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

Boundary Value Problems20132013:70

DOI: 10.1186/1687-2770-2013-70

Received: 25 July 2012

Accepted: 17 March 2013

Published: 2 April 2013

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq1_HTML.gif. 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 M p , λ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq2_HTML.gif are defined as the set of all functions f L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq3_HTML.gif 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 ) ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equa_HTML.gif

Under this definition, M p , λ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq2_HTML.gif becomes a Banach space; for λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq4_HTML.gif, it coincides with L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif and for λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq6_HTML.gif with L ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq7_HTML.gif.

We also denote by W M p , λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq8_HTML.gif the weak Morrey space of all functions f W L p loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq9_HTML.gif 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 ) ) < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equb_HTML.gif

where W L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq10_HTML.gif denotes the weak L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq11_HTML.gif-space.

Definition 1 Let φ ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq12_HTML.gif be a positive measurable function on R n × ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq13_HTML.gif and 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq14_HTML.gif. We denote by M p , φ M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq15_HTML.gif the generalized Morrey space, the space of all functions f L p loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq16_HTML.gif 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 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equc_HTML.gif
Also, by W M p , φ W M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq17_HTML.gif, we denote the weak generalized Morrey space of all functions f W L p loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq9_HTML.gif 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 ) ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equd_HTML.gif
According to this definition, we recover the spaces M p , λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq18_HTML.gif and W M p , λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq19_HTML.gif under the choice φ ( x , r ) = r λ n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq20_HTML.gif:
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Eque_HTML.gif
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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq21_HTML.gif. The fractional maximal operator M α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq22_HTML.gif and the Riesz potential I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq23_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equf_HTML.gif

If α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq24_HTML.gif, then M M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq25_HTML.gif is the Hardy-Littlewood maximal operator. In [2], Chiarenza and Frasca obtained the boundedness of M on M p , λ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq26_HTML.gif. In [3], Adams established the boundedness of I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq23_HTML.gif on M p , λ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq26_HTML.gif.

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 ( x y ) f ( y ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ1_HTML.gif
(1.1)
where K is a Calderón-Zygmund kernel (CZK). We say a kernel K C 1 ( R n { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq27_HTML.gif is a CZK if it satisfies
| K ( x ) | C | x | n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ2_HTML.gif
(1.2)
| K ( x ) | C | x | n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ3_HTML.gif
(1.3)
and
a < | x | < b K ( x ) d x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ4_HTML.gif
(1.4)

for all a, b with 0 < a < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq28_HTML.gif. Chiarenza and Frasca [2] showed the boundedness of T ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq29_HTML.gif on M p , λ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq26_HTML.gif.

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]
T f ( x ) = p . v . R n e i P ( x , y ) K ( x y ) f ( y ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ5_HTML.gif
(1.5)

where P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq30_HTML.gif is a real valued polynomial defined on R n × R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq31_HTML.gif, and K is a CZK.

It is well known that the oscillatory factor e i P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq32_HTML.gif makes it impossible to establish the L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq11_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq33_HTML.gif 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 , x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ6_HTML.gif
(1.6)
and
| x K ( x , y ) | + | y K ( x , y ) | C | x y | n + 1 , x y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ7_HTML.gif
(1.7)
The corresponding Calderón-Zygmund integral operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif and oscillatory integral operator S are defined by
S ˜ f ( x ) = p . v . R n K ( x , y ) f ( y ) d y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ8_HTML.gif
(1.8)
and
S f ( x ) = p . v . R n e i P ( x , y ) K ( x , y ) f ( y ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ9_HTML.gif
(1.9)
where P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq30_HTML.gif is a real valued polynomial defined on R n × R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq31_HTML.gif. In [7], Lu and Zhang proved that S is bounded on L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq11_HTML.gif with 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq35_HTML.gif. In [5], Ricci and Stein also introduced the standard fractional Calderón-Zygmund kernel (SFCZK) K α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq36_HTML.gif with 0 < α < n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq37_HTML.gif, where the conditions (1.6) and (1.7) were replaced by
| K α ( x , y ) | C | x y | n α , x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ10_HTML.gif
(1.10)
and
| x K α ( x , y ) | + | y K α ( x , y ) | C | x y | n + 1 α , x y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ11_HTML.gif
(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 ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ12_HTML.gif
(1.12)

where P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq30_HTML.gif is also a real valued polynomial defined on R n × R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq31_HTML.gif. Obviously, when α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq38_HTML.gif, S 0 = S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq39_HTML.gif and K 0 = K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq40_HTML.gif. 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 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq41_HTML.gif, and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfies the condition
r ess sup t < s < φ 1 ( x , s ) s n p t n p + 1 d t C φ 2 ( x , r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ13_HTML.gif
(1.13)

where C does not depend on x and t. If K is a SCZK and the operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif, then for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq44_HTML.gif and any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif the operator S is bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M p , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq47_HTML.gif.

Moreover, for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif and K is a CZK operator, the operator T is bounded from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M 1 , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq50_HTML.gif.

Theorem 1.2 Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq51_HTML.gif, 0 < α < n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq52_HTML.gif, 1 q = 1 p α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq53_HTML.gif, P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif is a polynomial, and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfies the condition
r ess sup t < s < φ 1 ( x , s ) s n p t n q + 1 d t C φ 2 ( x , r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ14_HTML.gif
(1.14)

where C does not depend on x and t. Then for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq54_HTML.gif the operator S α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq55_HTML.gif is bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq56_HTML.gif to M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq57_HTML.gif and for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif the operator S α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq55_HTML.gif is bounded from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq58_HTML.gif.

For a locally integrable function b, the commutator operator formed by S (or S α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq55_HTML.gif) and b are defined by
S b f ( x ) = b ( x ) S f ( x ) S ( b f ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equg_HTML.gif
and
S α , b f ( x ) = b ( x ) S α f ( x ) S α ( b f ) ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equh_HTML.gif
Theorem 1.3 Let 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq35_HTML.gif, b BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq59_HTML.gif and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfies the condition
r ( 1 + ln t r ) ess sup t < s < φ 1 ( x , s ) s n p t n p + 1 d t C φ 2 ( x , r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ15_HTML.gif
(1.15)

where C does not depend on x and t. If K is a SCZK and the operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif, then for any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif the operator S b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq60_HTML.gif is bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M p , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq47_HTML.gif.

Theorem 1.4 Let 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq61_HTML.gif, b BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq59_HTML.gif, 0 < α < n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq52_HTML.gif, 1 q = 1 p α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq62_HTML.gif, P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif is a polynomial, and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfies the condition
r ( 1 + ln t r ) ess sup t < s < φ 1 ( x , s ) s n p t n q + 1 d t C φ 2 ( x , r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ16_HTML.gif
(1.16)

where C does not depend on x and t. Then the operator S b , α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq63_HTML.gif is bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq57_HTML.gif.

2 Some known results in generalized Morrey spaces M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq64_HTML.gif

In [9, 10, 12, 13] and [14], there were obtained sufficient conditions on weights φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq65_HTML.gif and φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq66_HTML.gif for the boundedness of the singular operator T from M p , φ 1 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq67_HTML.gif to M p , φ 2 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq68_HTML.gif.

The following statements were proved by Nakai [14].

Theorem A Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq14_HTML.gif and φ ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq12_HTML.gif satisfy the conditions
c 1 φ ( x , r ) φ ( x , t ) c φ ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ17_HTML.gif
(2.1)
whenever r t 2 r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq69_HTML.gif, where c (≥1) does not depend on t, r and x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq70_HTML.gif and
r φ ( x , t ) p d t t C φ ( x , r ) p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ18_HTML.gif
(2.2)

where C does not depend on x and r. Then for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq54_HTML.gif the operators M and T are bounded in M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq71_HTML.gif and for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif, M and T are bounded from M 1 , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq72_HTML.gif to W M 1 , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq73_HTML.gif.

Theorem B Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq14_HTML.gif, 0 < α < n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq52_HTML.gif, 1 q = 1 p α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq53_HTML.gif and φ ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq74_HTML.gif satisfy the conditions (2.1) and
r φ ( x , t ) p d t t C φ ( x , r ) p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ19_HTML.gif
(2.3)

where C does not depend on x and r. Then for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq54_HTML.gif, the operators M α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq22_HTML.gif and I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq75_HTML.gif are bounded from M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq71_HTML.gif to M q , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq76_HTML.gif and for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif, M α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq22_HTML.gif and I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq75_HTML.gif are bounded from M 1 , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq72_HTML.gif to W M q , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq77_HTML.gif.

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

Theorem C Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq78_HTML.gif and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfy the condition
t φ 1 ( x , r ) d r r C φ 2 ( x , t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ20_HTML.gif
(2.4)

where C does not depend on x and t. Then the operators M and T are bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M p , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq47_HTML.gif for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq79_HTML.gif and from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M 1 , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq50_HTML.gif.

Theorem D Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq78_HTML.gif, 0 < α < n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq80_HTML.gif, 1 q = 1 p α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq81_HTML.gif and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfy the condition
r t α φ 1 ( x , t ) d t t C φ 2 ( x , r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ21_HTML.gif
(2.5)

where C does not depend on x and r. Then the operators M α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq82_HTML.gif and I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq75_HTML.gif are bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq57_HTML.gif for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq79_HTML.gif and from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq58_HTML.gif for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif.

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

Theorem E Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq78_HTML.gif and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfy the condition (2.4). Then the operators M and T are bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M p , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq47_HTML.gif for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq79_HTML.gif and from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M 1 , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq50_HTML.gif.

Theorem F Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq78_HTML.gif, 0 < α < n p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq80_HTML.gif, 1 q = 1 p α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq81_HTML.gif and ( φ 1 , φ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq42_HTML.gif satisfy the condition (1.14). Then the operators M α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq82_HTML.gif and I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq75_HTML.gif are bounded from M p , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq46_HTML.gif to M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq57_HTML.gif for p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq79_HTML.gif and from M 1 , φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq49_HTML.gif to W M q , φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq58_HTML.gif for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif.

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 r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq83_HTML.gif. Such inequalities are also of interest when they allow to impose different conditions as r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq83_HTML.gif and r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq84_HTML.gif; such a case was dealt with in [22, 23].

3 The fractional oscillatory integral operators in the spaces M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq85_HTML.gif

In this section, we are going to use the following statement on the boundedness of the Hardy operator:
( H g ) ( t ) : = 1 t 0 t g ( r ) d r , 0 < t < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equi_HTML.gif

Theorem G [24]

The inequality
ess sup t > 0 w ( t ) H g ( t ) c ess sup t > 0 v ( t ) g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equj_HTML.gif
holds for all non-negative and non-increasing g on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq86_HTML.gif if and only if
A : = sup t > 0 w ( t ) t 0 t d r ess inf 0 < s < r v ( s ) < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equk_HTML.gif

and c A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq87_HTML.gif.

Lemma 3.1 Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq51_HTML.gif, and K is a SCZK and the Calderón-Zygmund singular integral operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif. Then for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq44_HTML.gif and any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif 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 d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equl_HTML.gif

holds for any ball B ( x 0 , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq88_HTML.gif and for all f L p loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq16_HTML.gif.

Moreover, for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif 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 d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ22_HTML.gif
(3.1)

holds for any ball B ( x 0 , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq88_HTML.gif and for all f L 1 loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq89_HTML.gif.

Proof Let p ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq90_HTML.gif. For arbitrary x 0 R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq91_HTML.gif, set B = B ( x 0 , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq92_HTML.gif for the ball centered at x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq93_HTML.gif and radius r, 2 B = B ( x 0 , 2 r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq94_HTML.gif. 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equm_HTML.gif
and have
S f L p ( B ) S f 1 L p ( B ) + S f 2 L p ( B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equn_HTML.gif
It is known that (see [5], see also [7, 25, 26]), if K is a SCZK and the operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif, then for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq44_HTML.gif and any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif the operator S is bounded on L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif. Since f 1 L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq95_HTML.gif, S f 1 L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq96_HTML.gif and boundedness of S in L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif (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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equo_HTML.gif

where constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq97_HTML.gif is independent of f.

It is clear that x B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq98_HTML.gif, y ( 2 B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq99_HTML.gif implies 1 2 | x 0 y | | x y | 3 2 | x 0 y | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq100_HTML.gif. We get
| S f 2 ( x ) | c 0 ( 2 B ) | f ( y ) | | x 0 y | n d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equp_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ23_HTML.gif
(3.2)
Moreover, for all p [ 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq101_HTML.gif the inequality
S f 2 L p ( B ) r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ24_HTML.gif
(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 d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equq_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ25_HTML.gif
(3.4)
Hence,
S f L p ( B ) r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equr_HTML.gif
Let p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif. From the weak ( 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq102_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ26_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equs_HTML.gif
if p ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq90_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equt_HTML.gif

if p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq48_HTML.gif. □

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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equu_HTML.gif

 □

4 Commutators of fractional oscillatory integral operators in the spaces M p , φ ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq85_HTML.gif

Let T be a Calderón-Zygmund singular integral operator and b BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq59_HTML.gif. A well known result of Coifman, Rochberg and Weiss [27] states that the commutator operator [ b , T ] f = T ( b f ) b T f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq103_HTML.gif is bounded on L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq35_HTML.gif. 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq104_HTML.gif.

Definition 2 Suppose that f L 1 loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq89_HTML.gif, let
f = sup x R n , r > 0 1 | B ( x , r ) | B ( x , r ) | f ( y ) f B ( x , r ) | d y < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equv_HTML.gif
where
f B ( x , r ) = 1 | B ( x , r ) | B ( x , r ) f ( y ) d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equw_HTML.gif
Define
BMO ( R n ) = { f L 1 loc ( R n ) : f < } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equx_HTML.gif

If one regards two functions whose difference is a constant as one, then space BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq104_HTML.gif is a Banach space with respect to norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq105_HTML.gif.

Remark 1 (1) The John-Nirenberg inequality: there are constants C 1 , C 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq106_HTML.gif, such that for all f BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq107_HTML.gif and β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq108_HTML.gif
| { x B : | f ( x ) f B | > β } | C 1 | B | e C 2 β / f , B R n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equy_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ27_HTML.gif
(4.1)

for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq109_HTML.gif.

(3) Let f BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq110_HTML.gif. Then there is a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq97_HTML.gif such that
| f B ( x , r ) f B ( x , t ) | C f ln t r for  0 < 2 r < t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ28_HTML.gif
(4.2)

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

Lemma 4.1 Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq51_HTML.gif, b BMO ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq59_HTML.gif, K is a SCZK and the Calderón-Zygmund singular integral operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif. Then for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq44_HTML.gif and any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif 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 d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equz_HTML.gif

holds for any ball B ( x 0 , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq88_HTML.gif and for all f L p loc ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq16_HTML.gif.

Proof Let p ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq90_HTML.gif. For arbitrary x 0 R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq91_HTML.gif, set B = B ( x 0 , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq92_HTML.gif for the ball centered at x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq93_HTML.gif and radius r, 2 B = B ( x 0 , 2 r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq94_HTML.gif. 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equaa_HTML.gif
and have
S b f L p ( B ) S b f 1 L p ( B ) + S b f 2 L p ( B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equab_HTML.gif
It is known that (see [5], see also [7, 25, 26]), if K is a SCZK and the operator S ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq34_HTML.gif is of type ( L 2 ( R n ) , L 2 ( R n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq43_HTML.gif, then for 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq44_HTML.gif and any polynomial P ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq45_HTML.gif the commutator operator S b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq60_HTML.gif is bounded on L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif. Since f 1 L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq95_HTML.gif, S f 1 L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq96_HTML.gif and boundedness of S b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq60_HTML.gif in L p ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq5_HTML.gif (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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equac_HTML.gif

where constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq97_HTML.gif is independent of f.

For x B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq111_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equad_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equae_HTML.gif
Let us estimate I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq112_HTML.gif.
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equaf_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equag_HTML.gif
In order to estimate I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq113_HTML.gif note that
I 2 = ( B | b ( x ) b B | p d x ) 1 p ( 2 B ) | f ( y ) | | x 0 y | n d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equah_HTML.gif
By (4.1), we get
I 2 b r n p ( 2 B ) | f ( y ) | | x 0 y | n d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equai_HTML.gif
Thus, by (3.2)
I 2 b r n p 2 r f L p ( B ( x 0 , t ) ) t 1 n p d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equaj_HTML.gif
Summing up I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq112_HTML.gif and I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq113_HTML.gif, for all p ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_IEq114_HTML.gif 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 d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equ29_HTML.gif
(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 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equak_HTML.gif

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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-70/MediaObjects/13661_2012_Article_417_Equal_HTML.gif

 □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Nigde University

References

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