Endpoint Estimates for Fractional Hardy Operators and Their

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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. (𝐻𝑝 (R𝑛 ), πΏπ‘ž (R𝑛 )) 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 𝐻 (𝑓) (π‘₯) =

1 π‘₯ ∫ 𝑓 (𝑑) 𝑑𝑑, π‘₯ 0

(1)

where the function 𝑓 is nonnegative integrable on R+ = (0, ∞) and π‘₯ > 0. A classical inequality, due to Hardy et al. [1], states that 𝑝 σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© 󡄩𝑓󡄩 𝑝 + , 󡄩󡄩𝐻(𝑓)󡄩󡄩󡄩𝐿𝑝 (R+ ) ≀ (2) 𝑝 βˆ’ 1 σ΅„© 󡄩𝐿 (R ) holds for 1 < 𝑝 < ∞ and the constant 𝑝/(𝑝 βˆ’ 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 R𝑛 , 0 ≀ 𝛽 < 𝑛. In [5], Fu et al. defined 𝑛-dimensional fractional Hardy operators H𝛽 : H𝛽 𝑓 (π‘₯) =

1

𝑓 (𝑦) 𝑑𝑦. ∫ (3) |π‘₯|π‘›βˆ’π›½ |𝑦|0

1/π‘ž 1 󡄨󡄨 σ΅„¨π‘ž ∫ 󡄨󡄨𝑏 (π‘₯) βˆ’ 𝑏𝐡(0,π‘Ÿ) 󡄨󡄨󡄨 𝑑π‘₯) < ∞, |𝐡 (0, π‘Ÿ)| 𝐡(0,π‘Ÿ)

where the infimum has taken over all the decompositions of 𝑓 = βˆ‘π‘˜ πœ† π‘˜ π‘Žπ‘˜ as above. Given a positive integer π‘š and 1 ≀ 𝑖 ≀ π‘š, we denote by πΆπ‘–π‘š the family of all finite subsets 𝜎 = {𝜎(1), 𝜎(2), . . . , 𝜎(𝑖)} of {1, 2, . . . , π‘š} of 𝑗 different elements. For 𝜎 ∈ πΆπ‘–π‘š , set πœŽπ‘ = {1, . . . , π‘š}\𝜎. For 𝑏⃗ = (𝑏1 , 𝑏2 , . . . , π‘π‘š ) and 𝜎 = {𝜎1 , 𝜎2 , . . . , πœŽπ‘– } ∈ πΆπ‘—π‘š , set π‘πœŽβƒ— = (π‘πœŽ1 , . . . , π‘πœŽπ‘— ), π‘πœŽ = π‘πœŽ1 β‹… β‹… β‹… π‘πœŽπ‘— , and β€–π‘πœŽβƒ— β€–Lip𝛼 (R𝑛 ) = β€–π‘πœŽ1 β€–Lip (R𝑛 ) β‹… β‹… β‹… β€–π‘πœŽπ‘– β€–Lip (R𝑛 ) . 𝛼

(4)

Μ‡ π‘ž norm of where 𝑏𝐡(0,π‘Ÿ) = (1/|𝐡(0, π‘Ÿ)|) ∫𝐡(0,π‘Ÿ) 𝑏(π‘₯)𝑑π‘₯. The 𝐢𝑀𝑂 𝑏 is defined by ‖𝑏‖𝐢𝑀𝑂 Μ‡ π‘ž (R𝑛 )

𝛼

Definition 6. Let 𝑏𝑖 (𝑖 = 1, 2, . . . , π‘š) be a locally integrable functions, and 0 < 𝑝 ≀ 1. A bounded measurable function π‘Ž βƒ— if on R𝑛 is called π‘Ž(𝑝, 𝑏)-atom, (i) supp π‘Ž βŠ‚ 𝐡 = 𝐡(π‘₯0 , π‘Ÿ), (ii) β€–π‘Žβ€–πΏβˆž ≀ |𝐡(π‘₯0 , π‘Ÿ)|βˆ’1/𝑝 ,

1/π‘ž (5) 1 󡄨󡄨 σ΅„¨π‘ž ∫ 󡄨󡄨𝑏 (π‘₯) βˆ’ 𝑏𝐡(0,π‘Ÿ) 󡄨󡄨󡄨 𝑑π‘₯) < ∞. |𝐡 (0, π‘Ÿ)| 𝐡(0,π‘Ÿ)

(iii) ∫𝐡 π‘Ž(𝑦)𝑑𝑦 = ∫𝐡 π‘Ž(𝑦)βˆπ‘—βˆˆπœŽ 𝑏𝑗 𝑑𝑦 = 0 for any 𝜎 ∈ πΆπ‘–π‘š , 1 ≀ 𝑖 ≀ π‘š.

Μ‡ π‘ž (R𝑛 ); Remark 2. When 1 ≀ π‘ž < ∞, 𝐡𝑀𝑂(R𝑛 ) βŠ† 𝐢𝑀𝑂 π‘ž 𝑛 𝑝 𝑛 Μ‡ (R ). We choose Μ‡ (R ) βŠ† 𝐢𝑀𝑂 when 1 ≀ 𝑝 < π‘ž < ∞, 𝐢𝑀𝑂 to refer to papers [5, 9].

A temperate distribution (see [15, 16]) 𝑓 is said to belong 𝑝 to 𝐻 βƒ— (R𝑛 ), if, in the Schwartz distribution sense, it can be 𝑏 written as

Definition 3. The Lipschitz space Lip𝛼 (R𝑛 ) is the space of functions 𝑓 satisfying 󡄨 󡄨󡄨 󡄨𝑓 (π‘₯ + β„Ž) βˆ’ 𝑓 (π‘₯)󡄨󡄨󡄨 σ΅„©σ΅„© σ΅„©σ΅„© < ∞, (6) 󡄩󡄩𝑓󡄩󡄩Lip𝛼 (R𝑛 ) := sup 󡄨 |β„Ž|𝛼 π‘₯,β„ŽβˆˆR𝑛 ,β„Ž =ΜΈ 0

𝑓 (π‘₯) = βˆ‘πœ† 𝑖 π‘Žπ‘– (π‘₯) ,

= sup( π‘Ÿ>0

where 0 < 𝛼 ≀ 1. Remark 4. When 0 < 𝛼 < 1, Lip𝛼 (R𝑛 ) = βˆ§Μ‡ 𝛼 (R𝑛 ), where βˆ§Μ‡ 𝛼 (R𝑛 ) is the homogeneous Besov-Lipschitz space. Definition 5 (see [14]). Let 0 < 𝑝 ≀ 1; a function is called π‘Ž(𝑝, ∞, 𝑠)-atom, where 𝑠 β‰₯ [𝑛(1/𝑝 βˆ’ 1)], if it satisfies the following conditions: (1) supp (π‘Ž) βŠ‚ 𝐡 (π‘₯0 , π‘Ÿ) ; 󡄨 σ΅„¨βˆ’1/𝑝 ; (2) β€–π‘Žβ€–πΏβˆž ≀ 󡄨󡄨󡄨𝐡 (π‘₯0 , π‘Ÿ)󡄨󡄨󡄨 (3) ∫ π‘Ž (π‘₯) π‘₯𝛾 𝑑π‘₯ = 0, R𝑛

𝑝

(7)

󡄨 󡄨 where 0 ≀ 󡄨󡄨󡄨𝛾󡄨󡄨󡄨 ≀ 𝑠.

𝐻𝑝 (R𝑛 ) 𝑆󸀠 󡄨 󡄨𝑝 = {𝑓 ∈ 𝑆󸀠 (R𝑛 ) : 𝑓 (π‘₯) = βˆ‘πœ† π‘˜ π‘Žπ‘˜ (π‘₯) , βˆ‘σ΅„¨σ΅„¨σ΅„¨πœ† π‘˜ 󡄨󡄨󡄨 < ∞} , π‘˜

(8) where each π‘Žπ‘˜ is a (𝑝, ∞, 𝑠)-atom and 𝑓 is a tempered distribution. Set 𝐻𝑝 (R𝑛 ) norm of 𝑓 by 1/𝑝

} }, }

Remark 7. When π‘š = 1, 𝐻𝑏1βƒ— (R𝑛 ) = 𝐻𝑏1 (R𝑛 ). Let π΅π‘˜ = {π‘₯ ∈ R : |π‘₯| < 2π‘˜ } and πΈπ‘˜ = π΅π‘˜ \ π΅π‘˜βˆ’1 for π‘˜ ∈ Z. Denote πœ’π‘˜ = πœ’πΈπ‘˜ . Definition 8. Let 𝛾 ∈ R, 0 < 𝑝 and π‘ž ≀ ∞. (i) The homogeneous Herz space 𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) is defined by π‘ž σ΅„© σ΅„© 𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) = {𝑓 : 𝑓 ∈ 𝐿 loc (R𝑛 \ {0}) , 󡄩󡄩󡄩𝑓󡄩󡄩󡄩𝐾̇ π‘žπ›Ύ,𝑝 < ∞} , (11)

1/𝑝

As a proper subspace of 𝐿 (R ), the atomic Hardy space 𝐻𝑝 (R𝑛 ) is defined by

{ ∞ 󡄨 󡄨𝑝 σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑓󡄩󡄩𝐻𝑝 (R𝑛 ) := inf {(βˆ‘σ΅„¨σ΅„¨σ΅„¨πœ† π‘˜ 󡄨󡄨󡄨 ) { π‘˜

βƒ— πœ† 𝑖 ∈ 𝐢, and βˆ‘π‘– |πœ† 𝑖 |𝑝 < ∞. Moreover, where π‘Žπ‘– is (𝑝, 𝑏)-atom, 𝑝 1/𝑝 ‖𝑓‖𝐻𝑝 = inf(βˆ‘π‘– |πœ† 𝑖 | ) , where the infimum has taken over 𝑏⃗ all the decompositions of 𝑓 as above.

where

𝑛

π‘˜

(10)

𝑖

(9)

σ΅„©σ΅„© σ΅„©σ΅„© 𝛾,𝑝 󡄩𝑝 π‘˜π›Ύπ‘ σ΅„© 󡄩󡄩𝑓󡄩󡄩𝐾̇ π‘ž (R𝑛 ) = { βˆ‘ 2 σ΅„©σ΅„©σ΅„©π‘“πœ’π‘˜ σ΅„©σ΅„©σ΅„©πΏπ‘ž (R𝑛 ) }

.

(12)

π‘˜βˆˆZ

(ii) The nonhomogeneous Herz space πΎπ‘žπ›Ύ,𝑝 (R𝑛 ) is defined by π‘ž σ΅„© σ΅„© πΎπ‘žπ›Ύ,𝑝 (R𝑛 ) = {𝑓 : 𝑓 ∈ 𝐿 loc (R𝑛 \ {0}) , 󡄩󡄩󡄩𝑓󡄩󡄩󡄩𝐾̇ π‘žπ›Ύ,𝑝 < ∞} , (13)

where ∞

1/𝑝

σ΅„©σ΅„© σ΅„©σ΅„© 𝛾,𝑝 󡄩𝑝 π‘˜π›Ύπ‘ σ΅„© σ΅„©σ΅„©π‘“σ΅„©σ΅„©πΎπ‘ž (R𝑛 ) = { βˆ‘ 2 σ΅„©σ΅„©σ΅„©π‘“πœ’π‘˜ σ΅„©σ΅„©σ΅„©πΏπ‘ž (R𝑛 ) } π‘˜=0

(the usual modifications are made when 𝑝 = ∞).

(14)

Journal of Function Spaces

3

Remark 9. When 0 < π‘ž ≀ ∞, 𝐾̇ π‘ž0,π‘ž (R𝑛 ) = πΎπ‘ž0,π‘ž (R𝑛 ) = π‘ž πΏπ‘ž (R𝑛 ), and 𝐾̇ π‘žπ›Ύ/π‘ž (R𝑛 ) = 𝐿 |π‘₯|𝛾 (R𝑛 ). Definition 10 (see [15]). Let 𝛾 ∈ R, and 0 < 𝑝 < π‘ž < ∞. (i) The homogeneous Herz-type Hardy space 𝐻𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) is defined by 𝐻𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) = {𝑓 ∈ 𝑆󸀠 (R𝑛 ) : 𝐺 (𝑓) ∈ 𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 )} ,

(15)

and we define ‖𝑓‖𝐻𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) = ‖𝐺(𝑓)‖𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) .

0,𝑝

Remark 11. When 0 < 𝑝 < ∞, 𝐻𝐾̇ 𝑝 (R𝑛 ) = 𝐾𝑝 (R𝑛 ) = 𝛾/𝑝 𝑝 𝐻𝑝 (R𝑛 ) and 𝐻𝐾̇ 𝑝 (R𝑛 ) = 𝐿 |π‘₯|𝛾 (R𝑛 ). And when 1 < π‘ž < ∞, we know that 𝐻𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) = 𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) and π»πΎπ‘žπ›Ύ,𝑝 (R𝑛 ) = πΎπ‘žπ›Ύ,𝑝 (R𝑛 ), where βˆ’π‘›/π‘ž < 𝛾 < 𝑛(1 βˆ’ 1/π‘ž). However, when 𝛾 β‰₯ 𝑛(1βˆ’1/π‘ž), 𝐻𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) =ΜΈ 𝐾̇ π‘žπ›Ύ,𝑝 (R𝑛 ) and π»πΎπ‘žπ›Ύ,𝑝 (R𝑛 ) =ΜΈ πΎπ‘žπ›Ύ,𝑝 (R𝑛 ) (see [17, 18]).

2. (𝐻𝑝 (R𝑛 ), πΏπ‘ž (R𝑛 )) Bounds of Fractional Hardy Operators Theorem 12. Let 0 < 𝑝 ≀ 1 and 1/π‘ž = 1/𝑝 βˆ’ 𝛽/𝑛. H𝛽 maps 𝐻𝑝 (R𝑛 ) into πΏπ‘ž (R𝑛 ). Proof. Assume that π‘Ž is an atom of 𝐻𝑝 (R𝑛 ) and satisfies the following conditions: (i) supp(π‘Ž) βŠ‚ 𝐡(π‘₯0 , π‘Ÿ), (ii) β€–π‘Žβ€–πΏβˆž ≀ |𝐡(π‘₯0 , π‘Ÿ)|βˆ’1/𝑝 , and (iii) ∫ π‘Ž(π‘₯)π‘₯𝛾 𝑑π‘₯ = 0, where 0 ≀ |𝛾| ≀ 𝑠, 𝑠 β‰₯ [𝑛(1/𝑝 βˆ’ 1)]. We now take π‘ŽΜƒ(π‘₯) = π‘Ž(π‘₯ + π‘₯0 ); then π‘ŽΜƒ satisfies: (i) supp(Μƒ π‘Ž) βŠ‚ 𝐡(0, π‘Ÿ), (ii) β€–Μƒ π‘Žβ€–πΏβˆž ≀ |𝐡(0, π‘Ÿ)|βˆ’1/𝑝 , and (iii) ∫ π‘ŽΜƒ(π‘₯)𝑑π‘₯ = 0. Suppose that supp π‘ŽΜƒ βŠ‚ 𝐡(0, π‘Ÿ) for π‘Ÿ > 0. Consider σ΅„¨π‘ž 󡄨 π‘Ž) (π‘₯)󡄨󡄨󡄨󡄨 𝑑π‘₯ ∫ 󡄨󡄨󡄨󡄨H𝛽 (Μƒ 𝑛 R 󡄨󡄨 σ΅„¨σ΅„¨π‘ž 󡄨 1 󡄨 π‘ŽΜƒ (𝑦) 𝑑𝑦󡄨󡄨󡄨󡄨 𝑑π‘₯ = ∫ 󡄨󡄨󡄨󡄨 π‘›βˆ’π›½ ∫ 𝑛 󡄨󡄨 R 󡄨󡄨 |π‘₯| |𝑦|