Journal of Function Spaces
Volume 2017 (2017), Article ID 3295376, 9 pages
https://doi.org/10.1155/2017/3295376

Research Article

Boundedness of Fractional Oscillatory Integral Operators and Their Commutators in Vanishing Generalized Weighted Morrey Spaces

Bilal Çekiç¹ and Ayşegül Çelik Alabalık²

¹Department of Mathematics, Faculty of Science, Dicle University, 21280 Diyarbakir, Turkey
²Department of Mathematics, Institute of Natural and Applied Sciences, Dicle University, 21280 Diyarbakir, Turkey

Correspondence should be addressed to Bilal Çekiç; bilalcekic@gmail.com

Received 12 December 2016; Accepted 26 January 2017; Published 16 February 2017

Academic Editor: Yoshihiro Sawano

Copyright © 2017 Bilal Çekiç and Ayşegül Çelik Alabalık. 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.

In this article, we give the boundedness conditions in terms of Zygmund-type integral inequalities for oscillatory integral operators and fractional oscillatory integral operators on the vanishing generalized weighted Morrey spaces. Moreover, we investigate corresponding commutators.

1. Introduction

The classical Morrey spaces that play important role in the theory of partial differential equations were introduced by Morrey [1] in 1938. Since then Morrey spaces have been studied by various authors. We refer readers to the survey [2] and to the elegant book [3] for further information about these spaces and references on recent developments in this field.

So far various generalizations of Morrey spaces have been defined. Mizuhara [4] introduced the generalized Morrey space and Komori and Shirai [5] defined the weighted Morrey spaces . Guliyev [6] gave the notion of generalized weighted Morrey space which can be accepted as an extension of and . Eroglu [7] proved the boundedness of oscillatory integral operators, fractional oscillatory integral operators, and the corresponding commutators on and Shi et al. [8] proved the boundedness of these operators and commutators on . In [9], Lu et al. obtained the boundedness of sublinear operators with rough kernels on and in [10], Shi and Fu showed the boundedness of these operators on . The boundedness of some sublinear operators and their commutators on was obtained by Shi et al. [11] and the boundedness of sublinear operators on was proved by Mustafayev [12].

Vanishing Morrey spaces are subspaces of functions in Morrey spaces which were introduced by Vitanza [13] satisfying the condition

The properties and applications of vanishing Morrey Spaces were given in [14]. On vanishing Morrey spaces, the boundedness of commutators of the multidimensional Hardy type operators was proved in [15]. The vanishing generalized Morrey spaces were introduced and studied by Samko in [16].

In this study, as distinct from [8], we focus on vanishing generalized weighted Morrey spaces and give Zygmund-type conditions to prove the boundedness of oscillatory and the fractional oscillatory integral operators and their commutators in these spaces. In Section 2 we recall some definitions and necessary preliminaries and in Section 3 we give our main results; namely, we prove our theorems on vanishing generalized weighted Morrey spaces.

Throughout this paper, and so on are used as positive constant that can change from one line to another. means that with some positive constant . If and , then we say which means and are equivalent.

2. Preliminaries

Let be a weight function on , such that for almost every . is a measurable set with Lebesgue measure notated by and we define

For and we denote the weighted Lebesgue space by with the norm

Let , be a positive continuous function on and let be a weight function on . We show the generalized weighted Morrey space by , which is space of all functions with finite quasinorm

where is the open ball centered at of radius

The fractional integral operator (Riesz potential) and fractional Maximal operator , which play important roles in real and harmonic analysis, are defined by

where . If , then is the Hardy-Littlewood maximal operator.

The class of weights was introduced by Muckenhoupt in [17] to show that Hardy-Littlewood maximal function is bounded on weighted Lebesgue spaces if and only if

Now we define Muckenhoupt class Let . A weight is said to be an weight, if there exists a positive constant such that, for every ball ,

when , and for

for almost everywhere The smallest is shown by We define

A weight belongs to for if there exists such that

where . class was introduced by Muckenhoupt and Wheeden [18] to study weighted norm inequalities for fractional integral operators.

2.1. Vanishing Generalized Weighted Morrey Spaces

Let be an open set in and let be an arbitrary subset of . Let also be a measurable nonnegative function on and positive for all . Let be a weight function on ; then we notate by the vanishing weighted Morrey spaces which are defined as the spaces of functions with finite quasi norm:

such that

where and

Naturally, it is suitable to impose the function on the following conditions:

which makes nontrivial, since bounded functions which have compact support belong to this space.

Henceforth we denote by if is a nonnegative measurable function on and positive for all and satisfies conditions (10).

2.2. Oscillatory Integral Operators and Fractional Oscillatory Integral Operators

Oscillatory integral operators appear in many fields of mathematics and physics. Furthermore oscillatory integrals have been an essential part of harmonic analysis. Many important operators in harmonic analysis are some versions of oscillatory integrals, such as the Fourier transform, Bochner-Riesz means, and Radon transform. Properties of oscillatory integral operators have been studied by Stein in [19].

A distribution kernel is called a standard Calderón-Zygmund (in short C-Z) kernel when it satisfies the following conditions:

C-Z integral operator and the oscillatory integral operator are defined by

where is a real valued polynomial defined on Lu and Zhang [20] used -boundedness of to get -boundedness of with .

Ricci and Stein [21] introduced the standard fractional C-Z kernel, with , where

The fractional oscillatory integral operator is defined in [22] as

where is a real valued polynomial and defined on and is a standard fractional C-Z kernel; note that when , and

Lemma 1 (see [23]). If is a standard C-Z kernel and the C-Z singular integral operator is of type , then for any real polynomial and , there exists constants independent of the coefficients of such that .

Lemma 2 (see [24]). Let , , , and . Then the operators and are bounded from to , if

for every , and

where does not depend on and

2.3. The Commutator Operators and

Let ; we define

where

In harmonic analysis a function of bounded mean oscillation, also known as a BMO function, is a real valued function whose mean oscillation is bounded. BMO is the set of all locally integrable functions on with

Next we shall introduce the commutators of oscillatory integral operators and fractional oscillatory integral operators.

Let be a locally integrable function; the commutator operators and which are formed by and are defined by

Lemma 3 (see [25]). Suppose that is a standard C-Z kernel, , and the operator is of type Then for any , there exists constants independent on the coefficients of such that

Lemma 4 (see [18]). Let Then the norm of is equivalent to the norm of , where

Remark 5 (the John-Nirenberg inequality). There are possible constants such that for all and

for all The John-Nirenberg inequality implies that

for and

Lemma 6 (see [26]). (i) Let and let be a function in Let also , , and Then

where is independent of , , , and
(ii) Let and let be a function in Let also , , and Then

where is independent of , , , and

Lemma 7 (see [24]). Let , , and . Then and are bounded from to if

for every , and

where does not depend on and

3. Main Results

3.1. Boundedness of and in

Lemma 8. Let , and is a standard C-Z kernel and C-Z singular operator is of type . Then for any ball in and any real polynomial the following inequality holds:

where the constant does not depend on and

Proof. We split into two parts in a neighbourhood of point as follows:

By linearity of the operator we have

From the boundedness of on (see Lemma 1), it follows that

where constant is independent of and
Since , , we have

for It is obvious that and implies that . Thus we get

Since the right-hand side does not depend on we get

By Fubini’s theorem we have

By Hölder’s inequality, we get

Thus,

On the other hand,

By (31) and (38) we write

Then by (37) and (39) we get the inequality (29).

Remark 9. Note that Lemma 8 was proved in [7] in the case

Theorem 10. Let , , and The oscillatory integral operator is bounded from to for any real polynomial if is a standard C-Z kernel, the C-Z singular operator is of type :

for every , and

where does not depend on and

Proof. To estimate the norm of the operator we use (28) and get

By using (41) we can write

Thus,

Now we prove that it belongs to ; that is,

To show that for small , we split the right-hand side of (28):

where (we may take ) and

and it is supposed that
Now we fix such that , where and are constants from (41) and (46), respectively. Then we can write

By choosing sufficiently small and considering (40) we have

where is the constant from (40). Then by (10) we choose small enough such that

which completes the proof.

Theorem 11. Let be real polynomial; , , and . Then is bounded from to if

for every , and

where is not depending on and

Proof. Since and thanks to Lemma 2 proof is completed.

3.2. Boundedness of and in the Spaces

Lemma 12. Let , is a standard C-Z kernel, and the C-Z integral operator is type of Then for and polynomial the inequality

holds for any ball and for all

Proof. Let For any , we split into two parts in a neighbourhood of point such that

where , and by linearity of the operator we have

Moreover, from the boundedness of in (see Lemma 3), it follows that

where constant is independent of
For we have

Then

Let us estimate

Using Hölder’s inequality and (25), we have

In order to estimate note that

By (23) we get

Applying Hölder’s inequality, we get

Thus by (63) we write

Summing up and , for all we get

On the other hand,

Thus,

which along with (66) and (67) lead us to (53).

Theorem 13. Let and and . Then is bounded from to if

for every , and

where does not depend on and

Proof. Boundedness follows from Lemma 12 and the same procedure is argued in the proof of Theorem 10. We have to prove that

We suppose that In view of (53) we can write

To show that for small , we split the right side of (71)

where

Now we choose any fixed such that

where and are constants appearing in (72) and (69), respectively. This allows to estimate the first term uniformly in

The estimation of the second term can be obtained by choosing sufficiently small. Indeed, by (10), we have

where is the constant from (68). Then by (10) it suffices to choose small enough such that

which completes the proof.

Theorem 14. Let be a real valued polynomial, and . Then is bounded from to if

for every , and

where does not depend on and .

Proof. Since , from the boundedness of (see Lemma 7) proof is completed.

Competing Interests

The authors declare that they have no competing interests.

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