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Bilinear Calderón-Zygmund operators of type on non-homogeneous space
Journal of Inequalities and Applications volume 2014, Article number: 113 (2014)
Let be a geometrically doubling metric spaces and the measure μ satisfy the upper doubling condition. The aim of this paper, under this assumption, is to study the boundedness of the bilinear Calderón-Zygmund operator of type . As an application, we obtain the Morrey boundedness properties of the bilinear operator.
MSC:42B20, 42B25, 42B35.
1 Introduction and main results
In the last few decades, the classical theory of the singular integral has played an important role in harmonic analysis. One of the main features of these works is that the underlying spaces or domains possess the measure doubling property,
where μ is a Borel measure, denotes the ball with center x and radius . A metric space equipped with such a measure μ is called a space of homogeneous type. It is well known that the measure doubling condition in the analysis on spaces of homogeneous type is a key assumption, such as that Euclidean spaces with weighted measures satisfy the doubling property (1.1).
However, recently, some works indicated that the measure doubling condition is superfluous for most of the classical singular integral operator theory, and many results on the Calderón-Zygmund theory have been proved valid if the condition (1.1) is replaced by a mild volume growth condition,
where is a positive constant, d is a dimension of the underlying spaces, , . Such a measure does not satisfy the doubling condition. For example, Tolsa  established Calderón-Zygmund theory for a nondoubling measure and introduced the RBMO spaces, a variant of the space BMO, and he proved that Calderón-Zygmund operators are bounded from into . Nazarov et al.  showed that if T is a Calderón-Zygmund operator bounded on , then T is bounded on for all and from into .
Recently, Hytönen  gave a new class of metric measures spaces (instead of ), which are called non-homogeneous spaces, the measure μ satisfies the upper doubling condition (see definition 1.3). The new class of metric measures spaces are sufficiently general to include in a natural way both the space of homogeneous type and a metric space with the mild volume growth condition.
Anh and Doung  established the boundedness of Calderón-Zygmund operator on various function spaces on and they extended the work of Tolsa on the non-homogeneous spaces to a more general non-homogeneous spaces .
Meanwhile, multilinear Calderón-Zygmund theory has been studied by many researchers. The theory was introduced by Coifman and Meyer  in 1975 and it was further investigated by Grafakos and Torres . Chen and Fan [7, 8] obtained some estimates for the bilinear singular integral. Xu  obtained the boundedness of a multilinear Calderón-Zygmund operator on , .
Yabuta  introduced a generalized operator: a Calderón-Zygmund operator of type , which generalizes the classical Calderón-Zygmund operator. Maldonado and Naibo  developed a theory of the bilinear Calderón-Zygmund operator of type (see Definition 1.4) and extended some results of Yabuta.
Theorem A 
Consider , and let T be a bilinear Calderón-Zygmund operator of type in . If and such that , then T can be extended to a bounded operator from into , where or should be replaced by if or , respectively.
In this paper, we study the boundedness of a bilinear Calderón-Zygmund operator of type on a non-homogeneous metric space, where we only assume . And we also note that the condition of kernel (1.5) is more general than the size condition defined by Hu . So this is a new result, which generalizes some works of Maldonado and Naibo  and Anh and Doung  on . As an application, we investigate the boundedness of the bilinear Calderón-Zygmund operator of type over a Morrey space on .
Before stating our main results, we fix some notations and define some terminologies. Throughout this paper, a ball Q denotes which is equipped with a fixed center and radius . The center and radius of Q are denoted by and . For and , the notation stands for the concentric dilation of Q. For notational convenience, we will occasionally write . The following notions of geometrically doubling and upper doubling measures μ are originally from Hytönen .
We finally observe that in the sequel the letter C will be used to denote various constants which do not depend on the functions.
Definition 1.1 A metric space is called geometrically doubling if there exists a number such that any open ball can be covered by at most N balls .
Lemma 1.2 For a metric space , the following statements are equivalent:
is geometrically doubling.
For any , any ball can be covered by at most balls .
For every , any ball can contain at most centers of disjoint balls .
There exists such that any ball can contain at most M centers of disjoint balls .
Definition 1.3 A Borel measure μ in the metric space is said to be an upper doubling measures if there exists a dominating function and a constant such that:
For any fixed , is increasing.
The inequality holds for all , .
for all , and .
Remark 1 If we take the dominating function to be , then the measure doubling is a special case of upper doubling. On the other hand, a Radon measure μ as in (1.2) on is also an upper doubling measure by taking the dominating function .
In this paper, we assume that is a geometrically doubling metric spaces and the measure μ is an upper doubling measure. And we denote by for brevity.
We recall the Calderón-Zygmund operator defined by Anh and Doung . A kernel is called a Calderón-Zygmund kernel if it satisfies
for all with . There exists such that
A linear operator T is called a Calderón-Zygmund operator with satisfying the above conditions if for all with bounded support and ,
A new example of operators with kernel satisfying (1.3) and (1.4) is called Bergman-type operator; it appeared in .
For , we write if , ω is nondecreasing, concave, and
Now we define bilinear Calderón-Zygmund kernel of type and the corresponding bilinear Calderón-Zygmund operators.
Definition 1.4 Let be a nondecreasing function and be a locally integrable function defined away from the diagonal in . We say that is a bilinear Calderón-Zygmund kernel of type if it satisfies the size condition,
for some and all with for some i. We have the smoothness estimates,
A bilinear operator is said to be associated with a bilinear Calderón-Zygmund kernel of type , if
for all , and .
If the bilinear operator is associated with and admits some bounded extensions
for some () and with , or
for some () and , then is said to be a bilinear Calderón-Zygmund operator of type .
Note that for all , and . When , , the linear Calderón-Zygmund operator of type is the Calderón-Zygmund operator defined by Anh and Doung , so our results are more general.
Theorem 1.5 Consider , and let be a bilinear Calderón-Zygmund operator of type with . Assume , and with if . Suppose is a bounded operator from , then there exists a constant C such that
the constant C depends only on , , p.
Remark 2 The assumption of ω in  is , which is stronger than , and it is easy to see because is nondecreasing,
Next, we give the boundedness of the bilinear Calderón-Zygmund operator of type over Morrey space on (for the Morrey space see Definition 3.1).
Theorem 1.6 Assume that is a bilinear Calderón-Zygmund operator of type , let and for . Suppose is a bounded operator from to , then there exists a constant C such that
where and , .
2 Proof of the result
Before we prove Theorem 1.5, we need some notations and lemmas.
Definition 2.1 For any two balls , we define
For , a ball is said to be -doubling if .
Lemma 2.2 
If are balls in , then
If are of compatible size, then .
If are non--doubling balls (), then .
In what follows, unless α, β are specified otherwise, by a doubling ball we mean a -doubling with a fixed number , where n can be viewed as a geometric dimension of the spaces.
For any fixed ball , let be the smallest integer such that is doubling, we denote this ball by . Denote by the mean value of f on Q, namely, . Let be a fixed constant, we say that is in if there exists a constant such that
for any ball Q, and
for any two doubling balls . The minimal constant is the norm of f, and it will be denoted by .
We will prove Theorem 1.5 via the boundedness of sharp maximal estimates. Let f be a function in , the sharp maximal function of f is defined by
where the supremum is taking over all the balls Q containing the point x. In order to prove our results, we need a variant of (2.4)
For , we denote the non-centered Hardy-Littlewood maximal operator
which is bounded on for , we can find the proof in . We also need the following multilinear maximal operator:
which is introduced by Lerner  when μ is Lebesgue measure and . It obvious that the operator is strictly controlled by the 2-fold product of .
The non-centered doubling maximal operator is defined by
we denote . By the Lebesgue differential theorem, it is easy to see that for any and μ-a.e. .
Lemma 2.3 Let with the extra condition if . Assume that for some p, , . Then we have
The proof of Lemma 2.3 needs a slight modification of the proof of Theorem 4.2 in , so we omit the details.
In the following proofs we will employ several times the following simple Kolmogorov inequality. Let be a probability measure spaces and let , then there is a constant such that for any measurable function f,
Lemma 2.4 Let be a bilinear Calderón-Zygmund operator of type , . Suppose is bounded from to , then there exists a constant C such that
for any and for every .
hold for any balls with , Q is an arbitrary ball,
where we split each as , and , and we have
So we obtain
For the first term I, applying Kolmogorov’s inequality (2.6) with and , we derive
Next we consider II. Firstly, the condition holds since , , , then we have the following estimates:
Here the series is equivalent to , where . We use the estimate, since ,
We will use the same methods several times.
Similarly, we have
By the estimates above, we have
It remains to consider the term in . For , noting that for (), we use the condition of kernel (1.6) to obtain
A trivial computation now shows that
By the above estimate, we have
Fix any balls with , where Q is an arbitrary ball and R is a doubling ball. Noting that R is a doubling ball we have . We denote by N such that . Let , , , , and , write the difference in the following way:
For the term , we, firstly, deal with ; it follows from the size of kernel (1.5), for all ,
Using the analogous methods to deal with the term , we have
By an argument similar to the estimate for , we see that
For all , we have
Firstly, for , we note that , , so , . The properties of λ imply that
For , we have
In the following, we will estimate and , respectively:
The estimates of and imply that
Some estimates similar to yield
Finally, using a similar argument as that of II, which involves the kernel condition (1.6), we obtain
Combining all the estimates for with , we get (2.9).
Let us see how from (2.8) and (2.9) one gets (2.7). Use the definition of and the fact , , if Q is a doubling ball and , we have
and for all doubling balls with , we have
Since we proved (2.8) and (2.9), (2.7) holds obviously. □
Now we give the proof of Theorem 1.5.
Proof Since is dense in , , Lemma 2.4 holds for . Using Lemma 2.3, Hölder’s inequality, and the boundedness of , we get
3 Boundedness on Morrey spaces
We recall the definition of the Morrey space with non-doubling measure.
Definition 3.1 Let and ; the Morrey space is defined as
with the norm
As is easily seen, the space is a Banach space with its norm. The Morrey space norm reflects local regularity of f more precisely than the Lebesgue space norm. It is easy to see from Hölder’s inequality that whenever . Moreover, the definition of the spaces is independent of the constant , and the norms for different choice of are equivalent, see [15–18] for details. We will denote by .
For the proof of Theorem 1.6, we need some lemmas.
Lemma 3.2 
Let , satisfying conditions of definition (1.3), , , then
Lemma 3.3 Let and , for , we have
Proof Using Hölder’s inequality and Lemma 3.2, we get
Now we are ready to give the proof of Theorem 1.6.
Proof Fix a ball and we split each as , where and , and this yields
For the first term , using the results of Theorem 1.5, we have
Considering the case , firstly, we deal with . For , we have
For the term , it follows that
For the term , using Lemma 3.3, we get
According to the estimate above, we obtain
What remain to be considered are the terms in and . For , we use the size condition of kernel (1.5) and the property of , ,
Then using the above estimate, we get
Analogously, for , we have
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The research was partially supported by the National Nature Science Foundation of China under grant #11271330 and #11171306 and partially sponsored by the Science Foundation of Zhejiang Province of China under grant #Y604563 and the Education Foundation of Zhejiang Province under grant #Y201225707. Thanks are also given to the anonymous referees for careful reading and suggestions.
The authors declare that they have no competing interests.
TZ formulated the considered problem and gave the construction of Theorems 1.5 and 1.6. Additionally, TZ participated in the process of the proofs of Theorems 1.5 and 1.6. XT participated in the proof of Theorem 1.5. XW participated in the proof of Theorem 1.6. All authors read and approved the final manuscript.
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Zheng, T., Tao, X. & Wu, X. Bilinear Calderón-Zygmund operators of type on non-homogeneous space. J Inequal Appl 2014, 113 (2014). https://doi.org/10.1186/1029-242X-2014-113
- Calderón-Zygmund operators
- upper doubling measures
- Morrey space