CBMO Estimates for Multilinear Commutator of Marcinkiewicz Operator in Herz and Morrey-Herz Spaces

Transcription

1 SCIENTIA Series A: Matheatical Sciences, Vol ), 7 Universidad Técnica Federico Santa María Valparaíso, Chile ISSN c Universidad Técnica Federico Santa María 202 CBMO Estiates for Multilinear Coutator of Marcinkiewicz Operator in Herz and Morrey-Herz Spaces Xiaoing Huang, Chuangxia Huang and Lanzhe Liu Abstract In this paper, we establish CBMO estiates for the ultilinear coutator related to the Marcinkiewicz operator in Herz and Morrey-Herz spaces Introduction Let b BMOR n ) and I α be the fractional operator, the coutator b, I α ] generated by b and I α is defined by b, I α ]f) = bx)i α f)x) I α bf) A result of Chanillo see 3]) proved that the coutator b, I α ] is bounded fro L p R n ) to L q R n ), where < p < q < and /p /q = α/n Lu and Yang see 5]) introduced the central BMO space that is CBMO space Since it is obvious that BMOR n ) CBMO q R n ) for all q < However, we know the L p, L q ) boundedness fails with only the assuption b CBMO q R n ) Instead, certain boundedness properties on Herz spaces and Morrey-Herz spaces can be proved The purpose of this paper is to introduce the ultilinear operator associated to the Marcinkiewicz operator and establish CBMO estiates for the ultilinear coutator in Herz and Morrey-Herz spaces 2 Preliinaries and Theores First, let us introduce soe notations Definition Let q < A function f L q loc Rn ) is said to belong to the space CBMO q R n ) if f CBMOq = sup r>0 B0, r) B0,r) fx) f B0,r) q dx) /q <, 2000 Matheatics Subject Classification Priary 42B20 Secondary 42B25 Key words and phrases Multilinear coutator; Marcinkiewicz operator; CBMO; Herz space; Morrey-Herz space

2 2 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU where, B = B0, r) = x R n : x < r} and f B0,r) is the ean value of f on B0, r) Let q = q, q j ), for b j CBMO q j R n )j =,, ), set b CBMO q = b j CBMOq j Given a positive integer and j, we denote by Cj the faily of all finite subsets σ = σ),, σj)} of,, } of j different eleents For σ Cj, set σ c =,, } σ For b = b,, b ) and σ = σ),, σj)} C j, set b σ = b σ),, b σj) ), b σ = b σ) b σj) and b σ CBMO q = b σ) CBMOq b σj) CBMOq j Definition 2 Let α R, 0 < p and 0 < q < For k Z, set B k = x R n : x 2 k } and = B k B k Denote by χ k the characteristic function of and χ 0 the characteristic function of B 0 ) The hoogeneous Herz space is defined by where where K q α, p R n ) = f L q loc Rn 0}) : f K q α, p < }, f K α, p q = 2 kαp fχ k p L q ] /p 2) The nonhoogeneous Herz space is defined by Kq α, p R n ) = f L q loc Rn ) : f K α, p < }, q f K α, p q = ] /p 2 kαp fχ k p L + fχ q B 0 p L ; q k= And the usual odification is ade when p = q = Definition 3 Let α R, 0 λ <, 0 < p and 0 < q < α,λ hoogeneous Morrey-Herz space M K p,q R n ) is defined by where M α,λ K p,q R n ) = f L q loc Rn 0}) : f M K < }, p,q α,λ f M K α,λ p,q = sup k0 2 k0λ with the usual odifications ade when p = 2 kαp fχ k p L q ) /p Reark Copare the hoogeneous Morrey-Herz space M p,q R n ) with the α,p hoogeneous Herz space K q R n ) and the Morrey space Mq λ R n )see 6]), obviously, α,0 M K p,q R n α,p ) = K q R n ) and Mq λ R n α,0 ) M K p,q R n ) We can see that when λ = 0, α,0 M K p,q R n ) is just the hoogeneous Herz space Definition 4 Let 0 < δ < n, 0 < γ and Ω be hoogeneous of degree zero on R n such that Ωx )dσx ) = 0 Assue that Ω Lip S n γ S n ), that is there ; K α,λ The

3 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 3 exists a constant M > 0 such that for any x, y S n, Ωx) Ωy) M x y γ The Marcinkiewicz ultilinear coutator is defined by where F b t,δ f)x) = µ b Ω,δ f)x) = x y t 0 F b t,δ f)x) 2 dt ) /2 t 3, Ωx y) x y n δ b j x) b j y)) fy)dy Set we also define that F t,δ f)x) = x y t Ωx y) fy)dy, x y n δ µ Ω,δ f)x) = 0 F t,δ f)x) 2 dt ) /2 t 3, which is the Marcinkiewicz operatorsee 2]) Let H be the space H = h : h = ) ht) 2 dt /2 0 t < } Then, it is clear 3 that µ Ω,δ f)x) = F t,δ f)x) and µ b Ω,δ f)x) = F b t,δ f)x) Note that when b = = b, µ b Ω,δ is just the order coutator It is well known that coutators are of great interest in haronic analysis and have been widely studied by any authors see -3]7]0]) The purpose of this paper is to study the boundedness properties for the ultilinear coutator µ b Ω,δ in Herz spaces and Morrey-Herz spaces Now we state our theores as following Theore Let < q <, b CBMO q R n ), and µ b Ω,δ be defined as in Definition 4 with 0 < δ < n, < q < n/δ, < q 2 < If 0 < p, q 2 = q + q δ n, where /q = /q + /q j, and t = q + q <, u = q δ n, δ n q < α < n n q and α 2 = α n/q, then Ω,δ K α 2,p q 2 C b CBMO q f K α,p q Theore 2 Let λ 0, < q <, b CBMO q R n ), and µ b Ω,δ be defined as in Definition 4 with 0 < δ < n, < q < n/δ, < q 2 < If 0 < p, q 2 = q + q δ n, where /q = /q + /q j, and t = q + q <, u = q δ n, λ + δ n q < α < n n q + λ and α 2 = α n/q, then Ω,δ M K α 2,λ p,q 2 C b CBMO q f M K α,λ

4 4 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU 3 Proof of Theores To prove the theores, we need the following leas Lea see 7]) Suppose that f CBMO q R n ), q < and r, r 2 > 0 Then B0, r ) B0,r ) fx) f B0,r2)) q dx) /q + ln r r 2 ) ) f CBMOq Lea 2see ]) Let 0 < δ < n, < p < n/δ and /q = /p δ/n Then µ Ω,δ is bounded fro L p R n ) to L q R n ) α,p Proof of Theore We only consider the case 0 < p < Let f K q R n ) and decopose f into When =, we consider fx) = Ω,δ f) K α 2,p q 2 R n ) = C + C + C Let us first estiate E 2, note that fx)χ l x) = E + E 2 + E 3 f l x) Ω,δ f l)χ k p L q 2 R n ) k 3 k+2 ) /p Ω,δ f l)χ k L q 2 Rn ) Ω,δ f l)χ k L q 2 Rn ) Ω,δ f l)χ k L q 2 Rn ) We have µ b Ω,δ f l)χ k = b b Bk )µ Ω,δ f l )χ k + µ Ω,δ b b Bk )f l )χ k Ω,δ f l)χ k L q 2 b b Bk )µ Ω,δ f l )χ k L q 2 + µ Ω,δ b b Bk )f l )χ k L q 2 = J + J 2 For J, by Hölder s inequality, Lea and the boundedness of µ Ω,δ fro L q R n ) to L u R n ), we have ) /q ) /u J C b x) b Bk q dx µ Ω,δ f l ) u dx B k B k C B k /q b CBMOq f l L q

5 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 5 For J 2, by Hölder s inequality, Lea and the boundedness of µ Ω,δ fro L t R n ) to L q2 R n ), we have Therefore E 2 C ) /t J 2 C b b Bk )f l t dx B k ) /q C b x) b Bk q dx f l q dx B k B k ) /q C b x) b Bk q dx f l L q B l C B l /q b CBMOq f l L q k+2 J + C k+2 ) /q J 2 = I + I 2 For I, if < p <, by Minkowski s inequality and if 0 < p, by the inequality a i ) p a i p, we have I C k+2 2 kn/q b CBMOq f l L q l+2 ] /p, 2lαp f l p L q k=l 2 2k l)αp 0 < p ) ) p/p ] /p 2lαp f l p l+2 l+2 L q k=l 2 2k l)αp k=l 2 2k l)αp, < p < l+2 ] /p, 2lαp f l p L q k=l 2 2k l)αp 0 < p ) 2lαp f l p l+2 l+2 p/p ] /p /2) L q k=l 2 2k l)αp/2 k=l 2 2k l)αp, < p < f K α,p q 2 lαp f l p L q ) /p

6 6 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU For I 2, siilarly to the ethod estiating for I, we have I 2 C k+2 2 ln/q b CBMOq f l L q l+2 ] /p, 2lαp f l p n L q k=l 2 2k l)α q )p 0 < p ) 2lαp f l p l+2 n L q k=l 2 2k l)α q )p/2 l+2 p/p ] /p n k=l 2 2k l)α q /2) )p, < p < l+2 ] /p, 2lαp f l p n L q k=l 2 2k l)α q )p 0 < p ) 2lαp f l p l+2 n L q k=l 2 2k l)α q )p/2 l+2 p/p ] /p n k=l 2 2k l)α q /2) )p, < p < f K α,p q Thus, we deduce 2 lαp f l p L q ) /p E 2 f K α,p q Now let us turn to estiate E, choosing b ) B = B B b x)dx, by Minkowski s inequality, note that x, y A l, we have Ω,δ f l)χ k L q 2 Ωx y) 0 x y t x y n δ b x) b y))fy)dy 2 ] q2/2 } /q2 dt t 3 dx ) /2 ] q2 /q2 Ωx y) dt b x) b y)) fy) R x y n δ n x y t t 3 dy dx} /2 Ωx y) b x) b y)) fy) q2 } /q2 R x y n δ x y 2 dy] dx n ] q2 /q2 Ωx y) b x) b y)) fy)dy dx} R x y n δ n

7 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 7 ) q2 /q2 C B k δ/n b x) b y) fy) dy dx} A l ) q2 /q2 C B k A δ/n b x) b ) B q2 fy) dy dx) k A l ) /q2 + C B k δ/n b y) b ) B fy) dydx A l ) /q2 C B k δ/n f l L b x) b ) B q2 dx + C B k A δ/n +/q2 b y) b ) B fy) dy l ) /q C B k δ/n f l L q µb l ) A /q b x) b ) B q dx B k /q2 /q k ) /q ) /q + C B k A δ/n +/q2 b y) b ) B q dy fy) q dy B l /q /q l A l C B k δ/n +/q2 B l /q f l L q b CBMOq Therefore, we know E 2lαp f l p L q k=l+3 0 < p k 3 2 nkδ/n +/q2) 2 nl /q) f l L q 2 nkδ/n +/q2 /q) 2 nl /q) 2 k l)α ) p ] /p, 2 nkδ/n +/q2 /q) 2 nl /q) 2 k l)α ) p/2 ) B l αp f l p L q k=l+3 ) p k=l+3 2 /2) p/p ] /p nkδ/n +/q2 /q) 2 nl /q) 2 k l)α, < p < ) 2lαp f l p p ] /p L q k=l+3 2 nk/q ) 2 nl /q) 2 k l)α, 0 < p ) p/2 ) 2 nk/q ) 2 nl /q) 2 k l)α k=l+3 2lαp f l p L q ) p k=l+3 2 /2) p/p ] /p nk/q ) 2 nl /q) 2 k l)α, < p <

8 8 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU ] 2lαp f l p /p L q k=l+3 2k l)n/q n+α)p, 0 < p ) 2lαp f l p L q k=l+3 2k l)α + n q )p/2 p/p ] /p n k=l+3 2k l)α + q )p /2), < p < f K α,p q 2 lαp f l p L q ) /p Now let us turn to estiate E 3, by Hölder s inequality, we have Ω,δ f l)χ k L q 2 Ωx y) 0 x y t x y n δ b x) b y))fy)dy 2 ] q2/2 } /q2 dt t 3 dx ) /2 ] q2 /q2 Ωx y) dt b x) b y)) fy) R x y n δ n x y t t 3 dy dx} /2 Ωx y) b x) b y)) fy) q2 } /q2 R x y n δ x y 2 dy] dx n ] q2 /q2 Ωx y) b x) b y)) fy)dy dx} R x y n δ n ) q2 /q2 C B l δ/n b x) b y) fy) dy dx} A l ) q2 /q2 C B l A δ/n b x) b ) B q2 fy) dy dx) k A l ) /q2 + C B l δ/n b y) b ) B fy) dydx A l ) /q2 C B l δ/n f l L b x) b ) B q2 dx + C B l δ/n B k /q2 A l b y) b ) B fy) dy ) /q C B l δ/n f l L q B l A /q b x) b ) B q dx B k /q2 /q k ) /q ) /q + C B l δ/n B k A /q2 b y) b ) B q dy fy) q dy l A l B l /q /q C B l δ/n /q B k /q2 f l L q b CBMOq

9 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 9 Thus, in this case, we obtain E 3 2lαp f l p L q 0 < p 2 lnδ/n /q) 2 kn/q2) f l L q ) p ] /p l 3 2 lnδ/n /q) 2 kn/q2 /q) 2 k l)α, l 3 2 lnδ/n /q) 2 kn/q2 /q) 2 k l)α ) p/2 ) 2lαp f l p L q ) p l 3 2 /2) p/p ] /p lnδ/n /q) 2 kn/q2 /q) 2 k l)α, < p < 2lαp f l p L q 2lαp f l p L q l 3 /p n 2l k)δ α q )p], 0 < p ) l 3 2l k)δ α n q )p/2 l 3 p/p ] /p n 2l k)δ α q )p /2), < p < f K α,p q 2 lαp f l p L q ) /p This copletes the proof of the case = Now, we consider the case 2 We write, Ω,δ f) K α 2,p q 2 X) = C + C + C = G + G 2 + G 3 Ω,δ f l )χ k p L q 2 R n ) k 3 k+2 ) /p Ω,δ f l )χ k L q 2 Rn ) Ω,δ χ k L q 2 Rn ) Ω,δ χ k L q 2 R n )

10 0 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU Let us first estiate G 2, set b B = b ) B,, b ) B ), where b j ) B = B B b jx) dx, j, we have = = F b t,δ f)x) = j=0 σ Cj x y t b j x) b j ) B ) b j y) b j ) B )) Ωx y) fy) x y ) j bx) b B ) σ x y t n δ dy by) b B ) σ cfy)ωx y) x y n+δ dy b j x) b j ) B F t,δ f)x) + ) F t,δ b j y) b j ) B )f)x) + σ Cj ) j bx) b B ) σ F b σ c t,δ f)x), thus µ b Ω,δ f l )x) = F b t,δ f l )x) b j x) b j ) B )F t,δ f l )x) + + F t,δ b j b j ) B )f l )x) + σ Cj bx) b) B ) σ F b σ c t,δ f l)x) b j x) b j ) B ))µ Ω,δ f l )x) + ) µ Ω,δ b j b j ) B )) B )f l )x) σ Cj ) j bx) b B ) σ µ Ω,δ b b B ) σ cf l )x) = H + H 2 + H 3 For H, taking < q < n/δ and u such that /u = /q δ/n, choosing /q = /q + /q j, by Hölder s inequality and the boundedness of µ Ω,δ fro L q R n ) to L u R n ), we have b j x) b j ) B )µ Ω,δ f l )x)χ k L q 2 C b j x) b j ) B ) q ) /q ) /u dx µ Ω,δ fx) u dx

11 C C B k /q + /q j CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR b j x) b j ) B q C B k /q b j CBMOqj f l L q C B k /q b CBMO q f l L q ) /q j j dx Ak ) /q f l x) q dx ) /q ) b j x) b j ) B q j /q j dx f l x) q dx B k B k Ak For H 2, taking < t < n/δ and u such that /q 2 = /t δ/n, choosing /t = /q+/q, by Hölder s inequality and the boundedness of µ Ω,δ fro L t R n ) to L q2 R n ), we have ) µ Ω,δ b j b j ) B )))x)χ k L q 2 C b j b j ) B ))f l χ k L t C b j x) b j ) B )) C B k /q b CBMO q f l L q q ) /q ) /q dx f l x) q dx For H 3, choosing /q 2 = /q + /ω and /ω = /q 2 + /q δ/n, using Hölder s inequality and the boundedness of µ Ω,δ, we have σ Cj C ) j bx) b B ) σ µ Ω,δ b b B ) σ cf l )x)χ k L q 2 σ Cj C σ Cj f l x) p dx bx) b B ) σ q bx) b B ) σ q ) /q ) /q dx Ak ) /q dx Ak ) /ω µ Ω,δ b b B ) σ cf l )x) ω dx ) /q 2 bx) b B ) σ c q 2 dx C B k /q bσ CBMOq B k /q 2 bσ c CBMO q f l L q C B k /q b CBMO q f l L q

12 2 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU Then, siilarly to the ethod estiating for I, we get G 2 C b CBMOq f K α,p q Next, we estiate G, let τ, τ N such that τ + τ =, we have Ω,δ f l )χ k L q 2 0 x y t Ωx y) x y n δ b j x) b j y)fy)dy 2 ] q2/2 } /q2 dt dx ) /2 ] q2 /q2 Ωx y) dt b j x) b j y) fy) R x y n δ n x y t t 3 dy dx} /2 Ωx y) b j x) b j y) fy) q2 } /q2 R x y n δ x y 2 dy] dx n ] q2 /q2 Ωx y) b j x) b j y) fy)dy dx} R x y n δ n ) /q2 C B k δ/n bx) b B ) σ q2 dx by) b B ) σ c fy) dy A l C B k δ/n j=0 σ Cj j=0 σ Cj τ+τ = A l bx) b B ) σ c τ dx C B k δ/n +/q2 B l /q b CBMO q f l L q ) /τ bx) b B ) σ τ dx B k /q2 /τ ) /τ ) /q fy) q dy B l /τ /q A l t 3 Then, siilarly to the ethod estiating for E, we can obtain G C b CBMO q f K α,p q Finally, let us estiate G 3, since Ω,δ f l )χ k L q 2 0 R n R n R n x y t Ωx y) x y n δ b j x) b j y)fy)dy 2 ] q2/2 } /q2 dt dx Ωx y) b j x) b j y) fy) x y n δ x y t Ωx y) b j x) b j y) fy) x y n δ x y 2 ] q2 /q2 Ωx y) b j x) b j y) fy)dy dx} x y n δ t 3 ) /2 ] q2 /q2 dt t 3 dy dx} /2 q2 } /q2 dy] dx

13 C B l δ/n C B l δ/n CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 3 j=0 σ Cj ) /q2 bx) b B ) σ q2 dx bx) b B ) σ c fy) dy A l j=0 σ Cj τ+τ = A l by) b B ) σ c τ dx C B l δ/n /q B k /q2 b CBMO q f l L q ) /τ bx) b B ) σ τ dx B k /q2 /τ ) /τ ) /q fy) q dy B l /τ /q A l Then, siilarly to the ethod estiating for E 3, we can get G 3 C b CBMO q f K α,p q This copletes the proof of Theore Proof of Theore 2 Let f M When =, we consider fx) = α,λ K R n ) and decopose f into fx)χ l x) Ω,δ f) k0 M K α 2 = sup 2 k0λ,λ p,q 2 R n ) k0 C sup 2 k0λ k0 + C sup 2 k0λ k0 + C sup 2 k0λ = U + U 2 + U 3 f l x) Ω,δ f l)χ k p L q 2 R n ) k 3 k+2 ) /p Ω,δ f l)χ k L q 2 Rn ) Ω,δ f l)χ k L q 2 Rn ) Ω,δ f l)χ k L q 2 Rn ) Let us first estiate U 2, siilarly to the ethod estiating for E 2, we have k0 U 2 C sup 2 k0λ k0 +C sup 2 k0λ = V + V 2 k+2 k+2 J J 2

14 4 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU For V, we have k0 V C sup 2 k0λ k0 C sup 2 k0λ sup 2 k0λ k0 k+2 2 kλp k+2 k+2 2 kα n/q)p k0 sup 2 k0λ f M K α,λ Siilarly, for V 2, we have k0 V 2 C sup 2 k0λ k0 C sup 2 k0λ sup 2 k0λ k0 k+2 2 kλp B k /q b CBMOq f l L q 2 k l)α 2 l k)λ 2 lλ l k+2 k+2 2 kα n/q)p k0 sup 2 k0λ f M K α,λ k+2 2 kλp 2 kn/q b CBMOq f l L q i= 2 iαp f i p L q ) /p ] p } /p ] p } /p 2 k l)α λ) f M K α,p B l /q b CBMOq f l L q 2 k l)α 2 l k)n/q 2 l k)λ 2 lλ l k+2 2 kλp Therefore U 2 f M K α,p Then, let us estiate U, siilar to E, we get 2 ln/q b CBMOq f l L q i= 2 iαp f i p L q ) /p ] p } /p ] p } /p 2 k l)α n/q λ) f M K α,p U sup 2 k0λ k 3 B k δ/n +/q2 B l /q f l L q

15 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 5 sup 2 k0λ k 3 2 kα n/q)p sup 2 k0λ k0 k 3 2 kλp k0 sup 2 k0λ f M K α,λ 2 knδ/n +/q2) 2 ln /q) f l L q 2 k l)n/q n) 2 k l)α 2 l k)λ 2 lλ l k 3 2 kλp i= 2 iαp f i p L q ) /p ] p } /p ] p } /p 2 k l)α n+n/q λ) f M K α,p Last, let us estiate U 3, siilar to E 3, we get U 3 sup 2 k0λ B l δ/n /q B k /q2 f l L q sup 2 k0λ 2 kα n/q)p 2 lnδ/n /q) 2 kn/q2) f l L q sup 2 k0λ k0 2 kλp 2 l k)δ n/q) 2 k l)α 2 l k)λ 2 lλ l k0 sup 2 k0λ f M K α,λ 2 kλp i= 2 iαp f i p L q ) /p ] p } /p ] p } /p 2 l k)δ n/q α+λ) f M K α,p This copletes the proof of the case = When >, we consider µ k0 b Ω,δ f) M K α 2 = sup 2 k0λ,λ p,q 2 R n ) Ω,δ f l )χ k p L q 2 R n ) ) /p

16 6 XIAOMING HUANG, CHUANGXIA HUANG AND LANZHE LIU k0 C sup 2 k0λ k0 + C sup 2 k0λ k0 + C sup 2 k0λ = W + W 2 + W 3 For W 2, siilar to G 2, we have k 3 k+2 W 2 C k0 b CBMO q sup 2 k0λ C k0 b CBMO q sup 2 k0λ C b CBMO q f M K α,λ For W, siilar to G, we have W C b CBMO q sup 2 k0λ k0 k 3 C k0 b CBMO q sup 2 k0λ C b CBMO q f M K α,λ For W 3, siilar to G 3, we have W 3 C b CBMO q k+2 Ω,δ f l )χ k L q 2 Rn ) Ω,δ f l )χ k L q 2 Rn ) Ω,δ f l )χ k L q 2 Rn ) k+2 2 kλp B k /q f l L q ] p } /p 2 k l)α λ) f M K α,p B k δ/n +/q2 B l /q f l L q k 3 2 kλp C b CBMO q sup 2 k0λ k0 l 2 kλp 2 l k)δ n/q α+λ) 2 lλ C k0 b CBMO q sup 2 k0λ C b CBMO q f M K α,λ ] p } /p 2 k l)α n+n/q λ) f M K α,p B l δ/n /q B k /q2 f l L q 2 kλp i= 2 iαp f i p L q ) /p ] p } /p ] p } /p 2 l k)α n+n/q λ) f M K α,p

17 CBMO ESTIMATES FOR MULTILINEAR COMMUTATOR 7 This copletes the proof of Theore 2 References ] W G Chen, Besov, Estiates for a class of ultilinear singular integrals, Acta Math Sinica, 62000), ] S Chanillo, A note on coutators, Indiana Univ Math, J, 3982), 7 6 3] Y Koori, Notes on singular integrals on soe inhoogeneous Herz spaces, Taiwanese J Math, 82004), ] L Z Liu, Endpoint estiates for ultilinear integral operators, J Korean Math Soc, ), ] L Z Liu, The continuity of coutators on Triebel-Lizorkin spaces, Integral Equations and Operator Theory, 49)2004), ] L Z Liu and B S Wu, Weighted boundedness for coutator of Marcinkiewicz integral on soe Hardy spaces, Southeast Asian Bull of Math, ), ] S Z Lu and Q Wu, CBMO estiates for coutators and ultilinear singular integrals, Math Nachr, ), ] S Z Lu and D C Yang, The central BMO spaces and Littlewood-Paley operators, Approxiation Theory and its Applications, 995), ] S Z Lu, D C Yang and Z S Zhou, Sublinear operators with rough kernel on generalized Morrey spaces, Hokkaido Math J, 27998), ] R A Macias and C Segovia, Lipschitz functions on spaces of hoogeneous type, Adv in Math, 33979), ] E M Stein, Haronic analysis: Real variable ethods, orthogonality and oscillatory integrals, Princeton Univ Press, Princeton NJ, 993 2] C Q Tang, CBMO estiates for coutators of ultilinear fractional integral operators on Herz spaces, Integr Equ Oper Theory, ), ] A Torchinsky and S Wang, A note on the Marcinkiewicz integral, Colloq Math, 60/6990), Received , revised Departent of Matheatics Changsha University of Science and Technology Changsha, 40076, PR of China E-ail address: huangxiaoing26@26co, cxiahuang@26co, lanzheliu@63co