LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR

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1 ialian journal of pure and applied aheaics n (29 225) 29 LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR Ying Shen Lanzhe Liu Deparen of Maheaics Changsha Universiy of Science and Technology Changsha, 477 P.R. of China e-ail: lanzheliu@63.co Absrac. In his paper, we will sudy he coninuiy of ulilinear couaor generaed by Lilewood-Paley operaor and he funcions b j on Triebel-Lizorkin space, Hardy space and Herz-Hardy space, where he funcions b j belong o Lipschiz space. 2 Maheaics Subjec Classificaion: 42B2, 42B25. Keywords and phrases: Lilewood-Paley operaor; Mulilinear couaor; Triebel- Lizorkin space; Herz-Hardy space; Herz space; Lipschiz space.. Inroducion We know, he couaor [b, T ](f)(x) = b(x)t (f)(x) T (bf)(x) is bounded on L p (R n ) for < p < when T is he Calderón-Zygund operaor and b BMO(R n ). Janson and Paluszynski sudy he couaor for he Triebel- Lizorkin space and he case b Lip β (R n ), where Lip β (R n ) is he hoogeneous Lipschiz space. Chanillo (see [2]) proves a siilar resul when T is replaced by he fracional operaors. The ain purpose of his paper is o discuss he boundedness of Lilewood-Paley ulilinear couaor generaed by Lilewood-Paley operaor and Lipschiz funcions on Triebel-Lizorkin space, Hardy space and Herz- Hardy space. 2. Preliinaries and Definiions Throughou his paper, M(f) will denoe he Hardy-Lilewood axial funcion of f, and wrie M p (f) = (M(f p )) /p for < p <. will denoe a cube of R n wih side parallel ohe axes. Le f = f(x)dx and f # (x) = sup f(y) f dy denoe he x Hardy spaces by H p (R n ). I is well known ha H p (R n ) ( < p ) has he

2 2 ying shen, lanzhe liu aoic decoposiion characerizaion (see [], [6], [7]). For β > and p >, le F p β, (R n ) 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. (see [5]) For < β <, < p <, we have f F p β, sup f(x) f + β dx n L p sup inf f(x) c dx c + β n Lea 2. (see [5]) For < β <, p, we have f Lipβ sup sup L p f(x) f + β dx n ( /p f(x) f β dx) p. n Lea 3. (see [2]) For r < and β >, le M β,r (f)(x) = sup x βr n /r f(y) r dy suppose ha r < p < n/β, and / = /p β/n, hen M β,r (f) L C f L p.,. Lea 4. (see [5]) Le 2, hen f f 2 C f Λβ 2 β/n. Definiion. Le < 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 where K α,p (R n ) = {f L Loc(R n \{}) : f K α,p < }, f K α,p = 2 kαp fχ k p L /p ;

3 lipschiz esiaes for ulilinear couaor... 2 where 2) The nonhoogeneous Herz space is defined by K α,p (R n ) = {f L Loc(R n ) : f K α,p < }, f K α,p = [ Definiion 2. Le α R, < p, <. ] /p 2 kαp fχ k p L + f χ B p L. k= () 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 ; and (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 )}, 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 aoic decoposiion characerizaion. Definiion 3. Le α R, < <. A funcion a(x) on R n is called a cenral (α, )-ao (or a cenral (a, )-ao of resric ype), if ) suppa B(, r) for soe r > (or for soe r ), 2) a L B(, r) α/n, 3) a(x)x η dx = for η [α n( /)]. R n Lea 5. (see [6], [4]) Le < p <, < < and α n( /). α,p A eperae disribuion f belongs o H K (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(, 2 j ) and consans λ j, λ j p < such ha f = λ j a j j (or f = λ j a j ) in he S (R n ) sense, and j= f H K α,p (or f HK α,p) j /p λ j p.

4 22 ying shen, lanzhe liu Definiion 4. Le < δ < n, ε > and ψ be a fixed funcion which saisfies he following properies: ) ψ(x)dx =, R n 2) ψ(x) C( + x ) (n+ δ), 3) ψ(x + y) ψ(x) C y ε ( + x ) (n++ε δ) when 2 y < x. Le be a posiive ineger and b j ( j ) be he locally inegrable funcion, se b = (b,, b ). The ulilinear couaor of Lilewood-Paley operaor is defined by where and g b ψ,δ (f)(x) = F b (f)(x) = R n ( F b (x) 2 d ) /2, (b j (x) b j (y))ψ (x y)f(y)dy, j= ψ (x) = n+δ ψ(x/) for >. Se F (f) = ψ f. We also define ha g ψ,δ (f)(x) = ( F (f)(x) 2 d ) /2, which is he Lilewood-Paley g funcion (see [7]). Le H be he space ( /2 H(R n ) = {h : h = h() d/) 2 < }, hen, for each fixed x R n F (f)(x) ay be viewed as a apping fro [, + ) o H, and i is clear ha g ψ,δ (f)(x) = F (f)(x) and g b ψ,δ (f)(x) = F b (f)(x). Noe ha when b = = b, g b ψ,δ 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 (see [-4], [7-], [2], [5]). Our ain purpose is o esablish he boundedness of he ulilinear couaor on Triebel-Lizorkin space, Hardy space and Herz-Hardy space. Given a posiive ineger and j, we se b Lipβ = b j Lipβ j=

5 lipschiz esiaes for ulilinear couaor and denoe by C j he faily of all finie subses σ = {σ(),, σ(j)} of {,, } of j differen eleens. For σ C j, se σ c = {,, } \ σ. For b = (b,, b ) and σ = {σ(),, σ(j)} C j, se 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. Lea 6. (see []) Le < β, < δ < n, < p < n/β, / = /p β/n and b Lip β (R n ). Then g b ψ,δ is bounded fro L p (R n ) o L (R n ). 3. Theores and proofs Theore. Le < δ < n, < β < in(, ε/), < p <, b = (b,, b ) wih b j Lip β (R n ) for j and g b ψ,δ be he ulilinear couaor of Lilewood-Paley operaor as in Definiion 4. Then a) g b ψ,δ is bounded fro L p (R n ) o Fp β, (R n ) for <p<n/δ and /p /=δ/n. b) g b ψ,δ is bounded fro L p (R n ) o L (R n ) for /p / = β + δ/n and /p > β + δ/n. Proof. (a). Fixed a cube = (x, l) and x, see [] when =. Consider now he case 2. Se where b = ((b ),, (b ) ), (b j ) = b j (y)dy, j. Wriing f = f + f 2, where f = fχ 2, f 2 = fχ R n \2, we have 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) hen +( ) F ((b (b ) ) (b (b ) )f)(x) + ( ) j (b(x) b ) σ (b(y) j= σ C R b ) σ cψ (x y)f(y)dy j n = (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),

6 24 ying shen, lanzhe liu g b ψ,δ (f)(x) g ψ,δ (((b ) b ) ((b ) b )f 2 )(x ) F b (f)(x) F (((b ) b ) ((b ) b )f 2 )(x ) (b (x) (b ) ) (b (x) (b ) )F (f)(x) + j= σ Cj (b(x) b ) σ F ((b b ) σ cf)(x) so, + F ((b (b ) ) (b (b ) )f )(x) + F ((b (b ) ) (b (b ) )f 2 )(x) F ((b (b ) ) (b (b ) )f 2 )(x ) = I (x) + I 2 (x) + I 3 (x) + I 4 (x), β + n + g b ψ,δ (f)(x) g ψ,δ ((b ) b ) ((b ) b )f 2 )(x ) dx β + n β + n I (x)dx + I 3 (x)dx + = I + II + III + IV. For I, by using Lea 2, we have I β + n β + n β + n I 2 (x)dx I 4 (x)dx sup b (x) (b ) b (x) (b ) g ψ,δ (f)(x) dx x C b Lipβ β β n + n C b Lipβ M(g ψ,δ (f))( x). g ψ,δ (f)(x) dx For II, aking < r < p < < n/δ, / +/ =, /s +/s =, / = /p δ/n, ps = r by using he Hölder s ineualiy and he boundedness of g ψ,δ fro L p (R n ) o L (R n ) and Lea 2, we ge II j= σ Cj C j= σ Cj ( b(x) b +β/n ) σ g ψ,δ (( b b ) σ cf)(x) dx ( ) / ( β/n b(x) b ) σ dx ( ) / g ψ,δ (( b b ) σ cfχ )(x) dx R n

7 lipschiz esiaes for ulilinear couaor C C C j= σ Cj j= σ Cj ( j= σ Cj b β/n σ Lipβ σ β/ ( ( b(x) ) /p b / ) σ cfχ p dx R n β/n b σ Lipβ ( /)+(/ps )+( δps/n)/ps ) ( b(x) /ps b ) σ c ps C b Lipβ M r,δ (f)( x). δps n f(x) ps dx /ps β/n b σ Lipβ σ β/n b σ c Lipβ σc β/n M r,δ (f)( x) For III, we choose < r < p < < n/δ, < δ < n, / = /p δ/n, r = ps, by he boundness of g ψ,δ fro L p (R n ) o L (R n ) and Hölder s ineualiy wih /s + /s =, we ge III = +β/n C β/n C β/n C / g ψ,δ ((b (b ) ) (b (b ) )f )(x) dx g ψ,δ ( (b j (y) (b j ) )fχ )(x) dx R n j= /p (b j (y) (b j ) ) p fχ p dx R n j= ( (δps/n)/ps) β/n ( /)+/ps ( ) /ps δps/n C b Lipβ M r,δ (f)( x). f(x) ps dx / /ps (b j (y) (b j ) ) ps dx j= For IV, since x y x y for y (2) c, by Lea 4 and he condiion on ψ, we have F ((b (b ) ) (b (b ) )f 2 )(x) F ((b (b ) ) (b (b ) )f 2 )(x ) 2 /2 ψ (x y) ψ (x y) f(y) b j (y) (b j ) dy d (2) c j=

8 26 ying shen, lanzhe liu (2)c x x C ε 2 /2 ( + x y ) f(y) b n++ε δ j (y) (b j ) dy d j= C x x ε x y (n+ε δ) f(y) b j (y) (b j ) dy (2) c j= C x x ε x y (n+ε δ) 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 k= 2 kε 2 k+ δ/n 2 k+ δ/n b Lipβ M r,δ (f)( x) C b Lipβ β n Mr,δ (f)( x), hus IV C b Lipβ M r,δ (f)( x). We pu hese esiaes ogeher, by using Lea and aking he supreu over all such ha x, we obain This coplee he proof of (a). hus ψ,δ (f) F C b p β, Lipβ f L p. (b) By soe arguen as in proof of (a), we have g b ψ,δ (f)(x) g ψ,δ (((b ) b ) ((b ) b )f 2 )(x ) dx I (x)dx + I 2 (x)dx + I 3 (x)dx + I 4 (x)dx C b Lipβ (M β, (g ψ,δ (f)( x)) + M β+δ,r (f)( x)), (g b ψ,δ (f)) # C b Lipβ (M β, (g ψ,δ (f)( x)) + M β+δ,r (f)( x)). By using Lea 3 and he boundedness of g ψ,δ, we have ψ,δ (f) L C (g b ψ,δ (f)) # L C b Lipβ ( M β, (g ψ,δ (f)( x)) + M β+δ,r (f)( x)) L C b Lipβ f L p. This coplee he proof of (b) and he heore.

9 lipschiz esiaes for ulilinear couaor Theore 2. Le < δ < n, < β + δ/ < in(γ/, /2), n/(n + β + δ/) < p, / = /p (β+δ)/n, b = (b,, b ) wih b j Lip β (R n ) for j. Then g b ψ,δ is bounded fro H p (R n ) o L (R n ). Proof. I suffices o show ha here exiss a consan C > such ha for every H p -ao a, ψ,δ (a) L C. Le a be a H p -ao, ha is ha a suppored on a cube = (x, r), a L /p and a(x)x γ dx = for γ [n(/p )]. R n When =, see []. Now consider he case 2. Wrie ψ,δ (a)(x) L ( / ( ) / g b ψ,δ (a)(x) dx) + g b ψ,δ (a)(x) dx x x 2r x x >2r = I + II. For I, choose < p < n/(β + δ) and such ha / = /p β + δ/n. By he boundednss of g b ψ,δ fro L p (R n ) o L (R n ) (see Theore ), we ge I C ψ,δ (a) L (x, 2r) / C a L p b Lipβ / C b Lipβ /p+/p + / C b Lipβ. For II, le τ, τ N such ha τ + τ =, and τ. We ge F b (a)(x) (b (x) b (x )) (b (x) b (x )) (ψ (x y) ψ (x x ))a(y)dy B + (b(x) b(x )) σ c (b(y) b(x )) σ ψ (x y)a(y)dy j= σ Cj C b Lipβ x x β +C b Lipβ τ+τ = B B x x τβ C x x β b Lipβ ( + x x ) n++ε δ +C b Lipβ τ+τ = C b Lipβ ( + x x ) +C b Lipβ ( + x x ) ψ (x y) ψ (x x ) a(y) dy B y x τ β ψ (x y) a(y) dy B x y ε a(y) dy x x τβ ( + x x ) n+ δ rβ+ε+n( p ) n++ε δ rβ+n( p ), n+ δ B y x τ β a(y) dy

10 28 ying shen, lanzhe liu hus g b ψ,δ (a)(x) C ( b Lipβ + C ( b Lipβ ( + x x ) n++ε δ ( + x x ) n+ δ ) 2 d ) 2 d /2 /2 r β+ε+n( p ) r β+n( p ) so, C b Lipβ x x n+δ r β+n( p ), ( / II C b Lipβ r β+n( p ) x x dx) n+δ C b. Lipβ x x >2r This coplee he proof of Theore 2. Theore 3. Le < β, < δ < n, < p <, <, 2 <, / / 2 = β +δ/n, n( / ) α < n( / )+β +δ/, b = (b,, b ) wih b j Lip β (R n ) for j. Then g b α,p ψ,δ is bounded fro H K (R n ) o K α,p 2 (R n ). Proof. By Lea 5, le f H B(, 2 j ), a j be a cenral (α, ) ao, and ψ,δ (f) α,p C p K 2 + C = I + II. K α,p (R n ) and f = k 2 2 kαp 2 kαp j=k For II, by he boundedness of g b ψ,δ on (L, L 2 ), we have II C b p Lip β C b p Lip β C b p Lip β 2 kαp j=k 2 kαp j=k j=k 2 kαp j=k p λ j a j L p λ j 2 jα λ j p 2 (k j)αp, < p λ j p 2 jαp/2 λ j p <. We have λ j ψ,δ (a j )χ k L 2 λ j a j, supp a j = p p λ j ψ,δ (a j )χ k L 2 2 jαp /2 j=k p/p, < p <

11 lipschiz esiaes for ulilinear couaor C b p Lip β λ j p. For I, when =, we have F b (a j )(x) (b (x) b ()) (ψ (x y) ψ (x))a j (y)dy + ψ (b (y) b ())a j (y)dy hus g b ψ,δ (a j)(x) C b Lipβ + fro ha we have so, ( [ x β y ε C b Lipβ ( + x ) a j(y) dy n++ε δ y + Bj β ] ( + x y ) a j(y) dy n+ δ [ x β C b Lipβ y ε a ( + x ) n++ε δ j (y) dy ] + y ε a ( + x ) n+ δ j (y) dy [ x β ) α) C b Lipβ 2j(ε+n( ( + x ) n++ε δ ] ) α) + 2j(β+n(, ( + x ) n+ δ ( ( + x ) n+ δ ( + x ) n++ε δ ) 2 d /2 ) 2 /2 2 j(β+n( x β 2 j(ε+n( ) α) ) α) [ ] C b Lipβ x (n+ε δ) x β 2 j(ε+n( ) α) x n+δ 2 j(β+n( ) α) C b Lipβ x n+δ 2 j(β+n( ) α), g b ψ,δ (a j)χ k L 2 C b Lipβ 2 j(β+n( ) α) I C b p Lip β ( ) /2 B k x n 2+ 2 δ dx C b Lipβ 2 j(β+n( ) α) 2 kn( 2 )+kδ C b Lipβ 2 [j(β+n( ) α) k(β+n( ))], 2 kαp λ j 2 [j(β+n( ) α) k(β+n( p ))]

12 22 ying shen, lanzhe liu Then C b p Lip β C b p Lip β C b p Lip β k 2 k 2 2 kαp k 2 λ j p. λ j p λ j p 2 (j k)(β+n( ) α)p, < p λ j p 2 p 2 [j(β+n( ) α) k(β+n( 2 p 2 [j(β+n( ) α) k(β+n( k=j+2 λ j p k=j+2 g b ψ,δ (f) K α,p C b Lipβ ( 2 ))] 2 (j k)(β+n( ) α)p, < p 2 p 2 [(j k)(β+n( ) α)], ))] p/p, < p < < p < λ j p ) /p C f H K α,p. When >, we have F b (a j ))(x) (b (x) b ()) (b (x) b ()) (ψ (x y) ψ (x))a j (y)dy + (b(x) b()) σ c (b(y) b()) σ ψ (x y)a j (y)dy hus g b ψ,δ (a j )(x) = j= σ Cj C b Lipβ x β ψ (x y) ψ (x) a j (y) dy +C b Lipβ τ+τ = C b Lipβ x β ( + x ) n++ε δ +C b Lipβ τ+τ = C b Lipβ x β ( + x ) +C b Lipβ ( τ+τ = F b (a j )(x) 2 d ) /2 x τβ y τ β ψ (x y) a j (y) dy x τβ ( + x ) n+ δ y ε a j (y) dy ) α) 2j(ε+n( n++ε δ C b Lipβ x β 2 j(ε+n( ) α) y τ β a j (y) dy x τβ ( + x ) n+ δ 2j(τ β+n( ( ) α), ( + x ) n++ε δ ) 2 d /2

13 lipschiz esiaes for ulilinear couaor C b Lipβ τ+τ = x τβ 2 j(τ β+n( ) α) C b Lipβ x β x (n+ε δ) 2 j(ε+n( ) α) +C b Lipβ τ+τ = ( x τβ x n+δ 2 j(τ β+n( ) α) C b Lipβ x n+δ 2 j(β+n( ) α), ( + x ) n+ δ ) 2 d /2 hen ( ) ψ,δ (a j )χ k L 2 C /2 b Lipβ 2 j(β+n( ) α) x n 2+ 2 δ dx so, I C b p Lip β C b p Lip β C b Lipβ 2 [j(β+n( ) α) k(β+n( ))], k 2 2 kαp k 2 C b p Lip β k 2 2 kαp k 2 Fro I and II, we have λ j p. ψ,δ (f) C b Lipβ λ j 2 [j(β+n( ) α) k(β+n( p ))] λ j p 2 (j k)(β+n( ) α)p, < p λ j p 2 This coplees he proof of Theore 3. p 2 [ ( ) ( ( ))] j(β+n ) α k β+n 2 p 2 [j(β+n( ) α) k(β+n( λ j p /p ))] C f H K α,p. p/p, < p < Theore 4. Le < β < in(γ/, /2), < p, <, 2 <, < δ < n, / 2 = / (β + δ)/n, b = (b,, b ) wih b j Lip β (R n ) for j. Then g b ψ,δ aps H K n( / )+β+δ/,p (R n ) coninuously ino W K n( / )+β+δ/,p 2 (R n ). Proof. We wrie f = λ k a k,

14 222 ying shen, lanzhe liu where each a k is a cenral (n( / ) + β + δ/, ) ao suppored on B k and λ k p <. Wrie ψ,δ W K n( / )+β+δ/,p 2 sup λ 2 l(n( / )+β+δ/)p λ> l= + sup λ 2 l(n( / )+β+δ/)p λ> l= = G + G 2. x E l : g b ψ,δ λ k a k (x) > λ/2 p/ /p 2 k=l 3 x E l : l 4 g b ψ,δ λ k a k (x) > λ/2 p/ 2 By he (L, L 2 ) boundedness of g b ψ,δ and an esiae siilar o ha for I in Theore 3, we ge G p C 2 lp(n( / )+β+δ/) g b ψ,δ λ k a k (x)χ l p 2 C b p Lip β λ k p. l= l 3 To esiae G 2, le us now use he esiae g b ψ,δ (a k ) C b Lipβ x δ n (2 k ) β+n( / ) α, which we ge in he proof of Theore 3. Noe ha when x E l, α = n( / ) + β + δ/, λ < l 4 λ k ψ,δ (a k ) C l 4 b Lipβ C l 4 b Lipβ C l 4 b Lipβ λ k x δ n (2 k ) β+n( / ) α l 4 λ k 2 l δ n λ k (2 l ) (( )β+δ n δ/) C b Lipβ 2 l(( )β+δ n δ/) (2 k ) β+n( / ) α for λ >, le l λ be he axial posiive ineger saisfying 2 lλ(n+δ/ ( )β δ) C b Lipβ λ λ k p hen if l > l λ, we have {x E l : g b ψ,δ l 4 λ k a k > λ/2} =. /p λ k p, /p, /p

15 lipschiz esiaes for ulilinear couaor So, we obain G 2 sup λ> sup λ> λ l λ l= l λ λ l= 2 l(n( /)+β+δ/)p (2 l ) np/ 2 (2 l ) (n ( )β δ) /p sup λ2 lλ(n ( )β δ) C b Lipβ λ> /p λ k p Now, cobining he above esiaes for G and G 2, we obain ψ,δ (f) W K n( / )+β+δ/,p 2 C b Lipβ λ k p /p /p Theore 4 follows by aking he infiu over all cenral aoic decoposiions... References [] J. Alvarez, R. J. Babgy, D. S. Kurz and C. Pérez, C., Weighed esiaes for couaors of linear operaors, Sudia Mah., 4 (993), [2] Chanillo, S., A no on couaors, Indiana Univ. Mah. J., 3 (982), 7-6. [3] Chen, W,G., Besov esiaes for a class of ulilinear singular inegrals, Aca Mah. Sinica, 6 (2), [4] R. Coifan, R. Rochberg and G. Weiss, G., Facorizaion heores for Hardy spaces in several variables, Ann. of Mah., 3 (976), [5] Devore, R.A. and Sharply, R.C., Maxial funcions easuring soohness, Me. Aer. Mah. Soc., 47(984). [6] J. Garcia-Cuerva and M. J. L. Herrero, J.L., A heory of Hardy spaces associaed o Herz spaces, Proc. London Mah. Soc., 69 (994), [7] Janson, S., Mean Oscillaion and couaors of singular inegral operaors, Ark. Mah., 6 (978), [8] Liu, L.Z., Boundedness of ulilinear operaor on Triebel-Lizorkin spaces, Iner. J. of Mah. and Mah. Sci., 5 (24), [9] Liu, L.Z., The coninuiy of couaors on Triebel-Lizorkin spaces, Inegral Euaions and Operaor Theory, 49 (24),

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