- Research
- Open access
- Published:
Higher order commutators of Riesz transforms related to Schrödinger operators
Journal of Inequalities and Applications volume 2014, Article number: 466 (2014)
Abstract
Let be a Schrödinger operator on , , where the nonnegative potential V belongs to the reverse Hölder class for . Suppose that b belongs to a new space which is larger than the classical space. We obtain the boundedness of higher order commutators defined by , , , and T is any of the Riesz transforms or their conjugates associated to the Schrödinger operator . The range of p is related to the index q. Moreover, we prove that is bounded from the Hardy space into the space when T is the Riesz transform associated to the Schrödinger operator.
MSC:42B20, 35J10.
1 Introduction
Let be a Schrödinger operator on , , where is a nonnegative potential belonging to a reverse Hölder class for . Let , which is larger than the space . In this paper, we consider the Riesz transforms associated with the Schrödinger operator L defined by and the higher order commutator
where is the kernel of ℛ and .
We also consider its dual transforms associated with the Schrödinger operator L defined by and the higher order commutator
where is the kernel of and .
The commutators of singular integral operators have always been one of the hottest problems in harmonic analysis. Recently, some scholars have extended these results to the case of higher order commutators. Please refer to [1–6] and so on. Furthermore, the commutators of singular integral operators related to Schrödinger operators have been brought to many scholars’ attention. See, for example, [7–18] and the references therein. Motivated by the references, in this paper we aim to investigate the estimates and endpoint estimates for when .
Note that a nonnegative locally integrable function () on is said to belong to if there exists a constant such that
holds for every ball . It is known that implies for some . Therefore, under the assumption , we may conclude .
We introduce the auxiliary function ρ defined as, for ,
The class of locally integrable functions b is defined as follows:
for all and , where and . A norm for denoted by is given by the infimum of the constants satisfying (2) after identifying functions that differ upon a constant. Denote that . It is easy to see that for . Bongioanni et al. [8] gave some examples to clarify that the space is a subspace of .
Because and , the Schrödinger operator L generates a contraction semigroup . The maximal function associated with is defined by . The Hardy space associated with the Schrödinger operator L is defined as follows in terms of the maximal function mentioned.
Definition 1 A function is said to be in if the maximal function belongs to . The norm of such a function is defined by
Definition 2 Let . A measurable function a is called a -atom associated to the ball if and the following conditions hold:
-
(1)
;
-
(2)
;
-
(3)
if , .
The space admits the following atomic decomposition (cf. [19]).
Proposition 1 Let . Then if and only if f can be written as , where are -atoms and . Moreover,
where the infimum is taken over all atomic decompositions of f into -atoms.
Before stating the main theorems, we introduce the definition of the reverse Hölder index of V as (cf. [8]). In what follows, we state our main results in this paper.
Theorem 1 Let , and such that , where is the reverse Hölder index of V. If , then
where .
By duality, we immediately have the following theorem.
Theorem 2 Let , and such that , where is the reverse Hölder index of V. If , then
where .
Theorem 3 Suppose that for some . Let . Then, for any , we have
Namely, the commutator is bounded from into .
The proofs of Theorems 1 and 2 can be given by iterating m times starting from Lemmas 12 and 13. Please refer to Section 3 for details.
Throughout this paper, unless otherwise indicated, we always assume that for some . We will use C to denote a positive constant, which is not necessarily the same at each occurrence. By and , we mean that there exist some positive constants C, such that and , respectively.
2 Some lemmas
In this section, we collect some known results about the auxiliary function and some necessary estimates for the kernel of the Riesz transform in the paper (cf. [20] or [7]). In the end, we recall some propositions and lemmas for the spaces in [8].
Lemma 1 for some implies that V satisfies the doubling condition; that is, there exists a constant such that
Especially, there exist constants and C such that
holds for every ball and .
Lemma 2 Let . For the auxiliary function ρ, there exist and such that
for all .
In particular, if .
Lemma 3 If , then there exists such that
Moreover, if , then there exists such that
Lemma 4 For ,
It is easy to see that
Lemma 5 There exist constants and such that
Lemma 6 Let and . If , then
for all , with and , where and is the constant appearing in (4).
Lemma 7 Let , and . Then
for all with .
Lemma 8 If for , then we have the following:
-
(i)
for every N, there exists a constant such that
(8)and
-
(ii)
for every N, there exists a constant such that
(9)for some , whenever .
Lemma 9 If , then we have the following:
-
(i)
For every N, there exists a constant such that
(10)Moreover, the last inequality also holds with replaced by .
-
(ii)
For every N, there exists a constant such that
(11)whenever . Moreover, the last inequality also holds with replaced by .
-
(iii)
If denotes the vector-valued kernel of the adjoint of the classical Riesz operator, then for some ,
(12)whenever .
-
(iv)
When , the term involving V can be dropped from inequalities (10), (11) and (12).
Proposition 2 (cf. Theorem 0.5 in [20])
Suppose that for some , then
-
(i)
is bounded on for ;
-
(ii)
ℛ is bounded on for ,
where .
Proposition 3 (cf. Theorem 1 in [8])
Suppose that for some and , then
-
(i)
is bounded on for ;
-
(ii)
is bounded on for ,
where .
A ball is called critical. In [19], Dziubański and Zienkiewicz gave the following covering lemma on .
Lemma 10 There exists a sequence of points in such that the family of critical balls , , satisfies the following:
-
(i)
.
-
(ii)
There exists such that for every ,
Given that , we define the following maximal functions for and :
where .
Also, given a ball , for and , we define
where .
Lemma 11 (Fefferman-Stein type inequality, cf. Lemma 2 in [8])
For , there exist β and γ such that if is a sequence of balls as in Lemma 10, then
for all .
3 Proofs of the main results
Firstly, in order to prove the main theorems, we need the following lemmas. As usual, for , we denote by the p-maximal function which is defined as
Lemma 12 Let for some , , and . Then, for any , there exists a constant such that
for all and every ball .
Proof We only consider the case of because the proof of the case of can be easily deduced from that of the case of .
Following (4.5) in [21], we expand , where λ is an arbitrary constant, as follows:
Let and with , then we have to deal with the average on Q of each term.
Firstly, by the Hölder inequality with and Lemma 7,
As for , we split . Choosing and denoting and , using the boundedness of on and the Hölder inequality, we obtain
where in the last inequality we have used Lemma 6 for the remaining term. We firstly note the fact that and . Then we have to deal with
where
For , we have
where we have used the following inequality:
by using Lemma 7, and we choose N large enough. As for ,
Using the Hölder inequality and the boundedness of the fractional integral with , we obtain
Since ,
And when and , we also have
Choosing N large enough, we get
Therefore, this completes the proof. □
Remark 1 It is easy to check that if the critical ball Q is replaced by 2Q, the last lemma also holds.
Lemma 13 Let and . Then, for any and , there exists a constant such that
for all f and with . Additionally, if , the above estimate also holds for instead of .
Because the proof of this lemma is very similar to that of Lemma 6 in [8], we omit the details.
Proofs of Theorem 1 and Theorem 2 We will prove Theorem 1 via the mathematical induction and Theorem 2 follows by duality. When , we conclude that Theorem 1 is valid by Theorem 1 in [8]. Suppose that the boundedness of holds when , where . In what follows, we will prove that it is valid for .
We start with a function for , and we notice that due to Lemma 12 we have .
By using Lemma 11, Lemma 12 with and Remark 1, we have
where we use the finite overlapping property given by Lemma 10, the assumption on and the boundedness of in for .
Next, we consider the term . Our goal is to find a pointwise estimate of . Let and with such that . If with , then we write
Therefore, we need to control the mean oscillation on B of each term that we call , .
Let , by using the Hölder inequality and Lemma 6, we obtain
since .
As for , let . We split again . Choose and denote and . Using the boundedness of on and the Hölder inequality, we then get
For , by Lemma 13, we obtain
since the integral is clearly bounded by the left-hand side of (13).
Therefore, we have proved that
By the assumption on and the boundedness of , we obtain the desired result. □
Proof of Theorem 3 We will prove Theorem 3 using the mathematical induction. When , we conclude that Theorem 3 is valid by Theorem 5 in [17]. Suppose that Theorem 3 holds when . In what follows, we will prove that it is valid for .
Similarly, we only consider the case of . For , we can write , where each is a atom and . Suppose that with . Write
For the term , by Lemma 6 and Theorem 1, we obtain
since .
Secondly, we consider the term . It is easy to see that and
Note that . By the Hölder inequality and inequality (11), we obtain, for some ,
where . Via the above estimate, we have
if we choose N large enough.
Thirdly, we consider the term . Via the Hölder inequality and (12), we get, for some ,
where .
Similarly, via the above estimate and the vanishing moment of , we have
Thus, we have
Moreover, note that
where .
By the weak boundedness of ℛ, we get
Therefore,
This completes the proof of Theorem 3. □
References
Segovia C, Torrea JL: Higher order commutators for vector-valued Calderón-Zygmund operators. Trans. Am. Math. Soc. 1993,336(2):537-556.
Ding Y, Lu SZ: Higher order commutators for a class of rough operators. Ark. Mat. 1999, 37: 33-44. 10.1007/BF02384827
Ding Y, Lu SZ, Zhang P: Continuity of higher order commutators on certain Hardy spaces. Acta Math. Sin. Engl. Ser. 2002,18(2):391-404. 10.1007/s101140200160
Ding Y, Lu SZ:Weighted boundedness for higher order commutators of oscillatory singular integrals. Tohoku Math. J. 1996, 48: 437-449. 10.2748/tmj/1178225342
Pérez C: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 1995,128(1):163-185. 10.1006/jfan.1995.1027
Liu HP, Tang L: Compactness for higher order commutators of oscillatory singular integral operators. Int. J. Math. 2009,20(9):1137-1146. 10.1142/S0129167X09005698
Guo Z, Li P, Peng L: Boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 2008,341(1):421-432. 10.1016/j.jmaa.2007.05.024
Bongioanni B, Harboure E, Salinas O: Commutators of Riesz transforms related to Schrödinger operators. J. Fourier Anal. Appl. 2011,17(1):115-134. 10.1007/s00041-010-9133-6
Bongioanni B, Harboure E, Salinas O: Weighted inequalities for commutators of Schrödinger Riesz transforms. J. Math. Anal. Appl. 2012, 392: 6-22. 10.1016/j.jmaa.2012.02.008
Li PT, Peng LZ: Endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators. Bull. Aust. Math. Soc. 2010,82(3):367-389. 10.1017/S0004972710000390
Li PT, Peng LZ: Compact commutators of Riesz transforms associated to Schrödinger operator. Pure Appl. Math. Q. 2012,8(3):713-739. 10.4310/PAMQ.2012.v8.n3.a7
Li PT, Peng LZ: Boundedness of commutator operator associated with Schrödinger operators on Heisenberg group. Acta Math. Sci. 2012,32(2):568-578. 10.1016/S0252-9602(12)60039-3
Liu Y, Wang L, Dong J: Commutators of higher order Riesz transform associated with Schrödinger operators. J. Funct. Spaces Appl. 2013. Article ID 842375, 2013:
Liu Y:Commutators of functions and degenerate Schrödinger operators with certain nonnegative potentials. Monatshefte Math. 2012, 165: 41-56. 10.1007/s00605-010-0228-6
Liu Y, Huang JZ, Dong JF: Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators. Sci. China Math. 2013,56(9):1895-1913. 10.1007/s11425-012-4551-3
Liu Y, Sheng JL: Some estimates for commutators of Riesz transforms associated with Schrödinger operators. J. Math. Anal. Appl. 2014, 419: 298-328. 10.1016/j.jmaa.2014.04.053
Liu Y, Sheng JL, Wang LJ: Weighted endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators. Abstr. Appl. Anal. 2013. Article ID 281562, 2013:
Yu WX, Wang XY: Weighted sharp function inequalities and boundedness for commutator of Riesz transforms of Schrödinger operator. Integral Transforms Spec. Funct. 2014,25(10):765-776. 10.1080/10652469.2014.918612
Dziubański J, Zienkiewicz J:Hardy spaces associated to Schrödinger operators with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 1999,15(2):279-296.
Shen ZW: Estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 1995, 45: 513-546. 10.5802/aif.1463
Garcia-Cuerva J, Harboure E, Segovia C, et al.: Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J. 1991,40(4):1397-1420. 10.1512/iumj.1991.40.40063
Acknowledgements
The second author would like to thank Prof. Jie Xiao and the Department of Mathematics and Statistics of Memorial University of Newfoundland for their hospitality. This work is supported by the National Natural Science Foundation of China (Nos. 10901018, 11471018), the Fundamental Research Funds for the Central Universities (No. FRF-TP-14-005C1), Program for New Century Excellent Talents in University and the Beijing Natural Science Foundation under Grant (No. 1142005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wang, Y., Liu, Y. Higher order commutators of Riesz transforms related to Schrödinger operators. J Inequal Appl 2014, 466 (2014). https://doi.org/10.1186/1029-242X-2014-466
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-466