Commutators of intrinsic square functions on generalized Morrey spaces
© Wu and Zheng; licensee Springer. 2014
Received: 25 September 2013
Accepted: 13 March 2014
Published: 28 March 2014
In this paper, we obtain the boundedness of intrinsic square functions and their commutators generated with BMO functions on generalized Morrey spaces. Our theorems extend some well-known results.
Keywordsintrinsic square functions commutators generalized Morrey spaces BMO functions
where . Denote .
We refer for details to .
In , Wilson proved the following result.
Theorem A Let , , then is bounded from to itself.
were first defined by Guliyev in . When , . It is the classical Morrey space which was first introduced by Morrey in . There are many papers discussed the conditions on to obtain the boundedness of operators on the generalized Morrey spaces. For example, in , the function φ is supposed to be a positively growth function and satisfy the double condition: for all , , where is a constant independent of r. This type of conditions on φ is studied by many authors; see, for example, [11, 12]. In , the following statement was proved by Nakai for the Calderón-Zygmund singular integral operators T.
where c does not depend on x and r. Then the operator T is bounded on for and from to for .
where c does not depend on x and t. Then the operator T is bounded from to for and from to for .
By an easy computation, we can check that if the pair satisfies double condition, then it will satisfy condition (1). Moreover, if satisfies condition (1), it will also satisfy condition (2). But the opposite is not true. We refer to  and Remark 4.7 in  for details.
Our main results in this paper are stated as follows.
Theorem 1.1 Let , , let satisfy condition (2), then is bounded from to .
Theorem 1.2 Let , , let satisfy condition (2), then for , we have is bounded from to .
Theorem 1.3 Let , , , let satisfy condition (3), then is bounded from to .
Theorem 1.4 Let , , , let satisfy condition (3), then for , is bounded from to .
In , the author proved that the functions and are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.
Corollary 1.5 Let , , let satisfy condition (2), then is bounded from to .
Corollary 1.6 Let , , , and let satisfy condition (3), then is bounded from to .
Throughout this paper, we use the notation to mean that there is a positive constant C (≥1) independent of all essential variables such that . Moreover, C maybe different from place to place.
2 Proofs of main theorems
Before proving the main theorems, we need the following lemmas.
Lemma 2.1 ()
where is the Hardy operator , .
- (1)For ,
- (2)Let , , then
Then, by an easy computation, we get Lemma 2.3.
By a similar argument as in , we can easily get the following lemma.
Lemma 2.4 Let , , then the commutators is bounded from to itself whenever .
Now we are in a position to prove the theorems.
Since , by (7), (8) and (12), we have the desired theorem. □
Using an argument similar to the above proofs and that of Theorem 1.2, we can also show the boundedness of . □
This work was completed with the support of Scientific Research Fund of Zhejiang Provincial Education Department No. Y201225707.
- Wilson M: The intrinsic square function. Rev. Mat. Iberoam. 2007, 23: 771-791.MATHMathSciNetView ArticleGoogle Scholar
- Wilson M Lecture Notes in Math. 1924. In Weighted Littlewood-Paley Theory and Exponential-Square Integrability. Springer, Berlin; 2007.Google Scholar
- Huang JZ, Liu Y: Some characterizations of weighted Hardy spaces. J. Math. Anal. Appl. 2010, 363: 121-127. 10.1016/j.jmaa.2009.07.054MATHMathSciNetView ArticleGoogle Scholar
- Wang H: Boundedness of intrinsic square functions on the weighted weak Hardy spaces. Integr. Equ. Oper. Theory 2013, 75: 135-149. 10.1007/s00020-012-2011-7MATHView ArticleGoogle Scholar
- Wang H, Liu HP: Weak type estimates of intrinsic square functions on the weighted Hardy spaces. Arch. Math. 2011, 97: 49-59. 10.1007/s00013-011-0264-zMATHView ArticleGoogle Scholar
- Wang H: Weak type estimates for intrinsic square functions on weighted Morrey spaces. Anal. Theory Appl. 2013,29(2):104-119.MATHMathSciNetGoogle Scholar
- Wang H: Intrinsic square functions on the weighted Morrey spaces. J. Math. Anal. Appl. 2012, 396: 302-314. 10.1016/j.jmaa.2012.06.021MATHMathSciNetView ArticleGoogle Scholar
- Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. ICM-90 Conference Proceedings. In Harmonic Analysis. Edited by: Lgari S. Springer, Tokyo; 1991:183-189.Google Scholar
- Guliyev VS, Aliyev SS, Karaman T, Shukurov PS: Boundedness of sublinear operators and commutators on generalized Morrey spaces. Integr. Equ. Oper. Theory 2011, 71: 327-355. 10.1007/s00020-011-1904-1MATHMathSciNetView ArticleGoogle Scholar
- Morrey C: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 1938, 43: 126-166. 10.1090/S0002-9947-1938-1501936-8MathSciNetView ArticleGoogle Scholar
- Ding Y, Yang DC, Zhou Z:Boundedness of sublinear operators and commutators on . Yokohama Math. J. 1998, 46: 15-27.MATHMathSciNetGoogle Scholar
- Wang, H: Boundedness of intrinsic square functions on generalized Morrey spaces. arXiv:1103.1715v2Google Scholar
- Nakai E: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 1994, 166: 95-103. 10.1002/mana.19941660108MATHMathSciNetView ArticleGoogle Scholar
- Guliyev VS: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009., 2009: Article ID 503948Google Scholar
- Guliyev, VS: Integral operators on function spaces on the homogeneous groups and on domains in Rn, Doctor’s degree dissertation, Mat. Inst. Steklov, Moscow, 329 pp. (1994) (in Russian)Google Scholar
- Guliyev, VS: Function spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications, Cashioglu, Baku, 332 pp. (1999) (in Russian)Google Scholar
- Aliyev SS, Guliyev VS: Boundedness of the parametric Marcinkiewicz integral operator and its commutators on generalized Morrey spaces. Georgian Math. J. 2012, 19: 195-208.MathSciNetGoogle Scholar
- Carro M, Pick L, Soria J, Stepanow VD: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 2001,4(3):397-428.MATHMathSciNetGoogle Scholar
- John F, Nirenberg L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 1961, 14: 415-426. 10.1002/cpa.3160140317MATHMathSciNetView ArticleGoogle Scholar
- Ding Y, Lu SZ, Yabuta K: On commutators of Marcinkiewicz integrals with rough kernel. J. Math. Anal. Appl. 2002, 275: 60-68. 10.1016/S0022-247X(02)00230-5MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.