Commutators of Higher Order Riesz Transform Associated with Schrödinger Operators

Let L = −Δ + V be a Schrödinger operator on Rn (n ≥ 3), where V ̸ ≡ 0 is a nonnegative potential belonging to certain reverse Hölder class B s for s ≥ n/2. In this paper, we prove the boundedness of commutatorsR b f = bRHf −RH(bf) generated by the higher order Riesz transform RH = ∇2(−Δ + V)−1, where b ∈ BMO θ (ρ), which is larger than the space BMO(R). Moreover, we prove thatR b is bounded from the Hardy spaceH L (Rn) into weak L1weak(R n ).


Introduction
Let  = −Δ+ be a Schrödinger operator on R  ,  ≥ 3, where  ̸ ≡ 0 is a nonnegative potential belonging to the reverse Hölder class   for some  ≥ /2.In this paper, we will consider the higher order Riesz transforms associated with the Schrödinger operator  defined by R  ≐ ∇ 2  −1 and the commutator We also consider its dual higher order transforms associated with the Schrödinger operator  defined by R ≐  −1 ∇ 2 and the commutator where  ∈ BMO ∞ (), which is larger than the space BMO(R  ).
Because the investigation of commutators of singular integral operators plays an important role in Harmonic analysis and PDE, many authors concentrate on this topic.It is well known that Coifman et al. [1] proved that [, ] is a bounded operator on   for 1 <  < ∞ if and only if  ∈ BMO(R  ) when  is a Calderón-Zygmund operator.See [2,3] for the research development of the commutator   on Euclidean spaces R  and [4][5][6] on spaces of homogeneous type.
In recent years, singular integral operators related to Schrödinger operators and their commutators have been brought to many scholars attention.See, for example, [7][8][9][10][11][12][13][14][15][16][17][18][19] and their references.Especially, Guo et al. [12] investigated the boundedness of the commutators R   when  ∈ BMO(R  ).But their method is not valid to prove the boundedness of the commutators R   when  ∈ BMO ∞ ().In fact, since then R   may be written as follows: where  1 = ∇ 2 (−Δ) −1 and  2 =  − (−Δ + ) −1 .If  ∈ BMO(R  ), by using Corollary 1 in [12], we obtain the   boundedness of R   .But if  ∈ BMO  () and  ∉ BMO(R  ), it follows from [1] that [,  1 ] is not bounded on   , and then we cannot obtain the   boundedness of R   .Motivated by [12,15,17], our aim in this paper is to investigate the   estimates and endpoint estimates for R   when  ∈ BMO ∞ ().Different from the classical higher order Riesz transform, there exist some new problems for the higher order Riesz transform R  .We need to obtain some new estimates for R  when the potential  satisfies more stronger conditions.
A nonnegative locally   -integrable function  (1 <  < ∞) is called to belong to   if there exists a constant  > 0 such that the reverse Hölder inequality  ()  (6) holds for every (, ) in R  and 0 <  < ∞.
Obviously,   2 ⊂   1 , if  2 >  1 .But it is important that the   class has a property of "self-improvement"; that is, if  ∈   , then  ∈  + for some  > 0. Furthermore, it is easy to see that  ∞ ⊆   for any 1 <  < ∞.
The Hardy space  1  (R  ) associated with the Schrödinger operator  is defined as follows in terms of the maximal function mentioned earlier (cf.[20]).Definition 1.A function  ∈  1 (R  ) is said to be in  1   (R  ) if the semigroup maximal function    belongs to  1 (R  ).The norm of such a function is defined by We introduce the auxiliary function (, ) = () defined by It is known that 0 < () < ∞ for any  ∈ R  (from Lemma 8 in Section 2).Definition 2. Let 1 <  ≤ ∞.A measurable function  is called a (1, )  -atom associated to the ball (, ) if  < () and the following conditions hold: (i) supp  ⊂ (, ) for some  ∈ R  and  > 0, The space  1  (R  ) admits the following atomic decompositions (cf.[21]).Proposition 3. Let  ∈  1 (R  ).Then,  ∈ where the infimum is taken over all atomic decompositions of  into  1  -atoms.
Following [17], the class BMO  () of locally integrable function  is defined as follows: for all  ∈ R  and  > 0, where  > 0 and is given by the infimum of the constants satisfying (11), after identifying functions that differ upon a constant.If we let  = 0 in (11), then BMO  () is exactly the John-Nirenberg space BMO(R  ).Denote that BMO ∞ () = ⋃ >0 BMO  ().
It is easy to see that BMO(R  ) ⊂ BMO  () ⊂ BMO   () for 0 <  ≤   .Bongioanni et al. [17] gave some examples to clarify that the space BMO(R  ) is a subspace of BMO ∞ ().
Let  1 () be the auxiliary function of |∇()|.Our main results are given as follows.
Furthermore, we get the endpoint estimate for the commutator R   .
Then, for any  > 0, Namely, the commutator This paper is organized as follows.In Section 2, we collect some known facts about the auxiliary function () and some necessary estimates for the kernel of the higher order Riesz transform R  .In Section 3, we give the proof of Theorems 4 and 6.Section 4 gives the corresponding results when the potential  satisfies stronger conditions.In Section 5, we give some examples for the potentials  in Theorems 4 and 6.
Throughout this paper, unless otherwise indicated, we always assume that 0 ̸ ≡  ∈   for some  > .We will use  to denote the positive constants, which are not necessarily same at each occurrence even be different in the same line, and may depend on the dimension  and the constant in (5) or (6).By  ∼  and  ≲ , we mean that there exist some constants ,   such that 1/ ≤ / ≤  and  ≤   , respectively.

Some Lemmas
In this section, we collect some known results about auxiliary function () and some necessary estimates for the kernel of the higher order Riesz transform in the paper.Lemma 7.  ∈   ( ≥ /2) is a doubling measure; that is, there exists a constant  > 0 such that Especially, there exist constants  ≥ The previous facts had been obtained by Shen in [8].
We denote the fundamental solution of −Δ by Γ 0 (, ), which satisfies the following.
(i) There exists  > 0 such that     ∇ ()     . ( then, for  ∈ ( 0 , ), where we use Lemma 9 and (2) in Lemma 10 in the last step.Therefore, we complete the proof of the lemma.
Furthermore, we get the following corollary via the proof of Lemma 11.
Similarly, we have the following.
Remark 17.Following Remark 5 in [22], we know that if  is a nonnegative polynomial, condition (32) holds true.Therefore, Corollaries 15 and 16 also hold true.
(3) R is bounded on the space   (R  ) for   <  < ∞. Since , by using (1) in Lemma 18, we obtain the following.Lemma 19.Suppose that  ∈   for some  > .Then, for any 2.1.Some Lemmas Related to BMO Spaces   ().In this section, we recall some propositions and lemmas for the BMO spaces BMO  () in [17].
Given that  > 0, we define the following maximal functions for  ∈ for all  ∈  1  (R  ).

Proofs of the Main Results
Firstly, in order to prove Theorem 4, we need the following lemmas.As usual, for  ∈  1 loc (R  ), we denote by   the -maximal function which is defined as where we use the fact that R is bounded on   (R  ) with   <   1 <  < ∞.By Corollary 12 and the Hölder inequality, we have where () For  ∈ , note that () ∼ ( 0 ) by using Lemma 8. We also note that | − | ∼ | 0 − |.Then, ()      ) Using the Hölder inequality and the boundedness of the fractional integral I 1 with 1/ 1 = 1/  + 1/, we have where we use the assumption that ( 0 , |∇|) ≲ ( 0 , ) and (2) in Lemma 10.We also have Therefore, using the fact that / − /  1 = 1, we obtain where we choose  large enough such that the previous series converges and we use the fact that ( 0 ) ≲ 1.
Firstly, using the Hölder inequality and the boundedness of R on   (R  ), where where By the Hölder inequality and Lemma 22, we have ()      ) where 1/ + 1/  = 1, and we choose  large enough.The following estimate is similar to the estimate of  2 ().We repeat the previous method.Then, Using the Hölder inequality and the boundedness of the fractional integral I 1 with 1/ 1 = 1/ p + 1/, we have      ()      ) where p/ + p/] = 1.Therefore, using that / − /  1 = 1, we obtain where we choose  large enough such that the previous series converges and we use the fact that ( 0 ) ≲ 1.
Therefore, this completes the proof.
Remark 25.Similarly, we can conclude that the previous lemma also holds if the critical ball  is replaced by 2.
For  1 , by using the Hölder inequality and Lemma 22, we have where  0 is the least integer such that 2  0 ≥ (( 0 ))/.
To deal with  2 , using Lemma 22 and choosing  >   , we have where we use the fact that ( 0 )/2   ≤ 1/ when  >  0 .

Journal of Function Spaces and Applications
Furthermore, by using Lemma 7, where we use the fact that ( 0 , |∇|) ≲ ( 0 , ).Consider where we choose  large enough such that the previous series converges and we use the fact that ( 0 ) ≲ 1.
Proof of Theorem 4. We start with a function  ∈   (R  ) for where we choose  large enough. Similarly, where we choose  large enough and we use the fact that (  ) ≲ 1.