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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 572967, 6 pages
An Approximation Method for Convolution Calderón-Zygmund Operators
1Department of Mathematics, Xinxiang University, Xinxiang 453003, China
2Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, China
3Laboratoire de Mathématiques Raphaaël Salem, UMR 6085 CNRS-Université de Rouen, avenue de l'Université, BP 12, 76801 Saint-Etienne-du-Rouvray, France
Received 14 December 2012; Accepted 28 May 2013
Academic Editor: Youyu Wang
Copyright © 2013 Yunxia Yu and Zhanying Yang. 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.
We present an approximation method for convolution Calderón-Zygmund operators. We give a uniform approximation accuracy of the operators on the endpoint Triebel-Lizorkin space . Our proof mainly relies on the -dimensional Daubechies wavelet bases and the atomic-molecular approach.
1. Introduction and Main Result
For rapid application of dense matrices (or integral operators) to vectors, the celebrated work of Beylkin et al.  introduced a class of numerical algorithms which are based on the -dimensional wavelet bases with compact supports. These algorithms are also applicable to all Calderón-Zygmund operators and pseudodifferential operators. Since then, their algorithms are widely used in compression of matrices, operator approximation, and establishing boundedness of operators see [2–9]; In particular, Beylkin et al.  approximated a class of Calderón-Zygmund operators by banded operators and gave the approximation accuracy. It is intriguing to know whether we can get some similar approximation methods on some more general spaces. Notice that Yang  approximated the operators by compact operators and gave the approximation accuracy on .
In this paper, we are interested in considering the approximation method for a class of Calderón-Zygmund operators on Triebel-Lizorkin spaces. However, due to technical reasons, we can only get an approximation method for convolution Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces (see Theorem 1).
Now, we introduce a class of Calderón-Zygmund operators. Let denote the space of Schwartz test functions and the space of Schwartz distributions (the dual of ). Suppose that we have a linear continuous mapping associated with a kernel (in the sense that for test functions and with disjoint supports). We write if the following three conditions are satisfied.(I) is continuous on and satisfies for all with , where .(II)Weak boundedness condition: (III) condition: .
Convolution Calderón-Zygmund operators, such as Hilbert and Riesz operators, are commonly used in engineering. For a convolution operator , its kernel can be written as . In this case, the conditions for the operator are reduced to the following: for , where and for . For convenience, let denote the collection of all convolution operators in .
In what follows, we restrict our attention to the operator in . In general, the operator is analyzed by the -dimensional wavelet bases. However, Z. Y. Yang and Q. X. Yang , making use of the -dimensional Daubechies wavelet bases, approximated the operator by the banded operator and gave the approximation accuracy on the homogeneous Besov spaces . In this paper, we focus on an approximation method for the operator and obtaining the uniform approximation accuracy on the endpoint Triebel-Lizorkin spaces , whose definitions will be given in Section 2.
We first introduce some notations. Let and be the one-dimensional Daubechies father and mother wavelets, respectively. Assume that they are the real-valued and sufficiently regular functions. For and , denote . For any and , let We also put Then, forms orthonormal bases in , and it can be used to characterize general functions or distributions.
For any , let , then we have the representation in the sense of distribution. Now, we present the approximation of by the banded operator . For any integer and , we define Let be the annular operator associated to the kernel , and let Then, we can approximate by the banded operator and get the uniform approximation accuracy on the endpoint Triebel-Lizorkin space . Our result is stated as follows.
Theorem 1. Let and . If , then
Throughout this paper, the symbol denotes a constant that is independent of the main parameters involved but whose value may differ from line to line.
2. Endpoint Triebel-Lizorkin Spaces
Let be the space of tempered test functions. Let with supp and for ; let . Let , , and . Then, the homogeneous Triebel-Lizorkin space is defined as the collection of all (the tempered distributions modulo polynomials) such that with the usual modification made when ; see also Triebel . And it is well known that the homogeneous Triebel-Lizorkin space and Besov space are the same when .
For the homogeneous Triebel-Lizorkin space, Koskela et al. gave the characterization via grand Littlewood-Paley functions in  and gave the pointwise characterization in . Now, we recall its characterization based on wavelets. In fact, the Daubechies wavelet bases is an unconditional bases in the space . For any and , let and let be the characteristic function of the dyadic cube . For any , if we can define for all , then we have the representation in the sense of distribution. The space is characterized in terms of wavelets in the following way (for more details, see also [6, 7, 14]).
Proposition 2. For and , there exist two positive constants and such that
Triebel-Lizorkin spaces have been studied by means of the atomic and molecular decompositions. Next, we state the atomic-molecular decomposition for the endpoint space ; see Meyer and Yang  for more details. Let denote the collection of all dyadic cubes . Now, we recall the following two definitions which can be found in [6, 7, 15].
Definition 3. Let . is said to be a -atom in with the norm if there exists a ball such that(i)Supp ; (ii);(iii).
Definition 4. Let . is said to be a -molecule in with the norm if there exists a cube such that(i);(ii).
For convenience, in the above two definitions, is said to be a -atom or a -molecule when .
Proposition 5. Let . The following three conditions are equivalent: (i). (ii)There exist and -atoms such that .(iii)There exist and -molecules such that .
In fact, for any , we can write it as , where each is a -molecule and , and it can be verified that namely, for some fixed independent of . For more details, we refer the reader to [6, 15], This will play a key role in Section 3.
3. Estimate of and Proof of Theorem 1
To prove Theorem 1, we first estimate the annular operator . For any and integer , let
Lemma 6. For , let . One has
Proof. In terms of the molecular decomposition for the space , we only need to prove that for an arbitrary -molecule ,
where is independent of .
Denote , then we have Thus, In the following, we consider the estimates of and .
(i) For any , put , then For fixed and , let be the operator associated with the kernel By the properties of Daubechies wavelets, we can get that the operator is bounded on (see also [16, lemma 3.1]).
Set By means of the orthonormality of the wavelet bases, the right side of (23) is equal to Moreover, is bounded by Namely, we have Notice that the sum is adding among all with , we split into . Hence, we obtain Now, we consider and . As for , we use Proposition 2 to get that Let denote the Cartesian product . Let then the support of is contained in . Moreover, we can obtain that when . In other words, the number of the nonzero terms in the sum is a constant, which is independent of . By the inequality , we have From Hölder's inequality, we obtain Since , then we get that . Thus As for , we have By Definition 3, we can verify that is a -atom with the norm , see also  for more details. Furthermore, we obtain . Hence, By (29) and (35), it follows that
(ii) The estimate for can be treated as that for . For convenience of the reader, we repeat some details as follows. Let . Following the idea used to get (28), we have Splitting into and , then
The estimates of and can be obtained as we handle and , respectively. In conclusion, it follows that This completes the proof of Lemma 6.
In addition to the estimate of the annular operator , we need the estimate for wavelet coefficients of , which can be found in .
Proposition 7. Let be the distribution kernel associated to , then one has
We are now in a position to prove Theorem 1. By virtue of (42), we obtain that Together with Lemma 6, we have On the other hand, since belongs to the interval , then a direct computation leads to that This completes the proof of Theorem 1.
Remark 8. Notice that the atomic and molecular decompositions for the endpoint space play an important role in our proof. However, up to the best knowledge of the authors, it is unknown whether the Triebel-Lizorkin space has similar atomic and molecular decompositions. It would be interesting to know whether our method can be adjusted to get the approximation accuracy for the more general Triebel-Lizorkin space .
This research is supported by the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (no. CZY12015; no. CTZ13026), and the Research Fund for the Doctoral Program of Higher Education (no. 20090141120010).
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