# Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces

Ahmet Eroglu

Author Affiliations

Department of Mathematics, Nigde University, Nigde, Turkey

Boundary Value Problems 2013, 2013:70  doi:10.1186/1687-2770-2013-70

 Received: 25 July 2012 Accepted: 17 March 2013 Published: 2 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, it is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on generalized Morrey spaces . The corresponding commutators generated by BMO functions are also considered.

MSC: 42B20, 42B25, 42B35.

##### Keywords:
generalized Morrey space; oscillatory integral; commutator; BMO spaces

### 1 Introduction and main results

The classical Morrey spaces, were introduced by Morrey [1] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces.

Morrey spaces are defined as the set of all functions such that

Under this definition, becomes a Banach space; for , it coincides with and for with .

We also denote by the weak Morrey space of all functions for which

where denotes the weak -space.

Definition 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 :

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential and the singular integral operators etc., from one weighted Lebesgue space to another one is well studied. Let . The fractional maximal operator and the Riesz potential are defined by

If , then is the Hardy-Littlewood maximal operator. In [2], Chiarenza and Frasca obtained the boundedness of M on . In [3], Adams established the boundedness of on .

Here and subsequently, C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.

The Calderón-Zygmund singular integral operator is defined by

(1.1)

where K is a Calderón-Zygmund kernel (CZK). We say a kernel is a CZK if it satisfies

(1.2)

(1.3)

and

(1.4)

for all a, b with . Chiarenza and Frasca [2] showed the boundedness of on .

It is worth pointing out that the kernel in (1.1) is convolution kernel. However, there were many kinds of operators with non-convolution kernels, such as Fourier transform and Radon transform [4], which both are versions of oscillatory integrals. The object we consider in this paper is a class of oscillatory integrals due to Ricci and Stein [5]

(1.5)

where is a real valued polynomial defined on , and K is a CZK.

It is well known that the oscillatory factor makes it impossible to establish the norm inequalities of (1.5) by the method as in the case of Calderón-Zygmund operators or fractional integrals. In [6], Chanillo and Christ established the weak type estimate of T.

A distribution kernel K is called a standard Calderón-Zygmund kernel (SCZK) if it satisfies the following hypotheses:

(1.6)

and

(1.7)

The corresponding Calderón-Zygmund integral operator and oscillatory integral operator S are defined by

(1.8)

and

(1.9)

where is a real valued polynomial defined on . In [7], Lu and Zhang proved that S is bounded on with . In [5], Ricci and Stein also introduced the standard fractional Calderón-Zygmund kernel (SFCZK) with , where the conditions (1.6) and (1.7) were replaced by

(1.10)

and

(1.11)

The corresponding fractional oscillatory integral operator is defined by (see [8])

(1.12)

where is also a real valued polynomial defined on . Obviously, when , and . Partly motivated by the idea from [9,10] and the results of [11], we now give the results of this paper in the following.

Theorem 1.1Let, andsatisfies the condition

(1.13)

whereCdoes not depend onxandt. IfKis a SCZK and the operatoris of type, then forand any polynomialthe operatorSis bounded fromto.

Moreover, forandKis a CZK operator, the operatorTis bounded fromto.

Theorem 1.2Let, , , is a polynomial, andsatisfies the condition

(1.14)

whereCdoes not depend onxandt. Then forthe operatoris bounded fromtoand forthe operatoris bounded fromto.

For a locally integrable function b, the commutator operator formed by S (or ) and b are defined by

and

Theorem 1.3Let, andsatisfies the condition

(1.15)

whereCdoes not depend onxandt. IfKis a SCZK and the operatoris of type, then for any polynomialthe operatoris bounded fromto.

Theorem 1.4Let, , , , is a polynomial, andsatisfies the condition

(1.16)

whereCdoes not depend onxandt. Then the operatoris bounded fromto.

### 2 Some known results in generalized Morrey spaces

In [9,10,12,13] and [14], there were obtained sufficient conditions on weights and for the boundedness of the singular operator T from to .

The following statements were proved by Nakai [14].

Theorem ALetandsatisfy the conditions

(2.1)

whenever, wherec (≥1) does not depend ont, randand

(2.2)

whereCdoes not depend onxandr. Then forthe operatorsMandTare bounded inand for, MandTare bounded fromto.

Theorem BLet, , andsatisfy the conditions (2.1) and

(2.3)

whereCdoes not depend onxandr. Then for, the operatorsandare bounded fromtoand for, andare bounded fromto.

The following statements, containing Nakai results obtained in [13,14] was proved by Guliyev in [9,10] (see also [15,16]).

Theorem CLetandsatisfy the condition

(2.4)

whereCdoes not depend onxandt. Then the operatorsMandTare bounded fromtoforand fromto.

Theorem DLet, , andsatisfy the condition

(2.5)

whereCdoes not depend onxandr. Then the operatorsandare bounded fromtoforand fromtofor.

The following statements, containing Guliyev results obtained in [9,10] was proved by Guliyev et al. in [11,12].

Theorem ELetandsatisfy the condition (2.4). Then the operatorsMandTare bounded fromtoforand fromto.

Theorem FLet, , andsatisfy the condition (1.14). Then the operatorsandare bounded fromtoforand fromtofor.

Note that integral conditions of type (2.3) after the paper [17] of 1956 are often referred to as Bary-Stechkin or Zygmund-Bary-Stechkin conditions; see also [18]. The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see [19-21] and references therein, where the characterization of integral inequalities of such a kind was given in terms of certain lower and upper indices known as Matuszewska-Orlicz indices. Note that in the cited papers the integral inequalities were studied as . Such inequalities are also of interest when they allow to impose different conditions as and ; such a case was dealt with in [22,23].

### 3 The fractional oscillatory integral operators in the spaces

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

Theorem G[24]

The inequality

holds for all non-negative and non-increasinggonif and only if

and.

Lemma 3.1Let, andKis a SCZK and the Calderón-Zygmund singular integral operatoris of type. Then forand any polynomialthe inequality

holds for any balland for all.

Moreover, forandKis a CZK

(3.1)

holds for any balland for all.

Proof Let . For arbitrary , set for the ball centered at and radius r, . We represent f as

and have

It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator is of type , then for and any polynomial the operator S is bounded on . Since , and boundedness of S in (see [5]) it follows that

where constant is independent of f.

It is clear that , implies . We get

By Fubini’s theorem and applying Hölder inequality, we have

(3.2)

Moreover, for all the inequality

(3.3)

is valid. Thus,

On the other hand,

(3.4)

Hence,

Let . From the weak boundedness of T (see [6]) and (3.4), it follows that:

(3.5)

Then by (3.4) and (3.5), we get the inequality (3.1). □

Proof of Theorem 1.1

By Lemma 3.1 and Theorem G, we get

if , and

if . □

Proof of Theorem 1.2

The proof of Theorem 1.2 follows from Theorem F and the following observation:

□

### 4 Commutators of fractional oscillatory integral operators in the spaces

Let T be a Calderón-Zygmund singular integral operator and . A well known result of Coifman, Rochberg and Weiss [27] 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, for example, [2,28,29]).

First, we recall the definition of the space .

Definition 2 Suppose that , 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 1 (1) The John-Nirenberg inequality: there are constants , such that for all and

(2) The John-Nirenberg inequality implies that

(4.1)

for .

(3) Let . Then there is a constant such that

(4.2)

where C is independent of f, x, r and t.

Lemma 4.1Let, , Kis a SCZK and the Calderón-Zygmund singular integral operatoris of type. Then forand any polynomialthe inequality

holds for any balland for all.

Proof Let . For arbitrary , set for the ball centered at and radius r, . We represent f as

and have

It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator is of type , then for and any polynomial the commutator operator is bounded on . Since , and boundedness of in (see [5]) it follows that

where constant is independent of f.

For , we have

Then

Let us estimate .

Applying Hölder’s inequality and by (4.1), (4.2), we get

In order to estimate note that

By (4.1), we get

Thus, by (3.2)

Summing up and , for all we get

(4.3)

Finally,

and statement of Lemma 4.1 follows by (3.4). □

Proof of Theorem 1.3 The statement of Theorem 1.3 follows by Lemma 4.1 and Theorem G in the same manner as in the proof of Theorem G. □

Proof of Theorem 1.4 The proof of Theorem 1.4 follows from the Theorem 7.4 in [11] and the following observation:

□

### Competing interests

The author declares that they have no competing interests.

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