Abstract

It is proved that, for the nondivergence form elliptic equations 𝑛𝑖,𝑗=1𝑎𝑖𝑗𝑢𝑥𝑖𝑥𝑗=𝑓, if 𝑓 belongs to the weighted Herz spaces 𝐾𝑞𝑝(𝜑,𝑤), then 𝑢𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤), where 𝑢 is the 𝑊2,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 𝐾𝑞𝑝(𝜑,𝑤).

1. Introduction

For a sequence 𝜑={𝜑(𝑘)},𝜑(𝑘)>0, we suppose that 𝜑 satisfies doubling condition of order (𝑠,𝑡) and write 𝜑𝐷(𝑠,𝑡) if there exists 𝐶1 such that 𝐶12𝑠(𝑘𝑗)𝜑(𝑘)𝜑(𝑗)𝐶2𝑡(𝑘𝑗)for𝑘>𝑗.(1.1) Let 𝐵𝑘=𝐵(0,2𝑘)={𝑥𝑛|𝑥|2𝑘}, 𝐸𝑘=𝐵𝑘𝐵𝑘1 for 𝑘, and 𝜒𝑘=𝜒𝐸𝑘 be the characteristic function of the set 𝐸𝑘 for 𝑘. Suppose that 𝑤 is a weight function on 𝑛. For 1<𝑝<, 0<𝑞<, the weighted Herz space is defined by 𝐾𝑞𝑝(𝜑,𝑤)(𝑛)=𝑓𝑓isameasurablefunctionon𝑛,𝑓𝐾𝑞𝑝(𝜑,𝑤),<(1.2) where 𝑓𝐾𝑞𝑝(𝜑,𝑤)=𝑘=𝜑(𝑘)𝑞𝑓𝜒𝑘𝑞𝐿𝑝(𝑤)1/𝑞,𝑓𝐿𝑝(𝑤)=𝑛||||𝑓(𝑥)𝑝𝑤(𝑥)𝑑𝑥1/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 Calderón-Zygmund singular integral operators that will appear in the representation formula of the 𝑢𝑥𝑖𝑥𝑗 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 𝑛𝑖,𝑗=1𝑎𝑖𝑗𝑢𝑥𝑖𝑥𝑗=𝑓, and we prove that if 𝑓𝐾𝑞𝑝(𝜑,𝑤), then 𝑢𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤), where 𝑢 is the 𝑊2,𝑝-solution of the equations.

Throughout this paper, unless otherwise indicated, 𝐶 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(1𝑝<)). Let 𝑤(𝑥)0 and 𝑤(𝑥)𝐿1loc(𝑛). One says that 𝑤𝐴𝑝 for 1<𝑝< if there exists a constant 𝐶 such that for every ball 𝐵𝑛, sup𝐵1||𝐵||𝐵1𝑤(𝑥)𝑑𝑥||𝐵||𝐵𝑤(𝑥)1𝑝𝑑𝑥𝑝1𝐶(2.1) holds, here and below, 1/𝑝+1/𝑝=1. One says that 𝑤𝐴1 if there exists a positive constant 𝐶 such that 1||𝐵||𝐵𝑤(𝑥)𝑑𝑥𝐶essinf𝑥𝐵𝑤(𝑥).(2.2) The smallest constant appearing in (2.1) or (2.2) is called the 𝐴𝑝 constant of 𝑤, denoted by 𝐶𝑤.

Lemma 2.2. Let 1𝑝< and 𝑤𝐴𝑝. Then the following statements are true:(1)(strong doubling) there exists a constant 𝐶 such that 𝑤𝐵𝑘𝑤𝐵𝑗𝐶2𝑛𝑝(𝑘𝑗)for𝑘>𝑗,(2.3)(2)(centered reverse doubling) for some 𝛿>0, 𝑤𝑅𝐷(𝛿), that is, 𝑤𝐵𝑘𝑤𝐵𝑗𝐶2𝛿(𝑘𝑗)for𝑘>𝑗,(2.4)(3)for 1<𝑝<, one has 𝑤𝐴𝑝   for some 𝑝<𝑝,(4)there exist two constants 𝐶 and 𝛿>0 such that for any measurable set 𝐵𝐸, 𝑤(𝐵)||𝐵||𝑤(𝐸)𝐶||𝐸||𝛿.(2.5) If 𝑤 satisfies (2.5), one says 𝑤𝐴. Obviously, 𝐴=1𝑝<𝐴𝑝,(5)for all (1/𝑝)+(1/𝑝)=1, one has 𝑤1𝑝𝐴𝑝.

Remark 2.3. Note that 𝑤(𝐸)=𝐸𝑤(𝑥)𝑑𝑥 and 𝑤𝑝(𝐸)1/𝑝=(𝐸𝑤𝑝(𝑥)𝑑𝑥)1/𝑝.

Definition 2.4. Let Ω𝑛 be an open set. One says that any 𝑓𝐿1loc(Ω) is in the bounded mean oscillation spaces BMO(Ω) if sup𝛾>0,𝑥𝐵𝛾(𝑥)Ω1||𝐵𝛾(||𝑥)𝐵𝛾(𝑥)|||𝑓(𝑦)𝑓𝐵𝛾(𝑥)|||𝑑𝑦𝑓<,(2.6) where 𝑓𝐵𝛾(𝑥) is the average over 𝐵𝛾(𝑥) of 𝑓. Moreover, for any 𝑓BMO(Ω) and 𝑟>0, one sets sup𝛾𝑟,𝑥𝐵𝛾(𝑥)Ω1||𝐵𝛾(||𝑥)𝐵𝛾(𝑥)|||𝑓(𝑦)𝑓𝐵𝛾(𝑥)|||𝑑𝑦𝜂(𝑟).(2.7) One says that any 𝑓BMO(Ω) is in the vanishing mean oscillation spaces VMO(Ω) if 𝜂(𝑟)0 as 𝑟0 and refer to 𝜂(𝑟) as the modulus of 𝑓.

Remark 2.5. 𝑓BMO(𝑛) or VMO(𝑛) if 𝐵 ranges in the class of balls of 𝑛.

Lemma 2.6 (see [12, Theorem 5]). Let 𝑤𝐴. Then the norm of BMO(𝑤) is equivalent to the norm of BMO(𝑛), where BMO(𝑤)=𝑎𝑎,𝑤1=sup𝑤(𝐵)𝐵||𝑎(𝑥)𝑎𝐵,𝑤||,𝑎𝑤(𝑥)𝑑𝑥𝐵,𝑤=1𝑤(𝐵)𝐵𝑎(𝑧)𝑤(𝑧)𝑑𝑧.(2.8)

Definition 2.7. Let 𝐾𝑛{0}. One says that 𝐾(𝑥) is a constant Calderón-Zygmund kernel (constant 𝐶-𝑍 kernel) if(i)𝐾𝐶(𝑛{0}),(ii)𝐾 is homogeneous of degree 𝑛,(iii)𝑆𝑛1𝐾(𝑥)𝑑𝜎=0, 𝑆𝑛1={𝑥𝑛|𝑥|=1}.

Definition 2.8. Let Ω be an open set of 𝑛 and 𝐾Ω×{𝑛{0}}. One says that 𝐾(𝑥,𝑦) is a variable Calderón-Zygmund kernel (variable 𝐶-𝑍 kernel) on Ω if(i)𝐾(𝑥,) is a constant 𝐶-𝑍 kernel for a.e. 𝑥Ω,(ii)max|𝑗|2𝑛(𝜕𝑗/𝜕𝑧𝑗)𝐾(𝑥,𝑧)𝐿(Ω×𝑆𝑛1)𝑀<.
Let 𝐾 be a constant or a variable 𝐶-𝑍 kernel on Ω. One defines the corresponding 𝐶-𝑍 operator by 𝑇𝑓(𝑥)=P.V.𝑛𝐾(𝑥𝑦)𝑓(𝑦)𝑑𝑦or𝑇𝑓(𝑥)=P.V.Ω𝐾(𝑥,𝑥𝑦)𝑓(𝑦)𝑑𝑦.(2.9)

Lemma 2.9 (see [5, Theorem 3]). Let 1<𝑝<,0<𝑞<,𝛿>0. One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).
If 𝐾 is a constant or a variable 𝐶-𝑍 kernel on 𝑛 and 𝑇 is the corresponding 𝐶-𝑍 operator, then there exists a constant 𝐶 such that for all 𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛), 𝑇𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛)𝐶𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛).(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<𝑞<,𝛿>0, and Ω be an open set of n. One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).If 𝐾 is a constant or a variable 𝐶-𝑍 kernel on Ω, and 𝑇 is the corresponding 𝐶-𝑍 operator, then there exists a constant 𝐶 such that for all 𝑓𝐾𝑞𝑝(𝜑,𝑤)(Ω), 𝑇𝑓𝐾𝑞𝑝(𝜑,𝑤)(Ω)𝐶𝑓𝐾𝑞𝑝(𝜑,𝑤)(Ω).(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(𝑛) functions, where [𝑎,𝑇]𝑓(𝑥)=𝑇(𝑎𝑓)(𝑥)𝑎(𝑥)𝑇(𝑓)(𝑥). 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 𝑎BMO(𝑛). Then for any ball 𝐵𝑛, there exist constants 𝐶1,𝐶2 such that for all 𝛼>0, ||||𝑥𝐵𝑎(𝑥)𝑎𝐵||||>𝛼𝐶1||𝐵||𝑒𝐶2𝛼/𝑎.(3.1) The inequality (3.1) is also called John-Nirenberg inequality.

Theorem 3.2. Let 1<𝑝<,0<𝑞<,𝛿>0, and 𝑎BMO(𝑛). One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).If a linear operator 𝑇 satisfies ||||𝑇(𝑓)(𝑥)𝐶𝑛||||𝑓(𝑦)||||𝑥𝑦𝑛𝑑𝑦,𝑥supp𝑓,(3.2) for any 𝑓𝐿1loc(𝑛) and [𝑎,𝑇] is bounded on 𝐿𝑝(𝑤), then [𝑎,𝑇] is also bounded on 𝐾𝑞𝑝(𝜑,𝑤).

Proof. Let 𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛) and 𝑎BMO(𝑛), we write 𝑓(𝑥)=𝑗=𝑓(𝑥)𝜒𝑗(𝑥)=𝑗=𝑓𝑗(𝑥).(3.3) Then, we have []𝑎,𝑇𝑓𝐾𝑞𝑝(𝜑,𝑤)𝐶𝑘=𝜑(𝑘)𝑞𝑘2𝑗=[]𝑓𝑎,𝑇𝑗𝜒𝑘𝐿𝑝(𝑤)𝑞1/𝑞+𝐶𝑘=𝜑(𝑘)𝑞𝑘+1𝑗=𝑘1[]𝑓𝑎,𝑇𝑗𝜒𝑘𝐿𝑝(𝑤)𝑞1/𝑞+𝐶𝑘=𝜑(𝑘)𝑞𝑗=𝑘+2[]𝑓𝑎,𝑇𝑗𝜒𝑘𝐿𝑝(𝑤)𝑞1/𝑞=I+II+III.(3.4) For II, by the 𝐿𝑝(𝑤) boundedness of [𝑎,𝑇], we have II𝐶𝑘=𝜑(𝑘)𝑞𝑘+1𝑗=𝑘1𝑎𝑞𝑓𝑗𝜒𝑘𝐿𝑝(𝑤)𝑞1/𝑞𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(3.5) For I, note that when 𝑥𝐸𝑘,𝑦𝐸𝑗, and 𝑗𝑘2, |𝑥𝑦||𝑥|. So from the condition (3.2), we have ||[]𝑓𝑎,𝑇𝑗||𝐶𝑛||||𝑎(𝑥)𝑎(𝑦)||||𝑥𝑦𝑛||𝑓𝑗||(𝑦)𝑑𝑦𝐶2𝑛𝑘||𝑎(𝑥)𝑎𝐵𝑘,𝑤||𝑛||𝑓𝑗||(𝑦)𝑑𝑦+𝐶2𝑛𝑘|||𝑎𝐵𝑘,𝑤𝑎𝐵𝑗,𝑤|||𝑛||𝑓𝑗||(𝑦)𝑑𝑦+𝐶2𝑛𝑘𝑛|||𝑎(𝑦)𝑎𝐵𝑗,𝑤𝑓𝑗(|||𝑦)𝑑𝑦.(3.6) Thus, []𝑓𝑎,𝑇𝑗𝜒𝑘𝐿𝑝(𝑤)𝐶2𝑛𝑘𝑎(𝑥)𝑎𝐵𝑘,𝑤𝜒𝑘𝐿𝑝(𝑤)𝑛||𝑓𝑗(||𝑦)𝑑𝑦+𝐶2𝑛𝑘|||𝑎𝐵𝑘,𝑤𝑎𝐵𝑗,𝑤|||𝑤𝐵𝑘1/𝑝𝑛||𝑓𝑗||(𝑦)𝑑𝑦+𝐶2𝑛𝑘𝑤𝐵𝑘1/𝑝𝑛|||𝑎(𝑦)𝑎𝐵𝑗,𝑤𝑓𝑗|||=𝐽(𝑦)𝑑𝑦1+𝐽2+𝐽3.(3.7) According to Lemma 2.2, 𝑤𝐴𝑟 for some 𝑟<𝑟. By Hölder’s inequality and Lemma 2.6, 𝐽1𝐶2𝑛𝑘𝑎𝑤𝐵𝑘1/𝑝𝑓𝑗𝐿𝑝(𝑤)𝑤𝑝/𝑝𝐵𝑗1/𝑝=𝐶2𝑛𝑘𝑎𝑓𝑗𝐿𝑝(𝑤)𝑤𝑝/𝑝𝐵𝑗1/𝑝𝑤𝐵𝑗1/𝑝𝑤𝐵𝑘𝑤𝐵𝑗1/𝑝𝐶2𝑛𝑘𝑎𝑓𝑗𝐿𝑝(𝑤)||𝐵𝑗||2𝑛𝑟(𝑘𝑗)/𝑝𝐶2𝑘(𝑛+(𝑟𝑛/𝑝))2𝑛𝑗(1(𝑟/𝑝))𝑎𝑓𝑗𝐿𝑝(𝑤).(3.8) It is easy to see that |𝑎𝐵𝑘,𝑤𝑎𝐵𝑗,𝑤|𝐶(𝑘𝑗)𝑎. Therefore, similarly to 𝐽1, we have 𝐽2𝐶(𝑘𝑗)2𝑛𝑘𝑎𝑤𝐵𝑘1/𝑝𝑓𝑗𝐿𝑝(𝑤)𝑤𝑝/𝑝𝐵𝑗1/𝑝𝐶𝑘2𝑘(𝑛+(𝑟𝑛/𝑝))𝑗2𝑛𝑗(1(𝑟/𝑝))𝑎𝑓𝑗𝐿𝑝(𝑤).(3.9) Now, we establish the estimate for term 𝐽3, 𝐽3𝐶2𝑛𝑘𝑤𝐵𝑘1/𝑝𝑓𝑗𝐿𝑝(𝑤)𝐵𝑗|||𝑎(𝑦)𝑎𝐵𝑗,𝑤|||𝑝𝑤1𝑝(𝑦)𝑑𝑦1/𝑝.(3.10) For the simplicity of analysis, we denote 𝐻 as 𝐵𝑗|||𝑎(𝑦)𝑎𝐵𝑗,𝑤|||𝑝𝑤1𝑝(𝑦)𝑑𝑦1/𝑝.(3.11) By an elementary estimate, we have 𝐻𝐶𝐵𝑗|||𝑎(𝑦)𝑎𝐵𝑗,𝑤1𝑝|||+|||𝑎𝐵𝑗,𝑤1𝑝𝑎𝐵𝑗,𝑤|||𝑝𝑤1𝑝(𝑦)𝑑𝑦1/𝑝𝐶𝑎BMO(𝑤1𝑝)𝑤1𝑝𝐵𝑗1/𝑝+|||𝑎𝐵𝑗,𝑤1𝑝𝑎𝐵𝑗,𝑤|||𝑤1𝑝𝐵𝑗1/𝑝.(3.12) Note that |||𝑎𝐵𝑗,𝑤1𝑝𝑎𝐵𝑗,𝑤||||||𝑎𝐵𝑗,𝑤1𝑝𝑎𝐵𝑗|||+|||𝑎𝐵𝑗𝑎𝐵𝑗,𝑤|||=𝐽31+𝐽32.(3.13) Combining (2.5) with (3.1), 𝐽32=1𝑤𝐵𝑗𝐵𝑗|||𝑎(𝑦)𝑎𝐵𝑗|||=1𝑤(𝑦)𝑑𝑦𝑤𝐵𝑗0𝑤𝑥𝐵𝑗|||𝑎(𝑦)𝑎𝐵𝑗|||>𝛼𝑑𝛼𝐶0𝑒𝐶2𝛼𝛿/𝑎𝑑𝛼𝐶.(3.14) In the same manner, we can see that 𝐽31𝐶.(3.15) By Lemma 2.6, we get 𝐽3𝐶2𝑛𝑘𝑎𝑤𝐵𝑘1/𝑝𝑓𝑗𝐿𝑝(𝑤)𝑤𝑝/𝑝𝐵𝑗1/𝑝𝐶2𝑘(𝑛+(𝑟𝑛/𝑝))2𝑛𝑗(1(𝑟/𝑝))𝑎𝑓𝑗𝐿𝑝(𝑤).(3.16) Using hypotheses 𝜑𝐷(s,𝑡) and the estimates of 𝐽1,𝐽2, and 𝐽3, we obtain the following inequality: I𝐶𝑎𝑘2𝑘(𝑛+(𝑟𝑛/𝑝)+𝑡)𝑞𝑗𝑘22𝑛𝑗(1(𝑟/𝑝)(𝑡/𝑛))𝜑(𝑗)𝑓𝜒𝑗𝐿𝑝(𝑤)𝑞1/𝑞+𝐶𝑎𝑘𝑘2𝑘(𝑛+(𝑟𝑛/𝑝)+𝑡)𝑞𝑗𝑘2𝑗2𝑛𝑗(1(𝑟/𝑝)(𝑡/𝑛))𝜑(𝑗)𝑓𝜒𝑗𝐿𝑝(𝑤)𝑞1/𝑞=I1+I2.(3.17) When 𝑞1, we have I1𝐶𝑎𝑗2𝑛𝑗(1(𝑟/𝑝)(𝑡/𝑛))𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)𝑘=𝑗+22𝑘(𝑛+(𝑟𝑛/𝑝)+𝑡)𝑞1/𝑞𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤),(3.18) because 𝑛+𝑛𝑟/𝑝+𝑡0, that is, 𝑛+𝑛𝑟/𝑝+𝑡<0.
When 𝑞>1, we take 𝜀>0 such that 𝑛+𝑛𝑟/𝑝+𝑡+𝑛𝜀<0. Then I1𝐶𝑎𝑘2𝑘(𝑛+(𝑟𝑛/𝑝)+𝑡)𝑞𝑘2𝑗=2𝑛𝑗(1(𝑟/𝑝)(𝑡/𝑛)𝜀)𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)𝑘2𝑗=2𝑛𝜀𝑞𝑗𝑞/𝑞1/𝑞𝐶𝑎𝑘2𝑘(𝑛+(𝑟𝑛/𝑝)+𝑡+𝑛𝜀)𝑞𝑘2𝑗=2𝑛𝑗(1(𝑟/𝑝)(𝑡/𝑛)𝜀)𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)1/𝑞𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(3.19) Similar to I1, we have I2𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(3.20) Finally we estimate III. The proof of this part is analogue to I, so we just give out an outline. Note that 𝑗𝑘+2 and 𝑥𝐸𝑘,𝑦𝐸𝑗, |𝑥𝑦||𝑦|. So from the condition (3.2), we have []𝑓𝑎,𝑇𝑗𝜒𝑘𝐿𝑝(𝑤)𝐶2𝑛𝑗𝑎(𝑥)𝑎𝐵𝑘,𝑤𝐿𝑝(𝑤)𝑛||𝑓𝑗(||𝑦)𝑑𝑦+𝐶2𝑛𝑗|||𝑎𝐵𝑘,𝑤𝑎𝐵𝑗,𝑤|||𝑤𝐵𝑘1/𝑝𝑛||𝑓𝑗||(𝑦)𝑑𝑦+𝐶2𝑛𝑗𝑤𝐵𝑘1/𝑝𝑛|||𝑎(𝑦)𝑎𝐵𝑗,𝑤|||||𝑓𝑗||=𝐽(𝑦)𝑑𝑦1+𝐽2+𝐽3.(3.21) Using hypotheses (iii) for 𝑤 in place of strong doubling, 𝐽1𝐶2𝑗𝑛𝑎𝑓𝑗𝐿𝑝(𝑤)𝑤𝑝/𝑝𝐵𝑗1/𝑝𝑤𝐵𝑗1/𝑝𝑤𝐵𝑘𝑤𝐵𝑗1/𝑝𝐶2𝑘𝛿/𝑝2𝑗𝛿/𝑝𝑎𝑓𝑗𝐿𝑝(𝑤).(3.22) Similarly, 𝐽2𝐶𝑘2𝑘𝛿/𝑝𝑗2𝑗𝛿/𝑝𝑎𝑓𝑗𝐿𝑝(𝑤),𝐽3𝐶2𝑘𝛿/𝑝2𝑗𝛿/𝑝𝑎𝑓𝑗𝐿𝑝(𝑤).(3.23) Using hypotheses (i) for 𝑤, that is, 𝜑𝐷(𝑠,𝑡), we obtain the following inequality: III𝐶𝑎𝑘2𝑘(𝑠+𝛿/𝑝)𝑞𝑗𝑘+22𝑗(𝑠𝛿/𝑝)𝜑(𝑗)𝑓𝜒𝑗𝐿𝑝(𝑤)𝑞1/𝑞+𝐶𝑎𝑘𝑘2(𝑠+𝛿/𝑝)𝑞𝑗𝑘+2𝑗2𝑗(𝑠𝛿/𝑝)𝜑(𝑗)𝑓𝜒𝑗𝐿𝑝(𝑤)𝑞1/𝑞=III1+III2.(3.24) According to 𝑠+𝛿/𝑝>0, when 𝑞1, III1𝐶𝑎𝑗2𝑗(𝑠𝛿/𝑝)𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)𝑗2𝑘=2𝑘(𝑠+𝛿/𝑝)𝑞1/𝑞𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(3.25) When 𝑞>1, we take 𝜀>0 such that 𝑠+𝛿/𝑝𝜀>0. Then III1𝐶𝑎𝑘2𝑘(𝑠+𝛿/𝑝)𝑞𝑗𝑘+22𝑗(𝑠𝛿/𝑝+𝜀)𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)𝑗𝑘+22𝑗𝑞𝜀𝑞/𝑞1/𝑞𝐶𝑎𝑘2𝑘(𝑠+𝛿/𝑝𝜀)𝑞𝑗𝑘+22𝑗(𝑠𝛿/𝑝+𝜀)𝑞𝜑(𝑗)𝑞𝑓𝜒𝑗𝑞𝐿𝑝(𝑤)1/𝑞𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(3.26) Similar to III1, we have III2𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤).(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 𝐶-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<𝑞<,𝛿>0, and 𝑎BMO(𝑛). One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).If 𝐾 is a constant or a variable 𝐶-𝑍 kernel on 𝑛 and 𝑇 is the corresponding 𝐶-𝑍 operator, then there exists a constant such that for all 𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛), []𝑓𝑎,𝑇𝐾𝑞𝑝(𝜑,𝑤)(𝑛)𝐶𝑎𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝑛).(3.28)

From this and the extension theorem of BMO(Ω)-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<𝑞<, and 𝛿>0. Suppose that Ω is an open set of 𝑛 and 𝑎VMO(Ω). One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).If 𝐾 is a variable 𝐶-𝑍 kernel on Ω and 𝑇 is the corresponding 𝐶-𝑍 operator, then for any 𝜀>0, there exists a positive number 𝜌0=𝜌0(𝜀,𝜂) such that for any ball 𝐵𝑅 with the radius 𝑅(0,𝜌0), 𝐵RΩ and all 𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝐵𝑅), []𝑓𝑎,𝑇𝐾𝑞𝑝(𝜑,𝑤)(𝐵𝑅)𝐶𝜀𝑓𝐾𝑞𝑝(𝜑,𝑤)(𝐵𝑅),(3.29) where 𝐶=𝐶(𝑛,𝑝,𝑞,𝑎,𝜑,𝑀) is independent of 𝜀,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 𝑛3 and Ω is an open set of 𝑛. We are concerned with the nondivergence form elliptic equations 𝑢(𝑥)=𝑛𝑖,𝑗=1𝑎𝑖,𝑗(𝑥)𝑢𝑥𝑖𝑥𝑗=𝑓(𝑥),a.e.inΩ,(4.1) whose coefficients 𝑎𝑖𝑗 are assumed such that 𝑎𝑖𝑗(𝑥)=𝑎𝑗𝑖𝑎(𝑥),a.e.𝑥Ω,𝑖,𝑗=1,2,,𝑛,𝑖𝑗𝐿𝜇(Ω)VMO(Ω),1||𝜉||2𝑛𝑖,𝑗=1𝑎𝑖,𝑗(𝑥)𝜉𝑖𝜉𝑗||𝜉||𝜇2,𝜇>0,a.e.𝑥Ω,𝜉𝑛.(4.2) Let 1Γ(𝑥,𝑡)=(𝑛2)𝜔𝑛det𝑎𝑖𝑗(𝑥)1/2𝑛𝑖,𝑗=1𝐴𝑖𝑗(𝑥)𝑡𝑖𝑡𝑗(2𝑛)/2,Γ𝑖𝜕(𝑥,𝑡)=𝜕𝑡𝑖Γ(𝑥,𝑡),Γ𝑖𝑗𝜕(𝑥,𝑡)=2𝜕𝑡𝑖𝜕𝑡𝑗Γ(𝑥,𝑡),(4.3) for a.e. 𝑥𝐵 and 𝑡𝑛{0}, where the 𝐴𝑖𝑗 are the entries of the inverse of the matrix (𝑎𝑖𝑗)𝑖,𝑗=1,2,,𝑛.

From [13], we deduce the interior representation, that is, if 𝑢𝑊02,𝑝, 𝑢𝑥𝑖𝑥𝑗(𝑥)=P.V.𝐵Γ𝑖𝑗(𝑥,𝑥𝑦)𝑛,𝑙=1𝑎𝑙(𝑥)𝑎𝑙𝑢(𝑦)𝑥𝑥𝑙(𝑦)+𝑢(𝑦)𝑑𝑦+𝑢(𝑥)|𝑡|=1Γ𝑖(𝑥,𝑡)𝑡𝑗𝑑𝜎𝑡,a.e.for𝑥𝐵Ω,(4.4) where 𝐵 is a ball in Ω. We also set 𝑀max𝑖,𝑗=1,,𝑛max||𝛽||2𝑛𝜕𝛽𝜕𝑡𝛽Γ𝑖𝑗(𝑥,𝑡)𝐿(Ω×𝑆𝑛1)<.(4.5)

Theorem 4.1. Let 1<𝑝<,0<𝑞<, and 𝛿>0. Suppose that Ω is an open set of 𝑛 and 𝑎𝑖𝑗 satisfies (4.2) for 𝑖,𝑗=1,2,,𝑛. One assumes that(i)𝜑𝐷(𝑠,𝑡), where (𝛿/𝑝)<𝑠𝑡<𝑛(1(1/𝑝)),(ii)𝑤𝐴𝑟, where 𝑟=min(𝑝,𝑝(1(𝑡/𝑛))),(iii)𝑤𝑅𝐷(𝛿).Then there exists a constant 𝐶 such that for all balls 𝐵Ω and 𝑢𝑊02,𝑝, One has 𝑢𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤)(𝐵) and 𝑢𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤)(𝐵)𝐶𝑢𝐾𝑞𝑝(𝜑,𝑤)(𝐵).(4.6)

Proof. It is well known that Γ𝑖𝑗(𝑥,𝑡) are 𝐶-𝑍 kernels in the 𝑡 variable. Thus, using the technology of [13, 16] and the Corollaries 2.10 and 3.4, we deduce that, for any 𝜀>0, 𝑢𝑥𝑖𝑥𝑗𝐾q𝑝(𝜑,𝑤)(𝐵)𝑢𝐶𝜀𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤)(𝐵)+𝐶𝑢𝐾𝑞𝑝(𝜑,𝑤)(𝐵).(4.7) Choosing 𝜀 to be small enough (e.g.,𝐶𝜀<1), we obtain 𝑢𝑥𝑖𝑥𝑗𝐾𝑞𝑝(𝜑,𝑤)(𝐵)𝐶(1𝐶𝜀)𝑢𝐾𝑞𝑝(𝜑,𝑤)(𝐵).(4.8) This finishes the proof of Theorem 4.1.

Acknowledgment

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