Abstract
Let be a Schrödinger operator, where Δ is the Laplacian on and the nonnegative potential V belongs to the reverse Hölder class for . The Riesz transform associated with the operator L is denoted by and the dual Riesz transform is denoted by . In this paper, we establish the boundedness for the operator and its commutator on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class for , where and .
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1 Introduction
In this paper, we consider the Schrödinger operator
where is a nonnegative potential belonging to the reverse Hölder class for . The Riesz transform associated with the Schrödinger operator L is defined by and the commutator operator
where f is a suitable integral function. Also, the dual Riesz transform associated with the Schrödinger operator L is defined by and the commutator operator
First, Tang and Dong established the boundedness of some Schrödinger type operators on the Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [1]. Furthermore, Liu and Wang investigated the boundedness of the dual Riesz transforms and its commutators on the Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [2]. Recently, Pan and Tang established the boundedness of some Schrödinger type operators on weighted Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [3]. Motivated by [3], our aim is to establish the boundedness for the dual Riesz transform associated with Schrödinger operators and its commutators on weighted Morrey spaces related to the certain nonnegative potentials, where the condition on the potential is weaker than that in [3]. Our result is a nontrivial generalization of the main results in [3].
A nonnegative locally integrable function on is said to belong to () if there exists such that the reverse Hölder inequality
holds for every ball B in .
It is important that the class has a property of ‘self improvement’; that is, if , then for some (see [4]).
We assume the potential for throughout the paper. We introduce the auxiliary function defined by
It is well known that for any (cf. Lemma 1 in Section 2).
A kind of new Morrey spaces is established by Tang and Dong in [1]. Furthermore, the weighted Morrey space is introduced by Pan and Tang in [3]. Let , , and . For and (), we say (weighted Morrey spaces related to the nonnegative potential V) provided that
where denotes a ball with centered at and radius r, and the weight functions (see Section 2).
Now we are in a position to give the main results in this paper.
Theorem 1 Suppose for , , , and . Then, for and ,
where C is independent of f.
Theorem 2 Suppose for , , , , and so that . Then, for and ,
where C is independent of f.
We will use C to denote a positive constant, which is not necessarily same at each occurrence and even is different in the same line, and may depend on the dimension n and the constant in (3). By , we mean that there exists a constant C such that .
2 Some lemmas
In this section, we collect some known results proved in [4] in order to prove the main results in this paper.
Lemma 1 There exist constants such that
In particular, if .
Lemma 2 (1) For ,
and
-
(2)
There exist and such that
Let be the kernel of T and be the kernel of .
Lemma 3 If for , then for every N there exists a constant such that
Moreover, the last inequality also holds with replaced by .
In this paper, we always write , where ; and r denote the center and radius of B, respectively.
A weight will always mean a nonnegative function which is locally integrable. As in [5], we say that a weight ω belongs to the class for , if there is a positive constant C such that for the whole ball
We also say that a nonnegative function ω satisfies the condition if there exists a positive constant C, for all balls B
where
Since , obviously, for , where denote the classical Muckenhoupt weights (see [6]). It follows from [7] that for . For convenience, we always assume that denotes , , and .
Lemma 4 ([7])
Let , then:
-
(i)
If , then .
-
(ii)
if and only if , where .
-
(iii)
If for , then there exists a constant such that for any
Lemma 5 ([8])
Let , . If , then there exist positive constants δ, η, and C such that
for all ball .
As a consequence of Lemma 5, we have the following result.
Corollary 1 ([8])
Let , . If , then there exist positive constants , η, and C such that
for any measurable subset E of a ball .
Bongioanni et al. [9] introduced a new space defined by
where and , , and .
In particularly, Bongioanni et al. [9] proved the following results for .
Proposition 1 Let and . If , then
holds for all , with and , where and is the constant appearing in Lemma 1.
Proposition 2 Let , , and . Then
for all , with and the constant is given as in Proposition 1.
Obviously, the classical BMO space is properly contained in ; for more examples please see [9]. For convenience, we let .
From Corollary 2.2 in [3], the following result holds true.
Corollary 2 If and , then there exist positive constants C and η such that for every ball , we have
where .
3 The proof of our main results
Proof of Theorem 1 Without loss of generality, we may assume that and . Pick any ball , and write
where . Hence, we have
By the boundedness of (see Theorem 3 in [5]), we obtain
Now, for and using Lemma 3, we have
where
and
Then
By the proof of Theorem 1.1 in [3], we have
Next we deal with . For , . We get
Let . By simple computation, . By the definition of ,
where and .
Using Hölder’s inequality, (10), and the boundedness of the fractional integral with , for , we have
For , using Lemma 2, we get
It is easy to check that . Furthermore, using Corollary 1, we have
Therefore, by (13),
where we choose N large enough so that the above series converges.
From (6)-(14), we obtain
Thus, Theorem 1 is proved. □
Proof of Theorem 2 During the proof of Theorem 2, we always denote . Without loss of generality, we may assume that , , and . Pick any ball , and write
where . Hence, we have
By the boundedness of (see Theorem 2 in [8]), we obtain
Set . Write . Then
By (8) in the proof of Theorem 1, we obtain
Let . By simple computation, . By Lemma 4, . Then
By Lemma 1 and Corollary 2, as well as Lemma 3, we have
where we choose N large enough so that the above series converges.
For , we assume due to Lemma 3. Then, since , we also have . Then
By (11) and (12) in the proof of Theorem 1, we obtain
if we choose N large enough.
Now, for and using Lemma 3, we have
where
and
Then,
Firstly, we consider . By Proposition 2 and (10), for , we have
Then we get
where we choose N large enough so that the above series converges.
For , then for . Using Lemma 2, we get
Let . We choose u such that and .
Let . By simple computation,
Finally, we deal with . Using Hölder’s inequality, (10), and the boundedness of the fractional integral , for , we have
Then
where we choose N large enough so that the above series converges.
From (15)-(27), we obtain
Thus, we complete the proof of Theorem 2. □
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Acknowledgements
The authors would like to thank the referee for carefully reading which made the presentation more readable. This paper was supported by the National Natural Science Foundation of China under grant No. 10901018, the Fundamental Research Funds for the Central Universities, Program for New Century Excellent Talents in University and Beijing Natural Science Foundation under grant No. 1142005.
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Liu, Y., Wang, L. Boundedness for Riesz transform associated with Schrödinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials. J Inequal Appl 2014, 194 (2014). https://doi.org/10.1186/1029-242X-2014-194
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DOI: https://doi.org/10.1186/1029-242X-2014-194