ε C b Lipβ x 2 j(ε+n( ) α) x τβ ( + x ) n+ 2j(τ β+n( ) α). 0 ( 0 ( + x ) n++ε + C ( b Lipβ x τβ 2 j(τ β+n( ) α) τ+τ = 0 C b Lipβ x x (n+ε) 2 j(ε+n( ) α) + C b Lipβ τ+τ = x τβ x n 2 j(τ β+n( ) α) C b Lipβ x n 2 j(+n( ) α). ) 2 d /2 ( + x ) n+ ) 2 d /2 Then, ( ) g b ψ (a j )χ k L 2 C /2 b Lipβ 2 j(+n( ) α) x n 2 dx C b Lipβ 2 [j(+n( ) α) k(+n( ))], so I C b p Lip β C b p Lip β C b p Lip β k 2 2 kαp k= j= k= k 2 ( k= 2 kαp k 2 j= Fro I and II, we have λ j 2 [j(+n( ) α) k(+n( p ))] j= λ j p 2 (j k)(+n( ) α)p, 0 < p j= λ j p 2 p 2 [j(+n( ( k 2 j= 2 p 2 [j(+n( λ j p. g b ψ (f) C b Lipβ j= ) p/p ) α) k(+n( ))] λ j p /p ) ) α) k(+n( ))] C f H K α,p., < p <

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