Commutators of Littlewood-Paley Operators on the Generalized Morrey Space
© Yanping Chen et al. 2010
Received: 6 May 2010
Accepted: 11 July 2010
Published: 28 July 2010
Many important operators gave a characterization of BMO space. In 1976, Coifman et al.  gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo  studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space.
The main result in this paper is as follows.
It is easy to check that (see, e.g., the proof of ( ) in [15, page 89]), we therefore give only the proofs of Theorem 1.2 for and Theorem 1.3 for .
It is easy to see that the condition (1.15) is weaker than for . In the proof of Theorems 1.2 and 1.3, we will use some ideas in . However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof.
2. Proof of Theorem 1.2
Let us begin with recalling some known conclusion.
Similar to the proof of , we can easily get the following.
3. Proof of Theorem 1.3
Similar to the proof of Theorem 1.2, we only give the outline.
The authors wish to express their gratitude to the referee for his/her valuable comments and suggestions. The research was supported by NSF of China (Grant nos.: 10901017, 10931001), SRFDP of China (Grant no.: 20090003110018), and NSF of Zhenjiang (Grant no.: Y7080325).
- Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Transactions of the American Mathematical Society 1958, 88: 430–466. 10.1090/S0002-9947-1958-0112932-2MathSciNetView ArticleMATHGoogle Scholar
- Ding Y, Fan D, Pan Y: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana University Mathematics Journal 1999, 48(3):1037–1055.MathSciNetView ArticleMATHGoogle Scholar
- Chang S-YA, Wilson JM, Wolff TH: Some weighted norm inequalities concerning the Schrödinger operators. Commentarii Mathematici Helvetici 1985, 60(2):217–246.MathSciNetView ArticleMATHGoogle Scholar
- Kenig CE: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics. Volume 83. American Mathematical Society, Washington, DC, USA; 1994:xii+146.View ArticleGoogle Scholar
- Xue Q, Ding Y: Weighted boundedness for parametrized Littlewood-Paley operators. Taiwanese Journal of Mathematics 2007, 11(4):1143–1165.MathSciNetMATHGoogle Scholar
- Sakamoto M, Yabuta K: Boundedness of Marcinkiewicz functions. Studia Mathematica 1999, 135(2):103–142.MathSciNetMATHGoogle Scholar
- Torchinsky A, Wang SL: A note on the Marcinkiewicz integral. Colloquium Mathematicum 1990, 60–61(1):235–243.MathSciNetMATHGoogle Scholar
- Stein EM: The development of square functions in the work of A. Zygmund. Bulletin of the American Mathematical Society 1982, 7(2):359–376. 10.1090/S0273-0979-1982-15040-6View ArticleMathSciNetMATHGoogle Scholar
- Ding Y, Lu S, Yabuta K: On commutators of Marcinkiewicz integrals with rough kernel. Journal of Mathematical Analysis and Applications 2002, 275(1):60–68. 10.1016/S0022-247X(02)00230-5MathSciNetView ArticleMATHGoogle Scholar
- Ding Y, Xue Q: Endpoint estimates for commutators of a class of Littlewood-Paley operators. Hokkaido Mathematical Journal 2007, 36(2):245–282.MathSciNetView ArticleMATHGoogle Scholar
- Coifman R, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. The Annals of Mathematics 1976, 103(3):611–635. 10.2307/1970954MathSciNetView ArticleMATHGoogle Scholar
- Chanillo S: A note on commutators. Indiana University Mathematics Journal 1982, 31(1):7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleMATHGoogle Scholar
- Mizuhara T: Commutators of singular integral operators on Morrey spaces with general growth functions. Sūrikaisekikenkyūsho Kōkyūroku 1999, (1102):49–63. Proceedings of the Coference on Harmonic Analysis and Nonlinear Partial Differential Equations, Kyoto, Japan, 1998Google Scholar
- Komori Y, Mizuhara T: Factorization of functions in and generalized Morrey spaces. Mathematische Nachrichten 2006, 279(5–6):619–624. 10.1002/mana.200310381MathSciNetView ArticleMATHGoogle Scholar
- Stein EM: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30. Princeton University Press, Princeton, NJ, USA; 1970:xiv+290.Google Scholar
- Uchiyama A: On the compactness of operators of Hankel type. Tôhoku Mathematical Journal 1978, 30(1):163–171.MathSciNetView ArticleMATHGoogle Scholar
- Ding Y: A note on end properties of Marcinkiewicz integral. Journal of the Korean Mathematical Society 2005, 42(5):1087–1100.MathSciNetView 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.