Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 835938, 17 pages
doi:10.1155/2008/835938
Research Article
Boundedness on Hardy-Sobolev Spaces for
Hypersingular Marcinkiewicz Integrals with
Variable Kernels
Xiangxing Tao,1 Xiao Yu,1, 2 and Songyan Zhang1
1
Department of Mathematics, Faculty of Science, Ningbo University, Ningbo,
Zhejiang Province 315211, China
2
Department of Mathematics, Faculty of Science, Zhejiang University, Hangzhou,
Zhejiang Province 315211, China
Correspondence should be addressed to Songyan Zhang, zhangsongyan@nbu.edu.cn
Received 3 August 2008; Accepted 20 October 2008
Recommended by Nikolaos Papageorgiou
The existence and boundedness on Sobolev spaces and Hardy-Sobolev spaces for the hypersingular Marcinkiewicz integrals with variable kernels are derived.
Copyright q 2008 Xiangxing Tao et al. 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.
1. Introduction
The function Ωx, z defined on Rn × Rn is said to belong to L∞ Rn × Lq Sn−1 , if it satisfies
the following two conditions:
1 Ωx, λz Ωx, z, for any x, z ∈ Rn and any λ > 0;
1/q
2 ΩL∞ Rn ×Lq Sn−1 : supr≥0, y∈Rn Sn−1 |Ωrz y, z |q dσz < ∞.
Let α ≥ 0 and q > max{1, 2n − 1/n 2α}, and let Ω ∈ L∞ Rn × Lq Sn−1 satisfy the
following cancellation property:
Sn−1
Ω x, z Ym z dσ z 0,
1.1
for all spherical harmonic polynomials Ym z with degree ≤ α and for any x ∈ Rn . We
consider the hypersingular Marcinkiewicz integral μΩ,α fx defined by
2
Journal of Inequalities and Applications
∞ μΩ,α fx 0
Ωx, x − y
|x−y|≤t
|x − y|n−1
1/2
2
dt
fydy 32α
,
t
1.2
for f ∈ SRn , the Schwartz class. In general, the way in which the integrals were given
∞
1/2
has to be interpreted. Let H {ht : hH 0 |ht|2 dt/t
< ∞}, and let hf t, x n−1
−1−α
t
Ωx, x − y/|x − y| fydy. We have to show hf ·, x ∈ H for every x ∈ Rn and
|x−y|≤t
f ∈ S, and so that μΩ,α fx hf ·, xH can be regarded as a well-defined Hilbert-valued
function. To see this, applying the Taylor’s expansion, we can write
fx − y κ
|κ|≤α
:
Cκ y D f x κ
Cκ y
|κ|α1
aκ xy bx, y|y|
κ
κ
1
1 − sα Dκ f x − syds
0
1.3
α1
|κ|≤α
with some bounded functions aκ x and bx, y, where κ denotes the multi-indices. By
cancellation property 1.1 of Ω, we have
hf t, x t−1−α
t Sn−1
0
t
−1−α
t
α1
s
≤ Ct
Sn−1
0
1α−α
Ω x, y f x − sy dσ y ds
Ω x, y b x, sy dσ y ds
1.4
.
Simultaneously, by choosing r so that 1 < r < q and r sufficiently closing to 1, the Hölder
inequality for integrals implies
hf t, x ≤ t
−1−α
Ωx, yr
fLr |y|≤t
|y|n−1r
1/r
dy
≤ Ct−α−nn/r ,
1.5
where 1/r 1/r 1. Hence, noting 1 α − α > 0 and −α − n n/r < 0, one can easily
check that hf ·, xH ≤ C for any test function f and every x ∈ Rn , and so μΩ,α fx is well
defined for f ∈ SRn and α ≥ 0.
Historically, as high dimension corresponding to the Littlewood-Paley g-function,
Stein 1 first introduced the classical Marcinkiewicz integral with convolution kernel
Ωx, z ≡ Ωz and in the special case α 0. The Lp H p boundedness, 1 < p ≤ 2, or p ≤ 1
but is close to 1, for the classical Marcinkiewicz integrals was extended by many authors
to various kernels, for example, 1–4. On the other hand, because of connecting with the
problems about the partial differential equations with variable coefficients, more attentions
are focused on the variable kernels.
A fresh look at this problem is due to Ding et al. 5–7, where they showed the
following theorem.
Xiangxing Tao et al.
3
Theorem 1.1 see 7. Let α 0, and Ωx, z ∈ L∞ Rn × Lq Sn−1 , q > 2n − 1/n, satisfy
cancellation property 1.1. Then, the operator μΩ,0 is bounded from L2 Rn to L2 Rn .
Moreover, if Ω satisfies the L1 -Dini condition, then μΩ,0 is bounded from Hardy space H 1 Rn 1
to L Rn .
Our main results are as follows.
Theorem 1.2. Let α > 0, and Ωx, z ∈ L∞ Rn × Lq Sn−1 , q > max{1, 2n − 1/n 2α}, satisfy
cancellation property 1.1. Then,
μΩ,α f 2 n ≤ Cf 2 n
Lα R L R 1.6
with the constant C independent of any f ∈ L2α Rn ∩ SRn .
Theorem 1.3. Let α > 0, and Ωx, z ∈ L∞ Rn × Lq Sn−1 , q > max{1, 2n − 1/n 2α}, satisfy
cancellation property 1.1. Then, for n/n α < p ≤ 1,
μΩ,α f p n ≤ Cf p n
Hα R L R 1.7
p
with the constant C independent of any f ∈ Hα Rn ∩ SRn .
By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator,
p
we obtain the Lp − Lα boundedness of μΩ,α .
Corollary 1.4. Let α > 0, and Ωx, z ∈ L∞ Rn × Lq Sn−1 , q > max{1, 2n − 1/n 2α},
satisfy cancellation property 1.1. Then,
μΩ,α f p n ≤ Cf p n
Lα R L R for 1 < p ≤ 2
1.8
p
with the constant C independent of any f ∈ Lα Rn ∩ SRn .
p
Further, we can derive the boundedness of μΩ,α on Hα for some p ≤ n/n α under
an additional assumption, the L1,β -Dini condition, on Ω as follows:
1
δ
dδ < ∞,
1β
0 δ
1.9
where δ is the integral modulus of continuity of Ω defined by
δ sup
r>0, |O|<δ
Sn−1
Ω rz , Oz − Ω rz , z dσ z ,
where O is a rotation in Rn with |O| O − I, and I is an identical operator.
1.10
4
Journal of Inequalities and Applications
Theorem 1.5. Let Ωx, z ∈ L∞ Rn ×Lq Sn−1 , q > max{1, 2n−1/n2α}, satisfy cancellation
property 1.1 and the L1,β -Dini condition 1.9 with β ≥ 0. Then, for max{n/n α β, 2n/2n 2α 1} < p ≤ n/n α, 0 < α < n/2, one has
μΩ,α f p n ≤ Cf p n
Hα R L R 1.11
p
with the constant C independent of any f ∈ Hα Rn ∩ SRn .
We will give the definitions for Sobolev spaces and Hardy-Sobolev spaces, and prove
the key lemma Lemma 2.2 for Bessel functions in the next section. The proof of Theorem 1.2
will be given in Section 3, and Theorems 1.3 and 1.5 will be proved in Section 4. Finally, as
an application of Theorem 1.2, in Section 5 we will get the L2α − L2 boundedness for a class of
the Littlewood-Paley type operators μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0,
which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, respectively.
Here we point out that, by the same arguments in the paper, our theorems are also
hold for the following hypersingular parametric Marcinkiewicz integrals
ρ
μΩ,α fx
1/2
∞ 2
dt
1
Ωx, x − y
ρ
n−ρ fydy 12α
t
0 t
|x−y|≤t |x − y|
1.12
with the complex parameter ρ with Reρ > 0.
Throughout the paper, C always denotes a positive constant not necessarily the same
at each occurrence. we use a ∼
b to mean the equivalence of a and b; that is, there exists a
positive constant C independent of a, b such that C−1 a ≤ b ≤ Ca.
2. Some notations and lemmas
Let us first recall the Triebel-Lizorkin space. Fix a radial function ϕx ∈ C∞ satisfying
suppϕ ⊆ {x : 1/2 < |x| ≤ 2} and 0 ≤ ϕx ≤ 1, and ϕx > c > 0 if 3/5 ≤ |x| ≤ 5/3.
Let ϕj x ϕ2j x. Define the function ψj x by ψj ξ ϕj ξ. For 0 < p, q < ∞ and α ∈ R,
α,q
the homogeneous Triebel-Lizorkin space Ḟp is the set of all distributions f satisfying
α,q Ḟp Rn
f ∈S R
n
: fḞpα,q
1/q q
−αk
2 ψk ∗f
<∞ .
k
2.1
p
p
p
For 1 < p < ∞, the homogeneous Sobolev spaces Lα Rn are defined by Lα Rn namely, fLpα fḞpα,2 . From 8, we know that for any f ∈ L2α Rn ,
Ḟpα,2 Rn ,
fL2α Rn ∼
Rn
2 |ξ|2α dξ
fξ
1/2
,
2.2
p
and if α is a nonnegative integer, then for any f ∈ Lα Rn ,
fLpα Rn ∼
Dτ f p n .
L R |τ|α
2.3
Xiangxing Tao et al.
5
p
p
For 0 < p ≤ 1, we define the homogeneous Hardy-Sobolev space Hα Rn by Hα Rn It is well known that H p Rn Ḟp0,2 Rn for 0 < p ≤ 1, one can refer to 8 for the
details.
Next we give two lemmas, which play important roles in the paper and are valuable
in other applications of the Bessel functions.
Ḟpα,2 Rn .
Lemma 2.1 see 9. Suppose n ≥ 2 and f ∈ L1 Rn ∩ L2 Rn has the form fx f0 |x|P x,
where P x is a solid spherical harmonic polynomial of degree m. Then, the Fourier transform of f has
F0 |ξ|P ξ, where
the form fξ
F0 r 2πi
−m −n2m−2/2
∞
r
f0 sJn2m−2/2 2πrssn2m/2 ds,
2.4
0
and r |ξ|, Jm s is the Bessel function.
Lemma 2.2. For λ ≥ 0, there exists a constant C > 0 depending only on λ such that
t
Jmλ ρ ≤ C
dρ
mλ
λ
ρ
0
for 0 < t < ∞ , m 1, 2, . . . .
2.5
Proof. Let us write ν m λ. In case 0 < t ≤ ν, since Jν ρ > 0 for 0 < ρ < ν, it follows from
10, inequality 6.1 that
t
t
Jν ρ Jν ρ Cν
C
dρ ≤ ν
dρ ≤ 1λ ≤ λ .
λ
1λ
m
m
0 ρ
0 ρ
2.6
In case ν < t ≤ 2ν, the second mean value theorem and 10, inequality 6.2 yield
t
Jν ρ
dρ ν−λ
λ
ν ρ
ν
h
O ν
ν < h ≤ t < 2ν
ν
−λ1
Jν ρdρ
h
Jν νρdρ
1<h<2
2.7
1
−λ
,
where the big oh is an absolute one. Thus,
t
Jν ρ ν Jν ρ t Jν ρ ≤
≤ C .
dρ
dρ
dρ
mλ
λ
λ
λ
0 ρ
0 ρ
ν ρ
2.8
In case t > 2ν, we use the differential equation of Jν which may be written as see 10,
page 221
Jν ρ −
ρJν ρ ρ2 Jν ρ
−
,
ρ2 − ν 2 ρ2 − ν 2
2.9
6
Journal of Inequalities and Applications
which implies
t
t
Jν ρ t
Jν ρ
Jν ρ
: I1 I2 .
≤
dρ
dρ
dρ
λ
λ−1
2
2
λ−2
2
2
ρ −ν
ρ −ν
2ν ρ
2ν ρ
2ν ρ
2.10
−1
Since |Jν ρ| ≤ 1 and that ρλ 1 − ν/ρ2 is a decreasing function for λ ≥ 0 and ρ ≥ 2ν, we
obtain by the second mean value theorem
I2 ≤
4
32ν
t1
Jν ρdρ
λ
2ν ≤ t1 ≤ t
2ν
J t − Jν 2ν ≤ C .
≤
λ ν 1
νλ
32ν
4
2.11
To estimate I1 , if λ > 0, one can easily see that
I1 ≤
t
2ν ρ
dρ
4
≤
3
ρ2 − ν 2
λ−1
t
dρ
C
≤ λ.
1λ
ρ
ν
2ν
2.12
−1
If λ 0, then ν m is an integer. Since |Jm ρ| ≤ 1 see 9, page 137, and ρ1 − ν/ρ2 decreasing for ρ ≥ 2ν, we apply the second mean value theorem again to obtain
t
Jm
ρ
dρ
I1 −1
2
2
2ν ρ ρ − ν 2 t2 ≤
J
ρdρ
m
3ν 2ν
≤
2ν ≤ t2 ≤ t
is
2.13
2 Jν t2 − Jν 2ν ≤ C.
3ν
Now we have gotten
t
Jν ρ 2ν Jν ρ t Jν ρ ≤
≤ C ,
dρ
dρ
dρ
mλ
λ
λ
λ
ρ
ρ
ρ
0
0
2ν
2.14
which implies the lemma.
3. The boundedness on Sobolev spaces
In this section, we derive the boundedness for the Marcinkiewicz integral μΩ,α on Sobolev
spaces, and give the proof of Theorem 1.2.
Xiangxing Tao et al.
7
As in 9, let {Ym,1 , Ym,2 , . . . , Ym,Nm } denote an orthonormal basis of the space of
normalized spherical harmonics of degree m, then by using 1.1 and the spherical harmonic
development, see 11 for instance, we can decompose Ωx, z as follows:
∞
Ω x, z Nm
am,j xYm,j z ,
3.1
Ω x, z Ym,j z dσ z .
3.2
mα1 j1
where Nm ∼
mn−2 and
am,j x Sn−1
Denote
1/2
Nm
2
am,j x
am x ,
bm,j x j1
then
Nm
j1
am,j x
,
am x
3.3
2
bm,j
x 1. We also let
∞ μm,j,α fx 0
Ym,j x − y
|x − y|n−1
|x−y|≤t
1/2
2
dt
fydy 32α
.
t
3.4
Then, applying the Hölder inequality twice, we have for any 0 < ε < 1
μΩ,α fx2 2
∞ Nm
∞
dt
Ym,j x − y
am x
bm,j x
fydy 32α
n−1
t
|x − y|
0
|x−y|≤t mα1
j1
∞
≤
a2m xm−ε12α
mα1
∞
mε12α
mα1
2
∞ Nm
dt
Ym,j x − y
·
bm,j x
fydy 32α
n−1
t
|x − y|
0
|x−y|≤t j1
∞
≤
a2m xm−ε12α
mα1
j1
∞
mα1
mε12α
mα1
∞ Nm
·
0
∞
Ym,j x − y
|x−y|≤t
2
dt
fydy 32α
t
|x − y|n−1
∞
a2m xm−ε12α
mα1
mε12α
Nm
μm,j,α fx2 .
j1
3.5
8
Journal of Inequalities and Applications
By 11, we can observe that the series in the first parenthesis on the right-hand side
of the inequality above is equal to Ωx, ·2L2 Sn−1 , where L2−γ Sn−1 is the Sobolev space on
−γ
Sn−1 with γ ε1/2 α for any 0 < ε < 1. So if we take ε sufficiently close to 1, then by the
Sobolev imbedding theorem Lq ⊂ L2−γ , we have
m
a2m xm−ε12α ≤ CΩ2L∞ Rn ×Lq Sn−1 3.6
with q > max{1, 2n − 1/n 2α}. Therefore, to prove the theorem, it remains to show that
for ε sufficiently close to 1, we have
∞
mε12α
Nm
Rn
j1
mα1
μm,j,α fx2 dx ≤ Cf2 2 .
Lα
3.7
Put Pm,j x Ym,j x|x|m and ϕm,j,t x t−1−α Pm,j x|x|−n−m1 χ{|x|≤t} . One can deduce
from Lemma 2.1 that
m ϕ
m,j,t ξ F0 |ξ| Pm,j |ξ| Ym,j ξ |ξ| F0 |ξ| ,
3.8
where, by the changing of variable,
F0 r i
−m
2π
n/2−1 −m−1 −1−α
r
2πrt
t
0
Jmn−2/2 ρ
dρ.
ρn−2/2
3.9
Hence by Plancherel theorem, we have
Nm
j1
Rn
Nm
μm,j,α fx2 dx j1
R
∞ n
0
Nm
∞ j1
0
Nm
∞ j1
0
R
n
2
dt
dx
ϕm,j,t x − yfydy
t
Rn
2
dt
ϕm,j,t x − yfydy dx
t
Rn
2 F0 |ξ|2 dξ dt .
Ym,j ξ 2 |ξ|2m fξ
t
Rn
3.10
Xiangxing Tao et al.
9
By 11, we know that
Nm
j1
|Ym,j z |2 ∼
mn−2 , thus 3.10 is bounded by
2π|ξ|t
2
Jmn−2/2 ρ
dt
1
Cm
dρ
dξ
12α
n−2/2
2π|ξ|t
t
n
ρ
0
0
R
m/2 r
2
dr
1 Jmn−2/2 ρ
2 |ξ|2α
fξ
≤ Cmn−2
dρ
dξ
12α
n−2/2
r
r
n
ρ
0
0
R
∞ r
2
dr
1 Jmn−2/2 ρ
2 |ξ|2α
fξ
Cmn−2
dρ
dξ
12α
n−2/2
r
ρ
m/2 r 0
Rn
n−2
2
fξ
∞ 3.11
: I1 I2 .
r
Let ηr 1/r 0 Jmλ ρ/ρλ dρ. Using 12, Lemma 2.1 the proof of Lemma 2.1b and the
estimation in 12, page 565, we have ηr ≤ Jmλ r/r λ and |Jν r| ≤ r/2ν /Γν 1. Thus,
by Stirling’s formula, we have for m ≥ α 1 that
m/2
ηr2
0
dr
r 12α
≤
m/2
r 2m−1−2α
2mλ Γm λ 1
2
m/2mα
≤C m
2 Γm λ 1
0
≤
C
m2λ2−2α
2 dr
3.12
.
Now we let λ n − 2/2 and use the inequality above to see that
I1 ≤ Cm−2−2α
Rn
2 |ξ|2α dξ ≤ Cm−2−2α f2 2 .
fξ
Lα
3.13
r
To estimate I2 , using Lemma 2.2 that | 0 Jmλ t/tλ dt| ≤ C/mλ , we obtain
2 |ξ|2α
fξ
I2 ≤ Cmn−2
Rn
∞ m/2
C
2
mn−2/2
dr
dξ
r 32α
2 |ξ|2α m−2−2α dξ
fξ
≤C
3.14
Rn
≤ Cm−2−2α f2L2α .
Combining the estimates of I1 and I2 , we get from 3.10 and 3.11 that
∞
mα1
mε12α
Nm
j1
Rn
∞
μm,j,α fx2 dx ≤ C
mε12α−2−2α f2L2α ,
mα1
which implies 3.7. Then, we complete the proof of the theorem.
3.15
10
Journal of Inequalities and Applications
4. The boundedness on Hardy-Sobolev spaces
In order to show the boundedness for the operator μΩ,α on Hardy-Sobolev spaces, and prove
Theorems 1.3 and 1.5, we first recall the atomic decomposition for Hardy-Sobolev spaces.
Definition 4.1 see 13. For α ≥ 0, the function ax is called a p, 2, α atom if it satisfies the
following three conditions:
1 suppa ⊂ B with a ball B ⊂ Rn ;
2 aL2α ≤ |B|1/2−1/p ;
3 Rn axP x 0, for any polynomial P x of degree ≤ N n1/p − 1α.
p
By 13, we know that every f ∈ Hα Rn can be written as a sum of p, 2, α atoms
p
n
aj x, that is, f j λj aj which converges in Hα R and in the sense of distributions.
p ∼
Moreover, fH p j |λj |p .
α
To consider the hypersingular Marcinkiewicz integral μΩ,α f on Hardy-Sobolev
p
p
spaces Hα Rn , we can start with f in a nice dense class of function, say f ∈ Hα Rn ∩ SRn .
Then, by the Lebesgue dominated convergence theorem, we have
λj · ha ·, x .
hf ·, x ≤
j
H
H
4.1
j
Recalling μΩ,α fx hf ·, xH , so we obtain for 0 < p ≤ 1
p μΩ,α fp p n ≤
λj μΩ,α aj p p n .
L R L R 4.2
j
Now we are in the position to prove the following theorem.
Proof of Theorem 1.3. Because of 4.2, it is sufficient to show that
μΩ,α ap p ≤ C
L
4.3
with the constant C independent of any p, 2, α atom a. By translation argument, we can
assume that suppa ⊂ B0, ρ. We first note that
μΩ,α ap p ≤
L
|x|≤8ρ
μΩ,α ap dx |x|>8ρ
μΩ,α ap dx : J1 J2 .
4.4
By Theorem 1.2 and size condition 2 of a, it is not difficult to deduce that
p
p
J1 ≤ CμΩ,α aL2 ρn1−p/2 ≤ CaL2 ρn1−p/2 ≤ C.
α
Next we estimate J2 according to three cases.
4.5
Xiangxing Tao et al.
11
Case 1. 0 < α < n/2. In this case, one may refer to 14, Lemma 5.5 to get
axdx ≤ Cρn−n/pα .
4.6
B
Since 0 < p ≤ 1, the Minkowski inequality and the Hölder inequality give that
J2 ∞ Ωx, x − y
p/2
2
dt
aydy 32α
dx
t
|x − y|n−1
p
Ωx, x − y ≤
n |x − y|nα ay dy dx
|x|>8ρ
R
p
∞ Ωx, x − y
nα |ay|dy dx
j
j1
Rn |x − y|
j3 2 ρ<|x|<2 ρ
p
∞
Ωx, x − y j n1−p
aydy dx
≤
2ρ
nα
2j ρ<|x|<2j1 ρ Rn |x − y|
j3
∞
Ωx, x − y p
j n1−p
ay dy
≤
2ρ
nα dx
B
2j ρ<|x|<2j1 ρ |x − y|
j3
|x|>8ρ
0
|x−y|≤t
p
≤ CΩL∞ ×L1
aydy
4.7
p ∞
j n1−p−αp
·
.
2ρ
B
j3
Thus, by 4.6 and the condition p > n/n α, we get
p
J2 ≤ CΩL∞ ×L1
∞
2jn−np−αp ≤ C.
4.8
j3
Case 2. α > n/2 and α − n/2 is not an integer. Thus there is an integer k ≥ 1 such that
n/2 − 1 < α − k < n/2. By Pitt’s theorem, one has the following.
Lemma 4.2 see 15. Let ax be a p, 2, α atom with support in B B0, ρ, β denotes a multi 1
index with |β| k, then 0 B |Dβ asy|dy ds ≤ Cρn−n/pα−k .
Now using Taylor’s expansion on ay at y 0, and applying the cancellation property
of Ω, we see that
ha t, x t−1−α
|β|k
Cβ
Ωx, x − y
|x−y|≤t
|y|<ρ
|x − y|n−1
1
0
1 − sk−1 yβ Dβ asyds dy,
4.9
12
Journal of Inequalities and Applications
and so, by the Minkowski inequality,
μΩ,α ax ≤ Cρ
k
|β|k
Ωx, x − y 1 Dβ asyds dy.
nα
0
|y|<ρ |x − y|
4.10
Hence, by the same argument as in 4.7 and using Lemma 4.2, we have
J2 ≤ Cρ
kp
p
ΩL∞ ×L1
p
≤ CΩL∞ ×L1
∞
p
∞
1
j n1−p−αp
Dβ asyds dy
2ρ
·
B |β|k
0
j3
4.11
2jn−np−αp ≤ C.
j3
Case 3. α ≥ n/2 and α − n/2 is a nonnegative integer. In this case, we can take a small positive
real 0 < ε < 1 satisfying n/n α − ε < p and α ε α, then by Cases 1 and 2, we have
μΩ,α−ε f p ≤ Cf p ,
Hα−ε
L
μΩ,αε f p ≤ Cf p .
Hαε
L
4.12
By interpolation, see 15, we get μΩ,α fLp ≤ CfHαp .
Next we are going to proof the following.
Proof of Theorem 1.5. By similar arguments, it is enough to show the uniform boundedness of
μΩ,α aLp for any p, 2, α-atom a with the support B0, ρ. Write
μΩ,α ap p
|x|≤8ρ
μΩ,α axp dx |x|>8ρ
μΩ,α axp dx : W V.
4.13
It is easy to see that W ≤ C. For V , we decompose it further into
p/2
2
∞ dt Ωx,
x
−
y
V ≤
aydy 32α dx
n−1
t
|x|>8ρ 0
|x−y|≤t |x − y|
≤
|x|>8ρ
|x|2ρ 0
Ωx, x − y
|x−y|≤t
|x − y|n−1
p/2
2
dt aydy 32α dx
t
p/2
2
∞ dt Ωx, x − y
aydy 32α dx
n−1
t
|x|>8ρ |x|2ρ
|x−y|≤t |x − y|
: V1 V2 .
4.14
Xiangxing Tao et al.
13
Note that for |x| > 8ρ, |y| ≤ ρ, we have |x−y| ∼
|x| ∼
|x|2ρ, so the mean value theorem
−2−2α
−2−2α
−3−2α
− |x − y|
| ≤ Cρ|x − y|
. Thus, by the Minkowski inequality for
gives ||x| 2ρ
integrals,
V1 ≤ Cρ
p/2
∞ j3
p
Ωx, x − y aydy dx.
n1/2α
2j ρ≤|x|<2j1 ρ
B |x − y|
4.15
Now, like 4.7, we write V1 as the following sum, and apply the Hölder inequality for
integrals to obtain
V1 ≤ Cρ
p/2
∞
j
2ρ
n1−p
j3
p
≤ Cρp/2 ΩL∞ ×L1
p
≤ CΩL∞ ×L1
∞
2j ρ
p
Ωx, x − y aydydx
|x − y|n1/2α
2j ρ≤|x|<2j1 ρ
B
−1/2αp 2j ρ
n1−p
aydy
p
B
j3
4.16
∞
j −1/2αpn1−p
≤ C,
2
j3
because of p > n/n α 1/2 and the inequality 4.6.
To deal with V2 , we note that B0, ρ ⊂ {y : |x − y| ≤ t} for t ≥ |x| 2ρ and |x| > 8ρ. So
by the cancelation of the atom ax, we have
V2 |x|>8ρ
∞ |x|2ρ
|x−y|≤t
Ωx, x − y
|x − y|
n−1
Ωx, x
|x − y|n−1
−
−
Ωx, x
|x − y|n−1
Ωx, x
|x|n−1
p/2
2
dt
dx
· aydy 32α
t
p/2
∞ 2 dt Ωx, x − y
Ωx,
x
aydy
≤
−
t32α dx
n−1
|x − y|n−1
|x|>8ρ |x|2ρ
|x−y|≤t |x − y|
p/2
∞ 2 dt Ωx, x
Ωx,
x
aydy
−
t32α dx
n−1
|x|n−1
|x|>8ρ |x|2ρ
|x−y|≤t |x − y|
: V21 V22 .
4.17
14
Journal of Inequalities and Applications
Applying the Minkowski inequality, decomposition of integral domain, the Hölder
inequality, and the Fubini theorem successively, we can obtain
V21 ≤
≤
j3
p
Ωx, x − y − Ωx, x aydy dx
|x − y|nα
2j ρ≤|x|<2j1 ρ
B
∞
j
∞ 2ρ
n1−p
2j ρ≤|x|<2j1 ρ
j3
∞
j
2ρ
n1−p
ay
Ωx, x − y − Ωx, x
|x − y|nα
2j ρ≤|x|<2j1 ρ
B
j3
p
Ωx, x − y − Ωx, x aydy dx
|x − y|nα
B
4.18
p
dx dy .
Since |y| < ρ and |x| > 2ρ, we have |x − y| ≥ 1/2|x| and so |x − y/|x − y| − x/|x|| ≤ 4|y|/|x|.
Thus by using the L1,β -Dini condition of Ω, the inner integral on the right-hand side of the
integral inequality above is controlled by
2j1 ρ 2jρ
2j1 ρ
4|y|
−1−α
Ω rx , rx − y − Ω rx , x dσx r −1−α dr ≤ C
dr
r
1
rx − y
r
jρ
n−1
S
≤C
2
4|y|/2j ρ
4|y|/2j1 ρ
|y|−α
−jαβ −α
≤ C2
1 δ αβ
δ dδ
δ1β
1
1 δ
dδ
1β
0 δ
ρ
≤ C2−jαβ ρ−α .
4.19
From this, and using 4.6 and the condition p > n/n α β, we can thus deduce that
V21 ≤ C
∞
j
2ρ
≤C
ay2−jαβ ρ−α dy
p
B
j3
∞
n1−p
4.20
jn1−p−αβp
2
≤ C.
j3
Finally, we give the estimate of V22 . Obviously, for |y| < ρ and |x| > 2ρ, we have
n−1
− 1/|x|n−1 | ≤ C|y|/|x|n .
|x| ∼
|x − y| ∼
|x| 2ρ, and by mean value theorem, |1/|x − y|
Therefore, by Minkowski’s inequality for integrals, the inequality 4.6 and the condition
Xiangxing Tao et al.
15
n/n 1 α < p < 1, we can obtain
p
Ωx, x
Ωx, x 1
ay ≤C
−
dy dx
n−1
n−1 |x|1α
|x
−
y|
|x|
|x|>8ρ
B
Ωx, x · |y| p
ay
≤C
dy dx
|x|n1α
|x|>8ρ
B
p Ωx, xp
p
aydy
≤ Cρ
dx
n1αp
B
|x|>8ρ |x|
V22
4.21
≤ ρp ρpn−n/pα ρ−n1αpn ΩL∞ ×Lp ≤ C.
Combining 4.13, 4.14, 4.16, 4.20, and 4.21, we have proved that
μΩ,α ap p ≤ W V1 V21 V22 ≤ C,
L
4.22
which is our desired inequality. The proof of the theorem is finished.
5. Final remark
By similar argument as that for μΩ,α , we can consider the Littlewood-Paley type operators
μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral
and the Littlewood-Paley gλ∗ function, as follows:
μΩ,α,S fx μ∗Ω,α,λ fx Γx
Rn1
Ωy, y − z
|y−z|≤t
|y − z|n−1
t
t |x − y|
λn 1/2
2
dy dt
fzdz n32α
,
t
Ωy, y − z
|y−z|≤t
|y − z|n−1
1/2
2
dy dt
fzdz n32α
t
5.1
with λ > 1, where Γx {y, t ∈ Rn1
: |x − y| < t}.
As an application of Theorem 1.2, we have the following conclusion.
Theorem 5.1. Let α ≥ 0, and assume that Ωx, z ∈ L∞ Rn × Lq Sn−1 , q > max{1, 2n − 1/n 2α}, satisfy 1.1. Then,
μΩ,α,S f 2 n ≤ 2λn μ∗ f
≤ CfL2α Rn Ω,α,λ
L R L2 Rn with the constant C independent of any f ∈ L2α Rn ∩ SRn .
5.2
16
Journal of Inequalities and Applications
The proof of the theorem is based on the following lemma.
Lemma 5.2. Let λ > 1, there exists a positive constant Cλ,n such that for any nonnegative and locally
integrable function ϕ,
2
μ∗Ω,α,λ fx ϕxdx
Rn
≤ Cλ,n
2
μΩ,α fx Mϕxdx,
Rn
5.3
where M denotes the Hardy-Littlewood maximal operator.
Proof. Directly from the definition, we have
Rn
λn
FΩ,t y2 dy dt ϕxdx
tn32α
Rn
Rn1
∞
λn
2 dt
1
t
FΩ,t y 32α sup
≤
ϕx n dx dy
t
t
t>0 Rn t |x − y|
Rn 0
2
≤ Cλ,n
μΩ,α fy Mϕydy,
2
μ∗Ω,α,λ fx ϕxdx t
t |x − y|
Rn
5.4
which implies the lemma.
Now, if we take the supremum in inequality 5.3 over all ϕ satisfying ϕL∞ ≤ 1, and
apply Theorem 1.2, we can see
2
∗
μ
Ω,α,λ f L2 Rn ≤ Cλ,n
≤
sup MϕL∞
ϕL∞ ≤1
2
μΩ,α fx dx
Rn
5.5
Cλ,n f2L2α Rn .
Moreover, by the observation that μΩ,α,S fx ≤ 2λn μ∗Ω,α,λ fx, we have obtained
Theorem 5.1.
Acknowledgments
This work was supported partly by National Natural Science Foundation of China under
Grant no. 10771110, and sponsored by K. C. Wong Magna Fund in Ningbo University.
References
1 E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Transactions of the
American Mathematical Society, vol. 88, no. 2, pp. 430–466, 1958.
2 A. Al-Salman, H. Al-Qassem, L. C. Cheng, and Y. Pan, “Lp bounds for the function of Marcinkiewicz,”
Mathematical Research Letters, vol. 9, no. 5-6, pp. 697–700, 2002.
3 A. Benedek, A. P. Calderón, and R. Panzone, “Convolution operators on Banach space valued
functions,” Proceedings of the National Academy of Sciences of the United States of America, vol. 48, no.
3, pp. 356–365, 1962.
Xiangxing Tao et al.
17
4 Y. Ding, S. Lu, and Q. Xue, “Marcinkiewicz integral on Hardy spaces,” Integral Equations and Operator
Theory, vol. 42, no. 2, pp. 174–182, 2002.
5 Y. Ding, C.-C. Lin, and S. Shao, “On Marcinkiewicz integral with variable kernels,” Indiana University
Mathematics Journal, vol. 53, no. 3, pp. 805–821, 2004.
6 Y. Ding, C.-C. Lin, and S. Shao, “Erratum: “on Marcinkiewicz integral with variable kernels”,” Indiana
University Mathematics Journal, vol. 56, no. 2, pp. 991–994, 2007.
7 S.-Y. Zhang, R.-Y. Wei, and X.-X. Tao, “On boundedness of parametric Marcinkiewicz operators
with nonhomogeneous kernel on Morrey-Herz spaces,” Journal of Mathematical Sciences: Advances and
Applications, vol. 1, no. 2, pp. 359–371, 2008.
8 M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, vol. 79 of
CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA,
1991.
9 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical
Series, no. 32, Princeton University Press, Princeton, NJ, USA, 1971.
10 A. P. Calderón and A. Zygmund, “On a problem of Mihlin,” Transactions of the American Mathematical
Society, vol. 78, no. 1, pp. 209–224, 1955.
11 A. P. Calderón and A. Zygmund, “On singular integrals with variable kernels,” Applicable Analysis,
vol. 7, no. 3, pp. 221–238, 1978.
12 N. E. Aguilera and E. O. Harboure, “Some inequalities for maximal operators,” Indiana University
Mathematics Journal, vol. 29, no. 4, pp. 559–576, 1980.
13 Y.-S. Han, M. Paluszyński, and G. Weiss, “A new atomic decomposition for the Triebel-Lizorkin
spaces,” in Harmonic Analysis and Operator Theory (Caracas, 1994), vol. 189 of Contemporary Mathematics,
pp. 235–249, American Mathematical Society, Providence, RI, USA, 1995.
14 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.
15 D. Chen, J. Chen, and D. Fan, “Rough singular integral operators on Hardy-Sobolev spaces,” Applied
Mathematics, vol. 20, no. 1, pp. 1–9, 2005.