Research Article Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel

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1 Hindawi Publishing Corporation Abstract and Applied Analysis Volume 23, Article ID 43692, pages Research Article Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel Xiao Yu and Shanzhen Lu 2 Department of Mathematics, Shangrao Normal University, Shangrao 334, China 2 School of Mathematical Science, Beijing Normal University, Beijing 875, China Correspondence should be addressed to Xiao Yu; xyuzju@63.com Received 7 July 23; Accepted October 23 Academic Editor: Csaba Varga Copyright 23 X. Yu and S. Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. The authors consider the generalized commutator of fractional Hardy operator with a rough ernel as follows: H m Ω,A,β fx = / x n β y < x Ωx y/ x y m R m A; x, yfydy, where Ω L r S n, β < n,andr m A; x, y = Ax γ <m /γ!d γ Ayx y γ with m Z +. The authors prove that H m Ω,A,β is bounded on Herz type space and λ-central Morrey space with m, which is an open problem for m>2.. Introduction It is well nown that the C-Z singular integrals and their commutators have been studied a lot by many mathematicians; please see [] or[2] for more details. For the generalizations of the commutators of singular integrals, Cohen [3] studied the following generalized commutator T 2 A which is defined by T 2 A f x = Ωx y R n x y A x Ay n+ A y x y f y dy, where Ω L S n is homogeneous of degree zero and satisfies the moment condition S n Ω x x γ dσ x = 2 with γ =. Cohen[3] provedthatifω Lip S n and A BMO, then T 2 A is bounded on Lp R n with <p<.later,cohenandgosselin[4] considered another type of generalized commutator as follows: T m A f x = Ωx y R n x y n+m R m A; x, y f y dy, 3 where R m A; x, ym Z + is defined by R m A; x, y = Ax γ <m /γ!d γ Ayx y γ,themth remainder of Taylor series of the function A at y about x, andω satisfies the following moment condition: S n Ω x x γ dσ x =, 4 with γ = m. Obviously,ifwechoosem =, T m A becomes [A, T], the commutator of T generalized by A and T.Furthermore,T m A becomes T2 A if we choose m=2. Cohen and Gosselin proved that if m 2,Ω Lip S n, and the function A has derivatives of order m in BMOR n, then the operator T m A is bounded on Lp R n for <p<. Later, T m A was studied by many mathematicians; please see [5, 6] or[7] for more details. Recently, Wang and Zhang [8] gave a new proof of Wu s theorem in [9] byusingthe W,p estimate for the elliptic euation of divergence form with partially BMO coefficients and the L p boundedness of thecohen-gosselintypegeneralizedcommutatorsproved by Yan in [6]. Furthermore, the method used in [8] ismuch simpler than that in [9]. Recently, Yu and Tao [7]provedthat T m A is bounded on λ-central Morrey space.

2 2 Abstract and Applied Analysis Let f be a nonnegative integral on R + : then the Hardy operator is defined by Hf x = x x f t dt, x =. 5 In 92, Hardy []provedthefollowingineuality: Hf L p R + p p f L p R +, 6 where < p < and the constant p/p is the best possible. In 27, Fu et al. [] introduced the n-dimensional fractional type Hardy operator H β as follows: H β f x = x f t dt, x R n \ {}, n β 7 t < x where n<β<nand f is a locally integrable function on R n. Obviously, when β =, H is just the n-dimensional Hardy operator H which was proposed by Christ and Grafaos in [2]. In [], the authors gave the characterization of the CBMO R n by the boundedness of the commutator of the fractional type Hardy operator [H β,b]on Herz type spaces. Here the CBMO R n spaceisdefinedbythefollowing. Definition see [3]. Let <.Afunctionf L loc Rn is said to belong to the homogeneous Central BMO space CBMO R n if b C BMO R n := sup r> B, r B,r fx f B / dx <, where f B = / B, r B,r fxdx. From [4], we now that BMOR n CBMO R n for <. The CBMO R n space can be regarded as the space of bounded mean oscillation, a local version of BMOR n atthe origin. But the famous John-Nirenberg ineuality no longer holds in CBMO R n. Now we are interested in the following generalized commutator of Hardy operator: H m Af x = x n y < x x y m fyr m A; x, y dy, 9 where R m A;x,y=Ax γ <m /γ!d γ Ayx y γ and m Z +. In 2, Lu and Zhao [5]provedthatwhenm=2, H 2 A is bounded on Herz type space and Morrey-Herz type space. Later, Gao and Yu [6] provedthath 2 A is bounded on λ- Central Morrey spaces. However, we would lie to point out 8 that the method used in [5, 6] cannot apply to the case when m>2. An interesting uestion is whether the boundedness of H m A on Herz type space or λ-central Morrey space still holds with m > 2. In this paper, we will use a different method to answer this uestion. Furthermore, we will consider the generalized commutator of fractional Hardy operator with a rough ernel as follows: H m Ω,A,βf x = x Ωx y n β y < x x y m fyr m A; x, y dy, where m Z +, β<n,andω L r S n. In [7], we prove that H m Ω,A,β is bounded from Lp to L with /p / = β/n. Furthermore, we study the endpoint estimates of H m Ω,A,β on H spaces with Ω Lip S n in [7]. In this paper, we will prove that H m Ω,A,β is bounded on Herz type space and λ-central Morrey space when Ω L r S n. 2. Boundedness of H m Ω,A,β on Herz Type Spaces In this section, we will give the boundedness of H m Ω,A,β on Herz type spaces. First we introduce some notations that will be used throughout this paper. Let B ={x R n : x 2 }, C =B \B,andχ =χ C for Z:hereχ C is the characteristic function of the set C. Definition 2 see [8]. Let α R, < p,. Thenthe homogeneous Herz type space K α,p R n is defined by K α,p where f R n = {f L loc Rn \ {} : f K α,p R n < }, K α,p R n f K α,p is defined as /p R n ={ 2 αp fχ p L R n } with the usual modifications made when p= or =. Nowweshowourmainresultsinthissection. 2 Theorem 3. Suppose m 2, Ω L r S n with <r<, and A has derivatives of order m in CBMO p 2 with n < p 2 <.Let<s p<,<,p <,/=/p + /p 2 β/n with β<n.if /r / β/n >, r > p, α 2 =α +n/p 2 and α 2 satisfies the following condition: α 2 +n/ n/r /r n/p 2 +β<, 3 then there exists a constant C,suchthat Hm Ω,A,β f K α,s Dγ A C BMO p 2 f γ =m K α 2,p. p 4 For m =, H Ω,A,β is just the commutator of Hardy operator; that is, H Ω,A,β = HA Ω,β fx = AxH Ω,βfx H Ω,β Afx. WehavethefollowingtheoremofH A Ω,β on Herz type space.

3 Abstract and Applied Analysis 3 Theorem 4. Suppose Ω L r S n with < r < and A CBMO p 2.Let<s p<,<,p,p 2 <,/= /p +/p 2 β/n with β<n.if/r / β/n >, r > p, α 2 =α +n/p 2,andα 2 satisfies 3, then there exists a constant C,suchthat HA Ω,β f K α A,s C BMO p 2 f K α 2,p. p 5 Remar 5. Comparing Theorems 3 and 4, wefindthatthe restrictions on α and α 2 are more rigid in Theorem 4 than in Theorem 3, whichindicatesthath m Ω,A,β with m 2has better properties than the commutators. In order to prove Theorems 3 and 4,weneedthefollowing lemmas. Thus we obtain Hm Ω,A,β f x = x Ω x y f y n β y < x x y m R m A; x, y dy x n β fy Ωx y y < x γ =m γ =m D γ A x +D γ A y dy [D γ A x H Ω,β f x +H Ω,β D γ A f x]. 2 Lemma 6 see [9]. Let <p,p 2 <, β<n,and β/n = /p /p 2.IfΩ L r S n with r>p, then there exists a constant C independent of f,suchthat where H Ω,β is defined by H Ω,βf L p 2 f L p, 6 H Ω,β f x = x Ω x y f y dy. n β 7 y < x By checing [9] carefully, one can draw the conclusion that if one replaces H Ω,β fx by H Ω,β f x,then6 still holds. Lemma 7. Let m,<p <p 2 < and β<n.ifa has derivatives of order m in L r R n with /p 2 =/p + /r β/n and r>p, then one has Hm Ω,A,β f L p 2 Dγ A L r f L p, γ =m 8 By the above estimates, we can get Hm R n Ω,A,β f x / dx γ =m γ =m Dγ A x H Ω,β R f x / dx n + H Ω,β D γ A f x / R n I+II. 2 For the term I,let/ = /r + /l = /r + /p β/n;then by the Hölder ineuality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on L p spaces, we obtain D I R γ A x r /r dx R H Ω,β f x l /l dx n n Dγ A L rf L p where the constant C is independent of f and A. Proof. From [2,p.24],we have the following estimates: R m A; x, y x y m R m A; x, y x y m +C Dγ Ay γ =m γ =m D γ A x +D γ A y, 9 where m and f is the Hardy-Littlewood maximal function of f. Dγ A L r f L p. 22 For the term II, let/ = /t β/n = /r + /p β/n; then by the Hölder ineuality and Lemma 6,wehave II C Dγ A t /t x f x R dx n Dγ A L rf L p Dγ A L r f L p. 23 Combining the estimates of I and II, we finish the proof of Lemma 7.

4 4 Abstract and Applied Analysis Lemma 8 see [4]. Let b be a function on R n with mth order derivatives in L loc Rn for some >n.then R m b; x, y m,n x y m γ =m Qx,y Qx,y Dγ b z dz /, 24 where Qx, y is the cube centered at x having diameter 5 n x y. Lemma 9 see [5]. Suppose that f C BMO R n, <,andr,r 2 >;then B,r / f x f B,r 2 dx B,r + log r r 2 f C BMO. 25 Proof of Theorem 3. To prove Theorem 3,firstwespliteachf as then we have f x = Hm Ω,A,β fχ L = C B, x C x β n dx C i x β n dx C 3 f x χ i x = f i x ; 26 fy Ωx y x y m R m A;x,ydy Ωx yfy x y m R m A;x,ydy C i x β n dx +CC =CI +I 2. i= 2 Ωx yfy x y m R m A; x, y dy C i x β n dx Ω x y f y x y m R m A; x, y dy 27 For the term I, we denote A x = Ax γ =m /γ!m B D γ Ax γ ; then it is easy to chec R m A;x,y = R m A ;x,y.bythefactthatx C, y C i with i 3,wehave x y x 2.Asp 2 >n, then by Lemmas 8 and 9,weobtain R m A ;x,y R m A ;x,y + x y m x y m γ =m γ =m γ =m γ! Dγ A x x y m { Qx,y Qx,y Dγ A z p 2 dz + Dγ A y } Dγ A C + Dγ A y. /p 2 28 As x y x 2 and /p /p 2 /r = /r / β/n >, then by the Hölder ineuality, we have I C 3 γ =m x β n dx 2 n β C C i Ω x y f y Dγ A C + Dγ A y dy 3 C i fy Ωx y γ =m Dγ A C + Dγ A y dy dx 2 n β 3 C C fy p /p dy i γ =m C Ωx y r /r dy i C i Dγ A C + Dγ A y p 2 dy /p 2 C i /p /p 2 /r dx. 29

5 Abstract and Applied Analysis 5 As Thus by Lemma 7,wehave /p 2 C Dγ A C BMO p 2 + Dγ A y p 2 dy i I 2 i= 2 C x Ωx yf i y n β y < x x y m C i Dγ A C + Dγ A y m B D γ /p 2 A y 2dy p Dγ A C BMO p 2 C i i, 3 i= 2 Hm Ω,A φ,βfi i= 2 f i L p L γ =m R m A φ ;x,ydy dx Dγ A φ L p C i Ωx y r dy From [5, p.8], we have the following estimates: x +2 i r x 2 S Ωy dσ y r n dr C2 n +i, i n Dγ A φ L p 2n/p 2 2 Dγ A C BMO p 2, γ =m 34 we obtain the following estimates: I γ =m 3 Dγ A CBMO p 2 i 2 n+β 2 in /p /p 2 /r 2 n +i/r+in/p 2 3 f i L p dx C i 2 in /p /p 2 /r 2 n+β+n +i/r+in/p 2 +n/ f il p Dγ A C γ =m. 3 where C is dependent on L.Thusweget I 2 i= 2 f i L p 2 n/p 2 γ =m Dγ A CBMO. 35 p 2 As <s p<, we obtain the following estimates: { { 2 α s H m Ω,A,β fχ s L } /s γ =m { 2 α p Hm Ω,A,β fχ p Dγ A C 2 α p 3 L } /p i 2 in /p /p 2 /r f il p For the term I 2,wechooseφ C Rn satisfying suppφ B, 4 as well as φ in B, 2 and we set L = max{ D γ φ L, γ m }.Lety B +4 and A φ x = R m A ;x,y φ2 x. Thenitiseasytoseethat R m A;x,y = R m A φ ;x,y = R ma ;x,y for x B and y B i with 2 i.thusweget +C γ =m { 2 n+β+n +i/r+in/p 2+n/ p } Dγ A C 2 α p f i p L p 2 n/p2p } i= 2 /p /p H m Ω,A,β f i x = H m Ω,A φ,βf i x = H m Ω,A,β f i x. 32 =CA+B. 36

6 6 Abstract and Applied Analysis For the term A,wehave A = γ =m { Dγ A C 2 α p γ =m { 3 i 2 in /p /p 2 /r 2 n+β Dγ A C 3 2 n +i/r+in/p 2+n/ f i L p p } i 2 in/ n/r +β /r n/p 2 2 iα +n/p 2 2 iα +n/p 2 f il p p } /p. /p When p,bythehölder ineuality and condition 3, we have A p γ =m γ =m Dγ A p C 2 α p Dγ A p C 3 3 i 2 in/ n/r /r n/p 2 +β 2 n/p 2 f il p i 2 α 2 in/ n/r /r n/p 2 +β 2 n/p 2 f il p p p When <p, by condition 3, we get A p γ =m γ =m Dγ A p C Dγ A p C 2 iα 2p 3 2 in/ n/r /r n/p 2 +β 3 2 iα +n/p 2 i 2 iα +n/p 2 f il p i 2 in/ n/r /r n/p 2 +β+α 2 f il p Dγ A p CBMO p 2 f p γ =m p K α 2.,p p p γ =m γ =m Dγ A p C 3 γ =m 3 3 Dγ A p C 3 i 2 iα 2 2 in/ n/r /r n/p 2 +β+α 2 f il p p 2 iα 2p f i p L p 2 p in/ n/r /r n/p 2 +β+α 2 /p i p 2 i/p n/ n/r /r n/p 2 +β+α 2 p Dγ A p C =i+3 p/p 2 ipα +n/p 2 2 in/ n/r /r n/p 2 +β+α 2 f il p i p 2 in/ n/r /r n/p 2 +β+α 2 p/p

7 Abstract and Applied Analysis 7 γ =m Dγ A p C Dγ A p CBMO p 2 f p γ =m 2 iα 2p f i p L p K α 2.,p p For the term B, we have the following estimates: B γ =m γ =m Dγ A C Dγ A C { Dγ A C BMO p 2 f γ =m K α 2,p. p 2 α p 2 np/p 2 f i p L p } i= 2 2 iα 2p f i p L p /p 39 /p 4 Combining the estimates of A and B, wefinishtheproofof Theorem 3. Proof of Theorem 4. The proof of Theorem 4 is uite similar and much easier than Theorem 3 andweomitthedetailshere. 3. Boundedness of H m Ω,A,β on λ-central Morrey Spaces In [2], Wiener gave a way to describe the behavior of a function at the infinity.later,beurling[22] extended Weiner s idea and introduced a pair of dual Banach spaces, A and B, with / + / =.In[23], Feichtinger proved that B can be described as f B = sup 2 n/ fχ L <, 4 where χ is the characterization of the unit ball {x R n : x } and χ is defined as in Section 2. Now by duality, the space A,whichiscalledtheBeurling algebra, can be described by f A = = 2 n/ fχ L <. 42 Later, Chen and Lau [24] as well as García-Cuerva [25] introduced atomic spaces HA R n associated with the Buerling algebra A and the dual space of HA R n can be described by f CBMO := sup R B, R B,R f x f B,R / dx < ; 43 here the CBMO canberegardedastheinhomogeneous central BMO spaces. In 2, Alvarez et al. [26] introducedtheλ-central bounded mean oscillationspace and λ-central Morrey space, respectively. Definition see [26]. Given λ</n,<<,then afunctionf L loc Rn is said to belong to the λ-central boundedmeanoscillationspacecbmo,λ R n if f C BMO,λ := sup R> B, R +λ B,R / f x f B,R dx <. 44 Definition see [26]. Let λ Rand <<.Thenthe λ-central Morrey space E,λ R n is defined by f E,λ = sup R> B, R +λ B,R / f x dx <. 45 From [27], we now that if < < 2 <,weobtain E 2,λ E,λ for λ Rand CBMO 2,λ CBMO,λ for λ< /n. Furthermore,whenλ< /, E,λ reduces to {} and CBMO,λ reduces to the space of constant functions. When λ= /,CBMO,λ coincides with L R n modulo constant and E,λ =L. In 2, Fu et al. [9] proved the boundedness of the commutator of fractional Hardy operator with a rough ernel on λ-central Morrey space. Later, Fu et al. [28] proved the boundness of the weighted Hardy operator and its commutator on λ-centralmorreyspace. In thispaper, wewill give the boundedness of H m Ω,A,β on λ-central Morrey space with m. Ourresultscanbestatedasfollows. Theorem 2. Suppose m 2, n<p 2 <,<p <, / = /p +/p 2 β/n with β<nand λ=λ +λ 2 +β/n. Let Ω L r S n with /r > β/n + /,andahas derivatives of order m in CBMO p 2,λ 2.Ifr > p, λ > /p,and λ + >, then one has Hm Ω,A,β f E,λ Dγ A C BMO p 2,λ 2 f E p,λ, γ =m where the constant C is independent of f and A. For the case m=, we have the following theorem. 46 Theorem 3. Suppose <p,p 2 <, / = /p +/p 2 β/n with β<nand λ=λ +λ 2 + β/n.letω L r S n with /r > β/n + / and A CBMO p 2,λ 2.Ifλ > /p,r>,andλ + >, then one has p HA Ω,β f E,λ A CBMO p 2,λ 2 f E p,λ, 47 where the constant C is independent of f and A.

8 8 Abstract and Applied Analysis In order to prove Theorems 2 and 3, byastandard argument, we have the following lemma. R m A;x,y = R m A; x, y. Asp 2 8 and 9,wehave > n,thenbylemmas Lemma 4 see [6]. Suppose f C BMO p,λ, p<, λ</n,andr,r 2 R + ;then B,r +pλ B,r /p f x f p B,r 2 dx + log r r 2 f C BMO p,λ. 48 R m A; x, y R m A; x, y + γ! Dγ Ay x y m γ =m x y m γ =m [ Q λ 2 Dγ A C,λ 2 Proof of Theorem 2. For any R>,wedenoteB, R by B and B, R by B for any Z +. Thus we have the following estimates: B B B B = B B Hm Ω,A,β f x dx + B x n β B, x 2 B\2 B 2 B\2 B 2 B\2 B Ωx yfy x y m R m A; x, y dy dx x n β 3 x n β x n β i= 2 Ω x y f y 2 i B\2 i B x y m R m A; x, y dy dx Ω x y f y 2 i B\2 i B x y m R m A; x, y dy dx Ω x y f y 2 i B\2 i B x y m R m A; x, y dy dx + Dγ A y ], 5 where Qx, y is the cube centered at x and having diameter 5 n x y. As x 2 B\2 B and y 2 i B\2 i B with i 3,we have x y x 2 B /n and x y C 2 i B /n. Thus by the Hölder ineuality and the condition /p /p 2 /r = /r β/n / >,wehave I B 2 B\2 B B 2 B\2 B x n β 3 x n β 3 Ωx yfy 2 i B\2 i B x y m x y m λ 2 Q Dγ A C,λ 2 γ =m + Dγ Ay dy dx Ω x y f y 2 i B\2 i B =I+II. 49 λ 2 Q γ =m For the term I, let Ax = Ax γ =m /γ!m 2 B Dγ Ax γ. Then it is easy to see Dγ A C,λ 2 + Dγ Ay dy dx

9 Abstract and Applied Analysis 9 B 2 B\2 B x n β 3 Note the following fact: 2 i B\2 i B γ =m 2 i B\2 i B 2 i B\2 i B γ =m Ωx y r /r dy x y λ 2n fy p /p dy 2 i B\2 i B 2i /p /p 2 B /r Dγ A C,λ 2 + Dγ Ay p 2dy dx. x y λ 2n Dγ A C,λ 2 + Dγ Ay p 2dy /p 2 λ 2 [ Dγ A C BMO p 2,λ 2 2 B 2i /p 2 B γ =m γ =m /p Dγ Ay m Q Dγ p /p 2dy 2] A 2 i B Dγ A C,λ 2 2i /p 2 λ 2 B 2 B + 2i λ 2, B 52 where the last ineuality follows from Lemma 4 and the fact x y C 2 i B /n. Thus by the condition +λ > and λ +>λ +/p >,wehave I B β/n 2 B f 2 B\2 B E p,λ γ =m 3 f 3 f 2 B β/n 2 B 2i /p +λ B E p,λ γ =m 2i /p +λ B 2i /r B Dγ A C,λ 2 λ 2 2 B + 2i λ 2 B 2i /p 2 B 2i /p /p 2 /r B dx Dγ A CBMO p 2,λ 2 B λ 2 2 B 2i /p 2 B 2i B /r 2i /p /p 2 /r B E p,λ γ =m Dγ A C,λ 2 B +β/n +λ 2++λ + 2 nβ/n n+n+λ 2n+λ n+n Dγ A CBMO p 2,λ 2 f E p,λ B λ. γ =m 53 To estimate the term II, we adopt some basic ideas from the estimates of the term I 2 in Theorem 3. First,wedenote R m A;x,y = R m A φ ;x,y = R ma ;x,y for x 2 B and y 2 i B with 2 i,wherea φ x = R m A ;,x,y φ x y x with y 2 i B\2 i B.Here A x = Ax γ =m /γ!m 2 B Dγ Ax γ and φ is defined as in Section 2. As Ω L r S n with r>p,bylemma 7,weget II B C B i= 2 Hm Ω,A φ,βf i i= 2 f i L p γ =m Dγ A φ where f i is defined by f i x = fxχ 2 i B\2 i Bx. L p 2, 54

10 Abstract and Applied Analysis For D γ A φ,as x y y,thenbylemmas8 and 4, we have the following estimates: Dγ A φ x μ + ] =m C μ] R m μ D μ A ;x,y D] φ x y x x y ] μ + ] =m γ =m μ γ =m Thus we have γ =m x y m μ ] Qx,y χ x x y x Dγ A C Dγ A φ,λ 2 L p 2 Qx,y Dγ D μ p 2 A z dz x y λ 2n χ x y x. /p Dγ A CBMO p 2,λ 2 2i λ 2 B 2i /p 2. B γ =m By the above estimates, we obtain II C B i= 2 γ =m γ =m Dγ A C 2 in+λ i= 2 2 i B +p λ f x p /p dx 2 i B 2i /p +λ B,λ 2 2i λ 2 +/p 2 B Dγ A CBMO p 2,λ 2 f E p,λ B λ Dγ A CBMO p 2,λ 2 f E p,λ B λ. γ =m 57 Combining the estimates of I and II and by the definition of E,λ, we finish the proof of Theorem 2. Proof of Theorem 3. The proof of Theorem 3 is uite similar but much simpler than Theorem 2 and we omit the details here. Acnowledgments This wor was supported by the National Natural Science Foundation of China under Grant nos. 93, 2264, and 2775, the Natural Science Foundation of Jiangxi Province under Grant no. 24BAB27, and the Science Foundation of Jiangxi Education Department under Grant no. GJJ373. This wor was also supported by the Key Laboratory of Mathematics and Complex System Beijing Normal University, Ministry of Education, China. References [] A. P. Calderón and A. Zygmund, On singular integrals, American Journal of Mathematics,vol.78,pp ,956. [2]R.R.Coifman,R.Rochberg,andG.Weiss, Factorization theorems for Hardy spaces in several variables, Annals of Mathematics,vol.3,no.3,pp.6 635,976. [3] J. Cohen, A sharp estimate for a multilinear singular integral in R n, Indiana University Mathematics Journal, vol.3,no.5,pp , 98. [4] J. Cohen and J. Gosselin, A BMO estimate for multilinear singular integrals, Illinois Journal of Mathematics, vol.3,no. 3,pp ,986. [5] S. Lu and Q. Wu, CBMO estimates for commutators and multilinear singular integrals, Mathematische Nachrichten,vol. 276, pp , 24. [6] D. Y. Yan, Some problems on multilinear singular integral operators and multilinear oscillatory singular integral operators [Ph.D. thesis], Beijing Normal University, Beijing, China, 2. [7] X. Yu and X. X. Tao, Boundedness for a class of generalized commutators on λ-central Morrey space, Acta Mathematica Sinica. English Series,vol.29,no.,pp ,23. [8] C. Wang and Z. Zhang, A new proof of Wu s theorem on vortex sheets, Science China, vol. 55, no. 7, pp , 22. [9] S. Wu, Mathematical analysis of vortex sheets, Communications on Pure and Applied Mathematics,vol.59,no.8,pp.65 26, 26. [] G. H. Hardy, Note on a theorem of Hilbert, Mathematische Zeitschrift,vol.6,no.3-4,pp.34 37,92. [] Z.-W. Fu, Z.-G. Liu, S.-Z. Lu, and H.-B. Wang, Characterization for commutators of n-dimensional fractional Hardy operators, Science in China A, vol. 5, no., pp , 27. [2] M. Christ and L. Grafaos, Best constants for two nonconvolution ineualities, Proceedings of the American Mathematical Society,vol.23,no.6,pp ,995. [3] S. Lu and D. Yang, The central BMO spaces and Littlewood- Paley operators, Approximation Theory and its Applications,vol., no. 3, pp , 995. [4] Y. Komori, Notes on singular integrals on some inhomogeneous Herz spaces, Taiwanese Journal of Mathematics, vol. 8, no.3,pp ,24. [5] S.Z.LuandF.Y.Zhao, CBMOestimatesformultilinearHardy operators, Acta Mathematica Sinica. English Series,vol.26,no. 7,pp ,2. [6] G. L. Gao and X. Yu, Some estimates for the generalized Hardy operators on some function spaces, Acta Mathematica Sinica. Chinese Series, vol. 55, no. 6, pp., 22. [7] X. Yu and S. Lu, Endpoint estimates for generalized commutators of Hardy operators on H space, Journal of Function Spaces and Applications, vol. 23, Article ID 435, pages, 23.

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