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

and Applied Analysis 3 Theorem 4. Suppose Ω ∈ Lr(Sn−1) with 1 < r < ∞ and A ∈ CḂMOp . Let 0 < s ≤ p < ∞, 1 < q, p 1 , p 2 < ∞, 1/q = 1/p 1 +1/p 2 −β/nwith 0 ≤ β < n. If 1/r󸀠−1/q−β/n > 0, r > p󸀠 1 , α 2 = α 1 +n/p 2 , and α 2 satisfies (13), then there exists a constant C, such that 󵄩󵄩󵄩󵄩H A Ω,β f 󵄩󵄩󵄩󵄩?̇?α1,s q ≤ C‖A‖CḂMO2 󵄩󵄩󵄩f 󵄩󵄩󵄩?̇?α2,p p 1 . (15) Remark 5. Comparing Theorems 3 and 4, we find that the restrictions on α 1 and α 2 are more rigid in Theorem 4 than in Theorem 3, which indicates that Hm Ω,A,β with m ≥ 2 has better properties than the commutators. In order to proveTheorems 3 and 4, we need the following lemmas. Lemma 6 (see [19]). Let 1 < p 1 , p 2 < ∞, 0 ≤ β < n, and β/n = 1/p 1 − 1/p 2 . If Ω ∈ Lr(Sn−1) with r > p󸀠 1 , then there exists a constant C independent of f, such that 󵄩󵄩󵄩󵄩HΩ,βf 󵄩󵄩󵄩󵄩Lp2 ≤ C 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp1 , (16) whereH Ω,β is defined by H Ω,β f (x) = 1 |x|n−β ∫ |y|<|x| Ω(x − y)f (y) dy. (17) By checking [19] carefully, one can draw the conclusion that if one replacesH Ω,β f(x) byH |Ω|,β |f|(x), then (16) still holds. Lemma 7. Let m ≥ 1, 1 < p 1 < p 2 < ∞ and 0 ≤ β < n. If A has derivatives of order m − 1 in Lr(Rn) with 1/p 2 = 1/p 1 + 1/r − β/n and r > p󸀠 1 , then one has 󵄩󵄩󵄩󵄩H m Ω,A,β f 󵄩󵄩󵄩󵄩Lp2 ≤ C ∑ |γ|=m−1 󵄩󵄩󵄩D γA 󵄩󵄩󵄩Lr 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp1 , (18) where the constant C is independent of f and A. Proof. From [20, p. 241], we have the following estimates: R m (A; x, y) 󵄨󵄨󵄨x − y 󵄨󵄨󵄨 m−1 ≤ R m−1 (A; x, y) 󵄨󵄨󵄨x − y 󵄨󵄨󵄨 m−1 + C ∑ |γ|=m−1 󵄨󵄨󵄨D γA (y) 󵄨󵄨󵄨 ≤ C ∑ |γ|=m−1 ((DγA) ∗ (x) + (D γA) ∗ (y)) , (19) where m ≥ 1 and (f)∗ is the Hardy-Littlewood maximal function of f. Thus we obtain 󵄨󵄨󵄨󵄨H m Ω,A,β f (x) 󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 |x|n−β ∫ |y|<|x| Ω(x − y)f (y) 󵄨󵄨󵄨x − y 󵄨󵄨󵄨 m−1 R m (A; x, y) dy 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ C 1 |x|n−β ∫ |y|<|x| 󵄨󵄨󵄨f (y) 󵄨󵄨󵄨 󵄨󵄨󵄨Ω (x − y) 󵄨󵄨󵄨 × ∑ |γ|=m−1 ((DγA) ∗ (x) + (D γA) ∗ (y)) dy ≤ C ∑ |γ|=m−1 [(DγA) ∗ (x)H |Ω|,β 󵄨󵄨󵄨f 󵄨󵄨󵄨 (x) +H |Ω|,β ((DγA) ∗ 󵄨󵄨󵄨f 󵄨󵄨󵄨) (x)] . (20) By the above estimates, we can get (∫ Rn 󵄨󵄨󵄨󵄨H m Ω,A,β f (x) 󵄨󵄨󵄨󵄨 q dx) 1/q ≤ C ∑ |γ|=m−1 ((∫ Rn 󵄨󵄨󵄨󵄨(D γA) ∗ (x)H |Ω|,β 󵄨󵄨󵄨f 󵄨󵄨󵄨 (x) 󵄨󵄨󵄨󵄨 q dx) 1/q +(∫ Rn 󵄨󵄨󵄨󵄨H|Ω|,β ((D γA) ∗ 󵄨󵄨󵄨f 󵄨󵄨󵄨) (x) 󵄨󵄨󵄨󵄨 q ) 1/q ) ≤ C ∑ |γ|=m−1 (I + II) . (21) For the term I, let 1/q = 1/r + 1/l = 1/r + 1/p−β/n; then by the Hölder inequality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on Lp spaces, we obtain I ≤ (∫ Rn (DγA) ∗ (x) rdx) 1/r (∫ Rn 󵄨󵄨󵄨󵄨H|Ω|,β 󵄨󵄨󵄨f 󵄨󵄨󵄨 (x) 󵄨󵄨󵄨󵄨 l dx) 1/l ≤ C 󵄩󵄩󵄩󵄩(D γA) 󵄩󵄩󵄩󵄩Lr 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp ≤ C 󵄩󵄩󵄩D γA 󵄩󵄩󵄩Lr 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp . (22) For the term II, let 1/q = 1/t − β/n = 1/r + 1/p − β/n; then by the Hölder inequality and Lemma 6, we have II ≤ C(∫ Rn 󵄨󵄨󵄨󵄨(D γA) ∗ (x) f (x) 󵄨󵄨󵄨󵄨 t dx) 1/t ≤ C 󵄩󵄩󵄩󵄩(D γA) 󵄩󵄩󵄩󵄩Lr 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp ≤ C 󵄩󵄩󵄩D γA 󵄩󵄩󵄩Lr 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp . (23) Combining the estimates of I and II, we finish the proof of Lemma 7. 4 Abstract and Applied Analysis Lemma 8 (see [4]). Let b be a function on Rn withmth order derivatives in Lqloc(R n) for some q > n. Then 󵄨󵄨󵄨Rm (b; x, y) 󵄨󵄨󵄨 ≤ Cm,n 󵄨󵄨󵄨x − y 󵄨󵄨󵄨 m


Introduction
It is well known that the C-Z singular integrals and their commutators have been studied a lot by many mathematicians; please see [1] or [2] for more details.For the generalizations of the commutators of singular integrals, Cohen [3] studied the following generalized commutator  2   which is defined by where Ω ∈  1 (S −1 ) is homogeneous of degree zero and satisfies the moment condition with || = 1.Cohen [3] proved that if Ω ∈ Lip 1 (S −1 ) and ∇ ∈ BMO, then  2  is bounded on   (R  ) with 1 <  < ∞.Later, Cohen and Gosselin [4] considered another type of generalized commutator as follows: Ω ( − )      −      +−1   (; , )  () , where   (; , )( ∈  + ) is defined by   (; , ) = () − ∑ ||< (1/!)  ()( − )  , the mth remainder of Taylor series of the function  at  about , and Ω satisfies the following moment condition: with || =  − 1. Obviously, if we choose  = 1,    becomes [, ], the commutator of  generalized by  and .Furthermore,    becomes  2  if we choose  = 2. Cohen and Gosselin proved that if  ≥ 2, Ω ∈ Lip 1 (S −1 ), and the function  has derivatives of order −1 in BMO(R  ), then the operator    is bounded on   (R  ) for 1 <  < ∞.Later,    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] by using the  1, estimate for the elliptic equation of divergence form with partially BMO coefficients and the   boundedness of the Cohen-Gosselin type generalized commutators proved by Yan in [6].Furthermore, the method used in [8] is much simpler than that in [9].Recently, Yu and Tao [7] proved that     is bounded on -Central Morrey space.
In 2007, Fu et al. [11] introduced the -dimensional fractional type Hardy operator H  as follows: where − <  <  and  is a locally integrable function on R  .Obviously, when  = 0, H 0 is just the -dimensional Hardy operator H which was proposed by Christ and Grafakos in [12].
In [11], the authors gave the characterization of the C Ḃ MO  (R  ) by the boundedness of the commutator of the fractional type Hardy operator [H  , ] on Herz type spaces.
Here the C Ḃ MO  (R  ) space is defined by the following.
Definition 1 (see [13]).Let where From [14], we know that BMO(R The C Ḃ MO  (R  ) space can be regarded as the space of bounded mean oscillation, a local version of BMO(R  ) at the origin.But the famous John-Nirenberg inequality no longer holds in C Ḃ MO  (R  ).
Now we are interested in the following generalized commutator of Hardy operator: where   (; , ) = () − ∑ ||< (1/!)  ()( − )  and  ∈  + .In 2010, Lu and Zhao [15] proved that when  = 2, H 2  is bounded on Herz type space and Morrey-Herz type space.Later, Gao and Yu [16] proved that H 2  is bounded on -Central Morrey spaces.However, we would like to point out that the method used in [15,16] cannot apply to the case when  > 2.An interesting question is whether the boundedness of H   on Herz type space or -Central Morrey space still holds with  > 2. In this paper, we will use a different method to answer this question.Furthermore, we will consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: where  ∈  + , 0 ≤  < , and Ω ∈   (S −1 ).

Boundedness of H 𝑚 Ω,𝐴,𝛽 on Herz Type Spaces
In this section, we will give the boundedness of H  Ω,, on Herz type spaces.First we introduce some notations that will be used throughout this paper. Let and   =    for  ∈ : here    is the characteristic function of the set   .
Definition 2 (see [18]).Let  ∈ , 0 < ,  ≤ ∞.Then the homogeneous Herz type space K,  (R  ) is defined by where ‖ ‖ K,  (R  ) is defined as with the usual modifications made when  = ∞ or  = ∞.Now we show our main results in this section.
Thus we obtain By the above estimates, we can get For the term , let 1/ = 1/ + 1/ = 1/ + 1/ − /; then by the Hölder inequality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on   spaces, we obtain For the term , let 1/ = 1/ − / = 1/ + 1/ − /; then by the Hölder inequality and Lemma 6, we have Combining the estimates of  and , we finish the proof of Lemma 7.
Lemma 8 (see [4]).Let  be a function on R  with th order derivatives in where Q(, ) is the cube centered at  having diameter 5√| − |.
Lemma 9 (see [5]).Suppose that Proof of Theorem 3. To prove Theorem 3, first we split each  as For the term As then by the Hölder inequality, we have we obtain the following estimates: For the term  2 , we choose  ∈  ∞ 0 (R  ) satisfying supp ⊂ (0, 4) as well as  ≡ 1 in (0, 2) and we set (33) From [5, p.80], we have the following estimates: where  is dependent on .Thus we get As 0 <  ≤  < ∞, we obtain the following estimates: For the term , we have When 0 <  ≤ 1, by condition (13), we get When  ≥ 1, by the Hölder inequality and condition (13), we have For the term , we have the following estimates: Combining the estimates of  and , we finish the proof of Theorem 3.
Proof of Theorem 4. The proof of Theorem 4 is quite similar and much easier than Theorem 3 and we omit the details here.

Boundedness of H 𝑚 Ω,𝐴,𝛽 on 𝜆-Central Morrey Spaces
In [21], 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,   and    , with 1/ + 1/  = 1.In [23], Feichtinger proved that   can be described as where  0 is the characterization of the unit ball { ∈ R  : || ≤ 1} and   is defined as in Section 2. Now by duality, the space   , which is called the Beurling algebra, can be described by Later, Chen and Lau [24] as well as García-Cuerva [25] introduced atomic spaces   (R  ) associated with the Buerling algebra   and the dual space of   (R  ) can be described by (43) here the CBMO  can be regarded as the inhomogeneous central BMO spaces.
In 2011, Fu et al. [19] proved the boundedness of the commutator of fractional Hardy operator with a rough kernel on -Central Morrey space.Later, Fu et al. [28] proved the boundness of the weighted Hardy operator and its commutator on -Central Morrey space.In this paper, we will give the boundedness of H  Ω,, on -Central Morrey space with  ≥ 1.
Our results can be stated as follows.
For the case  = 1, we have the following theorem.
where the constant  is independent of  and .
Proof of Theorem 13.The proof of Theorem 13 is quite similar but much simpler than Theorem 12 and we omit the details here.