Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 272864, 9 pages http://dx.doi.org/10.1155/2014/272864
Research Article Endpoint Estimates for Fractional Hardy Operators and Their Commutators on Hardy Spaces Jiang Zhou and Dinghuai Wang Department of Mathematics, Xinjiang University, Urumqi 830046, China Correspondence should be addressed to Jiang Zhou;
[email protected] Received 19 October 2013; Revised 28 December 2013; Accepted 30 December 2013; Published 18 February 2014 Academic Editor: Yongqiang Fu Copyright Β© 2014 J. Zhou and D. Wang. 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 work is properly cited. (π»π (Rπ ), πΏπ (Rπ )) bounds of fractional Hardy operators are obtained. Moreover, the estimates for commutators of fractional Hardy operators on Hardy spaces are worked out. It is also proved that the commutators of fractional Hardy operators are mapped from the Herz-type Hardy spaces into the Herz spaces. The estimates for multilinear commutators of fractional Hardy operators are also discussed.
1. Introduction The most fundamental averaging operator is the Hardy operators defined by π» (π) (π₯) =
1 π₯ β« π (π‘) ππ‘, π₯ 0
(1)
where the function π is nonnegative integrable on R+ = (0, β) and π₯ > 0. A classical inequality, due to Hardy et al. [1], states that π σ΅©σ΅© σ΅©σ΅© σ΅© σ΅©σ΅© σ΅©πσ΅© π + , σ΅©σ΅©π»(π)σ΅©σ΅©σ΅©πΏπ (R+ ) β€ (2) π β 1 σ΅© σ΅©πΏ (R ) holds for 1 < π < β and the constant π/(π β 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 [2β4]. Let π be a locally integrable function on Rπ , 0 β€ π½ < π. In [5], Fu et al. defined π-dimensional fractional Hardy operators Hπ½ : Hπ½ π (π₯) =
1
π (π¦) ππ¦. β« (3) |π₯|πβπ½ |π¦|0
1/π 1 σ΅¨σ΅¨ σ΅¨π β« σ΅¨σ΅¨π (π₯) β ππ΅(0,π) σ΅¨σ΅¨σ΅¨ ππ₯) < β, |π΅ (0, π)| π΅(0,π)
where the infimum has taken over all the decompositions of π = βπ π π ππ as above. Given a positive integer π and 1 β€ π β€ π, we denote by πΆππ the family of all finite subsets π = {π(1), π(2), . . . , π(π)} of {1, 2, . . . , π} of π different elements. For π β πΆππ , set ππ = {1, . . . , π}\π. For πβ = (π1 , π2 , . . . , ππ ) and π = {π1 , π2 , . . . , ππ } β πΆππ , set ππβ = (ππ1 , . . . , πππ ), ππ = ππ1 β
β
β
πππ , and βππβ βLipπΌ (Rπ ) = βππ1 βLip (Rπ ) β
β
β
βπππ βLip (Rπ ) . πΌ
(4)
Μ π norm of where ππ΅(0,π) = (1/|π΅(0, π)|) β«π΅(0,π) π(π₯)ππ₯. The πΆππ π is defined by βπβπΆππ Μ π (Rπ )
πΌ
Definition 6. Let ππ (π = 1, 2, . . . , π) be a locally integrable functions, and 0 < π β€ 1. A bounded measurable function π β if on Rπ is called π(π, π)-atom, (i) supp π β π΅ = π΅(π₯0 , π), (ii) βπβπΏβ β€ |π΅(π₯0 , π)|β1/π ,
1/π (5) 1 σ΅¨σ΅¨ σ΅¨π β« σ΅¨σ΅¨π (π₯) β ππ΅(0,π) σ΅¨σ΅¨σ΅¨ ππ₯) < β. |π΅ (0, π)| π΅(0,π)
(iii) β«π΅ π(π¦)ππ¦ = β«π΅ π(π¦)βπβπ ππ ππ¦ = 0 for any π β πΆππ , 1 β€ π β€ π.
Μ π (Rπ ); Remark 2. When 1 β€ π < β, π΅ππ(Rπ ) β πΆππ π π π π Μ (R ). We choose Μ (R ) β πΆππ when 1 β€ π < π < β, πΆππ to refer to papers [5, 9].
A temperate distribution (see [15, 16]) π is said to belong π to π» β (Rπ ), if, in the Schwartz distribution sense, it can be π written as
Definition 3. The Lipschitz space LipπΌ (Rπ ) is the space of functions π satisfying σ΅¨ σ΅¨σ΅¨ σ΅¨π (π₯ + β) β π (π₯)σ΅¨σ΅¨σ΅¨ σ΅©σ΅© σ΅©σ΅© < β, (6) σ΅©σ΅©πσ΅©σ΅©LipπΌ (Rπ ) := sup σ΅¨ |β|πΌ π₯,ββRπ ,β =ΜΈ 0
π (π₯) = βπ π ππ (π₯) ,
= sup( π>0
where 0 < πΌ β€ 1. Remark 4. When 0 < πΌ < 1, LipπΌ (Rπ ) = β§Μ πΌ (Rπ ), where β§Μ πΌ (Rπ ) is the homogeneous Besov-Lipschitz space. Definition 5 (see [14]). Let 0 < π β€ 1; a function is called π(π, β, π )-atom, where π β₯ [π(1/π β 1)], if it satisfies the following conditions: (1) supp (π) β π΅ (π₯0 , π) ; σ΅¨ σ΅¨β1/π ; (2) βπβπΏβ β€ σ΅¨σ΅¨σ΅¨π΅ (π₯0 , π)σ΅¨σ΅¨σ΅¨ (3) β« π (π₯) π₯πΎ ππ₯ = 0, Rπ
π
(7)
σ΅¨ σ΅¨ where 0 β€ σ΅¨σ΅¨σ΅¨πΎσ΅¨σ΅¨σ΅¨ β€ π .
π»π (Rπ ) πσΈ σ΅¨ σ΅¨π = {π β πσΈ (Rπ ) : π (π₯) = βπ π ππ (π₯) , βσ΅¨σ΅¨σ΅¨π π σ΅¨σ΅¨σ΅¨ < β} , π
(8) where each ππ is a (π, β, π )-atom and π is a tempered distribution. Set π»π (Rπ ) norm of π by 1/π
} }, }
Remark 7. When π = 1, π»π1β (Rπ ) = π»π1 (Rπ ). Let π΅π = {π₯ β R : |π₯| < 2π } and πΈπ = π΅π \ π΅πβ1 for π β Z. Denote ππ = ππΈπ . Definition 8. Let πΎ β R, 0 < π and π β€ β. (i) The homogeneous Herz space πΎΜ ππΎ,π (Rπ ) is defined by π σ΅© σ΅© πΎΜ ππΎ,π (Rπ ) = {π : π β πΏ loc (Rπ \ {0}) , σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΎΜ ππΎ,π < β} , (11)
1/π
As a proper subspace of πΏ (R ), the atomic Hardy space π»π (Rπ ) is defined by
{ β σ΅¨ σ΅¨π σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©πσ΅©σ΅©π»π (Rπ ) := inf {(βσ΅¨σ΅¨σ΅¨π π σ΅¨σ΅¨σ΅¨ ) { π
β π π β πΆ, and βπ |π π |π < β. Moreover, where ππ is (π, π)-atom, π 1/π βπβπ»π = inf(βπ |π π | ) , where the infimum has taken over πβ all the decompositions of π as above.
where
π
π
(10)
π
(9)
σ΅©σ΅© σ΅©σ΅© πΎ,π σ΅©π ππΎπ σ΅© σ΅©σ΅©πσ΅©σ΅©πΎΜ π (Rπ ) = { β 2 σ΅©σ΅©σ΅©πππ σ΅©σ΅©σ΅©πΏπ (Rπ ) }
.
(12)
πβZ
(ii) The nonhomogeneous Herz space πΎππΎ,π (Rπ ) is defined by π σ΅© σ΅© πΎππΎ,π (Rπ ) = {π : π β πΏ loc (Rπ \ {0}) , σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΎΜ ππΎ,π < β} , (13)
where β
1/π
σ΅©σ΅© σ΅©σ΅© πΎ,π σ΅©π ππΎπ σ΅© σ΅©σ΅©πσ΅©σ΅©πΎπ (Rπ ) = { β 2 σ΅©σ΅©σ΅©πππ σ΅©σ΅©σ΅©πΏπ (Rπ ) } π=0
(the usual modifications are made when π = β).
(14)
Journal of Function Spaces
3
Remark 9. When 0 < π β€ β, πΎΜ π0,π (Rπ ) = πΎπ0,π (Rπ ) = π πΏπ (Rπ ), and πΎΜ ππΎ/π (Rπ ) = πΏ |π₯|πΎ (Rπ ). Definition 10 (see [15]). Let πΎ β R, and 0 < π < π < β. (i) The homogeneous Herz-type Hardy space π»πΎΜ ππΎ,π (Rπ ) is defined by π»πΎΜ ππΎ,π (Rπ ) = {π β πσΈ (Rπ ) : πΊ (π) β πΎΜ ππΎ,π (Rπ )} ,
(15)
and we define βπβπ»πΎΜ ππΎ,π (Rπ ) = βπΊ(π)βπΎΜ ππΎ,π (Rπ ) .
0,π
Remark 11. When 0 < π < β, π»πΎΜ π (Rπ ) = πΎπ (Rπ ) = πΎ/π π π»π (Rπ ) and π»πΎΜ π (Rπ ) = πΏ |π₯|πΎ (Rπ ). And when 1 < π < β, we know that π»πΎΜ ππΎ,π (Rπ ) = πΎΜ ππΎ,π (Rπ ) and π»πΎππΎ,π (Rπ ) = πΎππΎ,π (Rπ ), where βπ/π < πΎ < π(1 β 1/π). However, when πΎ β₯ π(1β1/π), π»πΎΜ ππΎ,π (Rπ ) =ΜΈ πΎΜ ππΎ,π (Rπ ) and π»πΎππΎ,π (Rπ ) =ΜΈ πΎππΎ,π (Rπ ) (see [17, 18]).
2. (π»π (Rπ ), πΏπ (Rπ )) Bounds of Fractional Hardy Operators Theorem 12. Let 0 < π β€ 1 and 1/π = 1/π β π½/π. Hπ½ maps π»π (Rπ ) into πΏπ (Rπ ). Proof. Assume that π is an atom of π»π (Rπ ) and satisfies the following conditions: (i) supp(π) β π΅(π₯0 , π), (ii) βπβπΏβ β€ |π΅(π₯0 , π)|β1/π , and (iii) β« π(π₯)π₯πΎ ππ₯ = 0, where 0 β€ |πΎ| β€ π , π β₯ [π(1/π β 1)]. We now take πΜ(π₯) = π(π₯ + π₯0 ); then πΜ satisfies: (i) supp(Μ π) β π΅(0, π), (ii) βΜ πβπΏβ β€ |π΅(0, π)|β1/π , and (iii) β« πΜ(π₯)ππ₯ = 0. Suppose that supp πΜ β π΅(0, π) for π > 0. Consider σ΅¨π σ΅¨ π) (π₯)σ΅¨σ΅¨σ΅¨σ΅¨ ππ₯ β« σ΅¨σ΅¨σ΅¨σ΅¨Hπ½ (Μ π R σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨ 1 σ΅¨ πΜ (π¦) ππ¦σ΅¨σ΅¨σ΅¨σ΅¨ ππ₯ = β« σ΅¨σ΅¨σ΅¨σ΅¨ πβπ½ β« π σ΅¨σ΅¨ R σ΅¨σ΅¨ |π₯| |π¦|