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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 717501, 13 pages
http://dx.doi.org/10.1155/2011/717501
Research Article

On Singular Integrals with Cauchy Kernel on Weight Subspaces: The Basicity Property of Sines and Cosines Systems in Weight Spaces

1Nakhchivan State University, University Campus, AZ 7000 Nakhchivan, Azerbaijan
2Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, AZ 1141 Baku, Azerbaijan

Received 3 February 2011; Accepted 28 March 2011

Academic Editor: Shyam Kalla

Copyright © 2011 Tofig Isa Najafov and Saeed Farahani. 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

A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is considered. A sufficient condition for a bounded action of this operator from a subspace to another subspace of weight Lebesgue space of functions is found. These conditions are not identical with Muckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are considered.

1. Introduction

Consider the following singular operator with Cauchy kernel: where , , is an appropriate density, is a weight function of the form ( for ), are real numbers.

Under we understand a Lebesgue weight space with the norm

Bounded action of the operator in the spaces plays an important role in many problems of mathematics including the theory of bases. This direction has been well developed and treated in the known monographs. We will need the following.

Statement 1. The operator is bounded in if and only if the inequalities are fulfilled.

Concerning this fact a one can see the monograph [1] and papers [24]. Inequalities (1.4) are Muckenhoupt condition with respect to the weight function with degrees . It is known that the classic system of exponents ( are integers) forms a basis in if and only if inequalities (1.4) hold (see, e.g., [3, 4]).

It turns that if you consider the singular operator acting on the subspace of the weighted Lebesgue space, then inequality (1.4) is not necessary for the bounded action. At different points of degeneration the change interval of the corresponding exponent is expanded. This paper is devoted to studying these issues.

2. Some Necessary Facts

Let , —a weight function of the form where , .

Denote the space of even (odd) functions in by (), that is, We'll need the following identity: Indeed, we have For compactness of the notation, we assume . Thus, From this identity, we can easily get the following relations: The authors of the papers [47] used these relations earlier while establishing the basicity criterion of the system of sines and cosines with linear phases in . Thus, the following is valid.

Lemma 2.1. The following identities are true:

In the similar way, we obtain Further, we must take into account the following relation: As a result, we have Assume Thus, In sequel the following lemma is valid.

Lemma 2.2. The following identities are true:

3. Boundedness of Singular Operators on Subspace of Even Functions

Let . We have where the kernels , , are determined by the expressions Continue the weight to the interval by parity and denote by : It is obvious that are the degeneration points of . Thus, , , that is, Accept the denotation It is easy to see that the following holds As a result, for , we get the representation It is obvious that the singular integral boundedly acts in if and only if it boundedly acts in . Statement 1 is valid also in the case if the Cauchy kernel is replaced by the Hilbert kernel . So, assume that the inequalities , , are fulfilled. Then, from Statement 1, we directly get that boundedly acts from to and so from to . Assume and consider the integral operator : with the kernel . We have : where Consequently, It is obvious that where . So, if the inequalities hold, then from Statement 1 we obtain where is a constant independent from (different in different places). As a result, we get that if the inequalities (3.14) hold, the operator boundedly acts from to . The same conclusion is true for the operator as well. As a result, we get that while fulfilling the conditions the operator boundedly acts from to . On the other hand, it is easily seen that . As a result, we get that the operator boundedly acts from to .

Now, consider the case when and , , satisfy conditions (3.16). Let and be a number conjugated to . It is obvious that the relations are fulfilled for sufficiently small . Then, from the previous reasonings we get that the operator boundedly acts from to . As a result, it follows from the Riesz-Torin theorem (see, e.g., [7, page 144]) that the operator boundedly acts from to . We get the following.

Statement 2. Let the inequalities be fulfilled. Then, the operator boundedly acts from to .

Now, consider the representation of the operator by the kernel . Having paid attention to the expression similar to the previous case we establish that the boundedness of the operator holds also in the case when the change interval of the exponent extends by . In the conclusion we get that the following main theorem is valid.

Theorem 3.1. Let the weight function be defined by the expression (2.1) and assume that the inequalities are fulfilled. Then the singular operator : with Cauchy kernel , boundedly acts from to , where , .

4. Boundedness of Singular Operators on Subspace of Odd Functions

Let the weight function be defined by expression (2.1) and assume , . Denote by the following Cauchy-type kernel Appropriate integral operator denote by : Let the following inequalities be fulfilled. We have

From (4.3) it follows that , , and as a result . As a result, from Statement 1 we obtain that the integral operator boundedly acts in , if the inequalities (4.3) hold. In particular, it follows that the operator boundedly acts from to , if the inequalities (4.3) are fulfilled, that is, where is a constant independent from . On the other hand for we have Pay an attention to the relation (2.13), we obtain that , . Then from (4.5) yields Now, let be fulfilled. Assume Denote the integral operator with kernel by , that is, Taking into account the relation (2.10), for we have that is, Take into account the nonparity on we obtain Let Consider the operator . We have Thus

Further, we must take into account the expression , . As a result, from the previous relation we have where It is clear that for weight function Muckenhoupt condition is fulfilled and applying Statement 1 to the expression (4.17) we obtain In the similar way we establish the validity of the inequality If the inequalities (4.8) hold, as a result, we have Consider the case Take sufficiently small and determine . Acting similarly to the case (par.3) and accept the Riesz-Torin theorem we obtain boundedly acting of the operator from to (since ). Thus, if the following inequalities are fulfilled, then the operator boundedly acts from to .

Using the identity (2.11) in the similar way we establish that the same conclusion with respect to the operator is true in the case when the change interval of the exponent is expanded on . As a result, we obtain the validity of the following theorem.

Theorem 4.1. Let the weight function be defined by the expression (2.1) and , . Assume that the inequalities are fulfilled. Then the singular operator : with Cauchy-type kernel , boundedly acts from to .

5. Completeness, Minimality, and Basicity of the System of Sines in Weight Space

Consider the system of sines . Let conditions (3.20) be fulfilled. It is easy to see that then the system is minimal in . The system , , is a biorthogonal system to it. Indeed, it is obvious that is a space conjugated to , and an arbitrary continuous functional on , generated by , realized by the formula where is a complex conjugation.

Take , . We have

Since as and as for every fixed , then from relation (5.2) follows that , if the inequalities are fulfilled.

Take and denote by generated by its functional, that is, It is clear that , , where is a Kronecker's symbol. Consider Thus, , if , , , . As a result, we obtain that if the inequalities hold, then the system is minimal in .

Now consider the completeness of the system in . Suppose that for some , holds. We have It is easy to see that if the inequalities are fulfilled, then . Then from the previous relation we get . Since, the system is complete in space of continuous on functions with sup-norm, which vanishes at the ends of the segment , then from (5.7) it follows that a.e. on . Consequently, the system is complete in . So, the following statement is true.

Statement 3. Let the weight function be defined by expression (2.1). The system of sines is minimal in , if the inequalities (5.6) are fulfilled. It is complete in , if the inequalities (5.9) are fulfilled. Moreover, it forms a basis in , if the inequalities hold.

The basicity of system of sines in , when the inequalities (5.10) hold, follows from the basicity of system of exponent in , where , . The basicity of these systems earlier considered in papers [3, 4, 8, 9].

In the similar way we prove the following statement.

Statement 4. Let the weight function defined by expression (2.1). The system of cosines is minimal (forms a basis) in , if the inequalities (5.10) are fulfilled. It is complete in , if the inequalities (5.9) holds.

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