Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 283730, 13 pages

doi:10.1155/2012/283730

Research Article

Weighted Herz Spaces and Regularity Results

Yuxing Guo and Yinsheng Jiang

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.

Abstract.

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 .

1. Introduction

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 [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 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.

2. Preliminaries

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

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 [13], 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, [13] as well as [14], then we consider important in themselves the related below results.

Lemma 3.1 (see [10, Theorem ]). Let . Then for any ball , there exist constants such that for all ,

The inequality (3.1) is also called John-Nirenberg inequality.

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

Thus,

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

Note that

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 [13], 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 ,

From this and the extension theorem of -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 , 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 [13], 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.

Acknowledgment

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

References

A. Beurling, “Construction and analysis of some convolution algebras,” Annales de l'Institut Fourier, vol. 14, no. 2, pp. 1–32, 1964.

C. S. Herz, “Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms,” vol. 18, pp. 283–323, 1968.

X. Li and D. Yang, “Boundedness of some sublinear operators on Herz spaces,” Illinois Journal of Mathematics, vol. 40, no. 3, pp. 484–501, 1996.

S. Lu and D. Yang, “The continuity of commutators on Herz-type spaces,” The Michigan Mathematical Journal, vol. 44, no. 2, pp. 255–281, 1997.

Y. Komori and K. Matsuoka, “Boundedness of several operators on weighted Herz spaces,” Journal of Function Spaces and Applications, vol. 7, no. 1, pp. 1–12, 2009.

S. Lu, K. Yabuta, and D. Yang, “Boundedness of some sublinear operators in weighted Herz-type spaces,” Kodai Mathematical Journal, vol. 23, no. 3, pp. 391–410, 2000.

S. Lu and S. Tao, “Global regularity in homogeneous Morrey-Herz spaces of solutions to nondivergence elliptic equations with VMO coefficients,” Journal of Partial Differential Equations, vol. 20, no. 2, pp. 97–113, 2007.

M. A. Ragusa, “Homogeneous Herz spaces and regularity results,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1909–e1914, 2009.

M. A. Ragusa, “Parabolic Herz spaces and their applications,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1270–1273, 2012.

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, USA, 2004.

A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123 of Pure and Applied Mathematics, Academic Press, San Diego, Calif, USA, 1986.

B. Muckenhoupt and R. L. Wheeden, “Weighted bounded mean oscillation and the Hilbert transform,” Studia Mathematica, vol. 54, no. 3, pp. 221–237, 1976.

F. Chiarenza, M. Frasca, and P. Longo, “Interior estimates for nondivergence elliptic equations with discontinuous coefficients,” Ricerche di Matematica, vol. 40, no. 1, pp. 149–168, 1991.

M. Ruggieri and A. Valenti, “Exact solutions for a nonlinear model of dissipative media,” Journal of Mathematical Physics, vol. 52, no. 4, Article ID 043520, 2011.

P. W. Jones, “Extension theorems for BMO,” Indiana University Mathematics Journal, vol. 29, no. 1, pp. 41–66, 1980.

G. Di Fazio and M. A. Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients,” Journal of Functional Analysis, vol. 112, no. 2, pp. 241–256, 1993.