Skip to main content

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

Abstract

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

MSC:42B20, 42B25, 42B35.

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 M p , λ ( R n ) are defined as the set of all functions f L p ( R n ) such that

f M p , λ f M p , λ ( R n ) = sup x , r > 0 r λ p f L p ( B ( x , r ) ) <.

Under this definition, M p , λ ( R n ) becomes a Banach space; for λ=0, it coincides with L p ( R n ) and for λ=1 with L ( R n ).

We also denote by W M p , λ the weak Morrey space of all functions fW L p loc ( R n ) for which

f W M p , λ f W M p , λ ( R n ) = sup x R n , r > 0 r λ p f W L p ( B ( x , r ) ) <,

where W L p denotes the weak L p -space.

Definition 1 Let φ(x,r) be a positive measurable function on R n ×(0,) and 1p<. We denote by M p , φ M p , φ ( R n ) the generalized Morrey space, the space of all functions f L p loc ( R n ) with finite quasinorm

f M p , φ f M p , φ ( R n ) = sup x R n , r > 0 φ ( x , r ) 1 | B ( x , r ) | 1 p f L p ( B ( x , r ) ) .

Also, by W M p , φ W M p , φ ( R n ), we denote the weak generalized Morrey space of all functions fW L p loc ( R n ) for which

f W M p , φ f W M p , φ ( R n ) = sup x R n , r > 0 φ ( x , r ) 1 | B ( x , r ) | 1 p f W L p ( B ( x , r ) ) <.

According to this definition, we recover the spaces M p , λ and W M p , λ under the choice φ(x,r)= r λ n p :

M p , φ | φ ( x , r ) = r λ n p = M p , λ , W M p , φ | φ ( x , r ) = r λ n p = W M p , λ .

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 f L 1 loc ( R n ). The fractional maximal operator M α and the Riesz potential I α are defined by

M α f ( x ) = sup t > 0 | B ( x , t ) | 1 + α n B ( x , t ) | f ( y ) | d y , 0 α < n , I α f ( x ) = R n f ( y ) d y | x y | n α , 0 < α < n .

If α=0, then M M 0 is the Hardy-Littlewood maximal operator. In [2], Chiarenza and Frasca obtained the boundedness of M on M p , λ ( R n ). In [3], Adams established the boundedness of I α on M p , λ ( R n ).

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

T ˜ f(x)= p . v . R n K(xy)f(y)dy,
(1.1)

where K is a Calderón-Zygmund kernel (CZK). We say a kernel K C 1 ( R n {0}) is a CZK if it satisfies

| K ( x ) | C | x | n ,
(1.2)
| K ( x ) | C | x | n + 1
(1.3)

and

a < | x | < b K(x)dx=0,
(1.4)

for all a, b with 0<a<b. Chiarenza and Frasca [2] showed the boundedness of T ˜ on M p , λ ( R n ).

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]

Tf(x)= p . v . R n e i P ( x , y ) K(xy)f(y)dy,
(1.5)

where P(x,y) is a real valued polynomial defined on R n × R n , and K is a CZK.

It is well known that the oscillatory factor e i P ( x , y ) makes it impossible to establish the L p 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 (1,1) type estimate of T.

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

| K ( x , y ) | C | x y | n ,xy
(1.6)

and

| x K ( x , y ) | + | y K ( x , y ) | C | x y | n + 1 ,xy.
(1.7)

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

S ˜ f(x)= p . v . R n K(x,y)f(y)dy
(1.8)

and

Sf(x)= p . v . R n e i P ( x , y ) K(x,y)f(y)dy,
(1.9)

where P(x,y) is a real valued polynomial defined on R n × R n . In [7], Lu and Zhang proved that S is bounded on L p with 1<p<. In [5], Ricci and Stein also introduced the standard fractional Calderón-Zygmund kernel (SFCZK) K α with 0<α<n, where the conditions (1.6) and (1.7) were replaced by

| K α ( x , y ) | C | x y | n α ,xy
(1.10)

and

| x K α ( x , y ) | + | y K α ( x , y ) | C | x y | n + 1 α ,xy.
(1.11)

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

S α f(x)= R n e i P ( x , y ) K α (x,y)f(y)dy,
(1.12)

where P(x,y) is also a real valued polynomial defined on R n × R n . Obviously, when α=0, S 0 =S and K 0 =K. 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 1.1 Let 1p<, and ( φ 1 , φ 2 ) satisfies the condition

r ess sup t < s < φ 1 ( x , s ) s n p t n p + 1 dtC φ 2 (x,r),
(1.13)

where C does not depend on x and t. If K is a SCZK and the operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )), then for 1<p< and any polynomial P(x,y) the operator S is bounded from M p , φ 1 to M p , φ 2 .

Moreover, for p=1 and K is a CZK operator, the operator T is bounded from M 1 , φ 1 to W M 1 , φ 2 .

Theorem 1.2 Let 1p<, 0<α< n p , 1 q = 1 p α n , P(x,y) is a polynomial, and ( φ 1 , φ 2 ) satisfies the condition

r ess sup t < s < φ 1 ( x , s ) s n p t n q + 1 dtC φ 2 (x,r),
(1.14)

where C does not depend on x and t. Then for p>1 the operator S α is bounded from M p , φ 1 to M q , φ 2 and for p=1 the operator S α is bounded from M 1 , φ 1 to W M q , φ 2 .

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

S b f(x)=b(x)Sf(x)S(bf)(x)

and

S α , b f(x)=b(x) S α f(x) S α (bf)(x).

Theorem 1.3 Let 1<p<, bBMO( R n ) and ( φ 1 , φ 2 ) satisfies the condition

r ( 1 + ln t r ) ess sup t < s < φ 1 ( x , s ) s n p t n p + 1 dtC φ 2 (x,r),
(1.15)

where C does not depend on x and t. If K is a SCZK and the operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )), then for any polynomial P(x,y) the operator S b is bounded from M p , φ 1 to M p , φ 2 .

Theorem 1.4 Let 1<p<, bBMO( R n ), 0<α< n p , 1 q = 1 p α n , P(x,y) is a polynomial, and ( φ 1 , φ 2 ) satisfies the condition

r ( 1 + ln t r ) ess sup t < s < φ 1 ( x , s ) s n p t n q + 1 dtC φ 2 (x,r),
(1.16)

where C does not depend on x and t. Then the operator S b , α is bounded from M p , φ 1 to M q , φ 2 .

2 Some known results in generalized Morrey spaces M p , φ ( R n )

In [9, 10, 12, 13] and [14], there were obtained sufficient conditions on weights φ 1 and φ 2 for the boundedness of the singular operator T from M p , φ 1 ( R n ) to M p , φ 2 ( R n ).

The following statements were proved by Nakai [14].

Theorem A Let 1p< and φ(x,r) satisfy the conditions

c 1 φ(x,r)φ(x,t)cφ(x,r)
(2.1)

whenever rt2r, where c (≥1) does not depend on t, r and x R n and

r φ ( x , t ) p d t t Cφ ( x , r ) p ,
(2.2)

where C does not depend on x and r. Then for p>1 the operators M and T are bounded in M p , φ ( R n ) and for p=1, M and T are bounded from M 1 , φ ( R n ) to W M 1 , φ ( R n ).

Theorem B Let 1p<, 0<α< n p , 1 q = 1 p α n and φ(x,t) satisfy the conditions (2.1) and

r φ ( x , t ) p d t t Cφ ( x , r ) p ,
(2.3)

where C does not depend on x and r. Then for p>1, the operators M α and I α are bounded from M p , φ ( R n ) to M q , φ ( R n ) and for p=1, M α and I α are bounded from M 1 , φ ( R n ) to W M q , φ ( R n ).

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

Theorem C Let 1p< and ( φ 1 , φ 2 ) satisfy the condition

t φ 1 (x,r) d r r C φ 2 (x,t),
(2.4)

where C does not depend on x and t. Then the operators M and T are bounded from M p , φ 1 to M p , φ 2 for p>1 and from M 1 , φ 1 to W M 1 , φ 2 .

Theorem D Let 1p<, 0<α< n p , 1 q = 1 p α n and ( φ 1 , φ 2 ) satisfy the condition

r t α φ 1 (x,t) d t t C φ 2 (x,r),
(2.5)

where C does not depend on x and r. Then the operators M α and I α are bounded from M p , φ 1 to M q , φ 2 for p>1 and from M 1 , φ 1 to W M q , φ 2 for p=1.

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

Theorem E Let 1p< and ( φ 1 , φ 2 ) satisfy the condition (2.4). Then the operators M and T are bounded from M p , φ 1 to M p , φ 2 for p>1 and from M 1 , φ 1 to W M 1 , φ 2 .

Theorem F Let 1p<, 0<α< n p , 1 q = 1 p α n and ( φ 1 , φ 2 ) satisfy the condition (1.14). Then the operators M α and I α are bounded from M p , φ 1 to M q , φ 2 for p>1 and from M 1 , φ 1 to W M q , φ 2 for p=1.

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 [1921] 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 r0. Such inequalities are also of interest when they allow to impose different conditions as r0 and r; such a case was dealt with in [22, 23].

3 The fractional oscillatory integral operators in the spaces M p , φ ( R n )

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

(Hg)(t):= 1 t 0 t g(r)dr,0<t<.

Theorem G [24]

The inequality

ess sup t > 0 w(t)Hg(t)c ess sup t > 0 v(t)g(t)

holds for all non-negative and non-increasing g on (0,) if and only if

A:= sup t > 0 w ( t ) t 0 t d r ess inf 0 < s < r v ( s ) <,

and cA.

Lemma 3.1 Let 1p<, and K is a SCZK and the Calderón-Zygmund singular integral operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )). Then for 1<p< and any polynomial P(x,y) the inequality

S f L p ( B ( x 0 , r ) ) r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt

holds for any ball B( x 0 ,r) and for all f L p loc ( R n ).

Moreover, for p=1 and K is a CZK

T f W L 1 ( B ( x 0 , r ) ) r n 2 r f L p ( B ( x 0 , t ) ) t 1 n dt
(3.1)

holds for any ball B( x 0 ,r) and for all f L 1 loc ( R n ).

Proof Let p(1,). For arbitrary x 0 R n , set B=B( x 0 ,r) for the ball centered at x 0 and radius r, 2B=B( x 0 ,2r). We represent f as

f= f 1 + f 2 , f 1 (y)=f(y) χ 2 B (y), f 2 (y)=f(y) χ ( 2 B ) (y)

and have

S f L p ( B ) S f 1 L p ( B ) + S f 2 L p ( B ) .

It is known that (see [5], see also [7, 25, 26]), if K is a SCZK and the operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )), then for 1<p< and any polynomial P(x,y) the operator S is bounded on L p ( R n ). Since f 1 L p ( R n ), S f 1 L p ( R n ) and boundedness of S in L p ( R n ) (see [5]) it follows that

S f 1 L p ( B ) S f 1 L p ( R n ) C f 1 L p ( R n ) =C f 1 L p ( 2 B ) ,

where constant C>0 is independent of f.

It is clear that xB, y ( 2 B ) implies 1 2 | x 0 y||xy| 3 2 | x 0 y|. We get

| S f 2 ( x ) | c 0 ( 2 B ) | f ( y ) | | x 0 y | n dy.

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

( 2 B ) | f ( y ) | | x 0 y | n d y ( 2 B ) | f ( y ) | | x 0 y | t 1 n d t d y 2 r 2 r < | x 0 y | < t | f ( y ) | d y t 1 n d t 2 r B ( x 0 , t ) | f ( y ) | d y t 1 n d t 2 r f L p ( B ( x 0 , t ) ) t 1 n p d t .
(3.2)

Moreover, for all p[1,) the inequality

S f 2 L p ( B ) r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt
(3.3)

is valid. Thus,

S f L p ( B ) f L p ( 2 B ) + r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt.

On the other hand,

f L p ( 2 B ) r n p f L p ( 2 B ) 2 r t 1 n p d t r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p d t .
(3.4)

Hence,

S f L p ( B ) r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt.

Let p=1. From the weak (1,1) boundedness of T (see [6]) and (3.4), it follows that:

T f 1 W L 1 ( B ) T f 1 W L 1 ( R n ) f 1 L 1 ( R n ) = f L 1 ( 2 B ) r n 2 r B ( x 0 , t ) | f ( y ) | d y d t t n + 1 .
(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

S f M p , φ 2 sup x R n , r > 0 φ 2 ( x , r ) 1 r f L p ( B ( x , t ) ) t 1 n p d t sup x R n , r > 0 φ 2 ( x , r ) 1 0 r n p f L p ( B ( x , t p n ) ) d t = sup x R n , r > 0 φ 2 ( x , r p n ) 1 0 r f L p ( B ( x , t p n ) ) d t sup x R n , r > 0 φ 1 ( x , r p n ) 1 r f L p ( B ( x , r p n ) ) = f M p , φ 1

if p(1,), and

T f W M 1 , φ 2 sup x R n , r > 0 φ 2 ( x , r ) 1 r f L 1 ( B ( x , t ) ) t 1 n d t sup x R n , r > 0 φ 2 ( x , r ) 1 0 r n f L 1 ( B ( x , t 1 n ) ) d t = sup x R n , r > 0 φ 2 ( x , r 1 n ) 1 0 r f L 1 ( B ( x , t 1 n ) ) d t sup x R n , r > 0 φ 1 ( x , r 1 n ) 1 r f L 1 ( B ( x , r 1 n ) ) = f M 1 , φ 1

if p=1. □

Proof of Theorem 1.2

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

| S α f ( x ) | I α ( | f | ) (x).

 □

4 Commutators of fractional oscillatory integral operators in the spaces M p , φ ( R n )

Let T be a Calderón-Zygmund singular integral operator and bBMO( R n ). A well known result of Coifman, Rochberg and Weiss [27] states that the commutator operator [b,T]f=T(bf)bTf is bounded on L p ( R n ) for 1<p<. 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 BMO( R n ).

Definition 2 Suppose that f L 1 loc ( R n ), let

f = sup x R n , r > 0 1 | B ( x , r ) | B ( x , r ) | f ( y ) f B ( x , r ) | dy<,

where

f B ( x , r ) = 1 | B ( x , r ) | B ( x , r ) f(y)dy.

Define

BMO ( R n ) = { f L 1 loc ( R n ) : f < } .

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

Remark 1 (1) The John-Nirenberg inequality: there are constants C 1 , C 2 >0, such that for all fBMO( R n ) and β>0

| { x B : | f ( x ) f B | > β } | C 1 |B| e C 2 β / f ,B R n .

(2) The John-Nirenberg inequality implies that

f sup x R n , r > 0 ( 1 | B ( x , r ) | B ( x , r ) | f ( y ) f B ( x , r ) | p d y ) 1 p
(4.1)

for 1<p<.

(3) Let fBMO( R n ). Then there is a constant C>0 such that

| f B ( x , r ) f B ( x , t ) |C f ln t r for 0<2r<t,
(4.2)

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

Lemma 4.1 Let 1p<, bBMO( R n ), K is a SCZK and the Calderón-Zygmund singular integral operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )). Then for 1<p< and any polynomial P(x,y) the inequality

S b f L p ( B ( x 0 , r ) ) b r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt

holds for any ball B( x 0 ,r) and for all f L p loc ( R n ).

Proof Let p(1,). For arbitrary x 0 R n , set B=B( x 0 ,r) for the ball centered at x 0 and radius r, 2B=B( x 0 ,2r). We represent f as

f= f 1 + f 2 , f 1 (y)=f(y) χ 2 B (y), f 2 (y)=f(y) χ ( 2 B ) (y)

and have

S b f L p ( B ) S b f 1 L p ( B ) + S b f 2 L p ( B ) .

It is known that (see [5], see also [7, 25, 26]), if K is a SCZK and the operator S ˜ is of type ( L 2 ( R n ), L 2 ( R n )), then for 1<p< and any polynomial P(x,y) the commutator operator S b is bounded on L p ( R n ). Since f 1 L p ( R n ), S f 1 L p ( R n ) and boundedness of S b in L p ( R n ) (see [5]) it follows that

S b f 1 L p ( B ) S b f 1 L p ( R n ) C b f 1 L p ( R n ) =C b f 1 L p ( 2 B ) ,

where constant C>0 is independent of f.

For xB, we have

| S b f 2 ( x ) | R n | b ( y ) b ( x ) | | x y | n | f ( y ) | d y ( 2 B ) | b ( y ) b ( x ) | | x 0 y | n | f ( y ) | d y .

Then

S b f 2 L p ( B ) ( B ( ( 2 B ) | b ( y ) b ( x ) | | x 0 y | n | f ( y ) | d y ) p d x ) 1 p ( B ( ( 2 B ) | b ( y ) b B | | x 0 y | n | f ( y ) | d y ) p d x ) 1 p + ( B ( ( 2 B ) | b ( x ) b B | | x 0 y | n | f ( y ) | d y ) p d x ) 1 p = I 1 + I 2 .

Let us estimate I 1 .

I 1 r n p ( 2 B ) | b ( y ) b B | | x 0 y | n | f ( y ) | d y r n p ( 2 B ) | b ( y ) b B | | f ( y ) | | x 0 y | d t t n + 1 d y r n p 2 r 2 r | x 0 y | t | b ( y ) b B | | f ( y ) | d y d t t n + 1 r n p 2 r B ( x 0 , t ) | b ( y ) b B | | f ( y ) | d y d t t n + 1 .

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

I 1 r n p 2 r B ( x 0 , t ) | b ( y ) b B ( x 0 , t ) | | f ( y ) | d y d t t n + 1 + r n p 2 r | b B ( x 0 , r ) b B ( x 0 , t ) | B ( x 0 , t ) | f ( y ) | d y d t t n + 1 r n p 2 r ( B ( x 0 , t ) | b ( y ) b B ( x 0 , t ) | p d y ) 1 p f L p ( B ( x 0 , t ) ) d t t n + 1 + r n p 2 r | b B ( x 0 , r ) b B ( x 0 , t ) | f L p ( B ( x 0 , t ) ) t 1 n p d t b r n p 2 r ( 1 + ln t r ) f L p ( B ( x 0 , t ) ) t 1 n p d t .

In order to estimate I 2 note that

I 2 = ( B | b ( x ) b B | p d x ) 1 p ( 2 B ) | f ( y ) | | x 0 y | n dy.

By (4.1), we get

I 2 b r n p ( 2 B ) | f ( y ) | | x 0 y | n dy.

Thus, by (3.2)

I 2 b r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p dt.

Summing up I 1 and I 2 , for all p(1,) we get

S b f 2 L p ( B ) b r n p 2 r ( 1 + ln t r ) f L p ( B ( x 0 , t ) ) t 1 n p dt.
(4.3)

Finally,

S b f L p ( B ) b f L p ( 2 B ) + b r n p 2 r ( 1 + ln t r ) f L p ( B ( x 0 , t ) ) t 1 n p dt,

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:

| S α , b f ( x ) | I α , b ( | f | ) (x).

 □

References

  1. Morrey CB: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 1938, 43: 126-166. 10.1090/S0002-9947-1938-1501936-8

    MathSciNet  Article  Google Scholar 

  2. Chiarenza F, Frasca M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl. 1987, 7: 273-279.

    MathSciNet  Google Scholar 

  3. Adams DR: A note on Riesz potentials. Duke Math. J. 1975, 42: 765-778. 10.1215/S0012-7094-75-04265-9

    MathSciNet  Article  Google Scholar 

  4. Phong DH, Stein EM: Singular integrals related to the Radon transform and boundary value problems. Proc. Natl. Acad. Sci. USA 1983, 80: 7697-7701. 10.1073/pnas.80.24.7697

    MathSciNet  Article  Google Scholar 

  5. Ricci F, Stein EM: Harmonic analysis on nilpotent groups and singular integrals I: oscillatory integrals. J. Funct. Anal. 1987, 73: 179-194. 10.1016/0022-1236(87)90064-4

    MathSciNet  Article  Google Scholar 

  6. Chanillo S, Christ M:Weak (1,1) bounds for oscillatory singular integral. Duke Math. J. 1987, 55: 141-155. 10.1215/S0012-7094-87-05508-6

    MathSciNet  Article  Google Scholar 

  7. Lu SZ, Zhang Y:Criterion on L p -boundedness for a class of oscillatory singular integrals with rough kernels. Rev. Mat. Iberoam. 1992, 8: 201-219.

    Google Scholar 

  8. Ding Y: L p -Boundedness for fractional oscillatory integral operator with rough kernel. Approx. Theory Appl. 1996, 12: 70-79.

    MathSciNet  Google Scholar 

  9. Guliyev, VS: Integral operators on function spaces on the homogeneous groups and on domains in R n . Doctor of Sciences, Mat. Inst. Steklova, Moscow (1994), 329 pp. (in Russian)

    Google Scholar 

  10. Guliyev VS: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009., 2009: Article ID 503948

    Google Scholar 

  11. Guliyev VS, Aliyev SS, Karaman T, Shukurov PS: Boundedness of sublinear operators and commutators on generalized Morrey space. Integral Equ. Oper. Theory 2011, 71: 327-355. 10.1007/s00020-011-1904-1

    MathSciNet  Article  Google Scholar 

  12. Akbulut A, Guliyev VS, Mustafayev R: On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces. Math. Bohem. 2012, 137(1):27-43.

    MathSciNet  Google Scholar 

  13. Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. ICM 90 Satellite Proceedings. In Harmonic Analysis. Edited by: Igari S. Springer, Tokyo; 1991:183-189.

    Google Scholar 

  14. Nakai E: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 1994, 166: 95-103. 10.1002/mana.19941660108

    MathSciNet  Article  Google Scholar 

  15. Sawano Y, Sugano S, Tanaka H: A note on generalized fractional integral operators on generalized Morrey spaces. Bound. Value Probl. 2009., 2009: Article ID 835865

    Google Scholar 

  16. Softova L: Singular integrals and commutators in generalized Morrey spaces. Acta Math. Sin. Engl. Ser. 2006, 22(3):757-766. 10.1007/s10114-005-0628-z

    MathSciNet  Article  Google Scholar 

  17. Bary NK, Stechkin SB: Best approximations and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483-522. (in Russian)

    Google Scholar 

  18. Guseinov AI, Mukhtarov KS: Introduction to the Theory of Nonlinear Singular Integral Equations. Nauka, Moscow; 1980. (in Russian)

    Google Scholar 

  19. Karapetiants NK, Samko NG:Weighted theorems on fractional integrals in the generalized Hölder spaces H 0 ω (ρ) via the indices m ω and M ω . Fract. Calc. Appl. Anal. 2004, 7(4):437-458.

    MathSciNet  Google Scholar 

  20. Samko N: Singular integral operators in weighted spaces with generalized Hölder condition. Proc. A. Razmadze Math. Inst. 1999, 120: 107-134.

    MathSciNet  Google Scholar 

  21. Samko N: On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class. Real Anal. Exch. 2004/2005, 30(2):727-745.

    MathSciNet  Google Scholar 

  22. Kokilashvili V, Samko S: Operators of harmonic analysis in weighted spaces with non-standard growth. J. Math. Anal. Appl. 2009, 352: 15-34. 10.1016/j.jmaa.2008.06.056

    MathSciNet  Article  Google Scholar 

  23. Samko N, Samko S, Vakulov B: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 2007, 335: 560-583. 10.1016/j.jmaa.2007.01.091

    MathSciNet  Article  Google Scholar 

  24. Carro M, Pick L, Soria J, Stepanov VD: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 2001, 4(3):397-428.

    MathSciNet  Google Scholar 

  25. Lu SZ: A class of oscillatory integrals. Int. J. Appl. Math. Sci. 2005, 2(1):42-58.

    Google Scholar 

  26. Lu SZ, Ding Y, Yan DY: Singular Integrals and Related Topics. World Scientific, Singapore; 2007.

    Book  Google Scholar 

  27. Coifman R, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103(2):611-635.

    MathSciNet  Article  Google Scholar 

  28. Chiarenza F, Frasca M, Longo P:Interior W 2 , p -estimates for nondivergence elliptic equations with discontinuous coefficients. Ric. Mat. 1991, 40: 149-168.

    MathSciNet  Google Scholar 

  29. Fazio GD, Ragusa MA: Interior estimates in Morrey spaces for strong solutions to nodivergence form equations with discontinuous coefficients. J. Funct. Anal. 1993, 112: 241-256. 10.1006/jfan.1993.1032

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ahmet Eroglu.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Eroglu, A. Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces. Bound Value Probl 2013, 70 (2013). https://doi.org/10.1186/1687-2770-2013-70

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-70

Keywords

  • generalized Morrey space
  • oscillatory integral
  • commutator
  • BMO spaces