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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 283730, 13 pages
http://dx.doi.org/10.1155/2012/283730
Research Article

Weighted Herz Spaces and Regularity Results

College of Mathematics and System Sciences, Xinjiang University, Xinjiang, Urumqi 830046, China

Received 13 April 2012; Accepted 20 July 2012

Academic Editor: Hans G.Β Feichtinger

Copyright Β© 2012 Yuxing Guo and Yinsheng Jiang. 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.

Abstract

It is proved that, for the nondivergence form elliptic equations βˆ‘π‘›π‘–,𝑗=1π‘Žπ‘–π‘—π‘’π‘₯𝑖π‘₯𝑗=𝑓, if 𝑓 belongs to the weighted Herz spaces πΎπ‘žπ‘(πœ‘,𝑀), then 𝑒π‘₯𝑖π‘₯π‘—βˆˆπΎπ‘žπ‘(πœ‘,𝑀), where 𝑒 is the π‘Š2,𝑝-solution of the equations. In order to obtain this, the authors first establish the weighted boundedness for the commutators of some singular integral operators on πΎπ‘žπ‘(πœ‘,𝑀).

1. Introduction

For a sequence πœ‘={πœ‘(π‘˜)}βˆžβˆ’βˆž,πœ‘(π‘˜)>0, we suppose that πœ‘ satisfies doubling condition of order (𝑠,𝑑) and write πœ‘βˆˆπ·(𝑠,𝑑) if there exists 𝐢β‰₯1 such that πΆβˆ’12𝑠(π‘˜βˆ’π‘—)β‰€πœ‘(π‘˜)πœ‘(𝑗)≀𝐢2𝑑(π‘˜βˆ’π‘—)forπ‘˜>𝑗.(1.1) Let π΅π‘˜=𝐡(0,2π‘˜)={π‘₯βˆˆβ„π‘›βˆΆ|π‘₯|≀2π‘˜}, πΈπ‘˜=π΅π‘˜β§΅π΅π‘˜βˆ’1 for π‘˜βˆˆβ„€, and πœ’π‘˜=πœ’πΈπ‘˜ be the characteristic function of the set πΈπ‘˜ for π‘˜βˆˆβ„€. Suppose that 𝑀 is a weight function on ℝ𝑛. For 1<𝑝<∞, 0<π‘ž<∞, the weighted Herz space is defined by πΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛)=π‘“βˆΆπ‘“isameasurablefunctiononℝ𝑛,β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀),<∞(1.2) where β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)=ξƒ©βˆžξ“π‘˜=βˆ’βˆžπœ‘(π‘˜)π‘žβ€–β€–π‘“πœ’π‘˜β€–β€–π‘žπΏπ‘(𝑀)ξƒͺ1/π‘ž,‖𝑓‖𝐿𝑝(𝑀)=ξ‚΅ξ€œβ„π‘›||||𝑓(π‘₯)𝑝𝑀(π‘₯)𝑑π‘₯1/𝑝.(1.3)

Beurling in [1] introduced the Beurling algebras, and Herz in [2] generalized these spaces; many studies have been done for Herz spaces (see, e.g, [3, 4]). Weighted Herz spaces are also considered in [5, 6]. Lu and Tao in [7] studied nondivergence form elliptic equations on Morrey-Herz spaces, which are more general spaces. Ragusa in [8, 9] obtained some regularity results to the divergence form elliptic and parabolic equations on homogeneous Herz spaces.

The paper is organized as follows. In Section 2, we give some basic notions. In this section, we recall also continuity results regarding the CalderΓ³n-Zygmund singular integral operators that will appear in the representation formula of the 𝑒π‘₯𝑖π‘₯𝑗 estimates. In Section 3, we prove the boundedness of the commutators of some singular integral operators on weighted Herz spaces. In Section 4, we study the interior estimates on weighted Herz spaces for the solutions of some nondivergence elliptic equations βˆ‘π‘›π‘–,𝑗=1π‘Žπ‘–π‘—π‘’π‘₯𝑖π‘₯𝑗=𝑓, and we prove that if π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀), then 𝑒π‘₯𝑖π‘₯π‘—βˆˆπΎπ‘žπ‘(πœ‘,𝑀), where 𝑒 is the π‘Š2,𝑝-solution of the equations.

Throughout this paper, unless otherwise indicated, 𝐢 will be used to denote a positive constant that is not necessarily the same at each occurrence.

2. Preliminaries

We begin this section with some properties of 𝐴𝑝 weights classes which play important role in the proofs of our main results. For more about 𝐴𝑝 classes, we can refer to [10, 11].

Definition 2.1 (𝐴𝑝weights(1≀𝑝<∞)). Let 𝑀(π‘₯)β‰₯0 and 𝑀(π‘₯)∈𝐿1loc(ℝ𝑛). One says that π‘€βˆˆπ΄π‘ for 1<𝑝<∞ if there exists a constant 𝐢 such that for every ball π΅βŠ‚β„π‘›, sup𝐡1||𝐡||ξ€œπ΅1𝑀(π‘₯)𝑑π‘₯ξ‚Όξ‚»||𝐡||ξ€œπ΅π‘€(π‘₯)1βˆ’π‘β€²ξ‚Όπ‘‘π‘₯π‘βˆ’1≀𝐢(2.1) holds, here and below, 1/𝑝+1/𝑝′=1. One says that π‘€βˆˆπ΄1 if there exists a positive constant 𝐢 such that 1||𝐡||ξ€œπ΅π‘€(π‘₯)𝑑π‘₯≀𝐢essinfπ‘₯βˆˆπ΅π‘€(π‘₯).(2.2) The smallest constant appearing in (2.1) or (2.2) is called the 𝐴𝑝 constant of 𝑀, denoted by 𝐢𝑀.

Lemma 2.2. Let 1≀𝑝<∞ and π‘€βˆˆπ΄π‘. Then the following statements are true:(1)(strong doubling) there exists a constant 𝐢 such that π‘€ξ€·π΅π‘˜ξ€Έπ‘€ξ€·π΅π‘—ξ€Έβ‰€πΆ2𝑛𝑝(π‘˜βˆ’π‘—)forπ‘˜>𝑗,(2.3)(2)(centered reverse doubling) for some 𝛿>0, π‘€βˆˆπ‘…π·(𝛿), that is, π‘€ξ€·π΅π‘˜ξ€Έπ‘€ξ€·π΅π‘—ξ€Έβ‰₯𝐢2𝛿(π‘˜βˆ’π‘—)forπ‘˜>𝑗,(2.4)(3)for 1<𝑝<∞, one has π‘€βˆˆπ΄π‘β€‰β€‰β€‰for some 𝑝<𝑝,(4)there exist two constants 𝐢 and 𝛿>0 such that for any measurable set π΅βŠ‚πΈ, 𝑀(𝐡)ξ‚΅||𝐡||𝑀(𝐸)≀𝐢||𝐸||𝛿.(2.5) If 𝑀 satisfies (2.5), one says π‘€βˆˆπ΄βˆž. Obviously, 𝐴∞=⋃1≀𝑝<βˆžπ΄π‘,(5)for all (1/𝑝)+(1/𝑝′)=1, one has 𝑀1βˆ’π‘ξ…žβˆˆπ΄π‘ξ…ž.

Remark 2.3. Note that βˆ«π‘€(𝐸)=𝐸𝑀(π‘₯)𝑑π‘₯ and 𝑀𝑝(𝐸)1/π‘βˆ«=(𝐸𝑀𝑝(π‘₯)𝑑π‘₯)1/𝑝.

Definition 2.4. Let Ξ©βŠ‚β„π‘› be an open set. One says that any π‘“βˆˆπΏ1loc(Ξ©) is in the bounded mean oscillation spaces BMO(Ξ©) if sup𝛾>0,π‘₯βˆˆπ΅π›Ύ(π‘₯)βŠ‚Ξ©1||𝐡𝛾(||ξ€œπ‘₯)𝐡𝛾(π‘₯)|||𝑓(𝑦)βˆ’π‘“π΅π›Ύ(π‘₯)|||π‘‘π‘¦β‰‘β€–π‘“β€–βˆ—<∞,(2.6) where 𝑓𝐡𝛾(π‘₯) is the average over 𝐡𝛾(π‘₯) of 𝑓. Moreover, for any π‘“βˆˆBMO(Ξ©) and π‘Ÿ>0, one sets supπ›Ύβ‰€π‘Ÿ,π‘₯βˆˆπ΅π›Ύ(π‘₯)βŠ‚Ξ©1||𝐡𝛾(||ξ€œπ‘₯)𝐡𝛾(π‘₯)|||𝑓(𝑦)βˆ’π‘“π΅π›Ύ(π‘₯)|||π‘‘π‘¦β‰‘πœ‚(π‘Ÿ).(2.7) One says that any π‘“βˆˆBMO(Ξ©) is in the vanishing mean oscillation spaces VMO(Ξ©) if πœ‚(π‘Ÿ)β†’0 as π‘Ÿβ†’0 and refer to πœ‚(π‘Ÿ) as the modulus of 𝑓.

Remark 2.5. π‘“βˆˆBMO(ℝ𝑛) or VMO(ℝ𝑛) if 𝐡 ranges in the class of balls of ℝ𝑛.

Lemma 2.6 (see [12, Theorem 5]). Let π‘€βˆˆπ΄βˆž. Then the norm of BMO(𝑀) is equivalent to the norm of BMO(ℝ𝑛), where ξ‚»BMO(𝑀)=π‘ŽβˆΆβ€–π‘Žβ€–βˆ—,𝑀1=supξ€œπ‘€(𝐡)𝐡||π‘Ž(π‘₯)βˆ’π‘Žπ΅,𝑀||ξ‚Ό,π‘Žπ‘€(π‘₯)𝑑π‘₯𝐡,𝑀=1ξ€œπ‘€(𝐡)π΅π‘Ž(𝑧)𝑀(𝑧)𝑑𝑧.(2.8)

Definition 2.7. Let πΎβˆΆβ„π‘›β§΅{0}→ℝ. One says that 𝐾(π‘₯) is a constant CalderΓ³n-Zygmund kernel (constant 𝐢-𝑍 kernel) if(i)𝐾∈𝐢∞(ℝ𝑛⧡{0}),(ii)𝐾 is homogeneous of degree βˆ’π‘›,(iii)βˆ«π‘†π‘›βˆ’1𝐾(π‘₯)π‘‘πœŽ=0, π‘†π‘›βˆ’1={π‘₯βˆˆβ„π‘›βˆΆ|π‘₯|=1}.

Definition 2.8. Let Ξ© be an open set of ℝ𝑛 and πΎβˆΆΞ©Γ—{ℝ𝑛⧡{0}}→ℝ. One says that 𝐾(π‘₯,𝑦) is a variable CalderΓ³n-Zygmund kernel (variable 𝐢-𝑍 kernel) on Ξ© if(i)𝐾(π‘₯,β‹…) is a constant 𝐢-𝑍 kernel for a.e. π‘₯∈Ω,(ii)max|𝑗|≀2𝑛‖(πœ•π‘—/πœ•π‘§π‘—)𝐾(π‘₯,𝑧)β€–πΏβˆž(Ξ©Γ—π‘†π‘›βˆ’1)≑𝑀<∞.
Let 𝐾 be a constant or a variable 𝐢-𝑍 kernel on Ξ©. One defines the corresponding 𝐢-𝑍 operator by ξ€œπ‘‡π‘“(π‘₯)=P.V.β„π‘›ξ€œπΎ(π‘₯βˆ’π‘¦)𝑓(𝑦)𝑑𝑦or𝑇𝑓(π‘₯)=P.V.Ω𝐾(π‘₯,π‘₯βˆ’π‘¦)𝑓(𝑦)𝑑𝑦.(2.9)

Lemma 2.9 (see [5, Theorem 3]). Let 1<𝑝<∞,0<π‘ž<∞,𝛿>0. One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).
If 𝐾 is a constant or a variable 𝐢-𝑍 kernel on ℝ𝑛 and 𝑇 is the corresponding 𝐢-𝑍 operator, then there exists a constant 𝐢 such that for all π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛), β€–π‘‡π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛)β‰€πΆβ€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛).(2.10)

From this lemma, by a proof similar to that of Theorem  2.11 in [13], we obtain the following corollary.

Corollary 2.10. Let 1<𝑝<∞,0<π‘ž<∞,𝛿>0, and Ξ© be an open set of ℝn. One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).If 𝐾 is a constant or a variable 𝐢-𝑍 kernel on Ξ©, and 𝑇 is the corresponding 𝐢-𝑍 operator, then there exists a constant 𝐢 such that for all π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(Ξ©), β€–π‘‡π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(Ξ©)β‰€πΆβ€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(Ξ©).(2.11)

3. Weighted Boundedness of Commutators

The aim of this section is to set up the weighted boundedness for the commutators formed by 𝑇 and BMO(ℝ𝑛) functions, where [π‘Ž,𝑇]𝑓(π‘₯)=𝑇(π‘Žπ‘“)(π‘₯)βˆ’π‘Ž(π‘₯)𝑇(𝑓)(π‘₯). This kind of operators is useful in lots of different fields, see, for example, [13] as well as [14], then we consider important in themselves the related below results.

Lemma 3.1 (see [10, Theorem 7.1.6 ]). Let π‘ŽβˆˆBMO(ℝ𝑛). Then for any ball π΅βŠ‚β„π‘›, there exist constants 𝐢1,𝐢2 such that for all 𝛼>0, ||ξ€½||π‘₯βˆˆπ΅βˆΆπ‘Ž(π‘₯)βˆ’π‘Žπ΅||ξ€Ύ||>𝛼≀𝐢1||𝐡||π‘’βˆ’πΆ2𝛼/β€–π‘Žβ€–βˆ—.(3.1) The inequality (3.1) is also called John-Nirenberg inequality.

Theorem 3.2. Let 1<𝑝<∞,0<π‘ž<∞,𝛿>0, and π‘ŽβˆˆBMO(ℝ𝑛). One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).If a linear operator 𝑇 satisfies ||||ξ€œπ‘‡(𝑓)(π‘₯)≀𝐢ℝ𝑛||||𝑓(𝑦)||||π‘₯βˆ’π‘¦π‘›π‘‘π‘¦,π‘₯βˆ‰supp𝑓,(3.2) for any π‘“βˆˆπΏ1loc(ℝ𝑛) and [π‘Ž,𝑇] is bounded on 𝐿𝑝(𝑀), then [π‘Ž,𝑇] is also bounded on πΎπ‘žπ‘(πœ‘,𝑀).

Proof. Let π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛) and π‘ŽβˆˆBMO(ℝ𝑛), we write 𝑓(π‘₯)=βˆžξ“π‘—=βˆ’βˆžπ‘“(π‘₯)πœ’π‘—(π‘₯)=βˆžξ“π‘—=βˆ’βˆžπ‘“π‘—(π‘₯).(3.3) Then, we have β€–[]π‘Ž,π‘‡π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)βŽ›βŽœβŽœβŽβ‰€πΆβˆžξ“π‘˜=βˆ’βˆžπœ‘(π‘˜)π‘žξƒ©π‘˜βˆ’2𝑗=βˆ’βˆžβ€–β€–ξ€·[]π‘“π‘Ž,π‘‡π‘—ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žβŽžβŽŸβŽŸβŽ 1/π‘žβŽ›βŽœβŽœβŽ+πΆβˆžξ“π‘˜=βˆ’βˆžπœ‘(π‘˜)π‘žξƒ©π‘˜+1𝑗=π‘˜βˆ’1β€–β€–ξ€·[]π‘“π‘Ž,π‘‡π‘—ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žβŽžβŽŸβŽŸβŽ 1/π‘žξƒ©+πΆβˆžξ“π‘˜=βˆ’βˆžπœ‘(π‘˜)π‘žξƒ©βˆžξ“π‘—=π‘˜+2β€–β€–ξ€·[]π‘“π‘Ž,π‘‡π‘—ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žξƒͺ1/π‘ž=I+II+III.(3.4) For II, by the 𝐿𝑝(𝑀) boundedness of [π‘Ž,𝑇], we have βŽ›βŽœβŽœβŽIIβ‰€πΆβˆžξ“π‘˜=βˆ’βˆžπœ‘(π‘˜)π‘žξƒ©π‘˜+1𝑗=π‘˜βˆ’1β€–π‘Žβ€–π‘žβˆ—β€–β€–π‘“π‘—πœ’π‘˜β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žβŽžβŽŸβŽŸβŽ 1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.5) For I, note that when π‘₯βˆˆπΈπ‘˜,π‘¦βˆˆπΈπ‘—, and π‘—β‰€π‘˜βˆ’2, |π‘₯βˆ’π‘¦|∼|π‘₯|. So from the condition (3.2), we have ||[]π‘“π‘Ž,𝑇𝑗||ξ€œβ‰€πΆβ„π‘›||||π‘Ž(π‘₯)βˆ’π‘Ž(𝑦)||||π‘₯βˆ’π‘¦π‘›||𝑓𝑗||(𝑦)𝑑𝑦≀𝐢2βˆ’π‘›π‘˜||π‘Ž(π‘₯)βˆ’π‘Žπ΅π‘˜,𝑀||ξ€œβ„π‘›||𝑓𝑗||(𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘˜|||π‘Žπ΅π‘˜,π‘€βˆ’π‘Žπ΅π‘—,𝑀|||ξ€œβ„π‘›||𝑓𝑗||(𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘˜ξ€œβ„π‘›|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀‖‖𝑓𝑗(|||𝑦)𝑑𝑦.(3.6) Thus, β€–β€–ξ€·[]π‘“π‘Ž,π‘‡π‘—ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)≀𝐢2βˆ’π‘›π‘˜β€–β€–ξ€·π‘Ž(π‘₯)βˆ’π‘Žπ΅π‘˜,π‘€ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)ξ€œβ„π‘›||𝑓𝑗(||𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘˜|||π‘Žπ΅π‘˜,π‘€βˆ’π‘Žπ΅π‘—,𝑀|||π‘€ξ€·π΅π‘˜ξ€Έ1/π‘ξ€œβ„π‘›||𝑓𝑗||(𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘˜π‘€ξ€·π΅π‘˜ξ€Έ1/π‘ξ€œβ„π‘›|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀‖‖𝑓𝑗|||=𝐽(𝑦)𝑑𝑦1+𝐽2+𝐽3.(3.7) According to Lemma 2.2, π‘€βˆˆπ΄π‘Ÿ for some π‘Ÿ<π‘Ÿ. By HΓΆlder’s inequality and Lemma 2.6, 𝐽1≀𝐢2βˆ’π‘›π‘˜β€–π‘Žβ€–βˆ—π‘€ξ€·π΅π‘˜ξ€Έ1/𝑝‖‖𝑓𝑗‖‖𝐿𝑝(𝑀)π‘€βˆ’π‘β€²/𝑝𝐡𝑗1/𝑝′=𝐢2βˆ’π‘›π‘˜β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀)π‘€βˆ’π‘β€²/𝑝𝐡𝑗1/𝑝′𝑀𝐡𝑗1/π‘ξƒ©π‘€ξ€·π΅π‘˜ξ€Έπ‘€ξ€·π΅π‘—ξ€Έξƒͺ1/𝑝≀𝐢2βˆ’π‘›π‘˜β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀)||𝐡𝑗||2π‘›π‘Ÿ(π‘˜βˆ’π‘—)/𝑝≀𝐢2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝))2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝))β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀).(3.8) It is easy to see that |π‘Žπ΅π‘˜,π‘€βˆ’π‘Žπ΅π‘—,𝑀|≀𝐢(π‘˜βˆ’π‘—)β€–π‘Žβ€–βˆ—. Therefore, similarly to 𝐽1, we have 𝐽2≀𝐢(π‘˜βˆ’π‘—)2βˆ’π‘›π‘˜β€–π‘Žβ€–βˆ—π‘€ξ€·π΅π‘˜ξ€Έ1/𝑝‖‖𝑓𝑗‖‖𝐿𝑝(𝑀)π‘€βˆ’π‘β€²/𝑝𝐡𝑗1/π‘β€²β‰€πΆπ‘˜2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝))𝑗2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝))β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀).(3.9) Now, we establish the estimate for term 𝐽3, 𝐽3≀𝐢2βˆ’π‘›π‘˜π‘€ξ€·π΅π‘˜ξ€Έ1/𝑝‖‖𝑓𝑗‖‖𝐿𝑝(𝑀)ξƒ©ξ€œπ΅π‘—|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀|||𝑝′𝑀1βˆ’π‘β€²ξƒͺ(𝑦)𝑑𝑦1/𝑝′.(3.10) For the simplicity of analysis, we denote 𝐻 as ξƒ©ξ€œπ΅π‘—|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀|||𝑝′𝑀1βˆ’π‘β€²ξƒͺ(𝑦)𝑑𝑦1/𝑝′.(3.11) By an elementary estimate, we have ξƒ©ξ€œπ»β‰€πΆπ΅π‘—ξ‚ƒ|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀′1βˆ’π‘|||+|||π‘Žπ΅π‘—,𝑀′1βˆ’π‘βˆ’π‘Žπ΅π‘—,𝑀|||𝑝′𝑀1βˆ’π‘β€²ξƒͺ(𝑦)𝑑𝑦1/π‘β€²β‰€πΆβ€–π‘Žβ€–BMO(𝑀′1βˆ’π‘)𝑀1βˆ’π‘β€²ξ€·π΅π‘—ξ€Έ1/𝑝′+|||π‘Žπ΅π‘—,𝑀′1βˆ’π‘βˆ’π‘Žπ΅π‘—,𝑀|||𝑀1βˆ’π‘β€²ξ€·π΅π‘—ξ€Έ1/𝑝′.(3.12) Note that |||π‘Žπ΅π‘—,𝑀′1βˆ’π‘βˆ’π‘Žπ΅π‘—,𝑀|||≀|||π‘Žπ΅π‘—,𝑀′1βˆ’π‘βˆ’π‘Žπ΅π‘—|||+|||π‘Žπ΅π‘—βˆ’π‘Žπ΅π‘—,𝑀|||=𝐽31+𝐽32.(3.13) Combining (2.5) with (3.1), 𝐽32=1π‘€ξ€·π΅π‘—ξ€Έξ€œπ΅π‘—|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—|||=1𝑀(𝑦)π‘‘π‘¦π‘€ξ€·π΅π‘—ξ€Έξ€œβˆž0𝑀π‘₯βˆˆπ΅π‘—βˆΆ|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—|||ξ€œ>π›Όξ‚‡ξ‚π‘‘π›Όβ‰€πΆβˆž0π‘’βˆ’πΆ2𝛼𝛿/β€–π‘Žβ€–βˆ—π‘‘π›Όβ‰€πΆ.(3.14) In the same manner, we can see that 𝐽31≀𝐢.(3.15) By Lemma 2.6, we get 𝐽3≀𝐢2βˆ’π‘›π‘˜β€–π‘Žβ€–βˆ—π‘€ξ€·π΅π‘˜ξ€Έ1/𝑝‖‖𝑓𝑗‖‖𝐿𝑝(𝑀)π‘€βˆ’π‘β€²/𝑝𝐡𝑗1/𝑝′≀𝐢2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝))2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝))β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀).(3.16) Using hypotheses πœ‘βˆˆπ·(s,𝑑) and the estimates of 𝐽1,𝐽2, and 𝐽3, we obtain the following inequality: Iβ‰€πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘˜2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝)+𝑑)π‘žβ‹…ξƒ©ξ“π‘—β‰€π‘˜βˆ’22𝑛𝑗(1βˆ’(π‘Ÿ/𝑝)βˆ’(𝑑/𝑛))β€–β€–πœ‘(𝑗)π‘“πœ’π‘—β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žξƒͺ1/π‘ž+πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘˜ξ‚€π‘˜2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝)+𝑑)ξ‚π‘žβ‹…ξƒ©ξ“π‘—β‰€π‘˜βˆ’2𝑗2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝)βˆ’(𝑑/𝑛))β€–β€–πœ‘(𝑗)π‘“πœ’π‘—β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žξƒͺ1/π‘ž=I1+I2.(3.17) When π‘žβ‰€1, we have I1β‰€πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘—2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝)βˆ’(𝑑/𝑛))π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)β‹…βˆžξ“π‘˜=𝑗+22π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝)+𝑑)π‘žξƒͺ1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀),(3.18) because βˆ’π‘›+π‘›π‘Ÿ/𝑝+𝑑≀0, that is, βˆ’π‘›+π‘›π‘Ÿ/𝑝+𝑑<0.
When π‘ž>1, we take πœ€>0 such that βˆ’π‘›+π‘›π‘Ÿ/𝑝+𝑑+π‘›πœ€<0. Then I1β‰€πΆβ€–π‘Žβ€–βˆ—βŽ‘βŽ’βŽ’βŽ£ξ“π‘˜2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝)+𝑑)π‘žβ‹…βŽ›βŽœβŽœβŽπ‘˜βˆ’2𝑗=βˆ’βˆž2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝)βˆ’(𝑑/𝑛)βˆ’πœ€)π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)β‹…ξƒ©π‘˜βˆ’2𝑗=βˆ’βˆž2π‘›πœ€π‘žβ€²π‘—ξƒͺπ‘ž/π‘žβ€²βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—ξƒ¬ξ“π‘˜2π‘˜(βˆ’π‘›+(π‘Ÿπ‘›/𝑝)+𝑑+π‘›πœ€)π‘žβ‹…ξƒ©π‘˜βˆ’2𝑗=βˆ’βˆž2𝑛𝑗(1βˆ’(π‘Ÿ/𝑝)βˆ’(𝑑/𝑛)βˆ’πœ€)π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)ξƒͺξƒ­1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.19) Similar to I1, we have I2β‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.20) Finally we estimate III. The proof of this part is analogue to I, so we just give out an outline. Note that 𝑗β‰₯π‘˜+2 and π‘₯βˆˆπΈπ‘˜,π‘¦βˆˆπΈπ‘—, |π‘₯βˆ’π‘¦|∼|𝑦|. So from the condition (3.2), we have β€–β€–ξ€·[]π‘“π‘Ž,π‘‡π‘—ξ€Έπœ’π‘˜β€–β€–πΏπ‘(𝑀)≀𝐢2βˆ’π‘›π‘—β€–β€–π‘Ž(π‘₯)βˆ’π‘Žπ΅π‘˜,𝑀‖‖𝐿𝑝(𝑀)ξ€œβ„π‘›||𝑓𝑗(||𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘—|||π‘Žπ΅π‘˜,π‘€βˆ’π‘Žπ΅π‘—,𝑀|||π‘€ξ€·π΅π‘˜ξ€Έ1/π‘ξ€œβ„π‘›||𝑓𝑗||(𝑦)𝑑𝑦+𝐢2βˆ’π‘›π‘—π‘€ξ€·π΅π‘˜ξ€Έ1/π‘ξ€œβ„π‘›|||π‘Ž(𝑦)βˆ’π‘Žπ΅π‘—,𝑀|||||𝑓𝑗||=𝐽(𝑦)π‘‘π‘¦ξ…ž1+π½ξ…ž2+π½ξ…ž3.(3.21) Using hypotheses (iii) for 𝑀 in place of strong doubling, π½ξ…ž1≀𝐢2βˆ’π‘—π‘›β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀)π‘€βˆ’π‘β€²/𝑝𝐡𝑗1/𝑝′𝑀𝐡𝑗1/π‘ξƒ©π‘€ξ€·π΅π‘˜ξ€Έπ‘€ξ€·π΅π‘—ξ€Έξƒͺ1/𝑝≀𝐢2π‘˜π›Ώ/𝑝2βˆ’π‘—π›Ώ/π‘β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀).(3.22) Similarly, π½ξ…ž2β‰€πΆπ‘˜2π‘˜π›Ώ/𝑝𝑗2βˆ’π‘—π›Ώ/π‘β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀),π½ξ…ž3≀𝐢2π‘˜π›Ώ/𝑝2βˆ’π‘—π›Ώ/π‘β€–π‘Žβ€–βˆ—β€–β€–π‘“π‘—β€–β€–πΏπ‘(𝑀).(3.23) Using hypotheses (i) for 𝑀, that is, πœ‘βˆˆπ·(𝑠,𝑑), we obtain the following inequality: IIIβ‰€πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘˜2π‘˜(𝑠+𝛿/𝑝)π‘žβ‹…ξƒ©ξ“π‘—β‰₯π‘˜+22𝑗(βˆ’π‘ βˆ’π›Ώ/𝑝)β€–β€–πœ‘(𝑗)π‘“πœ’π‘—β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žξƒͺ1/π‘ž+πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘˜ξ€·π‘˜2(𝑠+𝛿/𝑝)ξ€Έπ‘žβ‹…ξƒ©ξ“π‘—β‰₯π‘˜+2𝑗2𝑗(βˆ’π‘ βˆ’π›Ώ/𝑝)β€–β€–πœ‘(𝑗)π‘“πœ’π‘—β€–β€–πΏπ‘(𝑀)ξƒͺπ‘žξƒͺ1/π‘ž=III1+III2.(3.24) According to 𝑠+𝛿/𝑝>0, when π‘žβ‰€1, III1β‰€πΆβ€–π‘Žβ€–βˆ—ξƒ©ξ“π‘—2𝑗(βˆ’π‘ βˆ’π›Ώ/𝑝)π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)β‹…π‘—βˆ’2ξ“π‘˜=βˆ’βˆž2π‘˜(𝑠+𝛿/𝑝)π‘žξƒͺ1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.25) When π‘ž>1, we take πœ€>0 such that 𝑠+𝛿/π‘βˆ’πœ€>0. Then III1β‰€πΆβ€–π‘Žβ€–βˆ—βŽ‘βŽ’βŽ’βŽ£ξ“π‘˜2π‘˜(𝑠+𝛿/𝑝)π‘žβ‹…βŽ›βŽœβŽœβŽξ“π‘—β‰₯π‘˜+22𝑗(βˆ’π‘ βˆ’π›Ώ/𝑝+πœ€)π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)⋅𝑗β‰₯π‘˜+22βˆ’π‘—π‘žβ€²πœ€ξƒͺπ‘ž/π‘žβ€²βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—ξƒ¬ξ“π‘˜2π‘˜(𝑠+𝛿/π‘βˆ’πœ€)π‘žβ‹…ξƒ©ξ“π‘—β‰₯π‘˜+22𝑗(βˆ’π‘ βˆ’π›Ώ/𝑝+πœ€)π‘žπœ‘(𝑗)π‘žβ€–β€–π‘“πœ’π‘—β€–β€–π‘žπΏπ‘(𝑀)ξƒͺξƒ­1/π‘žβ‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.26) Similar to III1, we have III2β‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀).(3.27) This finishes the proof of Theorem 3.2.

The condition (3.2) in Theorem 3.2 can be satisfied by many operators such as Bochner-Riesz operators at the critical index, Ricci-Stein’s oscillatory singular integrals, Fefferman’s multiplier, and the 𝐢-𝑍 operators. From this theorem and Theorem 2.7 and  2.10 in [13], we easily deduce the following corollary.

Corollary 3.3. Let 1<𝑝<∞,0<π‘ž<∞,𝛿>0, and π‘ŽβˆˆBMO(ℝ𝑛). One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).If 𝐾 is a constant or a variable 𝐢-𝑍 kernel on ℝ𝑛 and 𝑇 is the corresponding 𝐢-𝑍 operator, then there exists a constant such that for all π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛), β€–[]π‘“β€–π‘Ž,π‘‡πΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛)β‰€πΆβ€–π‘Žβ€–βˆ—β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(ℝ𝑛).(3.28)

From this and the extension theorem of BMO(Ξ©)-functions in [15], by a procedure similar to Theorem  2.11 in [13] and Theorem  2.2 in [16], we can obtain the following corollary.

Corollary 3.4. Let 1<𝑝<∞,0<π‘ž<∞, and 𝛿>0. Suppose that Ξ© is an open set of ℝ𝑛 and π‘ŽβˆˆVMO(Ξ©). One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).If 𝐾 is a variable 𝐢-𝑍 kernel on Ξ© and 𝑇 is the corresponding 𝐢-𝑍 operator, then for any πœ€>0, there exists a positive number 𝜌0=𝜌0(πœ€,πœ‚) such that for any ball 𝐡𝑅 with the radius π‘…βˆˆ(0,𝜌0), 𝐡RβŠ†Ξ© and all π‘“βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(𝐡𝑅), β€–[]π‘“β€–π‘Ž,π‘‡πΎπ‘žπ‘(πœ‘,𝑀)(𝐡𝑅)β‰€πΆπœ€β€–π‘“β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡𝑅),(3.29) where 𝐢=𝐢(𝑛,𝑝,π‘ž,π‘Ž,πœ‘,𝑀) is independent of πœ€,𝑓, and 𝑅.

4. Interior Estimate of Elliptic Equation

In this section, we will establish the interior regularity of the strong solutions to elliptic equations in weighted Herz spaces by applying the estimates about singular integral operators and linear commutators obtained in the above section.

Suppose that 𝑛β‰₯3 and Ξ© is an open set of ℝ𝑛. We are concerned with the nondivergence form elliptic equations ℒ𝑒(π‘₯)=βˆ’π‘›ξ“π‘–,𝑗=1π‘Žπ‘–,𝑗(π‘₯)𝑒π‘₯𝑖π‘₯𝑗=𝑓(π‘₯),a.e.inΞ©,(4.1) whose coefficients π‘Žπ‘–π‘— are assumed such that π‘Žπ‘–π‘—(π‘₯)=π‘Žπ‘—π‘–π‘Ž(π‘₯),a.e.π‘₯∈Ω,𝑖,𝑗=1,2,…,𝑛,π‘–π‘—βˆˆπΏβˆžπœ‡(Ξ©)∩VMO(Ξ©),βˆ’1||πœ‰||2≀𝑛𝑖,𝑗=1π‘Žπ‘–,𝑗(π‘₯)πœ‰π‘–πœ‰π‘—||πœ‰||β‰€πœ‡2,βˆƒπœ‡>0,a.e.π‘₯∈Ω,πœ‰βˆˆβ„π‘›.(4.2) Let 1Ξ“(π‘₯,𝑑)=(π‘›βˆ’2)πœ”π‘›ξ€·detπ‘Žπ‘–π‘—ξ€Έ(π‘₯)1/2𝑛𝑖,𝑗=1𝐴𝑖𝑗(π‘₯)𝑑𝑖𝑑𝑗ξƒͺ(2βˆ’π‘›)/2,Ξ“π‘–πœ•(π‘₯,𝑑)=πœ•π‘‘π‘–Ξ“(π‘₯,𝑑),Ξ“π‘–π‘—πœ•(π‘₯,𝑑)=2πœ•π‘‘π‘–πœ•π‘‘π‘—Ξ“(π‘₯,𝑑),(4.3) for a.e. π‘₯∈𝐡 and βˆ€π‘‘βˆˆβ„π‘›β§΅{0}, where the 𝐴𝑖𝑗 are the entries of the inverse of the matrix (π‘Žπ‘–π‘—)𝑖,𝑗=1,2,…,𝑛.

From [13], we deduce the interior representation, that is, if π‘’βˆˆπ‘Š02,𝑝, 𝑒π‘₯𝑖π‘₯π‘—ξ€œ(π‘₯)=P.V.𝐡Γ𝑖𝑗(π‘₯,π‘₯βˆ’π‘¦)π‘›ξ“β„Ž,𝑙=1ξ€·π‘Žβ„Žπ‘™(π‘₯)βˆ’π‘Žβ„Žπ‘™ξ€Έπ‘’(𝑦)π‘₯β„Žπ‘₯π‘™ξƒ­ξ€œ(𝑦)+ℒ𝑒(𝑦)𝑑𝑦+ℒ𝑒(π‘₯)|𝑑|=1Γ𝑖(π‘₯,𝑑)π‘‘π‘—π‘‘πœŽπ‘‘,a.e.forπ‘₯βˆˆπ΅βŠ‚Ξ©,(4.4) where 𝐡 is a ball in Ξ©. We also set 𝑀≑max𝑖,𝑗=1,…,𝑛max||𝛽||≀2π‘›β€–β€–β€–πœ•π›½πœ•π‘‘π›½Ξ“π‘–π‘—β€–β€–β€–(π‘₯,𝑑)𝐿∞(Ξ©Γ—π‘†π‘›βˆ’1)<∞.(4.5)

Theorem 4.1. Let 1<𝑝<∞,0<π‘ž<∞, and 𝛿>0. Suppose that Ξ© is an open set of ℝ𝑛 and π‘Žπ‘–π‘— satisfies (4.2) for 𝑖,𝑗=1,2,…,𝑛. One assumes that(i)πœ‘βˆˆπ·(𝑠,𝑑), where βˆ’(𝛿/𝑝)<𝑠≀𝑑<𝑛(1βˆ’(1/𝑝)),(ii)π‘€βˆˆπ΄π‘Ÿ, where π‘Ÿ=min(𝑝,𝑝(1βˆ’(𝑑/𝑛))),(iii)π‘€βˆˆπ‘…π·(𝛿).Then there exists a constant 𝐢 such that for all balls π΅βŠ‚Ξ© and π‘’βˆˆπ‘Š02,𝑝, One has 𝑒π‘₯𝑖π‘₯π‘—βˆˆπΎπ‘žπ‘(πœ‘,𝑀)(𝐡) and ‖‖𝑒π‘₯𝑖π‘₯π‘—β€–β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡)β‰€πΆβ€–β„’π‘’β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡).(4.6)

Proof. It is well known that Γ𝑖𝑗(π‘₯,𝑑) are 𝐢-𝑍 kernels in the 𝑑 variable. Thus, using the technology of [13, 16] and the Corollaries 2.10 and 3.4, we deduce that, for any πœ€>0, ‖‖𝑒π‘₯𝑖π‘₯𝑗‖‖𝐾q𝑝(πœ‘,𝑀)(𝐡)β€–β€–π‘’β‰€πΆπœ€π‘₯𝑖π‘₯π‘—β€–β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡)+πΆβ€–β„’π‘’β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡).(4.7) Choosing πœ€ to be small enough (e.g.,πΆπœ€<1), we obtain ‖‖𝑒π‘₯𝑖π‘₯π‘—β€–β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡)≀𝐢(1βˆ’πΆπœ€)β€–β„’π‘’β€–πΎπ‘žπ‘(πœ‘,𝑀)(𝐡).(4.8) This finishes the proof of Theorem 4.1.

Acknowledgment

This research is supported by the NSF of China (no. 11161044).

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