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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 283730, 13 pages
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. Feichtinger
Copyright © 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.
It is proved that, for the nondivergence form elliptic equations , if belongs to the weighted Herz spaces , then , where is the -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 .
For a sequence , we suppose that satisfies doubling condition of order and write if there exists such that Let , for , and be the characteristic function of the set for . Suppose that is a weight function on . For , , the weighted Herz space is defined by where
Beurling in  introduced the Beurling algebras, and Herz in  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  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 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 , and we prove that if , then , where is the -solution of the equations.
Throughout this paper, unless otherwise indicated, will be used to denote a positive constant that is not necessarily the same at each occurrence.
Definition 2.1 (). Let and . One says that for if there exists a constant such that for every ball , holds, here and below, . One says that if there exists a positive constant such that The smallest constant appearing in (2.1) or (2.2) is called the constant of , denoted by .
Lemma 2.2. Let and . Then the following statements are true:(1)(strong doubling) there exists a constant such that (2)(centered reverse doubling) for some , , that is, (3)for , one has for some ,(4)there exist two constants and such that for any measurable set , If satisfies (2.5), one says . Obviously, ,(5)for all , one has .
Remark 2.3. Note that and .
Definition 2.4. Let be an open set. One says that any is in the bounded mean oscillation spaces if where is the average over of . Moreover, for any and , one sets One says that any is in the vanishing mean oscillation spaces if as and refer to as the modulus of .
Remark 2.5. or if ranges in the class of balls of .
Lemma 2.6 (see [12, Theorem 5]). Let . Then the norm of is equivalent to the norm of , where
Definition 2.7. Let . One says that is a constant Calderón-Zygmund kernel constant - kernel if(i),(ii) is homogeneous of degree ,(iii), .
Definition 2.8. Let be an open set of and . One says that is a variable Calderón-Zygmund kernel variable - kernel on if(i) is a constant - kernel for a.e. ,(ii).
Let be a constant or a variable - kernel on . One defines the corresponding - operator by
Lemma 2.9 (see [5, Theorem 3]). Let . One assumes that(i), where ,(ii), where ,(iii).
If is a constant or a variable - kernel on and is the corresponding - operator, then there exists a constant such that for all ,
From this lemma, by a proof similar to that of Theorem 2.11 in , we obtain the following corollary.
Corollary 2.10. Let , and be an open set of . One assumes that(i), where ,(ii), where ,(iii).If is a constant or a variable - kernel on , and is the corresponding - operator, then there exists a constant such that for all ,
3. Weighted Boundedness of Commutators
The aim of this section is to set up the weighted boundedness for the commutators formed by and functions, where . This kind of operators is useful in lots of different fields, see, for example,  as well as , then we consider important in themselves the related below results.
Theorem 3.2. Let , and . One assumes that(i), where ,(ii), where ,(iii).If a linear operator satisfies for any and is bounded on , then is also bounded on .
Proof. Let and , we write
Then, we have
For II, by the boundedness of , we have
For , note that when , and , . So from the condition (3.2), we have
According to Lemma 2.2, for some . By Hölder’s inequality and Lemma 2.6,
It is easy to see that . Therefore, similarly to , we have
Now, we establish the estimate for term ,
For the simplicity of analysis, we denote as
By an elementary estimate, we have
Combining (2.5) with (3.1),
In the same manner, we can see that
By Lemma 2.6, we get
Using hypotheses and the estimates of , and , we obtain the following inequality:
When , we have
because , that is, .
When , we take such that . Then Similar to , we have Finally we estimate . The proof of this part is analogue to , so we just give out an outline. Note that and , . So from the condition (3.2), we have Using hypotheses (iii) for in place of strong doubling, Similarly, Using hypotheses (i) for , that is, , we obtain the following inequality: According to , when , When , we take such that . Then Similar to , we have 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 - operators. From this theorem and Theorem 2.7 and 2.10 in , we easily deduce the following corollary.
Corollary 3.3. Let , and . One assumes that(i), where ,(ii), where ,(iii).If is a constant or a variable - kernel on and is the corresponding - operator, then there exists a constant such that for all ,
Corollary 3.4. Let , and . Suppose that is an open set of and . One assumes that(i), where ,(ii), where ,(iii).If is a variable - kernel on and is the corresponding - operator, then for any , there exists a positive number such that for any ball with the radius , and all , where is independent of , and .
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 and is an open set of . We are concerned with the nondivergence form elliptic equations whose coefficients are assumed such that Let for a.e. and , where the are the entries of the inverse of the matrix .
From , we deduce the interior representation, that is, if , where is a ball in . We also set
Theorem 4.1. Let , and . Suppose that is an open set of and satisfies (4.2) for . One assumes that(i), where ,(ii), where ,(iii).Then there exists a constant such that for all balls and , One has and
Proof. It is well known that are - kernels in the variable. Thus, using the technology of [13, 16] and the Corollaries 2.10 and 3.4, we deduce that, for any , Choosing to be small enough , we obtain This finishes the proof of Theorem 4.1.
This research is supported by the NSF of China (no. 11161044).
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