Multilinear Riesz Potential on Morrey-Herz Spaces with Non-Doubling Measures
© Yanlong Shi et al. 2010
Received: 19 November 2009
Accepted: 10 March 2010
Published: 14 March 2010
The authors consider the multilinear Riesz potential operator defined by , where denotes the -tuple , the nonnegative integers with , , and is a nonnegative -dimensional Borel measure. In this paper, the boundedness for the operator on the product of homogeneous Morrey-Herz spaces in nonhomogeneous setting is found.
Let denote a ball centered at with radius , and for any , will mean the ball with the same center as and with radius . A Borel measure on is called a doubling measure if it satisfies the so-called doubling condition; that is, there exists a constant such that
for every ball . The doubling condition is a key feature for a homogeneous (metric) measure space. Many classical theories in Fourier analysis have been generalized to the homogeneous setting without too much difficulties. In the last decade, however, some researchers found that many results are still true without the assumption of the doubling condition on (see, e.g., [1–4]). This fact has encouraged other researchers to study various theories in the nonhomogeneous setting. By a nonhomogeneous space we mean a (metric) measure space, here we will consider only , equipped with a nonnegative -dimensional Borel measure , that is, a measure satisfying the growth condition
was studied by García-Cuerva and Martell  in 2001. García-Cuerva and Martell proved that is bounded from to for all and and that is bounded from to . Here and denote the Lebesgue spaces and weak Lebesgue spaces with measure , respectively.
Simultaneously many classical multilinear operators on Euclidean spaces with Lebesgue measure have been generalized for nondoubling measures, that is, the case nonhomogeneous setting see [3, 4]. For example, based on the work of Kenig and Stein , Lian and Wu  studied multilinear Riesz potential operator
Proposition 1.1 (see ).
In addition, in the article [6–8], we have obtained the boundedness of the operator on the product of Morrey type spaces, (weak) homogeneous Morrey-Herz spaces, and Herz type Hardy spaces in the classical case and extended the result of Kenig and Stein. As a continuation of previous work in [4, 6–8], in this paper, we will study the operator in the product of (weak) homogeneous Morrey-Herz spaces in the nonhomogeneous setting.
The restriction in Theorem 1.3 cannot be removed see  for an counter-example when and .
In addition, we remark that the (weak) homogeneous Morrey-Herz spaces generalize the (weak) homogeneous Herz spaces. Particularly, and for and . Moreover, we have , the weighted spaces for and , for details, see Section 2.
Hence, it is easy to obtain the following corollaries from the theorems above.
2. The Definitions of Some Function Spaces
3. Proof of Theorems 1.2 and 1.3
To shorten the formulas below, we set
It is easy to see that the case for is analogous to the case for , the case for is similar to the case for , and the case for is analogous to the case for , respectively. Thus, by the symmetry of and in the operator , we will only discuss the cases for belong to , , , , and , respectively.
Combining inequalities (3.15) and (3.16), we obtain
Thus, we get
Therefore, the inequality above and inequalities (3.19), (3.20), and (3.24) yield
Thus, a similar argument shows that
This is the desired estimate of Theorem 1.2.
The proof of Theorem 1.2 is completed.
So we can show that
This work was supported partly by the National Natural Science Foundation of China under Grant no. 10771110 and sponsored by the Natural Science Foundation of Ningbo City under Grant no. 2009A610084.
- García-Cuerva J, Martell JM: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana University Mathematics Journal 2001, 50(3):1241–1280.MathSciNetView ArticleMATHGoogle Scholar
- Sawano Y, Tanaka H: Morrey spaces for non-doubling measures. Acta Mathematica Sinica 2005, 21(6):1535–1544. 10.1007/s10114-005-0660-zMathSciNetView ArticleMATHGoogle Scholar
- Xu J: Boundedness of multilinear singular integrals for non-doubling measures. Journal of Mathematical Analysis and Applications 2007, 327(1):471–480. 10.1016/j.jmaa.2006.04.049MathSciNetView ArticleMATHGoogle Scholar
- Lian J, Wu H: A class of commutators for multilinear fractional integrals in nonhomogeneous spaces. Journal of Inequalities and Applications 2008, 2008:-17.Google Scholar
- Kenig CE, Stein EM: Multilinear estimates and fractional integration. Mathematical Research Letters 1999, 6(1):1–15.MathSciNetView ArticleMATHGoogle Scholar
- Tao XX, Shi YL, Zhang SY: Boundedness of multilinear Riesz potential on the product of Morrey and Herz-Morrey spaces. Acta Mathematica Sinica. Chinese Series 2009, 52(3):535–548.MathSciNetMATHGoogle Scholar
- Shi Y, Tao X: Boundedness for multilinear fractional integral operators on Herz type spaces. Applied Mathematics, Journal of Chinese Universities B 2008, 23(4):437–446. 10.1007/s11766-008-1995-xMathSciNetView ArticleMATHGoogle Scholar
- Shi Y, Tao X: Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces. Hokkaido Mathematical Journal 2009, 38(4):635–662.MathSciNetView ArticleMATHGoogle Scholar
- Komori Y: Weak type estimates for Calderón-Zygmund operators on Herz spaces at critical indexes. Mathematische Nachrichten 2003, 259: 42–50. 10.1002/mana.200310093MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.