Journal of Inequalities and Applications Volume 2011, Article ID 479576,14pages doi:10.1155/2011/479576
Research Article
Hypersingular Marcinkiewicz Integrals along
Surface with Variable Kernels on Sobolev Space
and Hardy-Sobolev Space
Wei Ruiying and Li Yin
School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China
Correspondence should be addressed to Wei Ruiying,[email protected]
Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011
Academic Editor: Andrei Volodin
Copyrightq2011 W. Ruiying and L. Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let α ≥ 0, the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by μΦΩ,αfx
0∞||y|≤tΩx, y/|y|n−1fx−Φ|y|ydy|2
dt/t32α1/2, where Ωx, z ∈ L∞
Ê
n× Lq
Ë
n−1
withq > max{1,2n−1/n2α}. The authors prove that the operatorμΦΩ,αis bounded from Sobolev spaceLpαÊ
ntoLp
Ê
nspace for 1< p ≤2, and from Hardy-Sobolev spaceHp αÊ
nto
Lp
Ê
nspace forn/nα< p≤1. As corollaries of the result, they also prove the ˙L2
αRn−L2Rn boundedness of the Littlewood-Paley type operatorsμΦΩ,α,S andμ∗Ω,Φ,α,λwhich relate to the Lusin area integral and the Littlewood-Paleygλ∗function.
1. Introduction
Let Ê
n n ≥ 2 be the n-dimensional Euclidean space and Ë
n−1 be the unit sphere in Ê
n
equipped with the normalized Lebesgue measuredσdσ·. Forx∈Ê
n \ {0}, letxx/|x|. Before stating our theorems, we first introduce some definitions about the variable kernelΩx, z. A functionΩx, zdefined onÊ
n× Ê
nis said to be inL∞ Ê
n×Lq Ë
n−1,q≥1, ifΩx, zsatisfies the following two conditions:
1 Ωx, λz Ωx, z, for anyx, z∈Ê
n and anyλ >0;
2ΩL∞Ên×L q
Ën−1supr≥0, y∈ Ên
Ën−1|Ω
rzy, z|qdσz1/q<∞.
problem about the second-order linear elliptic equations with variable coefficients. In 2002, Tang and Yang 2 gave Lp boundedness of the singular integrals with variable kernels associated to surfaces of the form{x Φ|y|y}, whereyy/|y|for anyy∈Ê
n\ {0}n≥ 2. That is, they considered the variable Calder ´on-Zygmund singular integral operator TΩΦ
defined by
TΩΦfx p·v·
Ên
Ωx, y yn f
x−Φyydy. 1.1
On the other hand, as a related vector-valued singular integral with variable kernel, the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form
{x Φ|y|y}is considered. It is defined by
μΦΩfx
∞
0
FΩΦ,tx2dt t3
1/2
, 1.2
where
FΩΦ,tx
|y|≤t
Ωx, y yn−1 f
x−Φyydy, 1.3
Ën−1
Ωx, zdσz0. 1.4
If Φ|y| |y|, we put μΦΩ μΩ. Historically, the higher dimension Marcinkiewicz
integral operatorμΩ with convolution kernel, that isΩx, z Ωz, was first defined and
studied by Stein3 in 1958. See also4–6 for some further works onμΩ with convolution
kernel. Recently, Xue and Yabuta 7 studied theL2 boundedness of the operatorμΦ Ω with
variable kernel.
Theorem 1.1see7 . Suppose thatΩx, yis positively homogeneous inyof degree 0, and satisfies 1.4and
2supy∈ Ê
n
Ën−1|Ω
y, z|qdσz1/q < ∞, for someq > 2n−1/n. LetΦbe a positive and monotonic (or negative and monotonic)C1 function on 0,∞ and let it satisfy the following conditions:
iδ≤ |Φt/tΦt| ≤Mfor some0< δ≤M <∞; ii Φ
tis monotonic on0,∞.
Then there is a constant C such thatμΦΩf2≤Cf2, where constantCis independent off.
Since the condition 2 implies2, so the L2Ê
nboundedness ofμΦ
Ω holds if Ω ∈
L∞Ê
n×Lq Ë
n−1withq >2n−1/n.
LpαÊ
nfor 1< p≤2 and the homogeneous Hardy-Sobolev spaceHp αÊ
nfor somen/nα<
p≤1. LetFΩΦ,txbe as above, we then define the operatorsμΦΩ,αby
μΦΩ,αfx
∞
0
FΩΦ,tx2 dt t32α
1/2
, α≥0. 1.5
Our main results are as follows.
Theorem 1.2. Suppose thatα ≥ 0, Ωx, ysatisfies1.4andΩ ∈L∞Ê
n×Lq Ë
n−1with q > max{1,2n−1/n2α}. LetΦbe a positive and increasingC1function on0,∞and let it satisfy the following conditions:
i Φt tΦt;
ii0≤Φt≤Won0,∞.
Then there is a constant C such thatμΦΩ,αfL2
Ê
n≤CfL2
αÊn, where constantCis independent off.
Theorem 1.3. Suppose0 < α < n/2, and thatΩ ∈ L∞Ê
n×LqSn−1, withq > max{1,2n− 1/n2α}, and satisfies1.4. LetΦbe a positive and increasingC1function on0,∞and let it satisfy the following conditions:
i Φt tΦt;
ii0<Φt≤1,Φ0 0.
Then, forn/nα< p ≤1, there is a constant C such thatμΦΩ,αf
Lp Ên
≤CfHp αÊn, where constantCis independent of anyf ∈HαpÊ
n∩ S Ê
n.
Furthermore, our result can be extended to the Littlewood-Paley type operatorsμΦΩ,α,S
andμ∗,ΩΦ,α,λwith variable kernels and indexα≥0, which relate to the Lusin area integral and the Littlewood-Paleygλ∗ function, respectively. LetFΦΩ,txbe as above, we then define the operatorsμΦΩ,α,Sandμ∗,ΩΦ,α,λforf∈ SÊ
n, respectively by
μΦΩ,α,Sfx
Γx
FΦ
Ω,t
y2 dydt tn32α
1/2
,
μ∗,ΩΦ,α,λfx ⎛
⎝
Ê n1
t tx−y
λn
FΦ
Ω,t
y2 dydt tn32α
⎞ ⎠
1/2
,
1.6
withλ >1, whereΓx {y, t∈ Ê n1
: |x−y| < t}. As an application ofTheorem 1.2, we
have the following conclusion.
Theorem 1.4. Under the assumption of Theorem 1.2, thenTheorem 1.2 still holds for μΦΩ,α,S and
μ∗,ΩΦ,α,λ.
Corollary 1.5. Suppose0< α < n/2, and thatΩ∈L∞Ê
n×LqSn−1,q >max{1,2n−1/n 2α}, and satisfies1.4. LetΦbe given as inTheorem 1.3. Then, for1< p≤2, there exists an absolute positive constantCsuch that
μΦΩ,αf
Lp Ê
n ≤CfLpαÊn
, 1.7
for allf∈LpαÊ n∩ S
Ê n.
Remark 1.6. It is obvious that the conclusions ofTheorem 1.2 are the substantial improve-ments and extensions of Stein’s results in 3 about the Marcinkiewicz integral μΩ with
convolution kernel, and of Ding’s results in 8 about the Marcinkiewicz integralμΩ with variable kernels.
Remark 1.7. Recently, the authors in 9 proved the boundedness of hypersingular Marcinkiewicz integral with variable kernels on homogeneous Sobolev space LpαRn for 1< p≤2 and 0< α <1 without any smoothness onΩ. SoCorollary 1.5extended the results in9, Theorem 5 .
Throughout this paper, the letterCalways remains to denote a positive constant not necessarily the same at each occurrence.
2. The Bounedness on Sobolev Spaces
Before giving the definition of the Sobolev space, let us first recall the Triebel-Lizorkin space. Fix a radial functionϕx ∈ C∞ satisfying suppϕ ⊆ {x : 1/2 < |x| ≤ 2} and 0 ≤
ϕx≤1, andϕx> c >0 if 3/5≤ |x| ≤5/3. Letϕjx ϕ2jx. Define the functionψjxby
Fψjξ ϕjξ, such thatFψj∗fξ Ffξϕjξ.
For 0< p, q <∞, andα∈Ê, the homogeneous Triebel-Lizorkin space ˙F α,q
p is the set of all distributionsfsatisfying
˙
Fpα,qÊ n
⎧ ⎪ ⎨ ⎪ ⎩f∈ S
Ê
n:f ˙ Fα,qp
k
2−αkψk∗f q1/q
p
<∞ ⎫ ⎪ ⎬ ⎪
⎭. 2.1
Forp ≥ 1, the homogeneous Sobolev spacesLpαÊ
nis defined byLp αÊ
n F˙α,2 p Ê
n, namelyfLp
α fF˙pα,2. From10 we know that for anyf∈L 2 αÊ
n
f
L2
αÊ n∼
Ên
F
fξ2|ξ|2αdξ 1/2
, 2.2
and ifαis a nonnegative integer, then for anyf∈LpαÊ n
f
LpαÊn
∼
|τ|α
Dτf
For 0< p≤1, we define the homogeneous Hardy-Sobolev spaceHαpÊ
nbyHp αÊ
n ˙
Fpα,2Ê
n. It is well known thatHp Ê
n F˙0,2 p Ê
nfor 0 < p ≤ 1, one can refer10 for the details.
Next, let us give the main lemmas we will use in proving theorems.
Lemma 2.1 see 11 . Suppose that n ≥ 2 and f ∈ L1Ê
n∩ L2 Ê
n has the form fx
f0|x|Px wherePx is a solid spherical harmonic polynomial of degree m. Then the Fourier transform offhas the formFfx F0|x|Px, where
F0r 2πi−mr−n2m−2/2
∞
0
f0sJn2m−2/22πrssn2m/2ds, 2.4
andr |ξ|,Jmsis the Bessel function.
Lemma 2.2see12 . Forλ n−2/2, and−λ ≤ α ≤ 1, there existsC > 0such that for any
h≥0andm1,2, . . .,
h
0
Jmλt
tλα dt
≤
C
mλα. 2.5
Lemma 2.3. Letα≥0,λ n−2/2,Φis aC1function on0,∞and let it satisfy the conditions (i) and (ii) inTheorem 1.2.
Denotegαfx
∞
0 |Nεfx|
2dε/ε12α1/2, if
FNεf
ξ
Φε|ξ|
0
Jmλt
tλ1 dt· F
fξ. 2.6
Then there exists a constantCindependent ofm, such thatgαfL2 ≤ C/mλ1αfL2
α for every integerm∈Æ, m > α.
Proof. Letη|x| 0|x|Jmλt/tλ1dt, then we have
gα f22
Ê n
∞
0
Nεfx2 dε
ε12αdx
∞
0
Ên
ηΦε|ξ|F
fξ2dξ dε ε12α
∞
0
Ên
η
Φ
β
|ξ|
|ξ|
Ffξ
2
|ξ|2αdξ dβ β12α
Ên
∞
0
η
Φ
β
|ξ|
|ξ|
2 dβ
β12αF
fξ2|ξ|2αdξ.
2.7
ForI2, by usingLemma 2.2andΦt tΦt, we can get
I2
∞
m/2
Φβ/|ξ||ξ|
0
Jmλt
tλ1 dt
2
dβ β12α
≤ C
m2λ2
∞
m/2
dβ β12α
≤ C
m2λ22α.
2.8
For the other partI1, applying Stirling’s formula, we have
√
2πxx−1/2e−x ≤Γx≤2√2πxx−1/2e−x. 2.9
Also in13 , the authors proved the following inequality
|Jνt| ≤
t/2ν
Γν1. 2.10
So by2.9and2.10, 0≤α <α 1≤m, and noting thatΦt≤Wt, we have
I1
m/2
0
Φβ/|ξ||ξ|
0
Jmλt
tλ1 dt
2
dβ β12α
≤ m/2
0
Φβ/|ξ||ξ|
0
|Jmλt|
tλ1 dt
2
dβ β12α
≤ 1
22m2λΓ2mλ1
m/2
0
Φβ/|ξ||ξ|
0
tmλ
tλ1dt
2
dβ β12α
≤ 1
22m2λΓ2mλ1
m/2
0
Φβ |ξ|
2m dβ
β12α
≤ e2m2λ2
2π22m2λmλ12m2λ1
m/2
0 Φ
β
|ξ|
2m dβ
β12α−2m
≤C e
2m2λ2
2π22m2λmλ12m2λ1
m/2
0
dβ β12α−2m
≤Ce
4
2m2λ2 1
m2α2λ2
≤ C
m2α2λ2.
2.11
Proof ofTheorem 1.2. The basic idea of proof can go back to14 , for recently papers, one see8,
15 . By the same argument as in1 , let{Ym,j} m≥1, j 1,2, . . . , Dmdenote the complete system of normalized surface spherical harmonics. See14 for instance, we can decompose Ωx, yas following:
Ωx, y
∞
m1 Dm
j1
am,jxYm,j
yis a finite sum. 2.12
Denote
amx
⎛ ⎝Dm
j1
am,jx2
⎞ ⎠
1/2
, bm,jx
am,jx
amx
, 2.13
then we get
Dm
j1
b2m,jx 1, Ωx, y
∞
m1
amx Dm
j1
bm,jxYm,j
y. 2.14
Then, applying H ¨older inequality twice, we have for any 0< ε <1 that
μΦ
Ω,αfx 2
∞
0
|y|≤t
∞
m1
bm,jx
Ym,j
y yn−1 f
x−Φyydy
2
dt t32α
≤ ∞
m1
a2mxm−ε12α ∞
m1
mε12α
× ∞
0
|y|≤t Dm
j1
bm,jx
Ym,j
y yn−1 f
x−Φyydy
2
dt t32α
≤ ∞
m1
a2
mxm−ε12α
∞
m1
mε12α
∞
0
⎛ ⎝Dm
j1
b2 m,jx
⎞ ⎠
×Dm
j1
|y|≤t
Ym,j
y yn−1 f
x−Φyydy
2
dt t32α
∞ m1
a2mxm−ε12α
∞
m1
mε12α
× ∞
0 Dm
j1
|y|≤t
Ym,j
y yn−1 f
x−Φyydy
2
dt t32α.
By14, page 230, equation4.4 , we can observe that the series in the first parenthesis on the right-hand side of the inequality above, for eachxfixed, is equal toΩx,·2L2
−γËn−1 , whereL2
−γË
n−1is the Sobolev space on Ë
n−1withγ ε1/2 αfor any 0< ε <1. So if we takeεsufficiently close to 1, then by the Sobolev imbedding theoremLq⊂L2
−γ, we have
m
a2
mxm−ε12α
1/2
≤CΩL∞Ên×L q
Ën−1:CΩ 2.16
withq >max{1,2n−1/n2α}.
By Fourier transform and2.16, we get
μΦΩ,α
f2
2≤CΩ 2∞
m1
mε12α ∞ 0 Ên Dm
j1
|y|≤t Ym,j y yn−1 f
x−Φyydy
2
dx dt
t32α
≤CΩ2
∞
m1
mε12α
Dm
j1
∞ 0 Ên F |y|≤t Ym,j y yn−1 f
· −Φyydy
ξ 2 dξ dt
t32α
:CΩ2
∞
m1
mε12αDm j1
μΦΩ,j,αf2
2.
2.17
ForμΦΩ,j,αf, we have
μΦ
Ω,j,αf 2 2 ∞ 0 Ên |y|≤t Ên Ym,j y yn−1 f
x−Φyye−2πix·ξdx dy
2
dξ dt
t32α
∞ 0 Ên |y|≤t Ym,j y
yn−1 e−2πiΦ|y|y
·ξ
×
Ên
fx−Φyye−2πix−Φ|y|y·ξdx dy
2
dξ dt
t32α
∞ 0 Ê n |y|≤t Ym,j y
yn−1 e−2πiΦ|y|y
·ξ dy
2
F
fξ2dξ dt t32α
Ê n ∞ 0 1
t1α
|y|≤t
Ym,j
y
yn−1 e−2πiΦ|y|y
·ξ dy
2
dt t F
fξ2dξ.
For the integral on the right hand side of the above inequality, by changing of variable, we can get
1
t1α
|y|≤t
Ym,j
y yn−1 e
−2πiΦ|y|y·ξdy
1
t1α
t
0
Ë n−1
Ym,j
ye−2πiΦsy·ξdyds
1
t1α
Φt
0
Ë n−1
Ym,j
ye−2πiγ y·ξΦ−1γdydγ
1
t1α
|y|≤Φt
Ym,j
y yn−1 e
−2πiy·ξΦ−1ydy.
2.19
So we have
μΦΩ,j,αf2
2
Ên
∞
0
t11α
|y|≤Φt
Ym,j
y yn−1 e
−2πiy·ξΦ−1ydy
2
dt t F
fξ2dξ.
2.20
PutPm,jx Ym,jx|x|mandϕΦt,α,m,jx Pm,jx· |x|−n−m1χ|x|≤ΦtxΦ−1|x|t−1−α, we can deduce fromLemma 2.1that
FϕΦt,α,m,jξ Pm,j|ξ|·F0|ξ| Ym,j
ξ· |ξ|mF0|ξ|, 2.21
where
F0r 2πi−mr−n/2−m1
Φt
0
t−1−αs−n−m1Φ−1sJn/2m−12πrssn/2mds
2πi−mr−n/2−m1t−1−α Φt
0
s−n/21Jn/2m−12πrsd
Φ−1s
2πi−mr−n/2−m1t−1−α t
0
Jn/2m−1
2πrΦβ
Φβn/2−1 dβ
2πn/2i−mr−mt
−α
t t
0
Jn/2m−1
2πrΦβ
2πrΦβn/2−1 dβ.
Hence, we have
μΦΩ,j,αf2
2
∞
0
Ên
ϕΦt,α,m,j∗fx 2
dxdt t
∞
0
Ên
FϕΦt,α,m,j∗fξ2dξdt t
≤
Ên
∞
0
Ym,j
ξ|ξ|mi−m|ξ|−m2πn/2t
−α
t t
0
Jn/2m−1
2π|ξ|Φβ
2π|ξ|Φβn/2−1 dβ
2
dt t F
fξ2dξ
≤C
Ên
∞
0
Ym,j
ξ1
t t
0
Jn/2m−12π|ξ|Φβ
2π|ξ|Φβn/2−1 dβ
2
dt t12αF
fξ2dξ.
2.23
By14 , we know thatDmj1|Ym,jz|2∼mn−2. So we can get
Dm
j1
μΦ
Ω,j,αf 2 2≤Cm
n−2
Ên
∞
0
1t
t
0
Jn/2m−1
2π|ξ|Φβ
2π|ξ|Φβn/2−1 dβ
2
dt t12αF
fξ2dξ.
2.24
Setλ n/2−1, ρ2π|ξ|Φβand note thatΦt tΦt, we can deduce that
U: 1
t t
0
Jn/2m−12π|ξ|Φβ
2π|ξ|Φβn/2−1 dβ
1
t
2π|ξ|Φt
0
Jmλ
ρ ρλ
1
2π|ξ|ΦΦ−1ρ/2π|ξ|dρ
1
t
2π|ξ|Φt
0
Jmλ
ρ ρλ1 Φ
−1 ρ 2π|ξ|
dρ.
2.25
Noting thatΦtis increasing, by using the second mean-value theorem, we get, for some 0≤η <2π|ξ|Φt,
|U| ≤1 tΦ
−1Φt2π|ξ|Φt η
Jmλ
ρ ρλ1 dρ
≤
2π|ξ|Φt
0
Jmλ
ρ ρλ1 dρ
.
From2.26, it follows that
Dm
j1
μΦm,j,α
f2
2 ≤Cm n−2
Ên
∞
0
2π|ξ|Φt
0
Jmλ
ρ ρλ1 dρ· F
fξ
2
dt
t12αdξ. 2.27
Thus usingLemma 2.3, we can deduce the desired conclusion ofTheorem 1.2.
Proof ofTheorem 1.4. First, we know thatμΦΩ,α,Sfx≤2λnμ∗,Φ
Ω,α,λfx. On the other hand,
μ∗,ΩΦ,α,λf2
2
Ên
Rn1
t tx−y
λn
1t
|z|≤t
Ωx, z |z|n−1 f
x−Φ|z|zdz
2
dzdt tn12αdx
∞
0
Ên
⎛ ⎝1
tn
Ên
t tx−y
λn
dx ⎞ ⎠
1
t
|z|≤t
Ωx, z |z|n−1 f
x−Φ|z|zdz
2
dzdt t12α
≤CμΦΩ,αf2
2.
2.28
Thus, usingTheorem 1.2, we can finishTheorem 1.4.
3. The Bounedness on Hardy-Sobolev Spaces
In order to prove the boundedness for operator μΦΩ,α on Hardy-Sobolev spaces and prove
Theorem 1.3, we first introduce a new kind of atomic decomposition for Hardy-Sobolev space as following which will be used next.
Definition 3.1see16 . Forα≥0, the functionaxis called ap,2, αatom if it satisfies the following three conditions:
1suppa⊂Bwith a ballB⊂Ê n;
2aL2
α ≤ |B|
1/2−1/p;
3Ên
axPx 0, for any polynomialPxof degree≤N n1/p−1α .
By 16 , we have that everyf ∈ HαpÊ
ncan be written as a sum ofp,2, α atoms
ajx, that is,
f
j
Proof ofTheorem 1.3. Similar to the argument of Lemma 3.3 in17 and using above atomic decomposition, it suffices to show that
μΦ
Ω,αa p
Lp ≤C, 3.2
with the constantCindependent of anyp,2, αatoma. Assume suppa⊂B0, R. We first note that
μΦ
Ω,αa p
Lp ≤
|x|≤8R
μΦ
Ω,αax p
dx
|x|>8R
μΦ
Ω,αax p
dx
:U1U2.
3.3
ForU1, usingTheorem 1.2, it is not difficult to deduce that
U1≤CμΦΩ,αa p
L2R
n1−p/2≤Cap L2
αR
n1−p/2
≤CRnp/2−1Rn1−p/2≤C.
3.4
ForU2, we first consider the casen/nα< p <1, according to15, Lemma 5.5 , for 0< α < n/2 andp,2, αatomawith supportBB0, R, one has
B
|ax|dx≤CRn−n/pα. 3.5
Using Minkowski inequality and H ¨older inequality for integrals, and3.5, we can get
U2
|x|>8R
μΦ
Ω,αax p
dx
|x|>8R
⎛ ⎝∞
0
|y|≤t
Ωx, y yn−1 a
x−Φyydy
2
dt t32α
⎞ ⎠
p/2
dx
≤
|x|>8R
Ê n
Ω
x, y
ynα a
x−Φyydy
p
dx.
For the integral on the right hand side of the above inequality, by changing of variable and noting that 0<Φt≤1,Φ0 0, we can get
Ê n
Ω
x, y
ynα a
x−Φyydy
Ën−1
R
0
Ω x, y r1α a
x−Φrydr dy
Ë n−1
ΦR
0
Ω x, y
Φ−1γ1αa
x−γ y 1
ΦΦ−1
γdγ dy
Ën−1
ΦR
0
Ω x, y
Φ−1γ1αa
x−γ yΦ
−1γ
γ dγ dy
Ën−1
ΦR
0
Ω x, y
Φ−1γα
γa
x−γ ydγ dy
|y|≤ΦR
Ω
x, y
ynΦ−1
yαa
x−ydy
|x−y|≤ΦR
Ω
x, x−y x−ynΦ−1
x−yαa
ydy
≤
|x−y|≤ΦR
Ω
x, x−y x−ynα a
ydy.
3.7
By3.7, we can get
U2≤
∞
j3
2jR<|x|<2j1R
Ên
Ω
x, x−y x−ynα a
ydy
p
dx
≤∞
j3
2jRn1−p
2jR<|x|<2j1R
Ên
|Ωx, x−y| x−ynα a
ydy dx
p
≤∞
j3
2jRn1−p
B
ay
2jR<|x|<2j1R Ω
x, x−y x−ynα dx dy
p
≤CΩpL∞×L1
B
a
ydy
p
·∞
j3
2jR−αp2jRn1−p.
3.8
Thus by3.5and the conditionp > n/nα,
U2≤CΩpL∞×L1
∞
j3
As forp1, similar to the argument ofn/nα< p <1, we can easily getU2≤C. So far the proof ofTheorem 1.3has been finished.
Acknowledgments
This project supported by the National Natural Science Foundation of China under Grant no. 10747141, Zhejiang Provincial National Natural Science Foundation of China under Grant no. Y604056, and Science Foundation of Shaoguan University under Grant no. 200915001.
References
1 A. P. Calder ´on and A. Zygmund, “On a problem of Mihlin,”Transactions of the American Mathematical Society, vol. 78, pp. 209–224, 1955.
2 L. Tang and D. Yang, “Boundedness of singular integrals of variable rough Calder ´on-Zygmund kernels along surfaces,”Integral Equations and Operator Theory, vol. 43, no. 4, pp. 488–502, 2002. 3 E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,”Transactions of the
American Mathematical Society, vol. 88, pp. 430–466, 1958.
4 X. X. Tao and R. Y. Wei, “Boundedness of commutators related to Marcinkiewicz integrals with variable kernels in Herz-type Hardy spaces,”Acta Mathematica Scientia, vol. 29, no. 6, pp. 1508–1517, 2009.
5 D. Fan and S. Sato, “Weak type1,1estimates for Marcinkiewicz integrals with rough kernels,”The Tohoku Mathematical Journal, vol. 53, no. 2, pp. 265–284, 2001.
6 M. Sakamoto and K. Yabuta, “Boundedness of Marcinkiewicz functions,”Studia Mathematica, vol. 135, no. 2, pp. 103–142, 1999.
7 Q. Xue and K. Yabuta, “L2-boundedness of Marcinkiewicz integrals along surfaces with variable
kernels: another sufficient condition,” Journal of Inequalities and Applications, vol. 2007, Article ID 26765, 14 pages, 2007.
8 Y. Ding, C.-C. Lin, and S. Shao, “On the Marcinkiewicz integral with variable kernels,” Indiana University Mathematics Journal, vol. 53, no. 3, pp. 805–821, 2004.
9 J. Chen, Y. Ding, and D. Fan, “Littlewood-Paley operators with variable kernels,”Science in China Series A, vol. 49, no. 5, pp. 639–650, 2006.
10 M. Frazier, B. Jawerth, and G. Weiss,Littlewood-Paley Theory and the Study of Function Spaces, vol. 79 ofCBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 1991.
11 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, USA, 1971, Princeton Mathematical Series, No. 3.
12 Y. Ding and R. Li, “An estimate of Bessel function and its application,”Science in China Series A, vol. 38, no. 1, pp. 78–87, 2008.
13 N. E. Aguilera and E. O. Harboure, “Some inequalities for maximal operators,”Indiana University Mathematics Journal, vol. 29, no. 4, pp. 559–576, 1980.
14 A.-P. Calder ´on and A. Zygmund, “On singular integrals with variable kernels,”Applicable Analysis, vol. 7, no. 3, pp. 221–238, 1977-1978.
15 J. Chen, D. Fan, and Y. Ying, “Certain operators with rough singular kernels,”Canadian Journal of Mathematics, vol. 55, no. 3, pp. 504–532, 2003.
16 Y.-S. Han, M. Paluszy ´nski, and G. Weiss, “A new atomic decomposition for the Triebel-Lizorkin spaces,” inHarmonic Analysis and Operator Theory (Caracas, 1994), vol. 189 ofContemporary Mathematics, pp. 235–249, American Mathematical Society, Providence, RI, USA, 1995.
