- Research Article
- Open access
- Published:
Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures
Journal of Inequalities and Applications volume 2009, Article number: 175230 (2009)
Abstract
Let be a positive Radon measure on
which may be nondoubling. The only condition that
satisfies is
for all
,
, and some fixed constant
. In this paper, we introduce the operator
related to such a measure and assume it is bounded on
. We then establish its boundedness, respectively, from the Lebesgue space
to the weak Lebesgue space
, from the Hardy space
to
and from the Lesesgue space
to the space
. As a corollary, we obtain the boundedness of
in the Lebesgue space
with
.
1. Introduction
A positive Radon measure on
is said to be doubling if there exists some constant
such that
for all
,
. 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
is substituted by the growth condition as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ1_HTML.gif)
for all , where
is some fixed number with
. For example, In 2001, Tolsa in [1, 2] investigated the weak (1,1) inequality for singular integrals, the Littlewood-Paley theory and the
theorem with nondoubling measures. In 2002, GarcÃa-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 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 be a function on
such that there exist positive constants
,
,
, and
satisfying
(a) and
,
(b),
(c) for
.
For this , we define the Littlewood-Paley's
-function with nondoubling measures as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ2_HTML.gif)
where and
.
Note that if we replace by
in the above definition and when
is the Poisson kernel, we obtain classical
-function defined and studied by Stein [6] and later by Fefferman [7], where the weak (1,1) with
and weak
with
boundedness of
function were obtained. In the same paper, Fefferman [7] also established the
bounds of
for
and
. For the more generalized
-function defined by (1.1), the
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
-function systematically in his PhD thesis [9]. Later, in 2008, Lin and Meng [10] gave some results on parametric
-function with nondoubling measures. But their result only valid for
, one cannot obtain the results for classical operators even for
or in the classical case studied by Stein in 1961 [6].
In this paper, we will study the properties of operator with nondoubling measures on some function spaces under the conditions (a)–(c).
First, before stating our main results, we give some notation and definitions, let be a closed cube with sides parallel to the axes. Denote its side length by
and its center by
. Given
and
, we say
is
-doubling if
, where
is the cube concentric with
with side length
. If
are not specified, by a doubling cube we mean a
-doubling cube. For any cube
, we denote by
the smallest doubling cube which contains
and has the same center as
.
Given two cubes in
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ3_HTML.gif)
where is the first integer
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ4_HTML.gif)
with , where
is the first integer
such that
.
In the article of [1, page 95], we know that with constants that may depend on
and
. The following atomic Hardy space
was introduced by Tolsa in [11].
Definition 1.1.
For a fixed , a function
is called an atomic block if
(1)there exists some cube such that
;
(2);
(3)there are fucntions with supports in cubes
and numbers
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ5_HTML.gif)
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ6_HTML.gif)
We say that if there are atomic blocks
such that
with
. The
norm of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ7_HTML.gif)
where the infimum is taken over all the possible decompositions of in atomic blocks.
It was shown by Tolsa that the space was proved to be the Hardy space
in [11] with equivalent norms. We will denote the space
and the norm
, respectively, by
and
for convenience. He also proved that the dual space of
is the following space
.
Definition 1.2.
Let be a fixed constant. A function
is said to be in the space
if there exists some constant
such that for any cube
centered at some point of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ8_HTML.gif)
and for any two doubling cubes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ9_HTML.gif)
where denotes the mean value of
over cube
. The minimal constant
above is defined to be the norm of
in the space
and denoted by
.
Tolsa in [11] proved that the definition of the space and
are independent of the choice of
. The following space
was introduced in [12]. It is easy to see that
.
Definition 1.3.
A function is said to be in the space
if there exists some positive constant
such that for any
doubling cube
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ10_HTML.gif)
and for any two -doubling cubes
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ11_HTML.gif)
The minimal constant as above is defined to be the norm of
in the space
, we denote it by
.
In this paper, we always assume that and
are considered as they are defined at the beginning of this paper. Our main results are as follows.
Theorem 1.4.
Let be a function on
, satisfying (a)–(c),
,
. If
is bounded on
, then it is also bounded from
to
.
Theorem 1.5.
Let be a function on
, satisfying (a)–(c),
,
. If
is bounded on
, then it is also bounded from
to
.
Theorem 1.6.
Let be a function on
, satisfying (a)–(c),
,
. If
is bounded on
, then for
,
is either infinite everywhere or finite almost everywhere. More precisely, if
is finite at some point
, then
is
-finite almost everywhere and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ12_HTML.gif)
Corollary 1.7.
Let be a function on
, satisfying (a)–(c),
,
. If
is bounded on
, then it is also bounded on
for any
.
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 in (1.1) (in this case,
is defined and studied by [13]), from the results in [8], we know that it is impossible to give similar results as above for Littlewood-Paley
function even
for
. In fact, by the counter example in [13], even the
boundedness does not hold. In this sense, the condition we assumed on
is necessary and reasonable. On the other hand, in 2008, Lin and Meng [10] gave some results on parametric
-function with nondoubling measures. In fact the results in [10] are only valid for
. By the same reason as above, one cannot obtain the result when
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 is sharp for weak
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 will denote a positive constant that may vary at each occurrence but is independent of the essential variables.
will always denote that there exists a constant
, such that
.
2. Main Lemmas
We need two lemmas given by Tolsa.
Lemma 2.1 (see [11]).
If are concentric cubes such that there are no
-doubling cubes with
of the form
,
, with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ13_HTML.gif)
where depends only on
,
,
,
.
Lemma 2.2 (see [11]).
For any and any
then we have one has the following:
(a)there exists a family of almost disjoint cubes (that means
,
depends only on
) such that
(a.1);
(a.2) for any
;
(a.3) a.e.
on
,
(b)for each , let
be the smallest (
)-doubling cube of the form
,
, with that
, and let
, then there exists a family of functions
with
and with constant sign satisfying
(b.1);
(b.2), where
is some constant;
(b.3).
To prove our theorems, we prepare another two key lemmas.
For any subset , we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ14_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ15_HTML.gif)
Lemma 2.3.
Let and
be the same as in Lemma 2.2, and let
be
center. Then for any
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ16_HTML.gif)
Lemma 2.4.
Let and
be the same as in Lemma 2.2, and let
be
center. For any
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ17_HTML.gif)
Proof of Lemma 2.3.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ18_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ19_HTML.gif)
It is easy to see
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ20_HTML.gif)
We first estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ21_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ22_HTML.gif)
Note that and
, then
,
. These two inequalities will always be used in the following proof, so we will not mention them every time.
For any , we have that
,
,
,
,
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ23_HTML.gif)
For any , we get
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ24_HTML.gif)
For , we obtain that
,
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ25_HTML.gif)
For , we have that
,
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ26_HTML.gif)
Thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ27_HTML.gif)
Next we estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ28_HTML.gif)
For any , there exist two constants
,
such that
. Since
,
, we then have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ29_HTML.gif)
For any , the following inequalities hold:
,
, and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ30_HTML.gif)
Next we estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ31_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ32_HTML.gif)
For any , we can get that
and there exist two constants
and
such that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ33_HTML.gif)
For any , we can get
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ34_HTML.gif)
Next we estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ35_HTML.gif)
If , we obtain
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ36_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ37_HTML.gif)
If , we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ38_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ39_HTML.gif)
If , we have
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ40_HTML.gif)
The proof of Lemma 2.3 is finished.
Proof of Lemma 2.4.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ41_HTML.gif)
Let and divide
into four parts
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ42_HTML.gif)
Then .
Since and
, we have that
,
.
We first estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ43_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ44_HTML.gif)
For any , we have that
and
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ45_HTML.gif)
and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ46_HTML.gif)
For any , we have
,
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ47_HTML.gif)
If , we will get
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ48_HTML.gif)
If , we obtain that
. Thus we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ49_HTML.gif)
Next we estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ50_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ51_HTML.gif)
For any , the inequalities
and
hold, and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ52_HTML.gif)
For any , the inequalities
and
hold, and we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ53_HTML.gif)
Next we estimate . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ54_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ55_HTML.gif)
Since , we can choose
small enough such that
and
.
Now denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ56_HTML.gif)
and for a subset , we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ57_HTML.gif)
It is easy to see that .
For any , we have
,
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ58_HTML.gif)
The last inequality holds because and we choose
small enough such that
, and then we can get
, which leads to the above inequality. Using these inequalities, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ59_HTML.gif)
If , we get
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ60_HTML.gif)
Next we estimate . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ61_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ62_HTML.gif)
For any , we have
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ63_HTML.gif)
If , we get
and
. Then we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ64_HTML.gif)
If , we get
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ65_HTML.gif)
For any , the inequalities
,
and
hold, from which we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ66_HTML.gif)
If , we have
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ67_HTML.gif)
If , we have
and it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ68_HTML.gif)
If , we get
and
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ69_HTML.gif)
3. Proof of Theorems
Proof of Theorem 1.4.
To prove Theorem 1.4, we will choose and
.
Let and
. Applying Lemma 2.2 to
and
, we obtain a family of almost disjoint cubes
. With the notation
,
,
the same as in Lemma 2.2, we can decompose
, with that
and
. And
can be decomposed as
. By (a.1) of Lemma 2.2, we have
.
Thus, to prove that is of weak type
, we only need to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ70_HTML.gif)
Since and
, we only need to show that both
and
satisfy the inequality (2.4).
For , it follows from (b.1) of Lemma 2.2 that
, and we have
. Using
-boundedness of
and (a.3) and (b.2) from Lemma 2.2, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ71_HTML.gif)
To prove that satisfies inequality (3.1), it suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ72_HTML.gif)
Since , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ73_HTML.gif)
If we can prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ74_HTML.gif)
then we finish the proof of Theorem 1.4.
We first estimate and divide it into two parts
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ75_HTML.gif)
By the assumption of boundedness of
, and the fact that
is the
doubling cube, it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ76_HTML.gif)
Next we estimate . By the Minkowski's inequality, Fubini's theorem and condition (b) of
that
, we have the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ77_HTML.gif)
Choose small enough such that
,
and
.
For any subset , we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ78_HTML.gif)
Let be
center. Using the above notations, it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ79_HTML.gif)
By Lemma 2.3, we will have the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ80_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ81_HTML.gif)
where we used Lemma 2.1 to estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ82_HTML.gif)
We now estimate . Let
. Recall that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ83_HTML.gif)
We first estimate . By the Minkowski's inequality, Fubini's theorem and property (b) of
, we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ84_HTML.gif)
If we prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ85_HTML.gif)
for any and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ86_HTML.gif)
and by Lemma 2.2, we conclude that .
Now choose small enough such that
,
and
. Then for any
and
, note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ87_HTML.gif)
Next we estimate . By property (c) of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ88_HTML.gif)
since . Using the above inequality, Minkowski's inequality and Fubini's theorem, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ89_HTML.gif)
So by Lemma 2.4, for and any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ90_HTML.gif)
Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ91_HTML.gif)
and we will have . With the estimates of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ92_HTML.gif)
we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ93_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ94_HTML.gif)
Therefore we finish the proof of Theorem 1.4.
Proof of Theorem 1.5.
Note that the definition of is independent of the choice of the constant
, we can assume that
, still with
. By Theorem 1.4, the operator
is bounded from
to
. By a standard argument, we only need to prove that
for any atomic block
with supp
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ95_HTML.gif)
By definition of , we have
. By (3.25) in the proof of Theorem 1.4, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ96_HTML.gif)
To estimate , let
be as in Definition 1.1 and write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ97_HTML.gif)
By the assumption of boundedness of
and the Holder inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ98_HTML.gif)
By (3.11) in the proof of Theorem 1.4,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ99_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ100_HTML.gif)
where is a cube and has the same center as
and
,
is the center of
. From (3.29) to (3.30), we use the conclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ101_HTML.gif)
As a matter of fact, let and then by the definition of
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ102_HTML.gif)
Using (3.30) and the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ103_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ104_HTML.gif)
From (3.29) and (3.35), we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ105_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ106_HTML.gif)
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 such that for any
and
-doubling cube
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ107_HTML.gif)
To prove (3.38), for each fixed cube , let
be the smallest ball which contains
and has the same center as
. Then
. We decompose
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ108_HTML.gif)
By the Hölder's inequality and boundedness of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ109_HTML.gif)
We denote by the radius of
. Note that
for any
and for any
, then by Minkowski's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ110_HTML.gif)
Since we have the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ111_HTML.gif)
We claim
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ112_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ113_HTML.gif)
By (3.42), (3.44), and (3.45), we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ114_HTML.gif)
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 , the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ115_HTML.gif)
We note that can be looked as a vector valued Calderón-Zygmund singular integral operator in the following Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ116_HTML.gif)
In fact, can be written by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ117_HTML.gif)
where . Also note that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ118_HTML.gif)
where . We will divide
into four parts, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ119_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ120_HTML.gif)
This can be obtained from the same idea used before, see also the main step in [14], here we omit the proof of it.
From (3.38), we get that for , if
for some point
, then
is
-finite almost everywhere and in this case we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ121_HTML.gif)
provided that is a
-doubling cube.
To prove , we still need to prove that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ122_HTML.gif)
for any two -doubling cubes
.
Let and set
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ123_HTML.gif)
Again, we can get the conclusion under the nondoubling condition which is similar to the proof of (2.2) in [14] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ124_HTML.gif)
For any and each fixed
, similar to the proof of (2.4) from Lemma 2.3, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ125_HTML.gif)
Therefore we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ126_HTML.gif)
Next we estimate . By an estimate similar to (3.45), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ127_HTML.gif)
By (3.54), (3.55), (3.57), and (3.58), we can get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ128_HTML.gif)
Take mean value over for
and over
for
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ129_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ130_HTML.gif)
Since by Hölder's inequality, the boundedness assumption of
and the fact that
is
-doubling, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F175230/MediaObjects/13660_2009_Article_1908_Equ131_HTML.gif)
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.
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Acknowledgment
The first author was supported partly by NSFC (Grant: 10701010), NSFC (Key program Grant: 10931001), CPDRFSFP (Grant: 200902070) and SRF for ROCS, SEM.
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Xue, Q., Zhang, J. Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures. J Inequal Appl 2009, 175230 (2009). https://doi.org/10.1155/2009/175230
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DOI: https://doi.org/10.1155/2009/175230