Endpoint Estimates for Generalized Commutators of Hardy Operators on H 1 Space

We study the -boundedness of the generalized commutators of Hardy operator with a homogeneous kernel as follows: , where with , and . We prove that, when , is not bounded from to unless . Finally, we prove that is bounded from to with .


Introduction
Let  ∈   (R + ) with 1 <  < ∞; the classical Hardy operator is defined by A famous result proved by Hardy [1] can be stated as follows; Hardy [1] also pointed out the fact that the constant /( − 1) in ( 2) is the best possible.Later, Hardy operator was studied by many mathematicians; please see [2,3] for more details.
In 1995, Christ and Grafakos [4] studied the following dimensional Hardy operator; where ]  is the volume of the unit ball in R  , and they proved the following inequality: H      ≤   − 1           , 1 <  < ∞.
In 2007, Fu et al. [5] considered the following commutator of fractional Hardy operator: where H  () is defined by with − <  < .When  = 0, we simply denote H  0 by H  and H 0 is just the -dimensional Hardy operator proposed by Christ and Grafakos in [4] (without considering the constant ]  ).
Cohen and Gosselin [9] proved that    is bounded on   (R  ) for 1 <  < ∞ if Ω ∈ Lip 1 ( −1 ) and the function  has derivatives of order  − 1 in BMO(R  ).Later,    was studied by many mathematicians; for example, see [10,11] or [12] for more details.Particularly in [11], Lu and Wu studied the endpoint estimates of    on  1 space.It should be pointed out that the generalized commutators of some operators play an important role in the study of partial differential equation.Recently, 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 [12], Wang and Zhang [13] gave a new proof of Wu's theorem in [14].Here we would like to point out that the method used in [13] is much simpler than that in [14].
As the Hardy operator is controlled by the Hardy-Littlewood maximal function, we have where  Ω,, () is defined by and thus So we have where From [15, p. 222], we have the following lemma.
where the constant  is independent of  and .
Definition 3 (see [16]).One says a function () is an  1 atom if  satisfies the following conditions: It is well known that, if a function  belongs to  1 , then it can be written as  = ∑ ∞ =1     where each   is an  1 atom.Moreover, one has where the infimum is taken over all decompositions of .
In this paper, we would like to show that H  Ω,, is not bounded from  1 to  /(−) for all  ∈  + .Furthermore, we will prove that H  Ω,, is bounded from  1 to  /(−),∞ , where  /(−),∞ denotes the weak  /(−) space.Some ideas of this paper come from Zhao et al. [7].
Lemma 7 (see [9]).Let  be a function on R  with th order derivatives in where Q(, ) is the cube centered at  and having diameter 5√| − |.
Consequently, we have finished the proof of Theorem 8.
Before the proof of Theorems 10 and 11, we need the following lemma.
Proof of Theorem 10.Before giving the proof of Theorem 10, we introduce some notations that are very useful in this section.
For the term  2 , as  ∈ R  \ (0, 2), we have Thus by the vanishing condition of ã and the fact supp(ã) ⊂ (0, ), we can decompose H  Ω,, ã() as follows: Here we can simply denote each   by For  1 , by the fact that Ω ∈ Lip 1 ( where  1 is a constant only depending on , and Q is a cube centered at  and having diameter 5√||.
Thus we obtain For  3 , by the vanishing condition of ã(), we can split  3 as follows: For the term  32 , by the vanishing condition of ã and the fact Ω ∈ Lip 1 ( −1 ), we have    (72) Combining the estimates of  1 ,  2 ,  3 , and  4 , we finish the proof of Theorem 10.
Proof of Theorem 11.Theorem 11 was proved in [7] in the case  = 0 and Ω ≡ 1.For the case 0 ≤  <  and Ω ≡ 1, we can easily prove Theorem 11 by the proof of Theorem 5.3 in [7] with minor changes.Then for the case 0 ≤  <  and Ω ∈ Lip 1 ( −1 ), by the main idea used in the proof of Theorem 10, we can prove Theorem 11 easily and we omit the details here.