MO HUI-XIA,YU DONG-yAN AND SUI XIN

(School of Science,Beijing University of Posts and Telecommunications,Beijing,100876)

Communicated by Ji You-qing

Boundedness of Commutators Generated by Campanato-type Functions and Riesz Transforms Associated with Schr¨odinger Operators

MO HUI-XIA,YU DONG-yAN AND SUI XIN

(School of Science,Beijing University of Posts and Telecommunications,Beijing,100876)

Communicated by Ji You-qing

Let L=?Δ+V be a Schr¨odinger operator on Rn,n＞3,where Δ is the Laplacian on Rnand V?=0 is a nonnegative function satisfying the reverse H¨older's inequality.Let[b,T]be the commutator generated by the Campanatotype functionand the Riesz transform associated with Schr¨odinger operatorIn the paper,we establish the boundedness of[b,T]on Lebesgue spaces and Campanato-type spaces.

commutator,Campanato-type space,Riesz transform,Schr¨odinger operator

1 Introduction

Let L=?Δ+V be a Schr¨odinger operator on Rn,n＞3,where Δ is the Laplacian on Rnand V?=0 is a nonnegative locally integrable function.The problems related to the Schr¨odinger operators L have attracted much attention(see[1–3]for example).In particular, Fefferman[1],Shen[2]and Zhong[3]established some basic results about the fundamental solutions and the boundedness of Riesz transforms associated with the Schr¨odinger operator.

The commutators generated by the Riesz transform associated with Schr¨odinger operator and BMO functions or Lipschitz functions also attract much attention(see[4–7]for example).Chu[8]considered the boudedness of commutators generalized by the BMOLfunctionand the Riesz transformon Lebesgue spaces.And Jiang[9]investigates some

properties of the Riesz potentialon the Campanato-type spacesInspired by[4,6,8–9],in this paper we consider the boundedness of commutators generated by the Campanato-type functionand the Riesz transformon Lebesgue spaces and Campanato-type spaces.

Firstly,let us introduce some notations.A nonnegative locally Lq(Rn)integrable function V is said to belong to Bq(1＜q＜∞)if there exists a constant C=C(q,V)＞0 such that the reverse H¨older's inequality

holds for any ball B in Rn.

We also say a nonnegative function V∈B∞,if there exists a constant C＞0 such that

holds for any ball B in Rn.

By H¨older's inequality,we have Bq1?Bq2for q1＞q2＞1.One remarkable feature about the Bqclass is that if V ∈Bqfor some q＞1,then there exists an ε＞0 which depends only on n and the constant C in(1.1)such that V∈Bq+ε.It is also well known that if V∈Bq(q＞1),then V(x)dx is a doubling measure,namely,for any r＞0,x∈Rnand some constant C0,we have

∫

Definition 1.1[3]For x∈Rn,the function m(x,V)is defined by

Clearly,0＜m(x,V)＜1 for every x∈Rnand if r=m(x,V),then

Definition 1.3[8]Let f∈Lloc(Rn).Then the sharp maximal function associated with L=?Δ+V is defined by

Then the kernel K(x,y)of operatorsatisfies the following estimates:there exists a constant δ＞0 such that for any nonnegative integer i,

Hence,for 1＜p≤p0,there exists a constant C＞0 such that

Throughout this paper,C always remains to denote a positive constant that may vary at each occurrence but is independent of the essential variable.

2 Theorems and Lemmas

C＞0 and 1＜r＜p0such that

To prove the theorems,we need the following lemmas.

Lemma 2.1[8]Let 0＜p0＜∞,p0≤p＜∞ and δ＞0.If f satisfies the conditionthen there exists a constant C＞0 such that

Lemma 2.2[12]For 1≤γ＜∞and β＞0,let

Lemma 2.3 Let B=B(x,r)and 0＜r＜ρ(x).ThenHence

Proof.

Thus,

3 Proofs of Theorems 2.1–2.3

Proof of Theorem 2.1

Fix a ball B=B(x,r0).Let B?=2B=B(x,2r0)and denote f0=fχB?,f∞=f?f0for f∈Lloc(Rn).

Case I.When 0＜r0＜ρ(x),write

Taking

we get

Let us estimate every part.For E1,by H¨older's inequality,we obtain

For E3,recalling δ＞0,by(1.4),we get

For E31,by H¨older's inequality,we have

From Lemma 2.3,it follows that

Case II.When r0≥ρ(x),write[b,T]f=bT(f)?T(bf0)?T(bf∞).Then,

For F1,by H¨older's inequality

Taking p1,p′and r as in the estimate for E2,by H¨older's inequality,we have

For F3,by using(1.3)and i＞0,

Combining the estimates for E1,E2,E3,F1,F2and F3,we conclude that

This completes the proof of Theorem 2.1.

Proof of Theorem 2.2

Without loss of generality,we can assume that 1＜r＜p＜p0.Sinceis dense in Lp(Rn),applying Theorem 2.1,Lemmas 1.1,2.1 and 2.2,we derive the inequality as

follows:

The proof of Theorem 2.2 is completed.

Proof of Theorem 2.3

Case I.We first show that for all x∈Rn,0＜r0＜ρ(x)and B=B(x,r0),there exists a constant C＞0 such that

Applying(1.3),we have

From(1.4),it follows that

and

Case II.In the following,we show that for all x∈Rn,ρ(x)≤r0and B=B(x,r0), there exists a constant C＞0 such that

Take

For H1,by H¨older's inequality

Thus,combining the Cases I and II,we complete the proof of Theorem 2.3.

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A

1674-5647(2015)04-0289-09

10.13447/j.1674-5647.2015.04.01

Received date:Oct.31,2013.

The NSF(11161042,11471050)of China.

E-mail address:Huixmo@bupt.edu.cn(Mo H X).

2010 MR subject classification:42B20,42B30,42B35