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Commutators of intrinsic square functions on generalized Morrey spaces
Journal of Inequalities and Applications volume 2014, Article number: 128 (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.
For and , set
where . Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula
where . Denote .
This function is independent of any particular kernel, such as Poisson kernel. It dominates pointwise the classical square function (Lusin area integral) and its real-variable generalizations. Although the function depends on the kernels with uniform compact support, there is a pointwise relation between with different β ():
We refer for details to .
The intrinsic Littlewood-Paley g-function and the intrinsic -function are defined, respectively, by
In , Wilson proved the following result.
Theorem A Let , , then is bounded from to itself.
After that, Huang and Liu  studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In  and , Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In  and , Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let b be a locally integrable function on . Setting
the commutators are defined by
A function is said to be in if
In this paper, we will consider , , and their commutators on generalized Morrey spaces. Let be a positive measurable function on . For any , we denote by the generalized Morrey spaces, if
In , Mizuhara introduced these generalized Morrey spaces and discussed the boundedness of the Calderón-Zygmund singular integral operators. Note that the generalized Morrey spaces with normalized norm
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.
Theorem B Let and let satisfy the conditions
whenever , where c (≥1) does not depend on and
where c does not depend on x and r. Then the operator T is bounded on for and from to for .
Theorem C Let and let the pair satisfy the condition
where c does not depend on x and t. Then the operator T is bounded from to for and from to for .
Recently, in  and , Guliyev et al. introduced a weaker condition for the boundedness of Calderón-Zygmund singular integral operators from to : If , for any and , there exists a constant , such that
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.
In this paper, we will obtain the boundedness of the intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic function and their commutators on generalized Morrey spaces when the pair satisfies condition (2) or the following inequality:
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 ()
The inequality holds for all non-negative and non-increasing g on if and only if
where is the Hardy operator , .
Lemma 2.2 ()
Let , , then
Lemma 2.3 For , denote
Let , , then we have
From , we know that
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.
Proof of Theorem 1.1 The main ideas of these proofs come from . We decompose , where , , . Then
First, let us estimate I. By Theorem A, we obtain
Then let us estimate II. Recalling the properties of function ϕ, we know that
Since , and , we have
So, we obtain
By Minkowski’s inequality and , we have
The last inequality is due to Hölder’s inequality. Thus,
By combining (5) and (6), we have
So, let ; we have
Take , . Since satisfies condition (2), we can verify that , satisfy condition (4). Let . Obviously, it is decreasing on variable s. So, by Lemma 2.1, we can conclude the following estimates:
Proof of Theorem 1.2
First, let us estimate III:
Then let us estimate IV:
By Theorem 1.1, we have
To complete the proof, it suffices to estimate . Take , , . Then
For the first part, by Lemma 2.3, we obtain
For the other part, we know
Since , by Minkowski’s inequality, we get
For , we have . So by Fubini’s theorem and Hölder’s inequality, we obtain
Combining by (9), (10), and (11), we have
Thus, by substitution of variables and Lemma 2.1, we get
Since , by (7), (8) and (12), we have the desired theorem. □
Proof of Theorem 1.3 We decompose as in the proof of Theorem 1.2, where and . Then
By Lemma 2.4, we have
Next, we estimate the second part. We divide it into two parts. We have
First, for V, we find that
Following the proof in Theorem 1.1, we get
For VI, since , we get . Thus, by Minkowski’s inequality, we obtain
Since , by Fubini’s theorem, we get
For A, using Lemma 2.2 and Hölder’s inequality, we have
For B, we denote . Then, by Hölder’s inequality and Lemma 2.2, we get
This yields . Thus,
By a change of variables, we obtain
Let , . Since satisfies condition (3), by a similarly argument with Theorem 1.1, we conclude the following estimates:
Using an argument similar to the above proofs and that of Theorem 1.2, we can also show the boundedness of . □
Wilson M: The intrinsic square function. Rev. Mat. Iberoam. 2007, 23: 771-791.
Wilson M Lecture Notes in Math. 1924. In Weighted Littlewood-Paley Theory and Exponential-Square Integrability. Springer, Berlin; 2007.
Huang JZ, Liu Y: Some characterizations of weighted Hardy spaces. J. Math. Anal. Appl. 2010, 363: 121-127. 10.1016/j.jmaa.2009.07.054
Wang H: Boundedness of intrinsic square functions on the weighted weak Hardy spaces. Integr. Equ. Oper. Theory 2013, 75: 135-149. 10.1007/s00020-012-2011-7
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-z
Wang H: Weak type estimates for intrinsic square functions on weighted Morrey spaces. Anal. Theory Appl. 2013,29(2):104-119.
Wang H: Intrinsic square functions on the weighted Morrey spaces. J. Math. Anal. Appl. 2012, 396: 302-314. 10.1016/j.jmaa.2012.06.021
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.
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/s00020-011-1904-1
Morrey C: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 1938, 43: 126-166. 10.1090/S0002-9947-1938-1501936-8
Ding Y, Yang DC, Zhou Z:Boundedness of sublinear operators and commutators on . Yokohama Math. J. 1998, 46: 15-27.
Wang, H: Boundedness of intrinsic square functions on generalized Morrey spaces. arXiv:1103.1715v2
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.19941660108
Guliyev VS: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009., 2009: Article ID 503948
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)
Guliyev, VS: Function spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications, Cashioglu, Baku, 332 pp. (1999) (in Russian)
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.
Carro M, Pick L, Soria J, Stepanow VD: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 2001,4(3):397-428.
John F, Nirenberg L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 1961, 14: 415-426. 10.1002/cpa.3160140317
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)00230-5
This work was completed with the support of Scientific Research Fund of Zhejiang Provincial Education Department No. Y201225707.
The author declares that they have no competing interests.
All authors read and approved the final manuscript.
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Wu, X., Zheng, T. Commutators of intrinsic square functions on generalized Morrey spaces. J Inequal Appl 2014, 128 (2014). https://doi.org/10.1186/1029-242X-2014-128
- intrinsic square functions
- generalized Morrey spaces
- BMO functions