Bilinear Calderón-Zygmund operators of type on non-homogeneous space
© Zheng et al.; licensee Springer. 2014
Received: 31 October 2013
Accepted: 26 February 2014
Published: 12 March 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.
Keywordsω-type Calderón-Zygmund operators function upper doubling measures Morrey space
1 Introduction and main results
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).
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 .
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 .
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.
A new example of operators with kernel satisfying (1.3) and (1.4) is called Bergman-type operator; it appeared in .
Now we define bilinear Calderón-Zygmund kernel of type and the corresponding bilinear Calderón-Zygmund operators.
for all , and .
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.
the constant C depends only on , , p.
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).
where and , .
2 Proof of the result
Before we prove Theorem 1.5, we need some notations and lemmas.
For , a ball is said to be -doubling if .
Lemma 2.2 
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.
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 .
we denote . By the Lebesgue differential theorem, it is easy to see that for any and μ-a.e. .
The proof of Lemma 2.3 needs a slight modification of the proof of Theorem 4.2 in , so we omit the details.
for any and for every .
We will use the same methods several times.
Combining all the estimates for with , we get (2.9).
Since we proved (2.8) and (2.9), (2.7) holds obviously. □
Now we give the proof of Theorem 1.5.
3 Boundedness on Morrey spaces
We recall the definition of the Morrey space with non-doubling measure.
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 
Now we are ready to give the proof of Theorem 1.6.
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.
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