Endpoint Estimates for Fractional Hardy Operators and Their Commutators on Hardy Spaces

holds for 1 < p < ∞ and the constant p/(p − 1) is the best possible. The Hardy integral inequality has received considerable attention. A number of papers involved its alternative proofs, generalizations, variants, and applications. Among numerous papers dealing with such inequalities, we choose to refer to the papers [2–4]. Let f be a locally integrable function on R, 0 ≤ β < n. In [5], Fu et al. defined n-dimensional fractional Hardy operatorsHβ:


Introduction
The most fundamental averaging operator is the Hardy operators defined by where the function  is nonnegative integrable on R + = (0, ∞) and  > 0. A classical inequality, due to Hardy et al. [1], states that     ()      (R + ) ≤   − 1           (R + ) , holds for 1 <  < ∞ and the constant /( − 1) is the best possible.The Hardy integral inequality has received considerable attention.A number of papers involved its alternative proofs, generalizations, variants, and applications.Among numerous papers dealing with such inequalities, we choose to refer to the papers [2][3][4].
Let  be a locally integrable function on R  , 0 ≤  < .In [5], Fu et al. defined -dimensional fractional Hardy operators H  : When  = 0, H  is just -dimensional Hardy operators H; see [3] for more details.
This paper is organized as follows.In the second section, we give the (  (R  ),   (R  )) bounds of fractional Hardy operators.In the third section, we obtain the estimates for commutators of fractional Hardy operators on Hardy spaces.In Section 4, we consider the case on Herz-type Hardy spaces.In Section 5, we obtain the estimate for multilinear commutators of fractional Hardy operators.     () −  (0,)      ) where  (0,) = (1/|(0, )|) ∫ (0,) ().The  Ṁ  norm of  is defined by (5) We choose to refer to papers [5,9].
Definition 6.Let   ( = 1, 2, . . ., ) be a locally integrable functions, and A temperate distribution (see [15,16])  is said to belong to   ⃗  (R  ), if, in the Schwartz distribution sense, it can be written as where   is (, ⃗ )-atom,   ∈ , and , where the infimum has taken over all the decompositions of  as above.(i) The homogeneous Herz space K,  (R  ) is defined by where (ii) The nonhomogeneous Herz space  ,  (R  ) is defined by where (the usual modifications are made when  = ∞).
where we used the condition supp() ⊂ (0, ).For  1 , we have the following estimate where we used the condition ‖ã‖ The proof is completed.

Estimates for Commutators of Fractional Hardy Operators on Hardy Spaces
Definition 13 (see [5,19]).Let  be a locally integrable function on R  .The commutator of -dimensional Hardy operators is defined by Meanwhile, the commutators of -dimensional fractional Hardy operators are defined by In general, the properties of commutator are worse than those of the operators themselves (e.g., the Hardy operators [11] and the singular integral operators [20]).Therefore, when  is in Lip  (R  ), we prove that [, H  ] is not bounded from  1 (R  ) to  /(−−) (R  ).Furthermore, we conclude that the commutator maps from  1 (R  ) to  /(−−),∞ (R  ) and the commutator maps    (R  ) into   (R  ), where 0 <  ≤ 1 and 1/ = 1/ − ( + )/.
where  = /( −  − ).For the last term, by  ∈ R  \ (0, ), we have          1 So we obtain that Combining all the above estimates, we complete the proof of Theorem 15.

Estimates for Commutators of Fractional Hardy Operators on Herz-Type Hardy Spaces
The boundedness of [, H  ] on the Herz spaces K,  (R  ) has been obtained as the following, where  < (1 − 1/ 1 ).

Estimates for Multilinear Commutators of Fractional Hardy Operators
Definition 29.The multilinear commutator of fractional Hardy operators is defined by where The study of multilinear operators is motivated by a mere quest to generalize the theory of linear operators and by their natural appearance in analysis (see [23][24][25]).In this section, we consider the multilinear commutators of fractional Proof.It is similar to the proof of Theorem 15.
When  is in  Ṁ(R  ), we suppose the multilinear commutator of fractional Hardy operators maps  1 (R  ) into  /(−),∞ , but we cannot prove it.However, we get the following result.
Proof.Similar to the proof of Theorem 12, it is enough to prove that where ã is a center (, ⃗ )-atom supported on a ball  = (0, ).We write where  Combining all the above estimates, we complete the proof of Theorem 32.