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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 141379, 25 pages
doi:10.1155/2008/141379
Research Article
Boundedness of Parametrized Littlewood-Paley
Operators with Nondoubling Measures
Haibo Lin
1
and Yan Meng
2
1
School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
2
School of Information, Renmin University of China, Beijing 100872, China
Correspondence should be addressed to Yan Meng,
Received 2 April 2008; Accepted 30 July 2008
Recommended by Siegfried Carl
Let μ be a nonnegative Radon measure on
R
d
which only satisfies the following growth condition
that there exists a positive constant C such that μBx, r ≤ Cr
n
for all x ∈ R
d
,r > 0and
some fixed n ∈ 0,d. In this paper, the authors prove that for suitable indexes ρ and λ,the
parametrized g
∗
λ
function M
∗,ρ
λ
is bounded on L
p
μ for p ∈ 2, ∞ with the assumption that the
kernel of the operator M
∗,ρ
λ
satisfies some H
¨
ormander-type condition, and is bounded from L
1
μ
into weak L
1
μ with the assumption that the kernel satisfies certain slightly stronger H
¨
ormander-
type condition. As a corollary, M
∗,ρ
λ
with the kernel satisfying the above stronger H
¨
ormander-type
condition is bounded on L
p
μ for p ∈ 1, 2. Moreover, the authors prove that for suitable indexes
ρ and λ, M
∗,ρ
λ
is bounded from L
∞
μ into RBLOμthe space of regular bounded lower oscillation
functions if the kernel satisfies the H
¨
ormander-type condition, and from the Hardy space H
1
μ
into L
1
μ if the kernel satisfies the above stronger H
¨
ormander-type condition. The corresponding
properties for the parametrized area integral M
ρ
S
are also established in this paper.
Copyright q 2008 H. Lin and Y. Meng. 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
Let μ be a nonnegative Radon measure on R
d
which only satisfies the following growth
condition that for all x ∈ R
d
and all r>0:
μ
Bx, r
≤ C
0
r
n
, 1.1
where C
0
and n are positive constants and n ∈ 0,d,andBx, r is the open ball centered
at x and having radius r. Such a measure μ may be nondoubling. We recall that a measure
μ is said to be doubling, if there is a positive constant C such that for any x ∈ suppμ and
r>0, μBx, 2r ≤ CμBx, r. It is well known that the doubling condition on underlying
2 Journal of Inequalities and Applications
measures is a key assumption in the classical theory of harmonic analysis. However, in
recent years, many classical results concerning the theory of Calder
´
on-Zygmund operators
and function spaces have been proved still valid if the underlying measure is a nonnegative
Radon measure on R
d
which only satisfies 1.1see 1–8. The motivation for developing
the analysis with nondoubling measures and some examples of nondoubling measures can
be found in 9. We only point out that the analysis with nondoubling measures played a
striking role in solving the long-standing open Painlev
´
e’s problem by Tolsa in 10.
Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y}. Assume that there
exists a positive constant C such that for any x, y ∈ R
d
with x
/
y,
Kx, y
≤ C|x − y|
−n−1
, 1.2
and for any x, y, y
∈ R
d
,
|x−y|≥2|y−y
|
Kx, y − K
x, y
Ky, x − K
y
,x
1
|x − y|
dμx ≤ C. 1.3
The parametrized Marcinkiewicz integral M
ρ
f associated to the above kernel K and the
measure μ as in 1.1 is defined by
M
ρ
fx ≡
∞
0
1
t
ρ
|x−y|≤t
Kx, y
|x − y|
1−ρ
fydμy
2
dt
t
1/2
,x∈ R
d
, 1.4
where ρ ∈ 0, ∞. The parametrized area integral M
ρ
S
and g
∗
λ
function M
∗,ρ
λ
are defined,
respectively, by
M
ρ
S
fx ≡
∞
0
|y−x|<t
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
fzdμz
2
dμydt
t
n1
1/2
,x∈ R
d
,
1.5
M
∗,ρ
λ
fx ≡
R
d1
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
fzdμz
2
dμydt
t
n1
1/2
,x∈ R
d
,
1.6
where R
d1
{y, t : y ∈ R
d
,t > 0}, ρ ∈ 0, ∞,andλ ∈ 1, ∞. It is easy to verify that if μ is
the d-dimensional Lebesgue measure in R
d
,and
Kx, y
Ωx − y
|x − y|
d−1
1.7
with Ω homogeneous of degree zero and Ω ∈ Lip
α
S
d−1
for some α ∈ 0, 1, then K satisfies
1.2 and 1.3. Under these conditions, M
ρ
in 1.4 is just the parametrized Marcinkiewicz
integral introduced by H
¨
ormander in 11,andM
ρ
S
and M
∗,ρ
λ
as in 1.5 and 1.6, respectively,
are the parametrized area integral and the parametrized g
∗
λ
function considered by Sakamoto
and Yabuta in 12. We point out that the study of the Littlewood-Paley operators is motivated
by their important roles in harmonic analysis and PDE 13, 14. Since the Littlewood-Paley
Journal of Inequalities and Applications 3
operators of high dimension were first introduced by Stein in 15, a lot of papers focus on
these operators, among them we refer to 16–21 and their references.
When ρ 1, the operator M
ρ
as in 1.4 is just the Marcinkiewicz integral with
nondoubling measures in 22, where the boundedness of such operator in Lebesgue spaces
and Hardy spaces was established under the assumption that M
ρ
is bounded on L
2
μ.
Throughout this paper, we always assume that the parametrized Marcinkiewicz integral
with nondoubling measures M
ρ
as in 1.4 is bounded on L
2
μ. By a similar argument in
22, it is easy to obtain the boundedness of the parametrized Marcinkiewicz integral M
ρ
with ρ ∈ 0, ∞ from L
1
μ into weak L
1
μ, from the Hardy space H
1
μ into L
1
μ, and
from L
∞
μ into RBLOμthe space of regular bounded lower oscillation functions; see
Definition 2.5 below. As a corollary, it is easy to see that M
ρ
is bounded on L
p
μ with
p ∈ 1, ∞.
The main purpose of this paper is to establish some similar results for the parametrized
area integral M
ρ
S
and the parametrized g
∗
λ
function M
∗,ρ
λ
as in 1.5 and 1.6, respec-
tively.
This paper is organized as follows. In the rest of Section 1, we will make some
conventions and recall some necessary notation. In Section 2, we will establish the
boundedness of M
∗,ρ
λ
as in 1.6 in Lebesgue spaces L
p
μ for any p ∈ 1, ∞. And we
will also consider the endpoint estimates for the cases p 1andp ∞.InSection 3,
we will prove that M
∗,ρ
λ
as in 1.6 is bounded from H
1
μ into L
1
μ. And in the last
section, the corresponding results for the parametrized area function M
ρ
S
as in 1.5 are
established.
For a cube Q ⊂ R
d
we mean a closed cube whose sides parallel to the coordinate
axes and we denote its side length by lQ and its center by x
Q
.Letα>1andβ>α
n
.We
say that a cube Q is an α, β-doubling cube if μαQ ≤ βμQ, where αQ denotes the cube
with the same center as Q and lαQαlQ. For definiteness, if α and β are not specified,
by a doubling cube we mean a 2, 2
d1
-doubling cube. Given two cubes Q ⊂ R in R
d
,
set
K
Q,R
≡ 1
N
Q,R
k1
μ
2
k
Q
l
2
k
Q
n
, 1.8
where N
Q,R
is the smallest positive integer k such that l2
k
Q ≥ lRsee 23.
In what follows, C denotes a positive constant that is independent of main parameters
involved but whose value may differ from line t o line. Constants with subscripts, such as C
1
,
do not change in different occurrences. We denote simply by A B if there exists a positive
constant C such that A ≤ CB;andA∼B means that A B and B A. For a μ-measurable
set E, χ
E
denotes its characteristic function. For any p ∈ 1, ∞, we denote by p
its conjugate
index, namely, 1/p 1/p
1.
2. Boundedness of M
∗,ρ
λ
in Lebesgue spaces
This section is devoted to the behavior of the parametrized g
∗
λ
function M
∗,ρ
λ
in Lebesgue
spaces.
Theorem 2.1. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 1.3, and let M
∗,ρ
λ
be as in 1.6 with ρ ∈ 0, ∞ and λ ∈ 1, ∞. Then for any p ∈ 2, ∞, M
∗,ρ
λ
is bounded on L
p
μ.
4 Journal of Inequalities and Applications
To obtain the boundedness of M
∗,ρ
λ
in L
p
μ with p ∈ 1, 2, we introduce the following
condition on the kernel K, that is, for some fixed σ>2,
sup
r>0,y,y
∈R
d
|y−y
|≤r
∞
l1
l
σ
2
l
r<|x−y|≤2
l1
r
Kx, y − K
x, y
Ky, x − K
y
,x
1
|x − y|
dμx ≤ C,
2.1
which is slightly stronger than 1.3.
Theorem 2.2. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 2.1, and let M
∗,ρ
λ
be as in 1.6 with ρ ∈ n/2, ∞ and λ ∈ 2, ∞.ThenM
∗,ρ
λ
is bounded from
L
1
μ into weak L
1
μ, namely, there exists a positive constant C such that for any β>0 and any
f ∈ L
1
μ,
μ
x ∈ R
d
: M
∗,ρ
λ
fx >β
≤
C
β
f
L
1
μ
. 2.2
By the Marcinkiewicz interpolation theorem, and Theorems 2.1 and 2.2, we can
immediately obtain the L
p
μ-boundedness of the operator M
∗,ρ
λ
for p ∈ 1, 2.
Corollary 2.3. Under the same assumption of Theorem 2.2, M
∗,ρ
λ
is bounded on L
p
μ for any p ∈
1, 2.
Remark 2.4. We point out that it is still unclear if condition 2.1 in Theorem 2.2 and
Corollary 2.3 can be weakened.
Now we turn to discuss the property of the operator M
∗,ρ
λ
in L
∞
μ. To this end,
we need to recall the definition of the space RBLOμthe space of regular bounded lower
oscillation functions.
Definition 2.5. Let η ∈ 1, ∞.Aμ-locally integrable function f on R
d
is said to be in the space
RBLOμ if there exists a positive constant C such that for any η, η
d1
-doubling cube Q,
m
Q
f − ess inf
x∈Q
fx ≤ C, 2.3
and for any two η, η
d1
-doubling cubes Q ⊂ R,
m
Q
f − m
R
f ≤ CK
Q,R
. 2.4
The minimal constant C as above is defined to be the norm of f in the space RBLOμ and
denoted by f
RBLOμ
.
Remark 2.6. The space RBLOμ was introduced by Jiang in 24, where the η, η
d1
-doubling
cube was replaced by 4
√
d, 4
√
d
n1
-doubling cube. It was pointed out in 25 that it is
convenient in applications to replace 4
√
d, 4
√
d
n1
-doubling cubes by η, η
d1
-doubling
cubes with η ∈ 1, ∞ in the definition of RBLOμ. Moreover, it was proved in 25 that the
definition is independent of the choices of the constant η ∈ 1, ∞. The space RBLOμ is a
subspace of RBMOμ which was introduced by Tolsa in 23.
Journal of Inequalities and Applications 5
Theorem 2.7. Let K be a μ-locally integrable function on R
d
× R
d
\{x, y : x y} satisfying
1.2 and 1.3, and let M
∗,ρ
λ
be as in 1.6 with ρ ∈ 0, ∞ and λ ∈ 1, ∞. Then for any f ∈ L
∞
μ,
M
∗,ρ
λ
f is either infinite everywhere or finite almost everywhere. More precisely, if M
∗,ρ
λ
f is finite
at some point x
0
∈ R
d
,thenM
∗,ρ
λ
f is finite almost everywhere and
M
∗,ρ
λ
f
RBLOμ
≤ Cf
L
∞
μ
, 2.5
where the positive constant C is independent of f.
We point out that Theorem 2.7 is also new even when μ is the d-dimensional Lebesgue
measure on R
d
.
IntherestpartofSection 2, we will prove Theorems 2.1, 2.2,and2.7, respectively. To
prove Theorem 2.1, we first recall some basic facts and establish a technical lemma. For η>1,
let M
η
be the noncentered maximal operator defined by
M
η
fx ≡ sup
Qx
Q cube
1
μηQ
Q
fy
dμy,x∈ R
d
. 2.6
It is well known that M
η
is bounded on L
p
μ provided that p ∈ 1, ∞see 23.The
following lemma which is of independent interest plays an important role in our proofs.
Lemma 2.8. Let K be a μ-locally integrable function on R
d
× R
d
\{x, y : x y} satisfying
1.2 and 1.3, and η ∈ 1, ∞.LetM
ρ
be as in 1.4 and M
∗,ρ
λ
be as in 1.6 with ρ ∈ 0, ∞ and
λ ∈ 1, ∞. Then for any nonnegative function φ, there exists a positive constant C such that for all
f ∈ L
p
μ with p ∈ 1, ∞,
R
d
M
∗,ρ
λ
fx
2
φxdμx ≤ C
R
d
M
ρ
fx
2
M
η
φxdμx. 2.7
Proof. Notice that
R
d
M
∗,ρ
λ
fx
2
φxdμx
R
d
R
d1
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
fzdμz
2
dμydt
t
n1
φxdμx
≤
R
d
∞
0
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
fzdμz
2
dt
t
sup
t>0
R
d
t
t |x − y|
λn
φx
t
n
dμx
dμy
R
d
M
ρ
fy
2
sup
t>0
R
d
t
t |x − y|
λn
φx
t
n
dμx
dμy.
2.8
Thus, to prove Lemma 2.8,itsuffices to verify that for any y ∈ R
d
,
sup
t>0
R
d
t
t |x − y|
λn
φx
t
n
dμx M
η
φy. 2.9
6 Journal of Inequalities and Applications
For any fixed y ∈ R
d
and t>0, write
R
d
t
t |x − y|
λn
φx
t
n
dμx
|x−y|≤t
t
t |x − y|
λn
φx
t
n
dμx
|x−y|>t
t
t |x − y|
λn
φx
t
n
dμx
≡ E
1
E
2
.
2.10
Let Q
y
be the cube with center at y and side length lQ
y
2t. Obviously, {x : |x−y| <t}⊂Q
y
,
which leads to
E
1
≤
|x−y|≤t
φx
t
n
dμx
1
μ
ηQ
y
Q
y
φxdμx M
η
φy. 2.11
As for E
2
, a straightforward computation proves that
E
2
≤
∞
k0
2
k
t<|x−y|≤2
k1
t
t
t |x − y|
λn
φx
t
n
dμx
∞
k0
1
2
k
nλ
2
k1
t
n
1
2
k1
t
n
|x−y|≤2
k1
t
φxdμx
M
η
φy.
2.12
Combining the estimates for E
1
and E
2
yields 2.9, which completes the proof of Lemma 2.8.
Proof of Theorem 2.1. For the case of p 2, choosing φx1inLemma 2.8, then we easily
obtain that
R
d
M
∗,ρ
λ
fx
2
dμx
R
d
M
ρ
fx
2
dμx, 2.13
which, along with the boundedness of M
ρ
in L
2
μ, immediately yields that Theorem 2.1
holds in this case.
For the case of p ∈ 2, ∞,letq be the index conjugate to p/2. Then from Lemma 2.8
and the H
¨
older inequality, it follows that
M
∗,ρ
λ
f
2
L
p
μ
sup
φ≥0, φ
L
q
μ
≤1
R
d
M
∗,ρ
λ
fx
2
φxdμx
sup
φ≥0, φ
L
q
μ
≤1
R
d
M
ρ
fx
2
M
η
φxdμx
M
ρ
f
2
L
p
μ
sup
φ≥0, φ
L
q
μ
≤1
M
η
φ
L
q
μ
f
2
L
p
μ
sup
φ≥0, φx
L
q
μ
≤1
φ
L
q
μ
f
2
L
p
μ
,
2.14
which completes the proof of Theorem 2.1.
Journal of Inequalities and Applications 7
To prove Theorem 2.2, we need the following Calder
´
on-Zygmund decomposition with
nondoubling measures see 23 or 26.
Lemma 2.9. Let p ∈ 1, ∞. For any f ∈ L
p
μ and λ>0 (λ> 2
d1
f
L
1
μ
/μ if μ < ∞), one
has the following.
a There exists a family of almost disjoint cubes {Q
j
}
j
(i.e.,
j
χ
Q
j
≤ C) such that
1
μ
2Q
j
Q
j
fx
p
dμx >
λ
p
2
d1
,
1
μ
2ηQ
j
ηQ
j
fx
p
dμx ≤
λ
p
2
d1
∀η>2,
|fx|≤λμ-a.e. on R
d
\∪
j
Q
j
.
2.15
b For each j,letR
j
be the smallest 6, 6
n1
-doubling cube of the form 6
k
Q
j
, k ∈ N, and
let ω
j
χ
Q
j
/
k
χ
Q
k
. Then, there exists a family of functions ϕ
j
with suppϕ
j
⊂ R
j
satisfying
R
d
ϕ
j
xdμx
Q
j
fxω
j
xdμx,
j
ϕ
j
x
≤ Bλ 2.16
(where B is some constant), and when p 1,
ϕ
j
L
∞
μ
μ
R
j
≤ C
Q
j
fx
dμx; 2.17
when p ∈ 1, ∞,
R
j
ϕ
j
x
p
dμx
1/p
μ
R
j
1/p
≤
C
λ
p−1
Q
j
fx
p
dμx. 2.18
Remark 2.10. From the proof of the Calder
´
on-Zygmund decomposition with nondoubling
measures see 23 or 26, it is easy to see that if we replace R
j
with R
j
, the smallest
6
√
d, 6
√
d
n1
-doubling cube of the form 6
√
d
k
Q
j
k ∈ N, the above conclusions a and
b still hold. Here and hereafter, when we mention R
j
in Lemma 2.9 we always mean R
j
.
Proof of Theorem 2.2. Let f ∈ L
1
μ and β> 2
d1
f
L
1
μ
/μ note that if 0 <β≤
2
d1
f
L
1
μ
/μ, the estimate 2.2 obviously holds. Applying Lemma 2.9 to f at the level β,
we obtain fx ≡ gxbx with
gx ≡ fxχ
R
d
\
j
Q
j
x
j
ϕ
j
x,bx ≡
j
ω
j
xfx − ϕ
j
x
j
b
j
x, 2.19
where ω
j
, ϕ
j
, Q
j
,andR
j
are the same as in Lemma 2.9. It is easy to see that g
L
∞
μ
β and
g
L
1
μ
f
L
1
μ
. By the boundedness of M
∗,ρ
λ
in L
2
μ, we easily obtain that
μ
x ∈ R
d
: M
∗,ρ
λ
gx >β
≤ β
−2
M
∗,ρ
λ
g
2
L
2
μ
β
−1
f
L
1
μ
. 2.20
8 Journal of Inequalities and Applications
From a of Lemma 2.9, it follows that
μ
∪
j
2Q
j
β
−1
j
Q
j
fx
dμx β
−1
R
d
fx
dμx, 2.21
and therefore, the proof of Theorem 2.2 can be deduced to proving that
μ
x ∈ R
d
\∪
j
2Q
j
: M
∗,ρ
λ
bx >β
β
−1
R
d
fx
dμx. 2.22
For each fixed j,letR
∗
j
6
√
dR
j
.Noticethat
μ
x ∈ R
d
\∪
j
2Q
j
: M
∗,ρ
λ
bx >β
≤ β
−1
j
R
d
\R
∗
j
M
∗,ρ
λ
b
j
xdμx
j
R
∗
j
\2Q
j
M
∗,ρ
λ
b
j
xdμx
.
2.23
Thus, it suffices to prove that for each fixed j,
R
d
\R
∗
j
M
∗,ρ
λ
b
j
xdμx
Q
j
fx
dμx,
2.24
R
∗
j
\2Q
j
M
∗,ρ
λ
b
j
xdμx
Q
j
fx
dμx.
2.25
To verify 2.24, for each fixed j,letB
j
Bx
Q
j
, 2
√
dlR
j
, and write
R
d
\R
∗
j
M
∗,ρ
λ
b
j
xdμx
≤
R
d
\R
∗
j
|y−x|<t
t
t |x − y|
λn
|y−z|≤t
Ky, zb
j
z
|y − z|
1−ρ
dμz
2
dμydt
t
n2ρ1
1/2
dμx
R
d
\R
∗
j
⎡
⎣
|y−x|≥t
y∈B
j
t
t |x − y|
λn
|y−z|≤t
Ky, zb
j
z
|y − z|
1−ρ
dμz
2
dμydt
t
n2ρ1
⎤
⎦
1/2
dμx
R
d
\R
∗
j
⎡
⎣
|y−x|≥t
y∈R
d
\B
j
t
t |x − y|
λn
|y−z|≤t
Ky, zb
j
z
|y − z|
1−ρ
dμz
2
dμydt
t
n2ρ1
⎤
⎦
1/2
dμx
≡ F
1
F
2
F
3
.
2.26
For each fixed j, further decompose
F
1
≤
R
d
\R
∗
j
⎡
⎣
|y−x|<t
y∈4R
j
|y−z|≤t
Ky, z
|y − z|
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎤
⎦
1/2
dμx
R
d
\R
∗
j
⎡
⎣
|y−x|<t
y∈R
d
\4R
j
|y−z|≤t
Ky, z
|y − z|
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎤
⎦
1/2
dμx
≡ H
1
H
2
.
2.27
Journal of Inequalities and Applications 9
It is easy to see that for any x ∈ R
d
\R
∗
j
, y ∈ 4R
j
with |y−x| <tand z ∈ R
j
, |x−x
Q
j
|−2
√
dlR
j
≤
|x −y| <tand |y − z| < 4
√
dlR
j
. This fact along the Minkowski inequality and 1.2 leads to
H
1
≤
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎢
⎣
|y−x|<t
|y−z|≤t
y∈4R
j
|Ky, z|
2
|y − z|
2−2ρ
dμydt
t
n2ρ1
⎤
⎥
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
|y−z|<4
√
dlR
j
1
|y − z|
2n−2ρ
×
∞
|x−x
Q
j
|−2
√
dlR
j
dt
t
n2ρ1
dμy
1/2
dμzdμx
R
j
b
j
z
|y−z|<4
√
dlR
j
1
|y − z|
2n−2ρ
dμy
1/2
dμz
R
d
\R
∗
j
1
|x − x
Q
j
|
n2ρ/2
dμx
b
j
L
1
μ
.
2.28
As for H
2
, first write
H
2
≤
R
d
\R
∗
j
⎡
⎢
⎣
|y−x|<t, y∈R
d
\4R
j
t≤|y−x
Q
j
|2
√
dlR
j
|y−z|≤t
Ky, z
|y − z|
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎤
⎥
⎦
1/2
dμx
R
d
\R
∗
j
⎡
⎢
⎣
|y−x|<t, y∈R
d
\4R
j
t>|y−x
Q
j
|2
√
dlR
j
|y−z|≤t
Ky, z
|y − z|
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎤
⎥
⎦
1/2
dμx
≡ J
1
J
2
.
2.29
Notice that for any z ∈ R
j
, x ∈ R
d
\ R
∗
j
and y ∈ R
d
\ 4R
j
, |y − z|∼|y − x
Q
j
|, and |x − x
Q
j
| <
5
√
d|y − x
Q
j
|.Thus,by1.2 and the Minkowski inequality, we obtain that
J
1
R
d
\R
∗
j
R
j
b
j
z
R
d
\4R
j
1
|y − z|
2n−2ρ
|y−x
Q
j
|2
√
dlR
j
|y−z|
dt
t
n2ρ1
dμy
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
R
d
\4R
j
1
y − x
Q
j
n1/2
l
R
j
y − x
Q
j
2n1/2
dμy
1/2
dμzdμx
R
j
b
j
z
R
d
\4R
j
l
R
j
1/2
y − x
Q
j
n1/2
dμy
1/2
dμz
R
d
\R
∗
j
l
R
j
1/4
x − x
Q
j
n1/4
dμx
b
j
L
1
μ
.
2.30
10 Journal of Inequalities and Applications
On the other hand, it is easy to verify that for any y ∈ R
d
\ 4R
j
and t>|y − x
Q
j
| 2
√
dlR
j
,
R
j
⊂{z : |y − z|≤t} and |x − x
Q
j
| < 2t. Choose 0 <<min{1/2, λ − 2n/2,ρ− n/2,σ/2 − 1}
we always take to satisfy this restriction in our proof. The vanishing moment of b
j
on R
j
and the Minkowski inequality give us that
J
2
R
d
\R
∗
j
⎧
⎪
⎨
⎪
⎩
|y−x|<t,y∈R
d
\4R
j
t>|y−x
Q
j
|2
√
dlR
j
|y−z|≤t
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎫
⎬
⎭
1/2
dμx
≤
R
d
\R
∗
j
R
j
b
j
z
R
d
\4R
j
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
×
∞
|y−x
Q
j
|2
√
dlR
j
log
t/l
R
j
22
dt
t
2ρ−n1
t
2n
log
t/l
R
j
22
dμy
1/2
dμzdμx
R
d
\R
∗
j
1
|x − x
Q
j
|
n
log
x − x
Q
j
/l
R
j
1
R
j
b
j
z
×
R
d
\4R
j
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
×
∞
|y−x
Q
j
|2
√
dlR
j
log
t/l
R
j
22
t
2ρ−n1
dt
dμy
1/2
dμzdμx.
2.31
It follows from 27, Lemma 2.2 that for any y ∈ R
d
\ 4R
j
,
∞
|y−x
Q
j
|2
√
dlR
j
log
t/l
R
j
22
t
2ρ−n1
dt
log
y − x
Q
j
/l
R
j
2
√
d
22
y − x
Q
j
2
√
dl
R
j
2ρ−n
, 2.32
which, together with 2.1,leadsto
J
2
R
d
\R
∗
j
1
x − x
Q
j
n
log
x − x
Q
j
/l
R
j
1
×
R
j
b
j
z
R
d
\4R
j
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
×
log
y − x
Q
j
/l
R
j
2
√
d
22
y − x
Q
j
2
√
dl
R
j
2ρ−n
dμy
1/2
dμzdμx
Journal of Inequalities and Applications 11
R
d
\R
∗
j
1
x − x
Q
j
n
log
x − x
Q
j
/l
R
j
1
×
R
j
b
j
z
⎧
⎨
⎩
∞
k1
k 1
22
2
k
l
R
j
2ρ−n
×
⎡
⎣
2
k
lR
j
≤|y−x
Q
j
|<2
k1
lR
j
Ky, z − K
y, x
Q
j
2
|y − z|
2−2ρ
K
y, x
Q
j
2
1
|y−z|
1−ρ
−
1
y − x
Q
j
1−ρ
2
dμy
⎤
⎦
⎫
⎬
⎭
1/2
dμzdμx
R
d
\R
∗
j
1
x − x
Q
j
n
log
x − x
Q
j
/l
R
j
1
×
R
j
b
j
z
⎧
⎨
⎩
∞
k1
k 1
22
2
k
l
R
j
2ρ−n
×
⎡
⎣
2
k
lR
j
≤|y−x
Q
j
|<2
k1
lR
j
1
2
k
l
R
j
n−2ρ
Ky, z − K
y, x
Q
j
|y − z|
l
R
j
2
y − x
Q
j
2n−2ρ2
dμy
⎤
⎦
⎫
⎬
⎭
1/2
dμzdμx
R
d
\R
∗
j
1
x − x
Q
j
n
log
x − x
Q
j
/l
R
j
1
R
j
b
j
z
1
∞
k1
k 1
22
2
2k
1/2
dμzdμx
b
j
L
1
μ
.
2.33
Combining the estimates for H
1
,J
1
,andJ
2
yields
F
1
b
j
L
1
μ
Q
j
fx
dμx. 2.34
To estimate F
2
, first notice that for any y ∈ B
j
, x ∈ R
d
\R
∗
j
,andz ∈ R
j
, |y−x|≥|x−x
Q
j
|/2,
|y−z|≤4
√
dlR
j
,and|x−y|∼|x−x
Q
j
|. Thus, by the Minkowski inequality and 1.2, we easily
obtain that
F
2
≤
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎢
⎣
|y−x|≥t
|y−z|≤t
y∈B
j
t
t |x − y|
2n2
Ky, z
2
|y − z|
2−2ρ
dμydt
t
n2ρ1
⎤
⎥
⎥
⎦
1/2
dμzdμx
12 Journal of Inequalities and Applications
R
d
\R
∗
j
R
j
b
j
z
|y−z|≤4
√
dlR
j
1
x − x
Q
j
2n2
|y − z|
n−
×
|y−x|
0
t
−1
dt
dμy
1/2
dμzdμx
R
j
b
j
z
|y−z|≤4
√
dlR
j
1
|y − z|
n−
dμy
1/2
dμz
R
d
\R
∗
j
1
x − x
Q
j
n/2
dμx
b
j
L
1
μ
Q
j
fx
dμx.
2.35
It remains to estimate F
3
.By1.2, we can write
F
3
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎢
⎢
⎣
|y−z|≤t≤|y−x|,y∈R
d
\B
j
t≤|y−x
Q
j
|C
lR
j
|x−x
Q
j
|≤2|y−x
Q
j
|
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
⎤
⎥
⎥
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎢
⎢
⎣
|y−z|≤t≤|y−x|,y∈R
d
\B
j
t≤|y−x
Q
j
|C
lR
j
|x−x
Q
j
|>2|y−x
Q
j
|
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
⎤
⎥
⎥
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
⎡
⎢
⎣
t≤|y−x|,y∈R
d
\B
j
|y−x
Q
j
|C
lR
j
<t
t
t |x − y|
λn
|y−z|≤t
Ky, z
|y − z|
1−ρ
b
j
zdμz
2
dμydt
t
n2ρ1
⎤
⎥
⎦
1/2
dμx
≡ L
1
L
2
L
3
,
2.36
where C
8
√
de
22/
. Note that for any y ∈ R
d
\ B
j
and z ∈ R
j
with |y − z|≤t ≤|y − x|,
then |y − z|∼|y − x
Q
j
| and |y − x
Q
j
|≤t
√
dlR
j
. Consequently,
L
1
≤
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎣
y∈R
d
\B
j
|x−x
Q
j
|≤2|y−x
Q
j
|
|y−x
Q
j
|C
lR
j
|y−x
Q
j
|−
√
dlR
j
dt
t
n2ρ1
×
1
|y − z|
2n−2ρ
dμy
⎤
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎣
y∈R
d
\B
j
|x−x
Q
j
|≤2|y−x
Q
j
|
l
R
j
y − x
Q
j
3n1
dμy
⎤
⎥
⎦
1/2
dμzdμx
b
j
L
1
μ
.
2.37
Journal of Inequalities and Applications 13
A trivial computation involving the fact that |x − y| > |x − x
Q
j
|/2 for any x ∈ R
d
\ R
∗
j
and
y ∈ R
d
\ B
j
satisfying |x − x
Q
j
| > 2|y − x
Q
j
| proves that
L
2
≤
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎣
y∈R
d
\B
j
|x−x
Q
j
|>2|y−x
Q
j
|
t
t |x − y|
2n2
1
|y − z|
2n−2ρ
×
|y−x
Q
j
|C
lR
j
|y−x
Q
j
|−
√
dlR
j
dt
t
n2ρ1
dμy
⎤
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎣
y∈R
d
\B
j
|x−x
Q
j
|>2|y−x
Q
j
|
|y−x
Q
j
|C
lR
j
|y−x
Q
j
|−
√
dlR
j
1
t
2ρ−n−21
dt
×
1
|x − y|
2n2
|y − z|
2n−2ρ
dμy
⎤
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
R
d
\B
j
1
|y − z|
n−21
l
R
j
x − x
Q
j
2n2
dμy
1/2
dμzdμx
b
j
L
1
μ
.
2.38
Finally, let us estimate L
3
. It is easy to see that for any y ∈ R
d
\ B
j
and t>|y −x
Q
j
| C
lR
j
,
R
j
⊂{z : |y − z|≤t} and t |x − y|≥|x − x
Q
j
| C
lR
j
. Thus, from the vanishing moment of
b
j
on R
j
, it follows that
L
3
≤
R
d
\R
∗
j
R
j
b
j
z
⎡
⎢
⎢
⎢
⎣
y∈R
d
\B
j
t>|y−x
Q
j
|C
lR
j
|y−z|≤t≤|y−x|
t
t |x − y|
λn
×
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
dμydt
t
n2ρ1
⎤
⎥
⎥
⎦
1/2
dμzdμx
R
d
\R
∗
j
R
j
|b
j
z|
⎡
⎢
⎢
⎢
⎣
y∈R
d
\B
j
t>|y−x
Q
j
|C
lR
j
|y−z|≤t≤|y−x|
t
λn
t |x − y|
2n
log
t |x − y|
/l
R
j
22
×
log
t |x − y|
/l
R
j
22
t |x − y|
λn−2n
×
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
dμydt
t
n2ρ1
⎤
⎥
⎥
⎦
1/2
dμzdμx
14 Journal of Inequalities and Applications
R
d
\R
∗
j
R
j
|b
j
z|
x − x
Q
j
C
l
R
j
n
log
x − x
Q
j
C
l
R
j
/l
R
j
1
×
⎡
⎢
⎣
y∈R
d
\B
j
|y−x|≥|y−x
Q
j
|C
lR
j
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
×
|y−x|
|y−x
Q
j
|C
lR
j
t
λn
log
t |x − y|
/l
R
j
22
t |x − y|
λn−2n
×
1
t
n2ρ1
dt dμy
1/2
dμzdμx
R
d
\R
∗
j
R
j
b
j
z
x − x
Q
j
C
l
R
j
n
log
x − x
Q
j
C
l
R
j
/l
R
j
1
×
⎡
⎣
R
d
\B
j
Ky, z
|y − z|
1−ρ
−
K
y, x
Q
j
y − x
Q
j
1−ρ
2
×
log
y − x
Q
j
C
l
R
j
/l
R
j
22
y − x
Q
j
C
l
R
j
2ρ−n
dμy
⎤
⎦
1/2
dμzdμx,
2.39
where in the penultimate inequality, we have used the following inequality
|y−x|
|y−x
Q
j
|C
lR
j
log
t |x − y|
/l
R
j
22
t |x − y|
λn−2n
t
n2ρ1−λn
dt
log
y − x
Q
j
C
l
R
j
/l
R
j
22
y − x
Q
j
C
l
R
j
2ρ−n
,
2.40
which can be proved by the same way as in 28, page 357. Thus, by an argument similar to
the estimate of 2.33,weobtainthat
L
3
b
j
L
1
μ
. 2.41
Combining the estimates for L
1
,L
2
,andL
3
yields that
F
3
b
j
L
1
μ
Q
j
fx
dμx, 2.42
which along with the estimates for F
1
and F
2
leads to 2.24.
Nowweturntoprovetheestimate2.25. Observe that if supp h ⊂ I for some cube
I, then by 1.2, we have that for any s>1andanyx ∈ R
d
\ sI,
M
∗,ρ
λ
hx ≤
I
hz
|y−z|≤t
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
1/2
dμz
≤
I
hz
M
1
zM
2
zM
3
z
dμz,
2.43
Journal of Inequalities and Applications 15
where
M
1
z ≡
⎡
⎣
|y−z|≤t
2|y−z|>|x−z|
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
⎤
⎦
1/2
,
M
2
z ≡
⎡
⎣
|y−z|≤t,|y−x|<t
2|y−z|≤|x−z|
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
⎤
⎦
1/2
,
M
3
z ≡
⎡
⎣
|y−z|≤t,|y−x|≥t
2|y−z|≤|x−z|
t
t |x − y|
λn
1
|y − z|
2n−2ρ
dμydt
t
n2ρ1
⎤
⎦
1/2
.
2.44
Some trivial computation leads to that for any x ∈ R
d
\ sI and z ∈ I,
M
1
z ≤
2|y−z|>|x−z|
1
|y − z|
2n
∞
|x−z|/2
dt
t
n1
dμy
1/2
1
|x − z|
n
2|y−z|>|x−z|
1
|y − z|
2n
dμy
1/2
1
|x − x
I
|
n
.
2.45
As for M
2
z, notice that for any x, y, z ∈ R
d
satisfying |y − x| <tand 2|y − z|≤|x − z|,
|x − z|/2 <t. From this fact and ρ ∈ n/2, ∞, it follows that for any x ∈ R
d
\ sI and z ∈ I,
M
2
z ≤
2|y−z|≤|x−z|
1
|y − z|
2n−2ρ
∞
1/2|x−z|
dt
t
n2ρ1
dμy
1/2
1
|x − x
I
|
n
. 2.46
To estimate M
3
z, we first have that for any x, y, z ∈ R
d
satisfying 2|y −z|≤|x −z|,2|y −x|≥
|x − z|,and|y −x|≤3|x − z|/2. Consequently, for any x ∈ R
d
\ sI and z ∈ I,
M
3
z ≤
2|y−z|≤|x−z|
1
|y − z|
n−
1
|x − z|
2n2
|y−x|
0
dt
t
1−
dμy
1/2
2|y−z|≤|x−z|
|y − x|
|x − z|
2n2
1
|y − z|
n−
dμy
1/2
1
|x − x
I
|
n
.
2.47
Combining the estimates for M
1
z,M
2
z,andM
3
z, we obtain that for any x ∈ R
d
\ sI,
M
∗,ρ
λ
hx
1
x − x
I
n
I
hz
dμz. 2.48
On the other hand, it follows from 26, Lemma 2.3see also 23, Lemma 2.1 that
R
∗
j
\2Q
j
1
x − x
Q
j
n
dμx 1. 2.49
16 Journal of Inequalities and Applications
This fact together with 2.48 tells us that
R
∗
j
\2Q
j
M
∗,ρ
λ
ω
j
f
xdμx
R
∗
j
\2Q
j
1
x − x
Q
j
n
dμx
Q
j
fy
dμy
Q
j
fy
dμy.
2.50
The last estimate and the following trivial estimate that
R
∗
j
M
∗,ρ
λ
ϕ
j
xdμx ≤
R
∗
j
M
∗,ρ
λ
ϕ
j
x
2
dμx
1/2
μ
R
∗
j
1/2
R
∗
j
ϕ
j
x
2
dμx
1/2
μ
R
j
1/2
Q
j
fx
dμx,
2.51
which is obtained by the H
¨
older inequality and the L
2
μ-boundedness of M
∗,ρ
λ
,implythe
inequality 2.25. This finishes the proof of Theorem 2.2.
Proof of Theorem 2.7. Recalling that the definition of RBLOμ is independent of the choices of
the constant η ∈ 1, ∞, we choose η 16
√
d in our proof. Hence, to prove Theorem 2.7,itis
enough to prove for any f ∈ L
∞
μ,ifM
∗,ρ
λ
fx
0
< ∞ for some point x
0
∈ R
d
, then for any
16
√
d, 16
√
d
d1
-doubling cube Q x
0
,
m
Q
M
∗,ρ
λ
f
− ess inf
x∈Q
M
∗,ρ
λ
fx f
L
∞
μ
, 2.52
and for any two 16
√
d, 16
√
d
d1
-doubling cubes R ⊃ Q,
m
Q
M
∗,ρ
λ
f
− m
R
M
∗,ρ
λ
f
K
Q,R
f
L
∞
μ
. 2.53
We first verify 2.52. For each fixed cube Q,letB be the smallest ball which contains
Q and has the same center as Q. Denote by r the radius of B. Decompose f as
fxfxχ
8B
xfxχ
R
d
\8B
x ≡ f
1
xf
2
x, 2.54
and write
M
∗,ρ
λ
f
2
x ≤
r
0
R
d
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
f
2
zdμz
2
dμydt
t
n1
1/2
∞
r
R
d
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
f
2
zdμz
2
dμydt
t
n1
1/2
≡M
∗,ρ
λ,0
f
2
xM
∗,ρ
λ,∞
f
2
x.
2.55
Journal of Inequalities and Applications 17
Thus,
m
Q
M
∗,ρ
λ
f
− ess inf
x∈Q
M
∗,ρ
λ
fx
m
Q
M
∗,ρ
λ
f
1
m
Q
M
∗,ρ
λ,0
f
2
sup
x
∈Q
M
∗,ρ
λ,∞
f
x
−M
∗,ρ
λ,∞
f
2
x
1
μQ
sup
x
∈Q
M
∗,ρ
λ,∞
f
2
x −M
∗,ρ
λ,∞
f
2
x
dμx.
2.56
By the H
¨
older inequality and L
2
μ-boundedness of M
∗,ρ
λ
,weobtainthat
m
Q
M
∗,ρ
λ
f
1
≤
1
μQ
1/2
R
d
M
∗,ρ
λ
f
1
x
2
dμx
1/2
f
L
∞
μ
. 2.57
From 1.2 and the fact that for any x ∈ Q ⊂ B, z ∈ R
d
\ 8B, y ∈ R
d
satisfying |x − y| <r,and
t ≤ r, {z ∈ R
d
: z ∈ R
d
\ 8B}∩{z ∈ R
d
: |y − z|≤t} ∅, it follows that
M
∗,ρ
λ,0
f
2
x
r
0
|x−y|≥r
1
|x − y|
λn
|y−z|≤t
1
|y − z|
n−ρ
dμz
2
dμydt
t
n−λn2ρ1
1/2
f
L
∞
μ
|x−y|≥r
1
|x − y|
λn
dμy
r
0
1
t
n−λn1
dt
1/2
f
L
∞
μ
f
L
∞
μ
,
2.58
which gives us that
m
Q
M
∗,ρ
λ,0
f
2
f
L
∞
μ
. 2.59
Obviously, for any x
∈ Q, z ∈ 8B,andy ∈ R
d
with |x
− y| > 16r, |x
− y|∼|y − z|. Some
computation involving this fact and 1.2 yields that
sup
x
∈Q
M
∗,ρ
λ,∞
f
2
x
−M
∗,ρ
λ,∞
f
x
≤ sup
x
∈Q
∞
r
R
d
t
t
x
− y
λn
1
t
ρ
|y−z|≤t
Ky, zf
1
z
|y − z|
1−ρ
dμz
2
dμydt
t
n1
1/2
sup
x
∈Q
∞
r
|x
−y|≤16r
1
t
ρ
|y−z|≤t
|f
1
z|
|y − z|
n−ρ
dμz
2
dμydt
t
n1
1/2
sup
x
∈Q
⎛
⎝
∞
r
|x
−y|>16r
|x
−y|<t
1
t
ρ
|y−z|≤t
f
1
z
|y − z|
n−ρ
dμz
2
dμydt
t
n1
⎞
⎠
1/2
sup
x
∈Q
⎛
⎝
∞
r
|x
−y|>16r
|x
−y|≥t
1
x
− y
λn
1
t
ρ
|y−z|≤t
f
1
z
|y − z|
n−ρ
dμz
2
dμydt
t
n−λn1
⎞
⎠
1/2
f
L
∞
μ
sup
x
∈Q
∞
r
|x
−y|<t
8B
f
1
z
r
n−ρ
dμz
2
dμydt
t
n2ρ1
1/2
sup
x
∈Q
|x
−y|>16r
1
|x − y|
λn2n
|x
−y|
r
8B
f
1
z
dμz
2
dt dμy
t
n−λn1
1/2
f
L
∞
μ
.
2.60
18 Journal of Inequalities and Applications
Thus, the proof of the estimate 2.52 can be reduced to proving that for any x, x
∈ Q,
M
∗,ρ
λ,∞
f
2
x −M
∗,ρ
λ,∞
f
2
x
f
L
∞
μ
. 2.61
For any x, x
∈ Q, write
M
∗,ρ
λ,∞
f
2
x −M
∗,ρ
λ,∞
f
2
x
≤
∞
r
|x−y|>8r
t
t |x − y|
λn
−
t
t
x
− y
λn
×
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
f
2
zdμz
2
dμydt
t
n1
1/2
∞
r
|x−y|≤8r
t
t |x − y|
λn
−
t
t
x
− y
λn
×
1
t
ρ
|y−z|≤t
Ky, z
|y − z|
1−ρ
f
2
zdμz
2
dμydt
t
n1
1/2
≡ U
1
U
2
.
2.62
It follows from the mean value theorem that for any x, x
∈ Q ⊂ B and y ∈ R
d
with |x−y| > 8r,
t
t |x − y|
λn
−
t
t
x
− y
λn
x − x
t
t
t |x − y|
λn1
, 2.63
which, along with 1.2, tells us that
U
1
∞
r
|x−y|>8r
x − x
t |x − y|
λn1
|y−z|≤t
1
|y − z|
n−ρ
dμz
2
dμydt
t
n2ρ1−nλ
1/2
f
L
∞
μ
|x−y|>8r
r
|x − y|
λn1
|x−y|
r
1
t
n−λn1
dt dμy
∞
r
|x−y|≤t
r
t
n2
dμydt
1/2
f
L
∞
μ
f
L
∞
μ
.
2.64
As for U
2
, first note that for any x, y ∈ R
d
satisfying |y − x|≤8r and t>r, t |y − x|≤9t,and
then
t
t |x − y|
λn
−
t
t
x
− y
λn
x − x
t
. 2.65
Therefore,
U
2
∞
r
|x−y|≤8r
x − x
1
t
ρ
|y−z|≤t
1
|y − z|
n−ρ
dμz
2
dμydt
t
n2
1/2
f
L
∞
μ
∞
r
r
n1
t
n2
dt
1/2
f
L
∞
μ
f
L
∞
μ
.
2.66
TheestimatesforU
1
and U
2
yield 2.61.
Journal of Inequalities and Applications 19
Now we prove that M
∗,ρ
λ
f satisfies 2.53.LetQ ⊂ R be any two 16
√
d, 16
√
d
d1
-
doubling cubes. Set N ≡ N
Q,R
1. For any x ∈ Q and any y ∈ R, write
M
∗,ρ
λ
fx ≤M
∗,ρ
λ,0
f
1
xM
∗,ρ
λ,0
f
2
xM
∗,ρ
λ,∞
fχ
4Q
x
M
∗,ρ
λ,∞
N
Q,R
k2
fχ
2
k1
Q\2
k
Q
xM
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
y
M
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
x −M
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
y
.
2.67
By 1.2, we obtain that for any x ∈ Q,
M
∗,ρ
λ,∞
N
Q,R
k2
fχ
2
k1
Q\2
k
Q
x
≤
N
Q,R
k2
⎡
⎣
∞
r
2
k−1
Q
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
z∈2
k1
Q\2
k
Q
fz
|y − z|
n−ρ
dμz
2
dμydt
t
n1
⎤
⎦
1/2
⎡
⎣
N
Q,R
k2
∞
r
2
k2
Q\2
k−1
Q
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
z∈2
k1
Q\2
k
Q
fz
|y − z|
n−ρ
dμz
2
dμydt
t
n1
⎤
⎦
1/2
N
Q,R
k2
⎡
⎣
∞
r
R
d
\2
k2
Q
t
t |x − y|
λn
1
t
ρ
|y−z|≤t
z∈2
k1
Q\2
k
Q
fz
|y − z|
n−ρ
dμz
2
dμydt
t
n1
⎤
⎦
1/2
≡ V
1
V
2
V
3
.
2.68
The Minkowski inequality involving the fact that for any y ∈ 2
k−1
Q and z ∈ 2
k1
Q \ 2
k
Q,
|y − z|∼|z − x
Q
| and t ≥|y − z|≥2
k−2
lQ gives us that
V
1
f
L
∞
μ
N
Q,R
k2
2
k1
Q\2
k
Q
1
z − x
Q
n
∞
2
k−2
lQ
2
k−1
Q
1
t
n1
dμydt
1/2
dμz
f
L
∞
μ
N
Q,R
k2
2
k1
Q\2
k
Q
1
z − x
Q
n
dμz
K
Q,R
f
L
∞
μ
.
2.69
It is easy to verify that for any y ∈ 2
k2
Q \ 2
k−1
Q and x ∈ Q, |y −x|∼|y − x
Q
|, which leads to
V
2
N
Q,R
k2
2
k2
Q\2
k−1
Q
∞
|y−x|
1
t
n1
dt dμy
1/2
f
L
∞
μ
N
Q,R
k2
2
k2
Q\2
k−1
Q
1
|x − y|
λn
|x−y|
0
1
t
n−λn1
dt dμy
1/2
f
L
∞
μ
N
Q,R
k2
2
k2
Q\2
k−1
Q
1
y − x
Q
n
dμy
1/2
f
L
∞
μ
K
Q,R
f
L
∞
μ
.
2.70
20 Journal of Inequalities and Applications
To estimate V
3
, we first have that for any x ∈ Q, z ∈ 2
k1
Q \ 2
k
Q,andy ∈ R
d
\ 2
k2
Q,
|y − x
Q
|∼|y − x|∼|y − z| and |z − x
Q
| 2
k
lQ. This fact and the Minkowski inequality state
that
V
3
f
L
∞
μ
N
Q,R
k2
2
k1
Q\2
k
Q
R
d
\2
k2
Q
1
y − x
Q
2n
∞
|y−x|
1
t
n1
dt dμy
1/2
R
d
\2
k2
Q
1
y − x
Q
2nλn
|y−x|
0
1
t
n−λn1
dt dμy
1/2
dμz
f
L
∞
μ
N
Q,R
k2
2
k1
Q\2
k
Q
1
z − x
Q
n
dμz
K
Q,R
f
L
∞
μ
.
2.71
Combining the estimates for V
1
,V
2
,andV
3
yields that
M
∗,ρ
λ,∞
N
Q,R
k2
fχ
2
k1
Q\2
k
Q
x K
Q,R
f
L
∞
μ
. 2.72
An argument similar to the estimate of 2.60 shows that for any y ∈ R,
M
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
y ≤M
∗,ρ
λ
fyCf
L
∞
μ
. 2.73
By some estimate similar to that for 2.61, we easily obtain that for any x, y ∈ R,
M
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
x −M
∗,ρ
λ,∞
fχ
R
d
\2
N
Q
y
f
L
∞
μ
. 2.74
Therefore, for any x ∈ Q and y ∈ R,
M
∗,ρ
λ
fx −M
∗,ρ
λ
fy M
∗,ρ
λ,0
f
1
xM
∗,ρ
λ,∞
fχ
4Q
xK
Q,R
f
L
∞
μ
. 2.75
Taking mean value over Q for x, and over R for y, then yields
m
Q
M
∗,ρ
λ
f
− m
R
M
∗,ρ
λ
f
m
Q
M
∗,ρ
λ,0
f
1
m
Q
M
∗,ρ
λ,∞
fχ
4Q
K
Q,R
f
L
∞
μ
K
Q,R
f
L
∞
μ
,
2.76
where we used the fact that m
Q
M
∗,ρ
λ,0
f
1
f
L
∞
μ
and m
Q
M
∗,ρ
λ,∞
fχ
4Q
f
L
∞
μ
, which
can be proved by a way similar to that for the estimate 2.57. This finishes the proof of
Theorem 2.7.
Remark 2.11. From the proofs of Theorems 2.2 and 2.7, we can see that if we replace the
assumption that M
ρ
as 1.4 is bounded on L
2
μ by the one that M
∗,ρ
λ
is bounded on L
2
μ,
then Theorems 2.2 and 2.7 still hold. Therefore, applying the interpolation theorem see
23, Theorem 7.1 between the endpoint estimates that M
∗,ρ
λ
is bounded from L
∞
μ into
RBLOμ, which is a subspace of RBMOμ, and the boundedness of M
∗,ρ
λ
in L
2
μ, we can
Journal of Inequalities and Applications 21
obtain that M
∗,ρ
λ
as in 1.6 is bounded on L
p
μ for p ∈ 2, ∞ with the kernel satisfies 1.2
and 1.3. On the other hand, it follows from the Marcinkiewicz interpolation theorem that
M
∗,ρ
λ
as in 1.6 is also bounded on L
p
μ for p ∈ 1, 2 with the kernel satisfying 1.2 and
2.1.
3. Boundedness of M
∗,ρ
λ
in Hardy spaces
In this section, we will prove that the operator M
∗,ρ
λ
as in 1.6 is bounded from H
1
μ into
L
1
μ. To state our result, we first recall the definition of the space H
1
μ via the “grand”
maximal function characterization of Tolsa see 29.
Definition 3.1. Given f ∈ L
1
loc
μ,set
M
Φ
fx ≡ sup
ϕ∼x
R
d
fyϕydμy
, 3.1
where the notation ϕ∼x means that ϕ ∈ L
1
μ ∩ C
1
R
d
and satisfies
i ϕ
L
1
μ
≤ 1,
ii 0 ≤ ϕy ≤ 1/|y − x|
n
for all y ∈ R
d
,
iii |∇ϕy|≤1/|y − x|
n1
for all y ∈ R
d
.
Definition 3.2. The Hardy space H
1
μ is defined to be the set of all functions f ∈ L
1
μ
satisfying that
R
d
fdμ 0andM
Φ
f ∈ L
1
μ. Moreover, we define the norm of f ∈ H
1
μ by
f
H
1
μ
≡f
L
1
μ
M
Φ
f
L
1
μ
. 3.2
Theorem 3.3. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 2.1, and M
∗,ρ
λ
be as in 1.6 with ρ ∈ n/2, ∞ and λ ∈ 2, ∞. Then, M
∗,ρ
λ
is bounded from
H
1
μ into L
1
μ.
We begin with the proof of Theorem 3.3 with the atomic characterization of H
1
μ
established by Tolsa in 23.
Definition 3.4. Let η ∈ 1, ∞ and p ∈ 1, ∞.Afunctionb ∈ L
1
loc
μ is called to be an atomic
block if
i there exists some cube R such that suppb ⊂ R;
ii
R
d
bxdμx0;
iii there are functions a
j
with supports in cubes Q
j
⊂ R and numbers λ
j
∈ R such that
b ≡
j
λ
j
a
j
,and
a
j
L
∞
μ
≤
μ
ηQ
j
1/p−1
K
Q
j
,R
−1
. 3.3
Then, we define |b|
H
1,p
atb
μ
≡
j
|λ
j
|.
A function f ∈ L
1
μ is said to belong to the space H
1,p
atb
μ if there exist atomic
blocks b
i
such that f ≡
∞
i1
b
i
with
i
|b
i
|
H
1,p
atb
μ
< ∞.TheH
1,p
atb
μ norm of f is defined by
f
H
1,p
atb
μ
≡ inf
i
|b
i
|
H
1,p
atb
μ
, where the infimum is taken over all the possible decompositions
of f in atomic blocks.
22 Journal of Inequalities and Applications
Itwasprovedin23 that the definition of H
1,p
atb
μ is independent of the chosen
constant η ∈ 1, ∞, and for any p ∈ 1, ∞, all the atomic Hardy spaces H
1,p
atb
μ are just
the Hardy space H
1
μ with equivalent norms.
Proof of Theorem 3.3. By a standard argument, it suffices to verify that for any atomic block b
as in Definition 3.4 with η 2andp ∞,
M
∗,ρ
λ
b
L
1
μ
|b|
H
1,∞
atb
μ
. 3.4
Let all the notation be the same as in Definition 3.4.Write
R
d
M
∗,ρ
λ
bxdμx
R
d
\6
√
dR
M
∗,ρ
λ
bxdμx
6
√
dR
M
∗,ρ
λ
bxdμx ≡ W
1
W
2
.
3.5
By 2.24 and Definition 3.4, we have
W
1
b
L
1
μ
|b|
H
1,∞
atb
μ
. 3.6
To estimate the term W
2
,letb ≡
j
λ
j
a
j
be as in iii of Definition 3.4, and further write
W
2
≤
j
λ
j
2Q
j
M
∗,ρ
λ
a
j
xdμx
j
λ
j
6
√
dR\2Q
j
M
∗,ρ
λ
a
j
xdμx. 3.7
The L
2
μ-boundedness of M
∗,ρ
λ
via the H
¨
older inequality states that for each fixed j,
2Q
j
M
∗,ρ
λ
a
j
xdμx ≤
M
∗,ρ
λ
a
j
L
2
μ
μ
2Q
j
1/2
a
j
L
∞
μ
μ
2Q
j
1. 3.8
On the other hand, it follows from 2.48 that
6
√
dR\2Q
j
M
∗,ρ
λ
a
j
xdμx
6
√
dR\2Q
j
1
x − x
Q
j
n
dμx
a
j
L
1
μ
K
Q
j
,R
a
j
L
∞
μ
μ
Q
j
1.
3.9
Thus,
W
2
j
λ
j
|b|
H
1,∞
atb
μ
, 3.10
which completes the proof of Theorem 3.3.
Journal of Inequalities and Applications 23
4. Boundedness of M
ρ
S
in Lebesgue spaces and Hardy spaces
In this section, we will investigate the boundedness for the operator M
ρ
S
as in 1.5 in
Lebesgue spaces and Hardy spaces.
It is easy to verify that for any ρ ∈ 0, ∞, λ ∈ 1, ∞,andx ∈ R
d
,
M
ρ
S
fx ≤M
∗,ρ
λ
fx, 4.1
which, together with Theorems 2.1 and 2.2, gives us the boundedness of the operator M
ρ
S
in
L
p
μ for p ∈ 1, ∞ as follows.
Theorem 4.1. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 1.3, and M
ρ
S
be as in 1.5 with ρ ∈ 0, ∞. Then, for any p ∈ 2, ∞, M
ρ
S
is bounded on L
p
μ.
Theorem 4.2. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 2.1, and M
ρ
S
be as in 1.5 with ρ ∈ n/2, ∞. Then, M
ρ
S
is bounded from L
1
μ to weak L
1
μ.
By the Marcinkiewicz interpolation theorem, and Theorems 4.1 and 4.2, we easily
obtain the L
p
μ-boundedness of the operator M
ρ
S
for p ∈ 1, 2.
Corollary 4.3. Let K be a μ-locally integrable function on R
d
× R
d
\{x, y : x y} satisfying
1.2 and 2.1, and M
ρ
S
be as in 1.5 with ρ ∈ n/2, ∞. Then, M
ρ
S
is bounded on L
p
μ for any
p ∈ 1, 2.
For the case of p ∞, we also obtain the similar result for the operator M
∗,ρ
λ
.
Theorem 4.4. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 1.3, and M
ρ
S
be as in 1.5 with ρ ∈ 0, ∞. Then, for any f ∈ L
∞
μ, M
ρ
S
f is either infinite
everywhere or finite almost everywhere. More precisely, if M
ρ
S
f is finite at some point x
0
∈ R
d
,then
M
ρ
S
f is finite almost everywhere and
M
ρ
S
f
RBLOμ
≤ Cf
L
∞
μ
, 4.2
where the positive constant C is independent of f.
As for the behavior of the operator M
ρ
S
in Hardy spaces, we have the following
conclusion.
Theorem 4.5. Let K be a μ-locally integrable function on R
d
×R
d
\{x, y : x y} satisfying 1.2
and 2.1, and M
ρ
S
be as in 1.5 with ρ ∈ n/2, ∞. Then, M
ρ
S
is bounded from H
1
μ to L
1
μ.
We point out that Theorems 4.4 and 4.5 can not be easily deduced from 4.1,and
Theorems 2.7 and 3.3. However, using the same method, we can prove the above results
more easily than the corresponding results for M
∗,ρ
λ
. Here, we omit the proofs for brevity.
Acknowledgments
This work is supported by National Natural Science Foundation of China no. 10701078.The
authors want to express their deep thanks to Professor Dachun Yang for his useful advices.
The authors would like to thank the referee for his very careful reading and many useful
remarks.
24 Journal of Inequalities and Applications
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