Calderón-Zygmund Operators and Commutators on Morrey-Herz Spaces with Non-Homogeneous Metric Measure

SONG Yunchao(宋云超),ZHAO Kai(趙凱)(.Department of Mathematics and Physics,Qingdao Huanghai University,Qingdao 6647,China;.School of Mathematics and Statistics,Qingdao University,Qingdao 6607,China )

Abstract: Let (X,d,μ) be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling conditions.In this paper,the Morrey-Herz spaces on the non-homogeneous metric measure space are introduced.Then,by the properties of the non-homogeneous metric measure space,in particular the η-weak reverse doubling condition,the boundedness of Calderón-Zygmund operators and their commutators on the Morrey-Herz spaces with non-homogeneous metric measure is obtained.

Key words: Non-homogeneous metric measure space;Morrey-Herz space;Calderón-Zygmund operator;Commutator;Boundedness

1.Introduction

As we all know,the Herz spaces,the Herz type Hardy spaces and the Morrey-Herz spaces have been discussed by LU,YANG and other authors.And,some results of boundedness for singular integral operators on Herz type spaces are studied.[1?8]

Recall that a metric space equipped with a doubling condition measure is called a homogeneous space.In general,the doubling condition is important in the classical harmonic analysis.Nevertheless,it has been proved that the results in the classical function spaces and the boundedness of singular integrals onare remained valid if it is replaced by a non-doubling condition,some examples can follow in [9–15].

After that,Hyt?nen introduced the non-homogeneous metric measure space which satisfying geometrically doubling and the upper doubling conditions in [16].This kind of space contains both homogeneous type space and non-doubling metric measure space.Some results of non-homogeneous metric measure space and the boundedness of various operators on the spaces can be found in [17–24],etc.YANG and his collaborators[25?26]used the discrete coefficients to introduce the Hardy spaces on the non-homogeneous metric measure space,and discussed some characterizations of them.In [27],HAN and ZHAO introduced the Herz spaces and Herz type Hardy spaces on the non-homogeneous metric measure space.They also proved some characterizations for the spaces,and obtained some boundedness of singular operators on the Herz type Hardy spaces.

In this paper,motivated by the statements above,we investigate the Morrey-Herz spaces on the non-homogeneous metric measure space which satisfies the geometrically doubling and the upper doubling conditions.We obtain the boundedness of Calderón-Zygmund operators and their commutators on the Morrey-Herz spaces.

2.Preliminaries

For convenience,in this section,we recall some fundamental knowledge for the nonhomogeneous metric measure space.

The following is the geometrically doubling,which was original introduced by Coifman and Weiss in [9],and is also known as metrically doubling in [28].

3.Morrey-Herz Space and Main Result

In this section,we introduce the Morrey-Herz spaces on non-homogeneous metric measure space.And the boundedness of Calderón-Zygmund operators and commutators on the Morrey-Herz spaces with non-homogeneous metric measure is discussed.

The definition of Morrey-Herz spaces on non-homogeneous metric measure space is as follows.

Thus,by Definition 3.2 and the H?lder inequality,we conclude that

Proof of Theorem 3.2For anyThen,write

Therefore,by theη-weak reverse doubling condition,there is

Thus,it means that Theorem 3.2 is proved.