Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 175230, 28 pages
doi:10.1155/2009/175230
Research Article
Endpoint Estimates for a Class of Littlewood-Paley
Operators with Nondoubling Measures
Qingying Xue and Juyang Zhang
Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences,
Beijing Normal University, Ministry of Education, Beijing 100875, China
Correspondence should be addressed to Qingying Xue, qyxue@bnu.edu.cn
Received 18 April 2009; Accepted 18 August 2009
Recommended by Shusen Ding
Let μ be a positive Radon measure on Rd which may be nondoubling. The only condition that
μ satisfies is μBx, r ≤ C0 r n for all x ∈ Rd , r > 0, and some fixed constant C0 . In this paper,
∗
we introduce the operator gλ,μ
related to such a measure and assume it is bounded on L2 μ. We
then establish its boundedness, respectively, from the Lebesgue space L1 μ to the weak Lebesgue
space L1,∞ μ, from the Hardy space H 1 μ to L1 μ and from the Lesesgue space L∞ μ to the
∗
space RBLOμ. As a corollary, we obtain the boundedness of gλ,μ
in the Lebesgue space Lp μ
with p ∈ 1, ∞.
Copyright q 2009 Q. Xue and J. Zhang. 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
A positive Radon measure μ on Rd is said to be doubling if there exists some constant C such
that μBx, 2r ≤ CμBx, r for all x ∈ suppμ, r > 0. It is well known that the doubling
condition is an essential assumption in many results of classical Calderón-Zygmund theory.
However in the recent years, it has been shown that a big part of the classical theory remains
valid if the doubling assumption on μ is substituted by the growth condition as follows:
μBx, r ≤ C0 r n
1.1
for all x ∈ Rd , where n is some fixed number with 0 < n ≤ d. For example, In 2001,
Tolsa in 1, 2 investigated the weak 1,1 inequality for singular integrals, the LittlewoodPaley theory and the T 1 theorem with nondoubling measures. In 2002, Garcı́a-Cuerva and
Gatto 3 investigated the boundedness properties of fractional integral operators associated
to nondoubling measures. In 2005, Hu et al. 4 studied the multilinear commutators
2
Journal of Inequalities and Applications
of singular integrals with nondoubling measures. Since 2007, Hu et al. 5 have proved
some boundedness results of Marcinkiewicz integrals with nondoubling measures on some
function spaces.
On the other hand, let ψ be a function on Rd such that there exist positive constants
C0 , C1 , δ, and γ satisfying
a ψ ∈ L1 Rd and
Rd
ψxdμx 0,
b |ψx| ≤ C0 1 |x|−n−δ ,
c |ψx y − ψx| ≤ C1 |y|γ 1 |x|−n−γ−δ for 2|y| ≤ |x|.
∗
For this ψ, we define the Littlewood-Paley’s gλ,μ
-function with nondoubling measures as
follows:
∗
gλ,μ
⎛
f x ⎝
Rd1
t
t x − y
λn
⎞1/2
2 dμydt
ψt ∗ fy
⎠ ,
tn1
λ > 1,
1.2
where ψt x t−n ψx/t and ψt ∗ fy Rd ψt y − zfzdμz.
Note that if we replace dμy by dy in the above definition and when ψt Pt is the
Poisson kernel, we obtain classical gλ∗ -function defined and studied by Stein 6 and later by
Fefferman 7, where the weak 1,1 with λ > 2 and weak p, p with λ 2/p boundedness of
gλ∗ function were obtained. In the same paper, Fefferman 7 also established the Lp bounds of
gλ∗ for 1 < p < ∞ and λ > max{1, 2/p}. For the more generalized gλ∗ -function defined by 1.1,
the Lp boundedness is also well known see, e.g., 8, pages 309–318. On the other hand,
inspired by the works of Sakamoto and Yabuta in 1999, the first author in this paper studied
parametric gλ∗ -function systematically in his PhD thesis 9. Later, in 2008, Lin and Meng 10
gave some results on parametric gλ∗ -function with nondoubling measures. But their result
only valid for ρ > n/2, one cannot obtain the results for classical operators even for ρ 1 or
in the classical case studied by Stein in 1961 6.
∗
with nondoubling measures
In this paper, we will study the properties of operator gλ,μ
on some function spaces under the conditions a–c.
First, before stating our main results, we give some notation and definitions, let Q ⊂ Rn
be a closed cube with sides parallel to the axes. Denote its side length by lQ and its center
by xQ . Given α > 1 and β > αn , we say Q is α, β-doubling if μαQ ≤ βμQ, where αQ is the
cube concentric with Q with side length αlQ. If α, β are not specified, by a doubling cube
the smallest doubling
we mean a 2, 2d1 -doubling cube. For any cube Q, we denote by Q
cube which contains Q and has the same center as Q.
Given two cubes Q ⊂ R in Rd , set
SQ,R 1 N
Q,R
k1
μ 2k Q
n ,
l 2k Q
1.3
Journal of Inequalities and Applications
3
where NQ,R is the first integer k such that l2k Q ≥ lR and
μ αk Q
1
k n
k1 l α Q
α
NQ,R
SαQ,R
1.4
α
with α > 1, where NQ,R
is the first integer k such that lαk Q ≥ lR.
α
with constants that may
In the article of 1, page 95, we know that KQ,R ≈ KQ,R
1,∞
depend on α and C0 . The following atomic Hardy space Hatb
was introduced by Tolsa in
11.
Definition 1.1. For a fixed ρ > 1, a function b ∈ L1loc μ is called an atomic block if
1 there exists some cube R such that suppb ⊂ R;
2 Rd b dμ 0;
3 there are fucntions aj with supports in cubes Qj ⊂ R and numbers λj ∈ R such that
b
λj aj ,
j
1.5
aj ∞ ≤ μ ρQj SQ,R −1 .
L μ
Define
|b|H 1,∞ μ atb
λj .
1,∞
We say that f ∈ Hatb
μ if there are atomic blocks {bi }i such that f ∞. The
1,∞
Hatb
1.6
j
∞
i1
bi with
i
|bi |H 1,∞ <
atb
norm of f is defined by
f 1,∞ inf |bi | 1,∞ ,
H μ
H μ
atb
i
atb
1.7
where the infimum is taken over all the possible decompositions of f in atomic blocks.
1,∞
μ was proved to be the Hardy space H 1 μ
It was shown by Tolsa that the space Hatb
1,∞
μ and the norm · H 1,∞ μ ,
in 11 with equivalent norms. We will denote the space Hatb
atb
respectively, by H 1 μ and · H 1 μ for convenience. He also proved that the dual space of
H 1 μ is the following space RBMOμ.
Definition 1.2. Let ρ > 1 be a fixed constant. A function f ∈ L1loc μ is said to be in the space
RBMOμ if there exists some constant C > 0 such that for any cube Q centered at some point
of suppμ
1
μ ρQ
f y − mQ f dμ ≤ C
Q
1.8
4
Journal of Inequalities and Applications
and for any two doubling cubes Q ⊂ R
mQ f − mR f ≤ CSQ,R ,
1.9
where mQ f denotes the mean value of f over cube Q. The minimal constant C above is
defined to be the norm of f in the space RBMOμ and denoted by fRBMOμ .
1,∞
Tolsa in 11 proved that the definition of the space Hatb
μ and RBMOμ are
independent of the choice of ρ. The following space RBLOμ was introduced in 12. It is
easy to see that RBLOμ ⊂ RBMOμ.
Definition 1.3. A function f ∈ L1loc μ is said to be in the space RBLOμ if there exists some
√
√
positive constant C such that for any 4 d, 4 dn1 doubling cube Q,
mQ f − ess inf fx ≤ C
x∈Q
1.10
√
√
and for any two 4 d, 4 dn1 -doubling cubes Q ⊂ R,
mQ f − mR f ≤ CSQ,R .
1.11
The minimal constant C as above is defined to be the norm of f in the space RBLOμ, we
denote it by fRBLOμ .
In this paper, we always assume that μ and n are considered as they are defined at the
beginning of this paper. Our main results are as follows.
Theorem 1.4. Let ψ be a function on Rd , satisfying (a)–(c), λ > 2, 0 < γ < min{λ − 2n/2, δ}. If
∗
is bounded on L2 μ, then it is also bounded from L1 μ to L1,∞ μ.
gλ,μ
Theorem 1.5. Let ψ be a function on Rd , satisfying (a)–(c), λ > 2, 0 < γ < min{λ − 2n/2, δ}. If
∗
is bounded on L2 μ, then it is also bounded from H 1 μ to L1 μ.
gλ,μ
Theorem 1.6. Let ψ be a function on Rd , satisfying (a)–(c), λ > 2, 0 < γ < min{λ − 2n/2, δ}. If
∗
∗
is bounded on L2 μ, then for f ∈ L∞ μ, gλ,μ
f is either infinite everywhere or finite almost
gλ,μ
∗
∗
everywhere. More precisely, if gλ,μ
f is finite at some point x0 ∈ Rd , then gλ,μ
f is μ-finite almost
everywhere and
∗
gλ,μ f
RBLOμ
≤ Cf L∞ μ .
1.12
Corollary 1.7. Let ψ be a function on Rd , satisfying (a)–(c), λ > 2, 0 < γ < min{λ − 2n/2, δ}. If
∗
is bounded on L2 μ, then it is also bounded on Lp μ for any 1 < p < ∞.
gλ,μ
Remark 1.8. It is natural to consider the similar problems with more general rough kernels.
However, even in the doubling measure case, if we take ψx Ωx/|x|n−1 χ{|x|<1} in 1.1
in this case, gλ∗ is defined and studied by 13, from the results in 8, we know that it
Journal of Inequalities and Applications
5
∗
function even Ω ∈
is impossible to give similar results as above for Littlewood-Paley gλ,μ
Lipα 0 < α < 1 for n > 2. In fact, by the counter example in 13, even the Lp 1 < p <
2n/n 2 boundedness does not hold. In this sense, the condition we assumed on ψ is
necessary and reasonable. On the other hand, in 2008, Lin and Meng 10 gave some results
on parametric gλ∗ -function with nondoubling measures. In fact the results in 10 are only
valid for ρ > n/2. By the same reason as above, one cannot obtain the result when ρ 1
which in this case, the operator coincides with the classical operator studied by Torchinsky
and Wang in 13 and it is a generalization of the classical operators studied by Stein and
Fefferman.
Remark 1.9. Even in the classical case, the index λ > 2 is sharp for weak 1, 1 boundedness;
see 6 for detail.
We arrange our paper as follows, in Section 2, we give and prove some key lemmas.
The proof of our main theorems will be given in Section 3. Throughout this paper, the letter
C will denote a positive constant that may vary at each occurrence but is independent of the
essential variables. A B will always denote that there exists a constant C > 0, such that
A ≤ CB.
2. Main Lemmas
We need two lemmas given by Tolsa.
Lemma 2.1 see 11. If Q ⊂ R are concentric cubes such that there are no α, β-doubling cubes
with β > αn of the form αk Q, k ≥ 1, with Q ⊂ αk Q ⊂ R, then
1
n dμx ≤ C1 ,
R\Q x − xQ
2.1
where C1 depends only on α, β, n, C0 .
Lemma 2.2 see 11. For any f ∈ L1 μ and any λ > 0 λ > αd1 fL1 μ /μ, if μ < ∞
then we have one has the following:
a there exists a family of almost disjoint cubes {Qi }i (that means i χQi ≤ C, C depends only
on d) such that
a.1 1/μαQi Qi |f|dμ > λ/αd1 ;
a.2 1/μαηQi ηQi |f|dμ ≤ λ/αd1 for any η > 2;
a.3 |f| ≤ λ a.e. μ on Rd \ i Qi ,
b for each i, let Ri be the smallest (β, βn1 )-doubling cube of the form βk Qi , k ∈ N, with
that β ≥ 3α, and let wi χQi / k χQk , then there exists a family of functions ϕi with
supϕi ⊂ Ri and with constant sign satisfying
b.1 ϕi Qi fwi dμ;
b.2 i |ϕi | ≤ Bλ, where B is some constant;
b.3 ϕi L∞ μ μRi ≤ C Qi |f|dμ.
6
Journal of Inequalities and Applications
To prove our theorems, we prepare another two key lemmas.
For any subset E ⊂ Rd1 , we denote
T E :
E
T E :
E
t
t x − y
t
t x − y
2n2 λn
t
t y − z
2n2
dμ y dt
,
t3n1
dμ y dt
t2δ
.
2n2γ2δ
tn1
t y − xi 2.2
It is easy to see that
Rd1
t
t x − y
λn t
t y − z
2n2δ
dμ y dt
T Rd1
.
t3n1
2.3
Lemma 2.3. Let Qi and Ri be the same as in Lemma 2.2, and let xi be Qi s center. Then for any z ∈ Qi
and x ∈ αRi \ αQi ,
T Rd1
1
|x − xi |2n
2.4
.
Lemma 2.4. Let Qi and Ri be the same as in Lemma 2.2, and let xi be Qi s center. For any z ∈ Qi and
x ∈ αRi \ αQi , then
T
: y − xi > 2|xi − z| ≤
y, t ∈ Rd1
1
|x − xi |2n
2.5
.
Proof of Lemma 2.3. Denote
A :
t
t x − y
2n2
,
B :
t
t y − z
2n2
.
2.6
Set
D1 D2 D3 D4 : t > max y − x, y − z ,
y, t ∈ Rd1
,
y
−
z
<
t
<
y
−
x
:
y, t ∈ Rd1
y, t ∈
Rd1
: y − z < t < y − x ,
2.7
.
y
−
z
,
y
−
x
:
t
<
min
y, t ∈ Rd1
It is easy to see
T Rd1
T D 1 T D2 T D3 T D4 .
2.8
Journal of Inequalities and Applications
7
We first estimate T D1 . Set
D1,1 D1 ∩
D1,2 D1 ∩
D1,3 D1 ∩
D1,4 D1 ∩
: t < y − xi ,
y, t ∈ Rd1
: t > y − xi , |x − xi | > 2y − xi ,
y, t ∈ Rd1
: t > y − xi , |x − xi | ≤ 2y − xi , |xi − z| ≤ y − xi ,
y, t ∈ Rd1
2.9
: t > y − xi , |x − xi | ≤ 2y − xi , |xi − z| > y − xi .
y, t ∈ Rd1
We have
T D1 T D1,1 T D1,2 T D1,3 T D1,4 .
2.10
Note that z ∈ Qi and x ∈ αRi \ αQi , then |z − xi | ≤ ri , |x − xi | ≥ 16ri . These two inequalities will
always be used in the following proof, so we will not mention them every time.
For any y, t ∈ D1,1 , we have that |y − xi | ≥ 8ri , |y − z| ∼ |y − xi |, |y − xi | > 1/2|x − xi |,
t |x − xi |, t < |y − xi |. Then
A≤
t2n−2
|x − xi |
t2n2
B≤
.
y − xi 2n2
,
2n−2
2.11
For any y, t ∈ D1,2 , we get |x − xi | ≤ 2t. Then
A ≤ 1,
tn−
B≤
.
y − zn−
2.12
For y, t ∈ D1,3 , we obtain that |y − z| ≤ 2|x − xi |, t > 1/2|x − xi |. Therefore
A ≤ 1,
tn−
B≤
.
y − zn−
2.13
For y, t ∈ D1,4 , we have that |x − y| ∼ |x − xi |, t > ri . Therefore
A≤
t2n2
|x − xi |2n2
,
B ≤ 1.
2.14
8
Journal of Inequalities and Applications
Thus, we get
|y−xi |
T D1,1 ≤
T D1,2 ≤
T D1,3 ≤
|y−xi |>1/2|x−xi |
|x − xi |2n−2
0
t2n2
1
dt dμ y
y − xi 2n2 t3n1
1
,
|x − xi |2n
∞
1/2|x−xi |
∞
1/2|x−xi |
tn−
1
1
,
n− 3n1 dμ y dt t
|x − xi |2n
|y−z|≤t y − z
|y−xi |<ri
2.15
tn−
1
1
,
n− 3n1 dμ y dt |x − xi |2n
|y−z|≤2|x−xi | y − z t
∞
T D1,4 ≤
t2n−2
t2n2
|x − xi |
ri
1
2n2 t3n1
dt dμ y 1
|x − xi |2n
.
Next we estimate T D2 . Set
D2,1 D2 ∩
D2,2 D2 ∩
: y − xi > 2|x − xi | ,
y, t ∈ Rd1
2.16
: y − xi ≤ 2|x − xi | .
y, t ∈ Rd1
For any y, t ∈ D2,1 , there exist two constants C1 , C2 such that C1 |y − xi | < t < C2 |y − xi |. Since
A ≤ 1, B ≤ 1, we then have that
C2 |y−xi |
T D2,1 ≤
|y−xi |>2|x−xi |
C1 |y−xi |
1
t3n1
dt dμ y 1
|x − xi |2n
2.17
.
For any y, t ∈ D2,2 , the following inequalities hold: t < 3|x − xi |, |x − z| ∼ |x − xi |, and
t |y − z| > |x − z|. It follows that
A ≤ 1,
T D2,2 3|x−xi | 0
B
t2n2
|x − xi |2n2
t2n2
|y−x|<t |x − xi |
2n2
,
dμ y dt 3n1
1
t
1
|x − xi |2n
2.18
.
Next we estimate T D3 . Set
D3,1 D3 ∩
D3,2 D3 ∩
,
y
−
x
:
−
x
≤
2
y, t ∈ Rd1
|x
|
i
i
: |x − xi | > 2y − xi .
y, t ∈ Rd1
2.19
Journal of Inequalities and Applications
9
Then
T D3 T D3,1 T D3,2 .
2.20
For any y, t ∈ D3,1 , we can get that |y − z| ∼ |y − xi | and there exist two constants C1 and C2
such that C1 |y − xi | < t < C2 |y − xi |. Then
t2n2
,
B
y − xi 2n2
A ≤ 1,
C2 |y−xi |
T D3,1 1
t2n2
1
.
2n2 3n1 dt dμ y t
|x − xi |2n
C1 |y−xi | y − xi |y−xi |≥1/2|x−xi |
2.21
For any y, t ∈ D3,2 , we can get |x − y| ≥ 1/2|x − xi | and t < 3/2|x − xi |. Then
A
t2n2
|x − xi |
,
2n2
B ≤ 1,
3/2|x−xi | T D3,2 0
t2n2
|y−z|<t |x − xi |
2n2
dμ y dt.
2.22
Next we estimate T D4 . Set
D4,1 D4 ∩
D4,2 D4 ∩
D4,3 D4 ∩
: x − y > 2|x − z| ,
y, t ∈ Rd1
Rd1
y, t ∈
y, t ∈ Rd1
1
: |x − z| < x − y ≤ 2|x − z| ,
2
1
: x − y ≤ |x − z| .
2
2.23
If y, t ∈ D4,1 , we obtain |y − z| > |x − z|, |x − z| ∼ |x − xi | and
A
t2n−2
|x − z|
2n−2
,
t2n2
B≤
.
y − z2n2
2.24
Therefore
T D4,1 |y−z|
|y−z|>|x−z|
0
t2n−2
|x − z|2n−2
t2n2
1
1
.
2n2 3n1 dt dμ y y − z
t
|x − xi |2n
2.25
If y, t ∈ D4,2 , we have t < 2|x − z| and
A
t2n2
|x − z|
,
2n2
t2n−2
B≤
.
y − z2n−2
2.26
10
Journal of Inequalities and Applications
Therefore
2|x−z| T D4,2 t2n2
|y−z|>t |x
0
− z|
2n2
t2n−2
1
2n−2 3n1 dμ y dt.
y − z
t
2.27
If y, t ∈ D4,3 , we have |y − z| ∼ |x − z| and t < 1/2|x − z|. Then
t2n−2
,
A≤ x − y2n−2
T D4,3 1/2|x−z| 0
B
t2n2
|x − z|2n2
,
t2n−2
t2n2
1
1
dμ y dt .
2n−2
2n2
3n1
t
−
z|
−
xi |2n
|x
|x
|x−y|>t x − y
2.28
The proof of Lemma 2.3 is finished.
Proof of Lemma 2.4. Denote
A :
t
t |x − y|
λn
,
t2δ
2n2γ2δ .
t y − xi B : 2.29
Let K {y, t ∈ Rd1
: |y − xi | > 2|xi − z|} and divide K into four parts
K1 K ∩
K2 K ∩
K3 K ∩
K4 K ∩
: max y − x, y − z ≤ t ,
y, t ∈ Rd1
: y − x < t < y − z ,
y, t ∈ Rd1
: y − x ≤ t, y − xi < 4ri ,
y, t ∈ Rd1
2.30
y
−
x
≤
t,
y
−
x
≥
4r
.
:
y, t ∈ Rd1
i
i
Then T K T K1 T K2 T K3 T K4 .
Since x ∈ Rn \ αRi and z ∈ Ri , we have that |z − xi | ≤ ri , |x − xi | ≥ 16ri .
We first estimate T K1 . Set
K1,1 K1 ∩
K1,2 K1 ∩
K1,3 K1 ∩
3
y, t ∈ Rd1
: y − xi > ri , t < y − xi 2ri ,
2
y, t ∈
K1,4 K1 ∩
3
y
−
x
≤
r
y, t ∈ Rd1
:
,
i
i
2
Rd1
y, t ∈ Rd1
3
: y − xi > ri , t ≥ y − xi 2ri , |x − xi | ≤ 2 y − xi ,
2
3
: y − xi > ri , t ≥ y − xi 2ri , |x − xi | > 2 y − xi .
2
2.31
Journal of Inequalities and Applications
11
Then
T K1 T K1,1 T K1,2 T K1,3 T K1,4 .
2.32
For any y, t ∈ K1,1 , we have that |x − xi | ∼ |x − y| and t ≥ 1/2|x − xi |. Then we get
A ≤ 1,
B≤
t2δ
t2n2γ2δ
2.33
,
and therefore
T K1,1 ≤
∞
t2δ
|y−z|≤t t
1/2|x−xi |
2n2γ2δ
dμ y dt
1
.
tn1
|x − xi |2n2γ
2.34
For any y, t ∈ K1,2 , we have |x − xi | ≤ 3|y − xi |, 1/2|y − xi | < t < 2|y − xi |. It follows that
t2δ
B≤
,
xi − y2n2γ2δ
A ≤ 1,
T K1,2 ≤
dt dμ y
t2δ
1
.
2n2γ2δ
2n2γ
n1
t
−
x
|x
1/2|y−xi | xi − y
i|
2|y−xi |
|y−xi |>1/3|x−xi |
2.35
If y, t ∈ K1,3 , we will get |x − xi | ≤ |y − xi | ≤ 2t. It follows that
A ≤ 1,
B≤
1
t2n2γ
,
T K1,3 ≤
∞
1/2|x−xi |
1
|y−xi |<t t
2n2γ
dt dμ y
1
.
2n2γ
tn1
−
x
|x
i|
2.36
If y, t ∈ K1,4 , we obtain that t > 1/2|x − xi |. Thus we can get
A ≤ 1,
T K1,4 ≤
∞
1
1/2|x−xi | t
1
,
t2n2γ
dμ y dt
1
.
tn1
|x − xi |2n2γ
B≤
2n2γ
2.37
Next we estimate T K2 . Set
K2,1 K2 ∩
y
−
x
,
>
2|x
−
x
:
y, t ∈ Rn1
|
i
i
K2,2 K2 ∩
: y − xi ≤ 2|x − xi | .
y, t ∈ Rn1
2.38
Then
T K2 T K2,1 T K2,2 .
2.39
12
Journal of Inequalities and Applications
For any y, t ∈ K2,1 , the inequalities t > 1/2|y − xi | and t ≤ 2|y − xi | hold, and we have
A ≤ 1,
B≤
2|y−xi |
T K2,1 ≤
|y−xi |>2|x−xi |
1
,
t2n2γ
dt dμ y
.
tn1
1
1/2|y−xi | t
2n2γ
2.40
For any y, t ∈ K2,2 , the inequalities t < 3|x − xi | and t |y − xi | > |x − xi | hold, and we can get
A ≤ 1,
T K2,2
t2δ
B≤
,
|x − xi |2n2γ2δ
2|x−xi | dμ y dt
t2δ
1
≤
.
n1
2n2γ2δ
t
|x − xi |2n2γ
0
|y−x|<t |x − xi |
2.41
Next we estimate T K3 . Let
K3,1 K3 ∩
K3,2 K3 ∩
y, t ∈ Rd : t > y − xi ,
2.42
y, t ∈ Rd : t ≤ y − xi .
Then
2.43
T K3 T K3,1 T K3,2 .
Since 2n 2γ < λn, we can choose > 0 small enough such that 2n 2γ 2 < λn and
< 1/2n.
Now denote
:
A
t
t |x − y|
2n2γ2
2.44
,
and for a subset E ⊂ Rd , we denote
T E :
E
t
t |x − y|
It is easy to see that T E ≤ T E.
2n2γ2
t2δ
t |y − xi |
2n2γ2δ
dμ y dt
.
tn1
2.45
Journal of Inequalities and Applications
13
For any y, t ∈ K3,1 , we have |x − y| ∼ |x − xi |, t ≤ 2|x − xi |. It follows that
A
≤
AB
t2n2
|x − xi |2n2
t2n22γ
|x − xi |2n22γ
,
t2δ2γ
1
1
t2n2
n− n .
2n2γ2δ ≤
2n2 t
y − xi
|x − xi |
t y − xi
2.46
The last inequality holds because t > |y − xi | and we choose > 0 small enough such that
n − > 0, and then we can get t/|y − xi |n− > 1, which leads to the above inequality. Using
these inequalities, we obtain
T K3,1 2|x−xi |
|y−xi |<4ri
t2n2
1
n−
|x − xi |2n2 y − xi 0
1 dt dμ y
1
.
tn tn1
|x − xi |2n2γ
2.47
If y, t ∈ K3,2 , we get t ≤ |x − xi |. It follows that
≤
A
t2
|x − xi |2n2γ2
,
t2n2γ2δ
tnγδ
t2
,
2n2γ2δ ≤
2n2γ2
y − xi nγδ
|x − xi |
|x − xi |
t y − xi |x−xi | dμ y dt
t2
tnγδ
1
≤
×
.
nγδ
2n2γ2
n1
t
y − xi |x − xi |2n2γ
0
|y−xi |≥t |x − xi |
≤
AB
T K3,2
t2n2γ2
2n2γ2
2.48
Next we estimate T K4 . Set
K4,1 K4 ∩
K4,2 K4 ∩
K4,3 K4 ∩
K4,4 K4 ∩
K4,5 K4 ∩
K4,6 K4 ∩
K4,7 K4 ∩
: y − xi ≤ t < y − xi 2ri , 2y − xi > |x − xi | ,
y, t ∈ Rd1
y
−
x
y
−
x
y
−
x
≤
t
<
2r
≤
−
x
,
:
,
2
y, t ∈ Rd1
|x
|
i
i
i
i
i
: max y − xi , y − xi 2ri ≤ t, 2y − xi > |x − xi | ,
y, t ∈ Rd1
y
−
x
y
−
x
,
,
2r
≤
−
x
y
−
x
:
max
≤
t,
2
y, t ∈ Rd1
|x
|
i
i
i
i
i
y, t ∈ Rd1
y, t ∈
Rd1
y, t ∈ Rd1
: y − xi > t, x − y > 2|x − xi | ,
1
: y − xi > t, |x − xi | < x − y ≤ 2|x − xi | ,
2
1
: y − xi > t, x − y < |x − xi | .
2
2.49
14
Journal of Inequalities and Applications
Then
T K4 T K4,1 T K4,2 T K4,3 T K4,4 T K4,5 T K4,6 T K4,7 .
2.50
For any y, t ∈ K4,1 , we have 3/4|y − xi | < t < 3/2|y − xi |. Then we get
A ≤ 1,
3/2|y−xi |
T K4,1 ≤
3/4|y−xi | t
|y−xi |≥1/2|x−xi |
t2δ
B≤
t2n2γ2δ
t2δ
2n2γ2δ
,
dt dμ y
1
.
n1
t
|x − xi |2n2γ
2.51
If y, t ∈ K4,2 , we get |x − y| > 1/2|x − xi | and t < 2|x − xi |. Then we can obtain
t2n2γ2
<
A
T K4,2 ≤
|x − xi |
2|x−xi | t2n2γ2
|y−xi |<t |x
0
,
2n2γ2
− xi |2n2γ2
t2δ
B<
,
t2n2γ2δ
t2δ dμ y dt
1
.
n1
2n2γ2δ
t
t
|x − xi |2n2γ
2.52
If y, t ∈ K4,3 , we get t > 1/2|x − xi |. Then
1
t2δ
1
,
2n2γ ≤ 2δ y − xi 2n2γ
t y − xi t y − xi ∞
dt dμ y
1
1
.
2n2γ
n1
t
|x − xi |2n2γ
|y−xi |>1/2|x−xi | 1/2|x−xi | y − xi
A ≤ 1,
T K4,3
B
2.53
For any y, t ∈ K4,4 , the inequalities |x − y| ≥ 1/2|x − xi |, 0 < t < 2|x − xi |, and t > |y − xi |
hold, from which we obtain
A
T K4,4 t2n2γ2
|x − xi |2n2γ2
2|x−xi | t2n2γ2
|y−xi |≤t |x
0
1
,
t2n2γ
dμ y dt
1
.
tn1
|x − xi |2n2γ
B≤
,
1
− xi |2n2γ2 t2n2γ
2.54
If y, t ∈ K4,5 , we have |y − xi | ≥ |x − xi |. Then
A≤
t
t x − y
T K4,5 2n2γ−2
|y−xi |
|y−xi |>|x−xi |
0
t2n2γ−2
|x − xi |
t2n2γ−2
|x − xi |2n2γ−2
,
2n2γ−2
t2δ
B≤
,
y − xi 2n2γ2δ
dt dμ y
t2δ
1
.
2n2γ2δ
n1
t
y − xi |x − xi |2n2γ
2.55
Journal of Inequalities and Applications
15
If y, t ∈ K4,6 , we have t < 3|x − xi | and it follows that
A
T K4,6 t2n2γ2
|x − xi |
3|x−xi | t2n2γ2
|y−xi |>t |x
0
t2δ
B≤
,
y − xi 2n2γ2δ
dμ y dt
t2δ
1
.
2n2γ2δ
n1
t
y − xi |x − xi |2n2γ
,
2n2γ2
− xi |2n2γ2
2.56
If y, t ∈ K4,7 , we get |y − xi | ∼ |x − xi | and t |x − xi |. Then we have
A≤
T K4,7 t
x − y
3/2|x−xi | |y−x|>t
0
2n2γ
B
,
t
x − y
2n2γ
t2δ
|x − xi |2n2γ2δ
t2δ
|x − xi |2n2γ2δ
,
dμ y dt
1
.
n1
t
|x − xi |2n2γ
2.57
3. Proof of Theorems
√
Proof of Theorem 1.4. To prove Theorem 1.4, we will choose α 16 d and β 3α.
Let f ∈ L1 μ and λ > αd1 fL1 μ /μ. Applying Lemma 2.2 to f and λ, we obtain a
family of almost disjoint cubes {Qi }i . With the notation wi , ϕi , Ri the same as in Lemma 2.2,
we can decompose f g b, with that g fχRd \i Qi j ϕi and b j wj f − ϕj i bi .
And Rd can be decomposed as
Rd βRi c ∪ βR
i \ αQi ∪ αQi . By a.1 of Lemma 2.2, we
∞
∞ have μ i1 αQi ≤ C/λ i Qi |f|dμ ≤ C/λ |f|dμ.
∗
is of weak type 1, 1, we only need to prove
Thus, to prove that gλ,μ
μ x∈R \
d
∞
αQi :
∗
gλ,μ
f x > λ
i1
C f dμ.
≤
λ
3.1
∗
∗
∗
Since f g b and gλ,μ
f ≤ gλ,μ
g gλ,μ
b, we only need to show that both g and b satisfy
the inequality 2.4.
|f|dμ fL1 μ ,
For g, it follows from b.1 of Lemma 2.2 that ϕi L1 μ ≤
Qi
∗
2
and we have gL1 μ fL1 μ < ∞. Using L -boundedness of gλ,μ and a.3 and b.2 from
Lemma 2.2, we obtain
μ x ∈ Rd \
∞
i1
∗
αQi : gλ,μ
g x > λ
C
≤ 2
λ
f 1
2
C
L μ
g dμ g dμ ≤
.
λ
λ
3.2
16
Journal of Inequalities and Applications
To prove that b satisfies inequality 3.1, it suffices to show that
Since b j wj f
− ϕj Rd \ ∞
i1 αQi
∗
gλ,μ
bxdμ
Rd \ ∞
αQ
i
i1
i
bL1 μ f L1 μ .
3.3
bi , we have that
∗
gλ,μ
bxdμ ≤
Rd \αQi
i
∗
gλ,μ
bi xdμ
≤
Rd \αRi
i
:
∗
gλ,μ
bi xdμ αRi \αQi
∗
gλ,μ
bi xdμ
3.4
Ai Bi .
i
If we can prove
f dμ,
Ai f dμ,
Bi Qi
3.5
Qi
then we finish the proof of Theorem 1.4.
We first estimate Bi and divide it into two parts
Bi ≤
αRi \αQi
∗
gλ,μ
wi f xdμx αRi \αQi
∗
ϕi xdμx
gλ,μ
3.6
: Bi,1 Bi,2 .
∗
By the assumption of L2 boundedness of gλ,μ
, and the fact that Ri is the β, βn1 doubling
cube, it is easy to see that
Bi,2 ≤
αRi
∗
gλ,μ
ϕi xdμx ≤
2
ϕi dμx
αRi
αRi
2
∗
gλ,μ ϕi x dμx
μαRi 1/2
3.7
1/2
1/2
f dμ.
μRi 1/2 Qi
Journal of Inequalities and Applications
17
Next we estimate Bi,1 . By the Minkowski’s inequality, Fubini’s theorem and condition b of
ψ that |ψx| ≤ C0 1 |x|−n−δ , we have the following estimate:
Bi,1 αRi \αQi
⎡
⎢
⎣
⎤1/2
2
λn/2 dμydt
t
⎥
ψt y − zwi zfzdμz
⎦ dμx
t x − y
n1
d1 t
R
⎡
fz⎣
αRi \αQi
Qi
fz
αRi \αQi
Qi
Rd1
⎡
⎣
Rd1
t
t x − y
t
t x − y
λn
⎤1/2
2 dμydt
⎦ dμzdμx
ψt y − z
tn1
λn t
t y − z
2n2δ
⎤1/2
dμydt
⎦ dμxdμz.
t3n1
3.8
Choose > 0 small enough such that < δ, 2n 2 < λn and 2 < n.
For any subset E ⊂ Rd1 , we denote
T E :
E
t
t x − y
2n2 t
t y − z
2n2
dμ y dt
.
t3n1
3.9
Let xi be Qi s center. Using the above notations, it is easy to see that
Rd1
t
t x − y
λn t
t y − z
2n2δ
dμ y dt
T Rd1
.
t3n1
3.10
By Lemma 2.3, we will have the following inequalities:
∗
gλ,μ
fwi x fzdμz
Bi,1 Qi
1
|x − xi |n
fwi dμz,
Qi
1
n dμx αRi \αQi |x − xi |
fzdμz,
Qi
3.11
3.12
where we used Lemma 2.1 to estimate
1
dμx ≤
−
xi |n
|x
αRi \αQi
1
1
dμx
n dμx −
x
−
xi |n
|x
|
|x
i
αRi \Ri
Ri \βQi
1
n dμx ≤ Cα,β,n,C0 .
βQi \αQi |x − xi |
3.13
18
Journal of Inequalities and Applications
√
∗
bi xdμx, and
We now estimate Ai . Let ri d/2lRi . Recall that Ai Rd \αRi gλ,μ
Rd \αRi
∗
gλ,μ
bi xdμx
≤
Rd \αRi
⎡
⎢
⎣
⎡
⎣
⎤1/2
2
λn/2 dμydt ⎥
t
ψt y − z bi zdμz
⎦ dμx
tn1
|y−xi |≤2|xi −z| t x − y
Rd \αRi
λn/2
t
|y−xi |>2|xi −z| t x − y
2 ⎤1/2
dμ y dt ⎥
× ψt y − z − ψt y − xi bi zdμz
⎦ dμx
tn1
: A1i A2i .
3.14
We first estimate A1i . By the Minkowski’s inequality, Fubini’s theorem and property b of ψ,
we can obtain that
A1i
≤C
|bi z|
Ri
⎡
⎣
×
Rd \αRi
|y−xi |≤2|xi −z|
t
t x − y
λn t
t y − z
2n2δ
⎤1/2
dμydt
⎦ dμxdμz.
t3n1
3.15
If we prove
|y−xi |≤2|xi −z|
t
t x − y
λn t
t y − z
2n2δ
ri2
dμ y dt
,
t3n1
|x − xi |2n2
3.16
for any z ∈ Ri and x ∈ αRi c , then
⎡
⎣
Rd \αRi
|y−xi |≤2|xi −z|
t
t x − y
λn and by Lemma 2.2, we conclude that A1i t
t y − z
Ri
2n2δ
|bi z|dμz ≤
⎤1/2
dμ y dt
⎦ dμx 1,
t3n1
Qi
|fz|dμz.
3.17
Journal of Inequalities and Applications
19
Now choose > 0 small enough such that < δ, 2n 2 < λn and 2 < n. Then for any
z ∈ Ri and x ∈ αRi c , note that |x − y| ∼ |x − xi |
|y−xi |≤2|xi −z|
t
t x − y
≤
|y−xi |≤2|xi −z|
ri ≤
∞
ri
ri ≤
t
t x − y
0
|y−z|≤3ri
0
∞
ri
2n2 2n2 t
t x − y
t
x − y
|y−z|≤3ri
|y−z|≤3ri
t
t y − z
t
t x − y
|y−z|≤3ri
λn 2n2
dμ y dt
t3n1
t
t y − z
t
t y − z
2n2 2n2 t
x − y
2n2δ
2n2
2n2
t
t y − z
t
y − z
n−
dμ y dt
t3n1
dμ y dt
t3n1
2n2
dμ y dt
t3n1
3.18
dt dμ y
t3n1
ri2
dt dμ y
.
t3n1
|x − xi |2n2
Next we estimate A2i . By property c of ψ and 2|xi − z| ≤ |y − xi |, we have
ψt y − z − ψt y − xi ≤ 1
tn
|xi − z|
t
γ t
t y − xi nγδ
γ
≤
ri tδ
nγδ ,
t y − xi 3.19
since |xi − z| ≤ ri . Using the above inequality, Minkowski’s inequality and Fubini’s theorem,
we get
A2i
≤C
|bi z|
Ri
⎡
⎣
×
Rd \αRi
|y−xi |>2|xi −z|
t
t x − y
λn
⎤1/2
2γ
ri t2δ
dμydt
⎦ dμxdμz.
2n2γ2δ
tn1
t y − xi 3.20
So by Lemma 2.4, for x ∈ Rd \ αRi and any z ∈ Ri
|y−xi |>2|xi −z|
t
t x − y
λn
dμ y dt
t2δ
1
.
2n2γ2δ
n1
t
|x − xi |2n2γ
t y − xi
3.21
20
Journal of Inequalities and Applications
Then we get
⎡
⎣
Rd \αRi
|y−xi |>2|xi −z|
and we will have A2i t
t x − y
Ri
λn
|bi z|dμz ≤
⎤1/2
2γ
ri t2δ
dμ y dt
⎦ dμx 1,
2n2γ2δ
tn1
t y − xi Qi
|fz|dμz. With the estimates of
A1i fzdμz,
|bi z|dμz ≤
A2i 3.22
Ri
Qi
fzdμz,
|bi z|dμz ≤
Ri
3.23
Qi
we obtain that
Ai fzdμz.
|bi z|dμz ≤
Ri
Qi
3.24
Thus,
Rd \αRi
∗
gλ,μ
bi xdμx |bi z|dμz.
Ri
3.25
Therefore we finish the proof of Theorem 1.4.
Proof of Theorem 1.5. Note that the definition of H 1 μ is √
independent of the choice of the
constant ρ, we can assume that ρ α, still with α 16 d. By Theorem 1.4, the operator
∗
is bounded from L1 μ to L1,∞ μ. By a standard argument, we only need to prove that
gλ,μ
∗
bL1 μ ≤ |b|H 1 μ for any atomic block b with suppb ⊂ R. Write
gλ,μ
R
∗
gλ,μ
bxdμx
R\αR
∗
gλ,μ
bxdμx
αR
∗
gλ,μ
bxdμx
3.26
≡ J1 J2 .
By definition of b, we have bL1 μ ≤ |b|H 1 μ . By 3.25 in the proof of Theorem 1.4, we can
get
J1 Rd \αR
∗
gλ,μ
bxdμx R
|bz|dμz |b|H 1 μ .
3.27
Journal of Inequalities and Applications
To estimate J2 , let b j
λj aj be as in Definition 1.1 and write
J2 αR\∪αQj
∗
gλ,μ
bxdμx
λj ≤
αR\αQj
j
21
∗
gλ,μ
∪αQj
∗
gλ,μ
bxdμx
aj xdμx λj αQj
j
∗
gλ,μ
aj xdμx.
3.28
∗
and the Holder inequality,
By the assumption of L2 boundedness of gλ,μ
αQj
∗
gλ,μ
aj xdμx
∗ ≤ gλ,μ
L2 μ
1/2
1/2
∗ · μ αQj
≤ Cgλ,μ
· aj L2 μ · μ αQj
3.29
(−1 ' ≤ Caj L∞ · μ αQj ≤ C μ αQj SQj ,R
· μ αQj ≤ C.
By 3.11 in the proof of Theorem 1.4,
αR\αQj
∗
gλ,μ
aj xdμx 1
n
αR\αQj x − xj
aj L1 μ
aj L1 μ
aj y dμ y dμx
Qj
1
n dμx
x
−
xj αR\αQj
3.30
1
n dμx
αRj \αQj x − xj
aj L∞ · μ Qj · SQj ,R ,
3.31
where Rj is a cube and has the same center as Qj and lRj lR, xj is the center of Qj . From
3.29 to 3.30, we use the conclusion
1
n dμx SQj ,R .
αRj \αQj x − xj
3.32
22
Journal of Inequalities and Applications
As a matter of fact, let N NQj ,R and then by the definition of N, we get that
αRj \ αQj ⊂
N )
*
αk1 Qj \ αk Qj ,
k1
N
1
n dμx ≤
αRj \αQj x − xj
k1
αk1 Qj \αk Qj
1
dμx
x − xj n
N
N μ αk1 Q
μ αk1 Qj
j
≤
n k1 n
k
Qj
k1 1/2lα Qj k1 l α
3.33
SαQj ,R ≈ SQj ,R .
Using 3.30 and the fact that
aj ∞ ≤ μ ρQj SQ,R −1 ,
L μ
3.34
we have
αR\αQj
∗
gλ,μ
aj xdμx aj L∞ · μ Qj · SQj ,R
(−1 ' μ αQj SQj ,R
· μ Qj · SQj ,R 1.
3.35
From 3.29 and 3.35, we can obtain that
αR
∗
gλ,μ
bxdμx λj .
j
3.36
We have
J2 |b|H 1 μ .
3.37
Combining 3.27 and 3.37, we finish the proof of Theorem 1.5.
Proof of Theorem 1.6. Now we begin to prove Theorem√
1.6. First
√ we claim that there is a
positive constant C such that for any f ∈ L∞ μ and 4 d, 4 dn1 -doubling cube Q, the
following inequality holds:
1
μQ
Q
∗
∗
gλ,μ
f y dμ y ≤ Cf L∞ μ inf gλ,μ
f y .
y∈Q
3.38
Journal of Inequalities and Applications
23
To prove 3.38, for each fixed cube
√ Q, let B be the smallest ball which contains Q and has the
same center as Q. Then 2B ⊂ 4 dQ. We decompose f as
fx fxχ2B fxχRd \2B : f1 x f2 x.
3.39
∗
, we have
By the Hölder’s inequality and L2 μ boundedness of gλ,μ
1
μQ
Q
∗
f1 xdμx ≤
gλ,μ
1
μQ
∗ 2
gλ,μ f1 x dμx
Q
μ2B
f L∞ μ
μQ
1/2
1/2
μQ1/2 f1 2
L μ
μQ1/2
⎛ ) √ * ⎞1/2
μ 4 dQ
⎟
⎜
f L∞ μ ⎝
⎠
μQ
3.40
f L∞ μ .
We denote by r the radius of B. Note that |x − z| ≥ r for any x ∈ Q and for any z ∈ Rd \ 2B,
then by Minkowski’s inequality, we have
∗
f2 x
gλ,μ
⎛
⎜
⎝
⎞1/2
2
λn/2 dμydt
t
⎟
ψt y − zf2 zdμz
⎠
t x − y
n1
d1 t
R
|x−z|≥r
⎛
⎜
≤⎝
⎞1/2
2
λn/2 dμydt ⎟
t
ψt y − zfzdμz
⎠
t x − y
d1 tn1
R
|x−z|≥r
⎛
⎜
⎝
⎞1/2
2
λn/2 dμydt ⎟
t
ψt y − zf1 zdμz
⎠
tn1
t x−y
Rd1
|x−z|≤r
⎛
⎜
∗
≤ gλ,μ
f x ⎝
⎞1/2
2
λn/2 dμydt ⎟
t
ψt y − zf1 zdμz
⎠ .
t x − y
tn1
Rd1
|x−z|≤r
3.41
24
Journal of Inequalities and Applications
Since we have the following estimate:
⎛
⎜
⎝
⎞1/2
2
λn/2 dμydt ⎟
t
ψt y − zf1 zdμz
⎠
t x − y
d1 tn1
R
|x−z|≤r
f L∞ μ
⎛
⎝
r≤|x−z|≤3r
Rd1
t
t x − y
2n2 t
t y − z
2n2
⎞1/2
dμydt ⎠
dμz.
t3n1
3.42
We claim
Rd1
t
t x − y
2n2 t
t y − z
2n2
dμ y dt
1
.
t3n1
|z − x|2n
3.43
Then
⎛
⎝
r≤|x−z|≤3r
Rd1
t
t x − y
2n2 t
t y − z
2n2
⎞1/2
dμydt ⎠
dμz 1.
t3n1
3.44
By 3.42, 3.44, and 3.45, we can obtain
∗
∗
f2 x gλ,μ
f x f L∞ μ .
gλ,μ
3.45
The method to prove 3.43 is quite similar as to prove 2.4 in the proof of Theorem 1.4 and
we omit it.
Thus, to prove 3.38, we only need to prove for any x, y ∈ Q, the following inequality
holds:
∗ ∗
f2 y f L∞ μ .
gλ,μ f2 x − gλ,μ
3.46
∗
We note that gλ,μ
can be looked as a vector valued Calderón-Zygmund singular integral
operator in the following Hilbert space H:
H
h : hH ∞ 0
ht, y2 dμydt
tn1
Rd
1/2
<∞ .
3.47
Journal of Inequalities and Applications
25
∗
can be written by
In fact, gλ,μ
⎛
⎜
∗
gλ,μ fx ⎝
⎞1/2
2
λ/2 dμydt ⎟
t
ψt y − zfzdμz
⎠
t x − y
d1 tn1
R
3.48
: ψt,y fxH ,
where ψt,y fx t/t |x − y|λn/2 ψt y − zfzdμz. Also note that for u, v ∈ Q,
∗ ∗
f2 v ψt,y f2 uH − ψt,y f2 vH gλ,μ f2 u − gλ,μ
≤ ψt,y f2 u − ψt,y f2 vH
3.49
: I,
where ψt,y fx t/t |x − y|λn/2 ψt y − zfzdz. We will divide I into four parts,
namely,
I ≤ ψt,y f2 u − ψt,y f2 vχ{|u−y|≤t,|v−y|≤t} H
ψt,y f2 u − ψt,y f2 vχ{|u−y|>t,|v−y|≤t} H
ψt,y f2 u − ψt,y f2 vχ{|u−y|≤t,|v−y|>t} H
ψt,y f2 u − ψt,y f2 vχ{|u−y|>t,|v−y|>t} H
3.50
: I1 I2 I3 I4 .
Then
Ii ≤ CM f x,
for i 1, . . . , 4.
3.51
This can be obtained from the same idea used before, see also the main step in 14, here we
omit the proof of it.
∗
fx0 < ∞ for some point x0 ∈ Rd , then
From 3.38, we get that for f ∈ L∞ μ, if gλ,μ
∗
gλ,μ f is μ-finite almost everywhere and in this case we have
)
*
∗
∗
mQ gλ,μ
f − ess inf gλ,μ
f x f L∞ μ ,
x∈Q
√
√
provided that Q is a 4 d, 4 dn1 -doubling cube.
3.52
26
Journal of Inequalities and Applications
∗
∗
f ∈ RBLOμ, we still need to prove that gλ,μ
f satisfies
To prove gλ,μ
)
)
*
*
∗
∗
mQ gλ,μ
f − mR gλ,μ
f ≤ CSQ,R
3.53
√
√
for any two 4 √
d, 4 dn1 -doubling cubes Q ⊂ R.
Let α1 4 d and set N NQ,R 1, then
)
*
∗
∗
∗
f x ≤ Cgλ,μ
fχα1 Q gλ,μ
gλ,μ
fχαQ c
)
*
)
*
∗
∗
∗
fχαN1 Q\α1 Q gλ,μ
fχαN Qc
≤ Cgλ,μ
fχα1 Q gλ,μ
1
)
*
NQ,R ∗
∗
gλ,μ fχαk1 Q\αk Q x
≤ gλ,μ
fχα1 Q Σk1
1
3.54
1
)
*
)
* (
'
∗
∗
fχαN Qc x − gλ,μ
fχαN Qc y
gλ,μ
1
1
)
* ∗
fχαN Qc y .
gλ,μ
1
Again, we can get the conclusion under the nondoubling condition which is similar to
the proof of 2.2 in 14 that
)
*
)
* ∗
∗
fχαN Qc y f L∞ μ .
gλ,μ fχαN Qc x − gλ,μ
1
1
3.55
For any x ∈ Q and each fixed k ∈ N, similar to the proof of 2.4 from Lemma 2.3, we have
that
)
*
∗
fχαk1 Q\αk Q x
gλ,μ
1
1
≤C
k
αk1
1 Q\α1 Q
≤ Cf L∞ μ
≤ Cf L∞ μ
⎡
fz⎣
Rd1
⎡
⎣
k
αk1
1 Q\α1 Q
t
t x − y
Rd1
1
dμx.
|x − xi |n
2n2 t
t x − y
t
t y − z
2n2 2n2
t
t y − z
⎤1/2
dμydt
⎦ dμz
t3n1
2n2
⎤1/2
dμydt
⎦ dμz
t3n1
3.56
Journal of Inequalities and Applications
27
Therefore we have
N
Q,R
k1
)
*
∗
fχαk1 Q\αk Q x ≤ Cf L∞ μ SQ,R .
gλ,μ
1
3.57
1
∗
fχαN Qc y. By an estimate similar to 3.45, we have
Next we estimate gλ,μ
1
)
* ∗
∗
fχαN Qc y gλ,μ
gλ,μ
f y f L∞ μ .
1
3.58
By 3.54, 3.55, 3.57, and 3.58, we can get that
∗
∗
∗
gλ,μ
f x gλ,μ
fχα1 Q x f L∞ μ SQ,R f L∞ μ gλ,μ
f y .
3.59
Take mean value over Q for x and over R for y, we get
)
)
)
*
* *
∗
∗
∗
mQ gλ,μ
f mQ gλ,μ
fχα1 Q f L∞ μ SQ,R mR gλ,μ
f .
3.60
Therefore
)
)
*
*
∗
∗
mQ gλ,μ
f − mR gλ,μ
f
)
)
)
**
*
∗
∗
≤ Cf L∞ μ SQ,R mQ gλ,μ
fχαN1 Q .
fχα1 Q mR gλ,μ
3.61
∗
Since by Hölder’s inequality, the L2 μ boundedness assumption of gλ,μ
and the fact that Q is
α1 , αn1
1 -doubling, we have that
)
∗
mQ gλ,μ
fχα1 Q
*
≤
≤
C
∗
gλ,μ
μQ1/2 Q
Cfχα1 Q L2 μ
μQ1/2
≤ Cf L∞ μ ,
≤
1/2
2
fχα1 Q x dμx
Cf L∞ μ μα1 Q1/2
3.62
μQ1/2
which completes the proof of Theorem 1.6.
Using Theorems 1.4 and 1.5 and 4, Theorem 3.1, Corollary 1.7 is obvious.
Acknowledgment
The first author was supported partly by NSFC Grant: 10701010, NSFC Key program
Grant: 10931001, CPDRFSFP Grant: 200902070 and SRF for ROCS, SEM.
28
Journal of Inequalities and Applications
References
1 X. Tolsa, “A proof of the weak 1, 1 inequality for singular integrals with non doubling measures
based on a Calderón-Zygmund decomposition,” Publicacions Matemàtiques, vol. 45, no. 1, pp. 163–
174, 2001.
2 X. Tolsa, “Littlewood-Paley theory and the T 1 theorem with non-doubling measures,” Advances in
Mathematics, vol. 164, no. 1, pp. 57–116, 2001.
3 J. Garcı́a-Cuerva and A. E. Gatto, “Boundedness properties of fractional integral operators associated
to non-doubling measures,” Publicacions Matemàtiques, vol. 46, no. 2, pp. 405–434, 2002.
4 G. Hu, Y. Meng, and D. Yang, “Multilinear commutators of singular integrals with non doubling
measures,” Integral Equations and Operator Theory, vol. 51, no. 2, pp. 235–255, 2005.
5 G. Hu, H. Lin, and D. Yang, “Marcinkiewicz integrals with non-doubling measures,” Integral Equations
and Operator Theory, vol. 58, no. 2, pp. 205–238, 2007.
6 E. M. Stein, “On some funcions of Littlewood-Paley and Zygmund,” Bulletin of the American
Mathematical Society, vol. 67, pp. 99–101, 1961.
7 C. Fefferman, “Inequalities for strongly singular convolution operators,” Acta Mathematica, vol. 124,
pp. 9–36, 1970.
8 A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123 of Pure and Applied Mathematics,
Academic Press, Orlando, Fla, USA, 1986.
9 Q. Xue, Parametric Littlewood-Paley operators, Ph.D. thesis, Beijing Normal University, 2004.
10 H. Lin and Y. Meng, “Boundedness of parametrized Littlewood-Paley operators with nondoubling
measures,” Journal of Inequalities and Applications, vol. 2008, Article ID 141379, 25 pages, 2008.
11 X. Tolsa, “BMO, H 1 , and Calderón-Zygmund operators for non doubling measures,” Mathematische
Annalen, vol. 319, no. 1, pp. 89–149, 2001.
12 Y. Jiang, “Spaces of type BLO for non-doubling measures,” Proceedings of the American Mathematical
Society, vol. 133, no. 7, pp. 2101–2107, 2005.
13 A. Torchinsky and S. L. Wang, “A note on the Marcinkiewicz integral,” Colloquium Mathematicum, vol.
60-61, no. 1, pp. 235–243, 1990.
14 Q. Xue and Y. Ding, “Weighted estimates for the multilinear commutators of the Littlewood-Paley
operators,” Science in China. Series A, vol. 52, no. 9, pp. 1849–1868, 2009.