- Open Access
Higher order commutators of Riesz transforms related to Schrödinger operators
© Wang and Liu; licensee Springer. 2014
- Received: 8 September 2014
- Accepted: 12 November 2014
- Published: 26 November 2014
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.
- Schrödinger operators
- Riesz transform
where is the kernel of ℛ and .
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 .
holds for every ball . It is known that implies for some . Therefore, under the assumption , we may conclude .
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.  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.
if , .
The space admits the following atomic decomposition (cf. ).
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. ). In what follows, we state our main results in this paper.
By duality, we immediately have the following theorem.
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.
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.  or ). In the end, we recall some propositions and lemmas for the spaces in .
holds for every ball and .
for all .
In particular, if .
for all , with and , where and is the constant appearing in (4).
for all with .
- (i)for every N, there exists a constant such that(8)
- (ii)for every N, there exists a constant such that(9)
for some , whenever .
- (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)
When , the term involving V can be dropped from inequalities (10), (11) and (12).
Proposition 2 (cf. Theorem 0.5 in )
is bounded on for ;
ℛ is bounded on for ,
Proposition 3 (cf. Theorem 1 in )
is bounded on for ;
is bounded on for ,
A ball is called critical. In , Dziubański and Zienkiewicz gave the following covering lemma on .
- (ii)There exists such that for every ,
Lemma 11 (Fefferman-Stein type inequality, cf. Lemma 2 in )
for all .
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 .
Let and with , then we have to deal with the average on Q of each term.
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.
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 , 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 . 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 .
where we use the finite overlapping property given by Lemma 10, the assumption on and the boundedness of in for .
Therefore, we need to control the mean oscillation on B of each term that we call , .
since the integral is clearly bounded by the left-hand side of (13).
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 . Suppose that Theorem 3 holds when . In what follows, we will prove that it is valid for .
if we choose N large enough.
This completes the proof of Theorem 3. □
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).
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