Journal of Inequalities and Applications Volume 2008, Article ID 835938,17pages doi:10.1155/2008/835938
Research Article
Boundedness on Hardy-Sobolev Spaces for
Hypersingular Marcinkiewicz Integrals with
Variable Kernels
Xiangxing Tao,1Xiao Yu,1, 2 and Songyan Zhang1
1Department of Mathematics, Faculty of Science, Ningbo University, Ningbo,
Zhejiang Province 315211, China
2Department of Mathematics, Faculty of Science, Zhejiang University, Hangzhou,
Zhejiang Province 315211, China
Correspondence should be addressed to Songyan Zhang,[email protected]
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 hypersingu-lar Marcinkiewicz integrals with variable kernels are derived.
Copyrightq2008 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 functionΩx, zdefined onRn×Rnis said to belong toL∞Rn×LqSn−1, if it satisfies
the following two conditions:
1 Ωx, λz Ωx, z, for anyx, z∈Rnand anyλ >0;
2ΩL∞Rn×LqSn−1: supr≥0, y∈Rn
Sn−1|Ωrzy, z|qdσz1/q<∞.
Letα≥0 andq >max{1,2n−1/n2α}, and letΩ∈L∞Rn×LqSn−1satisfy the
following cancellation property:
Sn−1Ω
x, zYm
zdσz0, 1.1
for all spherical harmonic polynomials Ymz with degree ≤ αand for any x ∈ Rn. We
μΩ,αfx
∞
0
|
x−y|≤t
Ωx, x−y
|x−y|n−1 fydy
2t3dt2α
1/2
, 1.2
for f ∈ SRn, the Schwartz class. In general, the way in which the integrals were given
has to be interpreted. LetH {ht :hH ∞0|ht|2dt/t1/2 <∞}, and lethft, x
t−1−α
|x−y|≤tΩx, x−y/|x−y|
n−1fydy. We have to showh
f·, x∈ Hfor everyx∈Rnand
f ∈ S, and so thatμΩ,αfx hf·, xHcan be regarded as a well-defined Hilbert-valued
function. To see this, applying the Taylor’s expansion, we can write
fx−y
|κ|≤α
Cκyκ
Dκfx
|κ|α1 Cκyκ
1
0
1−sαDκfx−syds
:
|κ|≤α
aκxyκbx, y|y|α1
1.3
with some bounded functions aκx and bx, y, where κ denotes the multi-indices. By
cancellation property1.1ofΩ, we have
hft, x t−1−α t
0
Sn−1Ω
x, yfx−sydσyds
t−1−α
t
0 sα1
Sn−1
Ωx, ybx, sydσyds
≤Ct1α−α.
1.4
Simultaneously, by choosingr so that 1 < r < q andr sufficiently closing to 1, the H ¨older
inequality for integrals implies
hft, x≤t−1−αfLr
|y|≤t
Ωx, yr
|y|n−1r dy
1/r
≤Ct−α−nn/r, 1.5
where 1/r1/r 1. Hence, noting 1 α−α > 0 and−α−n n/r < 0, one can easily
check thathf·, xH≤Cfor any test functionfand everyx∈Rn, and soμΩ,αfxis well
defined forf∈ SRnandα≥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 ≡ Ωzand in the special caseα 0. TheLpHpboundedness, 1< p ≤2, orp ≤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
Theorem 1.1see7. Letα 0, andΩx, z ∈ L∞Rn×LqSn−1,q > 2n−1/n, satisfy cancellation property1.1. Then, the operatorμΩ,0is bounded fromL2RntoL2Rn.
Moreover, ifΩsatisfies theL1-Dini condition, thenμ
Ω,0is bounded from Hardy spaceH1Rn toL1Rn.
Our main results are as follows.
Theorem 1.2. Letα >0, andΩx, z∈L∞Rn×LqSn−1,q >max{1,2n−1/n2α}, satisfy cancellation property1.1. Then,
μΩ,αf
L2Rn≤CfL2
αRn 1.6
with the constantCindependent of anyf∈L2
αRn∩ SRn.
Theorem 1.3. Letα >0, andΩx, z∈L∞Rn×LqSn−1,q >max{1,2n−1/n2α}, satisfy cancellation property1.1. Then, forn/nα< p≤1,
μΩ,αfLpRn≤CfHαpRn 1.7
with the constantCindependent of anyf∈HαpRn∩ SRn.
By Theorems1.2and1.3and applying the interpolation theorem of sublinear operator,
we obtain theLp−Lp
αboundedness ofμΩ,α.
Corollary 1.4. Letα > 0, andΩx, z ∈ L∞Rn×LqSn−1,q > max{1,2n−1/n2α}, satisfy cancellation property1.1. Then,
μΩ,αf
LpRn≤CfLαpRn for1< p≤2 1.8
with the constantCindependent of anyf∈LpαRn∩ SRn.
Further, we can derive the boundedness ofμΩ,αonHαpfor somep≤ n/nαunder
an additional assumption, theL1,β-Dini condition, onΩas follows:
1
0 δ
δ1β dδ <∞, 1.9
whereδis the integral modulus of continuity ofΩdefined by
δ sup
r>0,|O|<δ
Sn−1
Ω
rz, Oz−Ωrz, zdσz, 1.10
Theorem 1.5. LetΩx, z∈L∞Rn×LqSn−1,q >max{1,2n−1/n2α}, satisfy cancellation property1.1and theL1,β-Dini condition1.9withβ≥0. Then, formax{n/nαβ,2n/2n
2α1}< p≤n/nα,0< α < n/2, one has
μΩ,αf
LpRn≤CfHαpRn 1.11
with the constantCindependent of anyf∈HαpRn∩ SRn.
We will give the definitions for Sobolev spaces and Hardy-Sobolev spaces, and prove
the key lemmaLemma 2.2for Bessel functions in the next section. The proof ofTheorem 1.2
will be given inSection 3, and Theorems1.3and 1.5will be proved inSection 4. Finally, as
an application ofTheorem 1.2, inSection 5we will get theL2α−L2boundedness for a class of
the Littlewood-Paley type operatorsμΩ,α,Sandμ∗Ω,α,λwith variable kernels and indexα≥0,
which relate to the Lusin area integral and the Littlewood-Paleygλ∗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
μρΩ,αfx
∞
0
t1ρ
|x−y|≤t
Ωx, x−y
|x−y|n−ρ fydy
2t1dt2α
1/2
1.12
with the complex parameterρwith Reρ>0.
Throughout the paper,Calways denotes a positive constant not necessarily the same
at each occurrence. we usea ∼ bto mean the equivalence of aandb; that is, there exists a
positive constantCindependent ofa, bsuch thatC−1a≤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, andϕx > c > 0 if 3/5 ≤ |x| ≤ 5/3. Letϕjx ϕ2jx. Define the functionψjxbyψjξ ϕjξ. For 0 < p, q < ∞andα∈R,
the homogeneous Triebel-Lizorkin space ˙Fpα,qis the set of all distributionsfsatisfying
˙
Fpα,q
Rn
f∈ SRn:fF˙α,qp
k
2−αkψk∗f q
1/q p
<∞
. 2.1
For 1 < p < ∞, the homogeneous Sobolev spaces LpαRnare defined by LpαRn
˙
Fpα,2Rn, namely,fLpα fF˙pα,2. From8, we know that for anyf∈L
2
αRn,
fL2 αRn ∼
Rn
fξ2|ξ|2αdξ
1/2
, 2.2
and ifαis a nonnegative integer, then for anyf∈LpαRn,
fLp αRn
∼
|τ|α
For 0< p≤1, we define the homogeneous Hardy-Sobolev spaceHαpRnbyHαpRn
˙
Fpα,2Rn. It is well known thatHpRn F˙p0,2Rnfor 0 < p ≤ 1, one can refer to8for the
details.
Next we give two lemmas, which play important roles in the paper and are valuable in other applications of the Bessel functions.
Lemma 2.1see9. Supposen≥2andf ∈L1Rn∩L2Rnhas the formfx f
0|x|Px, wherePxis a solid spherical harmonic polynomial of degreem. Then, the Fourier transform offhas the formfξ F0|ξ|Pξ, where
F0r 2πi−mr−n2m−2/2
∞
0
f0sJn2m−2/22πrssn2m/2ds, 2.4
andr|ξ|,Jmsis the Bessel function.
Lemma 2.2. Forλ≥0, there exists a constantC >0depending only onλsuch that
t
0
Jmλρ
ρλ dρ ≤ C
mλ for0< t <∞, m1,2, . . . . 2.5
Proof. Let us writeνmλ. In case 0< t≤ν, sinceJνρ>0 for 0< ρ < ν, it follows from
10, inequality6.1that
t
0 Jνρ
ρλ dρ ≤ν
t
0 Jνρ
ρ1λ dρ ≤ Cν
m1λ ≤
C
mλ. 2.6
In caseν < t≤2ν, the second mean value theorem and10, inequality6.2yield
t
ν
Jνρ
ρλ dρν
−λ h
ν
Jνρdρ
ν < h≤t <2ν
ν−λ1
h
1
Jννρdρ
1< h <2
Oν−λ,
2.7
where the big oh is an absolute one. Thus,
t
0 Jνρ
ρλ dρ ≤
ν
0 Jνρ
ρλ dρ
t
ν
Jνρ
ρλ dρ ≤ C
mλ. 2.8
In caset >2ν, we use the differential equation ofJνwhich may be written assee10,
page 221
Jνρ −
ρJνρ ρ2−ν2 −
ρ2J
νρ
which implies
t
2ν
Jνρ
ρλ dρ ≤
t
2ν
Jνρ ρλ−1ρ2−ν2dρ
t
2ν
Jνρ ρλ−2ρ2−ν2dρ
:I1I2. 2.10
Since|Jνρ| ≤1 and thatρλ1−ν/ρ2−1is a decreasing function forλ≥0 andρ≥2ν, we
obtain by the second mean value theorem
I2≤
4 32νλ
t1
2ν
Jνρdρ
≤ 4
32νλ
J
ν
t1
−Jν2ν≤ C νλ.
2ν≤t1≤t
2.11
To estimateI1, ifλ >0,one can easily see that
I1≤
t
2ν
dρ ρλ−1ρ2−ν2 ≤
4 3
t
2ν
dρ ρ1λ ≤
C
νλ. 2.12
Ifλ0, thenνmis an integer. Since|Jmρ| ≤1see9, page 137, andρ1−ν/ρ2−
1
is
decreasing forρ≥2ν, we apply the second mean value theorem again to obtain
I1
t
2ν
Jm ρ ρ−1ρ2−ν2dρ
≤ 2
3ν
t2
2ν
Jmρdρ
≤ 2
3νJν
t2
−Jν2ν≤C.
2ν≤t2≤t
2.13
Now we have gotten
t
0 Jνρ
ρλ dρ ≤
2ν
0 Jνρ
ρλ dρ
t
2ν
Jνρ
ρλ dρ ≤ C
mλ, 2.14
which implies the lemma.
3. The boundedness on Sobolev spaces
In this section, we derive the boundedness for the Marcinkiewicz integralμΩ,αon Sobolev
As in 9, let {Ym,1, Ym,2, . . . , Ym,Nm} denote an orthonormal basis of the space of
normalized spherical harmonics of degreem, then by using1.1and the spherical harmonic
development, see11for instance, we can decomposeΩx, zas follows:
Ωx, z
∞
mα1
Nm
j1
am,jxYm,j
z, 3.1
whereNm∼mn−2and
am,jx
Sn−1Ω
x, zYm,j
zdσz. 3.2
Denote
amx
Nm
j1
am,jx2 1/2
, bm,jx
am,jx
amx
, 3.3
thenNj1mb2
m,jx 1. We also let
μm,j,αfx
∞
0
|
x−y|≤t
Ym,jx−y
|x−y|n−1 fydy
2t3dt2α
1/2
. 3.4
Then, applying the H ¨older inequality twice, we have for any 0< ε <1
μΩ,αfx2 ∞
0
|x−y|≤t
∞
mα1 amx
Nm
j1
bm,jx
Ym,jx−y
|x−y|n−1 fydy
2 dt t32α
≤
∞
mα1
a2mxm−ε12α
∞
mα1
mε12α
·
∞
0
|x−y|≤t Nm
j1
bm,jx
Ym,jx−y
|x−y|n−1 fydy
2 dt t32α
≤
∞
mα1
a2mxm−ε12α
∞
mα1
mε12α
·
∞
0
Nm
j1
|x−y|≤t
Ym,jx−y
|x−y|n−1 fydy
2t3dt2α
∞
mα1
a2mxm−ε12α
∞
mα1
mε12α
Nm
j1
μm,j,αfx2.
By11, we can observe that the series in the first parenthesis on the right-hand side
of the inequality above is equal toΩx,·2L2
−γSn−1, whereL
2
−γSn−1is the Sobolev space on
Sn−1withγ ε1/2αfor any 0 < ε < 1. So if we takeεsufficiently close to 1, then by the
Sobolev imbedding theoremLq ⊂L2
−γ, we have
m
a2mxm−ε12α≤CΩ2L∞Rn×LqSn−1 3.6
withq >max{1,2n−1/n2α}. Therefore, to prove the theorem, it remains to show that
forεsufficiently close to 1, we have
∞
mα1
mε12α
Nm
j1
Rn
μm,j,αfx2
dx≤Cf2L2
α. 3.7
PutPm,jx Ym,jx|x|m andϕm,j,tx t−1−αPm,jx|x|−n−m1χ{|x|≤t}. One can deduce
fromLemma 2.1that
ϕm,j,tξ F0
|ξ|Pm,j
|ξ|Ym,j
ξ|ξ|mF0
|ξ|, 3.8
where, by the changing of variable,
F0r i−m2πn/2−1r−m−1t−1−α
2πrt
0
Jmn−2/2ρ
ρn−2/2 dρ. 3.9
Hence by Plancherel theorem, we have
Nm
j1
Rn
μm,j,αfx2
dx
Nm
j1
Rn
∞
0
Rn
ϕm,j,tx−yfydy 2dtt dx
Nm
j1
∞
0
Rn
Rn
ϕm,j,tx−yfydy 2dxdt
t
Nm
j1
∞
0
Rn
Ym,j
ξ2|ξ|2m fξ2F0|ξ|2dξ dt
t .
By11, we know thatNj1m|Ym,jz|2∼mn−2, thus3.10is bounded by
Cmn−2
Rn fξ2
∞
0
1 2π|ξ|t
2π|ξ|t
0
Jmn−2/2ρ ρn−2/2 dρ
2 dt t12αdξ
≤Cmn−2
Rn
fξ2|ξ|2α
m/2
0
1
r
r
0
Jmn−2/2ρ ρn−2/2 dρ
2 dr r12αdξ
Cmn−2
Rn
fξ2|ξ|2α
∞
m/2
1
r
r
0
Jmn−2/2ρ ρn−2/2 dρ
2 dr r12αdξ
:I1I2.
3.11
Letηr 1/rr0Jmλρ/ρλdρ. Using12, Lemma 2.1 the proof of Lemma 2.1band the
estimation in12, page 565, we haveηr≤Jmλr/rλand|Jνr| ≤r/2ν/Γν1. Thus,
by Stirling’s formula, we have form≥α 1 that
m/2
0
ηr2 dr r12α ≤
m/2
0
r2m−1−2α
2mλΓmλ12dr
≤C
m/2mα
2mΓmλ1
2
≤ C
m2λ2−2α.
3.12
Now we letλ n−2/2 and use the inequality above to see that
I1≤Cm−2−2α
Rn
fξ2|ξ|2αdξ≤Cm−2−2αf2L2
α. 3.13
To estimateI2, usingLemma 2.2that|
r
0Jmλt/tλdt| ≤C/mλ, we obtain
I2≤Cmn−2
Rn
fξ2|ξ|2α
∞
m/2
C mn−2/2
2 dr
r32αdξ
≤C
Rn
fξ2|ξ|2αm−2−2αdξ
≤Cm−2−2αf2L2 α.
3.14
Combining the estimates ofI1andI2, we get from3.10and3.11that
∞
mα1
mε12α
Nm
j1
Rn
μm,j,αfx2 dx≤C
∞
mα1
mε12α−2−2αf2L2
α, 3.15
4. The boundedness on Hardy-Sobolev spaces
In order to show the boundedness for the operatorμΩ,αon Hardy-Sobolev spaces, and prove
Theorems1.3and1.5, we first recall the atomic decomposition for Hardy-Sobolev spaces.
Definition 4.1see13. Forα≥0, the functionaxis called ap,2, αatom if it satisfies the following three conditions:
1suppa⊂Bwith a ballB⊂Rn;
2aL2 α ≤ |B|
1/2−1/p;
3RnaxPx 0, for any polynomialPxof degree≤N n1/p−1α.
By13, we know that everyf ∈ HαpRncan be written as a sum ofp,2, αatoms
ajx, that is, f jλjaj which converges in HαpRn and in the sense of distributions.
Moreover,fpHp α
∼
j|λj|p.
To consider the hypersingular Marcinkiewicz integral μΩ,αf on Hardy-Sobolev
spacesHαpRn, we can start withfin a nice dense class of function, sayf∈HαpRn∩ SRn.
Then, by the Lebesgue dominated convergence theorem, we have
hf·, xH ≤ j
λj·haj·, xH. 4.1
RecallingμΩ,αfx hf·, xH, so we obtain for 0< p≤1
μΩ,αfp LpRn≤
j λjp
μΩ,αajLppRn. 4.2
Now we are in the position to prove the following theorem.
Proof ofTheorem 1.3. Because of4.2, it is sufficient to show that
μΩ,αapLp ≤C 4.3
with the constantC independent of anyp,2, α atoma. By translation argument, we can
assume that suppa⊂B0, ρ. We first note that
μΩ,αap Lp ≤
|x|≤8ρ
μΩ,αap
dx
|x|>8ρ
μΩ,αap
dx:J1J2. 4.4
ByTheorem 1.2and size condition2ofa, it is not difficult to deduce that
J1≤CμΩ,αapL2ρn1−p/2≤Ca
p L2
αρ
n1−p/2≤C. 4.5
Case 1. 0< α < n/2. In this case, one may refer to14, Lemma 5.5to get
B
axdx≤Cρn−n/pα. 4.6
Since 0< p≤1, the Minkowski inequality and the H ¨older inequality give that
J2
|x|>8ρ ∞
0
|x−y|≤t
Ωx, x−y
|x−y|n−1 aydy
2t3dt2α
p/2 dx
≤
|x|>8ρ
Rn
Ωx, x−y
|x−y|nα aydy
pdx
∞
j3
2jρ<|x|<2j1ρ
Rn
Ωx, x−y
|x−y|nα |ay|dy
pdx
≤ ∞
j3
2jρn1−p
2jρ<|x|<2j1ρ
Rn
Ωx, x−y
|x−y|nα aydy dx
p
≤ ∞
j3
2jρn1−p
B
aydy
2jρ<|x|<2j1ρ
Ωx, x−y
|x−y|nα dx
p
≤CΩpL∞×L1
B
aydy
p
·∞
j3
2jρn1−p−αp.
4.7
Thus, by4.6and the conditionp > n/nα, we get
J2≤CΩpL∞×L1
∞
j3
2jn−np−αp≤C. 4.8
Case 2. α > n/2 and α−n/2 is not an integer. Thus there is an integerk ≥ 1 such that
n/2−1< α−k < n/2. By Pitt’s theorem, one has the following.
Lemma 4.2see15. Letaxbe ap,2, αatom with support inBB0, ρ,βdenotes a multi-index with|β|k, then10B|Dβasy|dy ds≤Cρn−n/pα−k.
Now using Taylor’s expansion onayaty0, and applying the cancellation property
ofΩ, we see that
hat, x t−1−α
|β|k
Cβ
|x−y|≤t
|y|<ρ
Ωx, x−y
|x−y|n−1
1
0
and so, by the Minkowski inequality,
μΩ,αax≤Cρk
|β|k
|y|<ρ
Ωx, x−y
|x−y|nα
1
0
Dβasyds dy. 4.10
Hence, by the same argument as in4.7and usingLemma 4.2, we have
J2≤CρkpΩpL∞×L1
B|β|k 1
0
Dβasyds dy
p
·∞
j3
2jρn1−p−αp
≤CΩpL∞×L1
∞
j3
2jn−np−αp≤C.
4.11
Case 3. α≥n/2 andα−n/2 is a nonnegative integer. In this case, we can take a small positive
real 0< ε <1 satisfyingn/nα−ε< pandαε α, then by Cases1and2, we have
μΩ,α−εf
Lp ≤CfHp
α−ε, μΩ,αεfLp ≤CfHαεp . 4.12
By interpolation, see15, we getμΩ,αfLp ≤CfHp α.
Next we are going to proof the following.
Proof ofTheorem 1.5. By similar arguments, it is enough to show the uniform boundedness of
μΩ,αaLp for anyp,2, α-atomawith the supportB0, ρ. Write
μΩ,αap p
|x|≤8ρ
μΩ,αaxp
dx
|x|>8ρ
μΩ,αaxp
dx:WV. 4.13
It is easy to see thatW≤C. ForV, we decompose it further into
V ≤
|x|>8ρ
∞
0
|x−y|≤t
Ωx, x−y
|x−y|n−1 aydy
2t3dt2α
p/2 dx
≤
|x|>8ρ |x|2ρ
0
|x−y|≤t
Ωx, x−y
|x−y|n−1 aydy
2t3dt2α
p/2 dx
|x|>8ρ
∞
|x|2ρ
|x−y|≤t
Ωx, x−y
|x−y|n−1 aydy
2t3dt2α
p/2 dx
:V1V2.
Note that for|x|>8ρ,|y| ≤ρ, we have|x−y| ∼|x| ∼|x|2ρ, so the mean value theorem gives||x|2ρ−2−2α − |x−y|−2−2α| ≤ Cρ|x−y|−3−2α. Thus, by the Minkowski inequality for integrals,
V1≤Cρp/2
∞
j3
2jρ≤|x|<2j1ρ
B
Ωx, x−y
|x−y|n1/2α
aydyp
dx. 4.15
Now, like 4.7, we write V1 as the following sum, and apply the H ¨older inequality for
integrals to obtain
V1≤Cρp/2
∞
j3
2jρn1−p
2jρ≤|x|<2j1ρ
B
Ωx, x−y
|x−y|n1/2αaydy dx
p
≤Cρp/2ΩpL∞×L1
∞
j3
2jρ−1/2αp2jρn1−p
B
aydy
p
≤CΩpL∞×L1
∞
j3
2j−1/2αpn1−p≤C,
4.16
because ofp > n/nα1/2and the inequality4.6.
To deal withV2, we note thatB0, ρ⊂ {y:|x−y| ≤t}fort≥ |x|2ρand|x|>8ρ. So
by the cancelation of the atomax, 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
·aydy 2 dt
t32α p/2
dx
≤
|x|>8ρ
∞
|x|2ρ
|x−y|≤t
Ω|xx, x−y|−n−y1 −
Ωx, x
|x−y|n−1
aydy 2 dt
t32α
p/2 dx
|x|>8ρ
∞
|x|2ρ
|x−y|≤t
|xΩ−x, xy|n−1 −
Ωx, x
|x|n−1
aydy 2 dt
t32α
p/2 dx
:V21V22.
Applying the Minkowski inequality, decomposition of integral domain, the H ¨older inequality, and the Fubini theorem successively, we can obtain
V21≤
∞
j3
2jρ≤|x|<2j1ρ
B
Ωx, x−y−Ωx, x
|x−y|nα aydy pdx
≤ ∞
j3
2jρn1−p
2jρ≤|x|<2j1ρ
B
Ωx, x−y−Ωx, x
|x−y|nα aydy dx p
∞
j3
2jρn1−p
B
ay
2jρ≤|x|<2j1ρ
Ωx, x−y−Ωx, x
|x−y|nα dx
dy
p
.
4.18
Since|y|< ρand|x|>2ρ, we have|x−y| ≥1/2|x|and so|x−y/|x−y|−x/|x|| ≤4|y|/|x|.
Thus by using theL1,β-Dini condition ofΩ, the inner integral on the right-hand side of the
integral inequality above is controlled by
2j1ρ
2jρ
Sn−1
Ω
rx, rx
−y
rx−y
−Ωrx, xdσxr−1−αdr≤C 2j1ρ
2jρ
r−1−α
1
4|y|
r
dr
≤C
4|y|/2jρ
4|y|/2j1ρ|
y|−α1δ δ1β δ
αβdδ
≤C2−jαβρ−α
1
0 1δ
δ1β dδ
≤C2−jαβρ−α.
4.19
From this, and using4.6and the conditionp > n/nαβ, we can thus deduce that
V21≤C
∞
j3
2jρn1−p
B
ay2−jαβρ−αdy
p
≤C
∞
j3
2jn1−p−αβp≤C.
4.20
Finally, we give the estimate ofV22. Obviously, for |y| < ρ and |x| > 2ρ, we have
|x| ∼ |x−y| ∼ |x|2ρ, and by mean value theorem,|1/|x−y|n−1−1/|x|n−1| ≤ C|y|/|x|n.
n/n1α< p <1, we can obtain
V22≤C
|x|>8ρ
B
ay Ωx, x
|x−y|n−1 −
Ωx, x
|x|n−1
|x|11αdy pdx
≤C
|x|>8ρ
B
ayΩx, x· |y|
|x|n1α dy pdx
≤Cρp
B
aydyp
|x|>8ρ
Ωx, xp
|x|n1αpdx
≤ρpρpn−n/pαρ−n1αpnΩL∞×Lp≤C.
4.21
Combining4.13,4.14,4.16,4.20, and4.21, we have proved that
μΩ,αapLp ≤WV1V21V22≤C, 4.22
which is our desired inequality. The proof of the theorem is finished.
5. Final remark
By similar argument as that forμΩ,α, we can consider the Littlewood-Paley type operators
μΩ,α,Sandμ∗Ω,α,λwith variable kernels and indexα≥0, which relate to the Lusin area integral
and the Littlewood-Paleygλ∗function, as follows:
μΩ,α,Sfx
Γx
|y−z|≤t
Ωy, y−z
|y−z|n−1 fzdz
2tdy dtn32α
1/2 ,
μ∗Ω,α,λfx
Rn1
t t|x−y|
λn
|y−z|≤t
Ωy, y−z
|y−z|n−1 fzdz
2tdy dtn32α
1/2 5.1
withλ >1, whereΓx {y, t∈Rn1
:|x−y|< t}.
As an application ofTheorem 1.2, we have the following conclusion.
Theorem 5.1. Letα≥0, and assume thatΩx, z∈L∞Rn×LqSn−1,q >max{1,2n−1/n
2α}, satisfy1.1. Then,
μΩ,α,SfL2Rn≤2λnμ∗Ω,α,λfL2Rn≤CfL2
αRn 5.2
with the constantCindependent of anyf∈L2
The proof of the theorem is based on the following lemma.
Lemma 5.2. Letλ >1, there exists a positive constantCλ,nsuch that for any nonnegative and locally
integrable functionϕ,
Rn
μ∗Ω,α,λfx2ϕxdx≤Cλ,n
Rn
μΩ,αfx 2
Mϕxdx, 5.3
whereMdenotes the Hardy-Littlewood maximal operator.
Proof. Directly from the definition, we have
Rn
μ∗Ω,α,λfx2ϕxdx
Rn
Rn1
t t|x−y|
λn
FΩ,ty2
dy dt
tn32αϕxdx
≤
Rn
∞
0
FΩ,ty2 dt t32α
sup
t>0
Rn
t t|x−y|
λn
ϕx1 tndx
dy
≤Cλ,n
Rn
μΩ,αfy 2
Mϕydy,
5.4
which implies the lemma.
Now, if we take the supremum in inequality5.3over allϕsatisfyingϕL∞ ≤1, and
applyTheorem 1.2, we can see
μ∗Ω,α,λf2
L2Rn≤Cλ,n
sup
ϕL∞≤1
MϕL∞
Rn
μΩ,αfx 2
dx
≤Cλ,nf2L2 αRn.
5.5
Moreover, by the observation that μΩ,α,Sfx ≤ 2λnμ∗Ω,α,λfx, 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.
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