#### Abstract

The authors study the boundedness for a large class of sublinear operator generated by Calderón-Zygmund operator on generalized Morrey spaces . As an application of this result, the boundedness of the commutator of sublinear operators on generalized Morrey spaces is obtained. In the case , and is a sublinear operator, we find the sufficient conditions on the pair () which ensures the boundedness of the operator from one generalized Morrey space to another . In all cases, the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on (), which do not assume any assumption on monotonicity of in . Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

#### 1. Introduction

For and , we denote by the open ball centered at of radius , and by denote its complement. Let be the Lebesgue measure of the ball .

Let . The Hardy-Littlewood maximal operator is defined by

Let be a Calderón-Zygmund singular integral operator, briefly a Calderón-Zygmund operator, that is, a linear operator bounded from to taking all infinitely continuously differentiable functions with compact support to the functions represented by Such operators were introduced in [1]. Here, is a continuous function away from the diagonal which satisfies the standard estimates: there exist and such that for all , , and whenever .

It is well known that maximal operator and Calderón-Zygmund operators play an important role in harmonic analysis (see [2–6]).

Suppose that represents a linear or a sublinear operator, which satisfies that for any with compact support and where is independent of and .

For a function , suppose that the commutator operator represents a linear or a sublinear operator, which satisfies that for any with compact support and where is independent of and .

We point out that the condition (1.5) was first introduced by Soria and Weiss in [7]. The condition (1.5) are satisfied by many interesting operators in harmonic analysis, such as the Calderón-Zygmund operators, Carleson's maximal operator, Hardy-Littlewood maximal operator, C. Fefferman's singular multipliers, R. Fefferman's singular integrals, Ricci-Stein's oscillatory singular integrals, and the Bochner-Riesz means (see [7, 8] for details).

In this work, we prove the boundedness of the sublinear operator satisfies the condition (1.5) generated by Calderón-Zygmund operator from one generalized Morrey space to another , , and from to the weak space . In the case , and the commutator operator is a sublinear operator, we find the sufficient conditions on the pair which ensures the boundedness of the operators from to . Finally, as applications, we apply this result to several particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

By , we mean that with some positive constant independent of appropriate quantities. If and , we write and say that and are equivalent.

#### 2. Morrey Spaces

The classical Morrey spaces were originally introduced by Morrey Jr. in [9] to study the local behavior of solutions to second-order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [9, 10].

We denote by the Morrey space, the space of all functions with finite quasinorm where and .

Note that and . If or , then , where is the set of all functions equivalent to 0 on .

We also denote by the weak Morrey space of all functions for which where denotes the weak -space of measurable functions for which Here, denotes the nonincreasing rearrangement of a function .

Chiarenza and Frasca [11] studied the boundedness of the maximal operator in these spaces. Their result can be summarized as follows.

Theorem 2.1. *Let and . Then, for the operator is bounded on and for is bounded from to .*

Di Fazio and Ragusa [12] studied the boundedness of the Calderón-Zygmund operators in Morrey spaces, and their results imply the following statement for Calderón-Zygmund operators .

Theorem 2.2. * Let , . Then, for Calderón-Zygmund operator is bounded on and for is bounded from to . *

Note that Theorem 2.2 was proved by Peetre [10] in the case of the classical Calderón-Zygmund singular integral operators.

#### 3. Generalized Morrey Spaces

We find it convenient to define the generalized Morrey spaces in the form as follows.

*Definition 3.1. *Let be a positive measurable function on and . We denote by the generalized Morrey space, the space of all functions with finite quasinorm
Also, by we denote the weak generalized Morrey space of all functions for which

According to this definition, we recover the spaces and under the choice :

In [13–19], there were obtained sufficient conditions on and for the boundedness of the maximal operator and Calderón-Zygmund operator from to , (see also [20–23]). In [19], the following condition was imposed on : whenever , where (≥1) does not depend on , and , jointly with the condition for the sublinear operator satisfies the condition (1.5), where (>0) does not depend on and .

#### 4. Sublinear Operators Generated by Calderón-Zygmund Operators in the Spaces

In [24] (see, also [25, 26]), the following statements was proved by sublinear operator satisfies the condition (1.5), containing the result in [18, 19].

Theorem 4.1. *Let and satisfy conditions (3.4) and (3.5). Let be a sublinear operator satisfies the condition (1.5) and bounded on . Then, the operator is bounded on .*

The following statements, containing results obtained in [18, 19] was proved in [13] (see also [14, 15]).

Theorem 4.2. * Let , and let satisfy the condition
**
where does not depend on and . Then, the operators and are bounded from to for and from to . *

In this section, we are going to use the following statement on the boundedness of the Hardy operator:

Theorem 4.3 (see [27]). *The inequality
**
holds for all nonnegative and nonincreasing on if and only if
**
and . *

Lemma 4.4. * Let , be a sublinear operator which satisfies the condition (1.5) bounded on for and bounded from to .**Then, for ,
**
holds for any ball and for all .**Moreover, for ,
**
holds for any ball and for all . *

*Proof. *Let . For arbitrary , set for the ball centered at and of radius , . We represent as
and have

Since , and from the boundedness of in , it follows that
where constant is independent of .

It is clear that , implies . We get
By Fubini's theorem, we have
Applying Hölder's inequality, we get

Moreover, for all ,
is valid. Thus,
On the other hand,
Thus,

Let . From the weak (1,1) boundedness of and (4.15), it follows that

Then, by (4.13) and (4.17), we get (4.6).

Theorem 4.5. *Let , and let satisfy the condition
**
where does not depend on and . Let be a sublinear operator which satisfies the condition (1.5) bounded on for and bounded from to . Then, the operator is bounded from to for and from to . Moreover, for **
and for *

*Proof. *By Lemma 4.4 and Theorem 4.3, we have for
and for

Corollary 4.6. *Let , and satisfies the condition (4.18). Then, the operators and are bounded from to for and bounded from to .*

Note that Corollary 4.6 was proved in [28].

#### 5. Commutators of Sublinear Operators Generated byCalderón-Zygmund Operators in the Spaces

Let be a Calderón-Zygmund singular integral operator and . A well-known result of Coifman et al. [29] states that the commutator operator is bounded on for . The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, e.g., [12, 30, 31]).

First, we introduce the definition of the space of .

*Definition 5.1. *Suppose that , and let
where
Define

If one regards two functions whose difference is a constant as one, then space is a Banach space with respect to norm .

*Remark 5.2. *
(1) The John-Nirenberg inequality: there are constants such that for all and ,

(2) The John-Nirenberg inequality implies that
for .

(3) Let . Then, there is a constant such that
where is independent of , , , and .

In [24], the following statement was proved for the commutators of sublinear operators, containing the result in [18, 19].

Theorem 5.3. * Let , which satisfies the conditions (3.4) and (3.5) and . Suppose that is a linear operator and satisfies the condition (1.5). If the operator is bounded on , then the operator is bounded on . *

*Remark 5.4. *Note that Theorem 5.3 in the following form is also valid. Let , satisfy the conditions (3.4) and (3.5) and . Suppose that is a sublinear operator satisfies the condition (1.6) and bounded on , then the operator is bounded on .

Lemma 5.5. * Let , , and a sublinear operator satisfies the condition (1.6) and bounded on .**Then,
**
holds for any ball and for all .*

*Proof. *, , and a sublinear operator satisfies the condition (1.6). For arbitrary , set for the ball centered at and of radius . Write with and . Hence,
From the boundedness of in , it follows that

For , we have

Then,
Let us estimate
Applying Hölder's inequality and by (5.5), (5.6), we get
In order to estimate , note that
By (5.5), we get
Thus, by (4.12),
Summing up and , for all , we get
Finally,
and statement of Lemma 5.5 follows by (4.15).

The following theorem is true.

Theorem 5.6. *Let , and satisfy the condition
**
where does not depend on and . Suppose that is a sublinear operator which satisfies the condition (1.6) and bounded on .**Then, the operator is bounded from to . Moreover,
*

*Proof. *The statement of Theorem 5.6 is followed by Lemma 5.5 and Theorem 4.3 in the same manner as in the proof of Theorem 4.5.

For the sublinear commutator of the maximal operator and for the linear commutator of the Calderón-Zygmund operator from Theorem 5.6, we get the following new results.

Corollary 5.7. * Let , satisfy the condition (5.19) and . Then, the sublinear commutator operator is bounded from to . *

Corollary 5.8. * Let , satisfy the condition (5.19) and . Then, Calderón-Zygmund singular integral exists for a.e. and the operator is bounded from to . *

Note that when the conditions of Corollary 5.8 are satisfied, the existence of for a.e. was proved in [28].

#### 6. Some Applications

In this section, we will apply Theorems 4.5 and 5.6 to several particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

##### 6.1. Pseudodifferential Operators

Pseudodifferential operators are generalizations of differential operators and singular integrals. Let be real number, and . Following [32, 33], a symbol in is a smooth function defined on such that for all multi-indices and the following estimate holds: where is independent of and . A symbol in is one which satisfies the above estimates for each real number .

The operator given by is called a pseudodifferential operator with symbol , where is a Schwartz function and denotes the Fourier transform of . As usual, will denote the class of pseudodifferential operators with symbols in .

Miller [34] showed the boundedness of pseudodifferential operators on weighted spaces whenever the weight function belongs to Muckenhoupt’s class . In [1], it is shown that pseudodifferential operators in are Calderón-Zygmund operators, then from Corollary 5.8, we get the following new results.

Corollary 6.1. *Let , and let satisfy the condition (4.18). If is a pseudodifferential operator of the Hörmander class , then the operator is bounded from to for and bounded from to . *

Corollary 6.2. *Let , satisfy the condition (5.19) and . Let also be a pseudodifferential operator of the Hörmander class . Then, the commutator operator is bounded from to .*

##### 6.2. Littlewood-Paley Operator

The Littlewood-Paley functions play an important role in classical harmonic analysis, for example, in the study of nontangential convergence of Fatou type and boundedness of Riesz transforms and multipliers [4–6, 35]. The Littlewood-Paley operator (see [6, 36]) is defined as follows.

*Definition 6.3. **Suppose that ** satisfies *
Then, the generalized Littlewood-Paley function is defined by
where for and .

The sublinear commutator of the operator is defined by
where

The following theorem is valid (see [3, Theorem 5.1.2]).

Theorem 6.4. * Suppose that satisfies (6.3) and the following properties:
**
where and are both independent of and . Then, is bounded on for all , and bounded from to . *

Let be the space , then for each fixed , may be viewed as a mapping from to , and it is clear that . In fact, by Minkowski inequality and the conditions on , we get Thus, we get the following.

Corollary 6.5. *Let , satisfies the condition (4.18) and satisfies (6.3) and (6.7). Then the operator is bounded from to for and bounded from to . *

Corollary 6.6. * Let , satisfies the condition (5.19), and satisfies (6.3) and (6.7). Then the operator is bounded from to .*

##### 6.3. Marcinkiewicz Operator

Let be the unit sphere in equipped with the Lebesgue measure . Suppose that satisfies the following conditions.(a) is the homogeneous function of degree zero on ; that is, (b) has mean zero on ; that is, (c), , that is there exists a constant such that

In 1958, Stein [35] defined the Marcinkiewicz integral of higher dimension as where

Since Stein's work in 1958, the continuity of Marcinkiewicz integral has been extensively studied as a research topic and also provides useful tools in harmonic analysis [3–6].

The sublinear commutator of the operator is defined by where

Let be the space . Then, it is clear that .

By Minkowski inequality and the conditions on , we get Thus, satisfies the condition (1.5). It is known that is bounded on for and bounded from to (see [37]), then from Theorems 4.5 and 5.6, we get the following collory.

Corollary 6.7. *Let and satisfy the condition (4.18), and let satisfy the conditions (a)–(c). Then, is bounded from to for and bounded from to .*

Corollary 6.8. * Let , satisfy the condition (5.19), , and satisfy the conditions (a)–(c). Then, is bounded from to .*

##### 6.4. Bochner-Riesz Operator

Let , and for . The maximal Bochner-Riesz operator is defined by (see [38, 39])

Let be the space , then it is clear that .

By the condition on (see [2]), we have Thus, satisfies the condition (1.5). It is known that is bounded on for , and bounded from to , then from Theorems 4.5 and 5.6, we get the following corollary.

Corollary 6.9. *Let , satisfy the condition (4.18) and . Then, the operator is bounded from to for and bounded from to .*

Corollary 6.10. * Let , satisfy the condition (5.19), and . Then, the operator is bounded from to . *

*Remark 6.11. *Recall that under the assumption that satisfies the conditions (3.4) and (3.5), the Corollaries 6.9 and 6.10 were proved in [38].

#### Acknowledgments

The authors express their thank to prof A. Serbetci for helpful comments on the paper of this paper. The authors would like to express their gratitude to the referees for his very valuable comments and suggestions. The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project no-01/023. The research of V. Guliyev and T. Karaman was partially supported by the grant of 2010-Ahi Evran University Scientific Research Projects (BAP FBA-10-05) and by the Scientific and Technological Research Council of Turkey (TUBITAK Project no: 110T695).