1 Introduction and results

Let L=−∆+V be a Schrödinger operator on Rn,where n≥3.The function V is nonnegative,and belongs to a reverse Hölder class RHq1 for some q1>n/2,that is to say,V satisfies the reverse Hölder inequality

for all ball B⊂Rn.We consider the Riesz transform Tα=Vα(−∆+V)−α,where 0<α≤1.

Many results about Tα=Vα(−∆+V)−α and its commutator have been obtained.Shen[1]established the Lp-boundedness of T1 and T1/2,Liu and Tang[2]showed that T1 and T1/2 are bounded on forFor 0<α≤1,Sugano[3]studied the Lp-boundedness and Hu and Wang[4]obtained the boundedness.When b ∈BMO,Guo,Li and Peng[5]obtained the Lp-boundedness of commutators[b,T1]and[b,T1/2],Li and Peng in[6]proved that[b,T1]and[b,T1/2]map continuously into weak L1(Rn).When b ∈BMOθ(ρ)and 0<α ≤1,the Lp-boundedness of[b,Tα]was investigated in[7]and the boundedness from into weak L1(Rn)given in[4].

In this paper,we are interested in the boundedness of[b,Tα]when b belongs to the new Campanato class Let us recall some concepts.

As in[1],for a given potential V∈RHq1 with q1>n/2,we define the auxiliary function

It is well known that 0<ρ(x)<∞for any x∈Rn.

Let θ>0 and 0<β<1,in view of[8],the new Campanato class consists of the locally integrable functions b such that

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由表1可以知，SPAD值与叶绿素含量呈正相关性。以“小叶茼蒿”为例，根据表1，分别将叶绿素a、叶绿素b及总叶绿素含量按线性函数进行统计分析，如图1。

Note that if θ=0,is the classical Campanato space;If is exactly the space BMOθ(ρ)introduced in[9].

We recall the Hardy space associated with Schrödinger operator L,which had been studied by Dziubański and Zienkiewicz in[10,11].Becausethe Schrödinger operator L generates a(C0)contraction semigroupThe maximal function associated with{:s>0}is defined by .we always denote δ'=min{1,2−n/q1}.ForWe say that f is an element of if the maximal function ML f belongs to Lp(Rn).The quasi-norm of f is defined by

We now formulate our main results as follows.

Theorem 1.1.Let V ∈RHq1 with q1>n/2,and letIf 0<α ≤1 and then

where 1/q=1/p−β/n,and

We immediately deduce the following result by duality.

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Corollary 1.1.Let V ∈RHq1 with q1>n/2,and let If 0<α≤1 and 11/α,then

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where 1/q=1/p−β/n.

Theorem 1.2.Let V ∈RHq1 with q1>n/2,and let 0<α≤1.Suppose and 0<β<δ'.Ifand then the commutator[b,Tα]is bounded from into

Proposition 2.2.There exists a sequence of points in Rn,so that the family of critical balls k≥1,satisfies

Then the commutator[b,Tα]is bounded from into weak L1(Rn).

where|y−z|≤|x−y|/4.

2 Some preliminaries

We recall some important properties concerning the auxiliary function.

Proposition 2.1(see[1]).Let V ∈RHn/2.For the function ρ there exist C and k0 ≥1 such that

for all x,y∈Rn.

Assume that Q=B(x0,ρ(x0)),for any x∈Q,Proposition 2.1 tell us that if|x−y|

Lemma 2.1.Let k∈N and Then we have

Lemma 2.2(see[11]).Suppose V∈RHq1,q1≥n/2.Then there exists constants C>0 and l0>0 such that

The following finite overlapping property given by Dziubański and Zienkiewicz in[10].

Theorem 1.3.Let V ∈RHq1 with q1>n/2,and let 0<α ≤1.Suppose 0<β<δ'.

(i)

(ii)There exists N=N(ρ)such that for every k∈N,card

△ABC在由△A2B2C2转动到△A3B3C3过程中，外心T(x,y)与动点A (a,1)，B (0,b)之间有 x≤a，y≥b，且满足

For α>0,and x∈Rn,we introduce the following maximal functions

where Bρ,α={B(z,r):z∈Rn and r≤αρ(y)}.

for all x∈Rn and r>0.A seminorm of denoted byis given by the infimum of the constants in the inequalities above.

We have the following Fefferman-Stein type inequality.

By s inequality and Lemma 2.3 we have

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for all

We have an inequality for the function

Lemma 2.3(see[8]).Let 1≤s<∞,,and B=B(x,r).Then

for all k∈N,where θ'=(k0+1)θ and k0 is the constant appearing in Proposition 2.1.

Let Kα be the kernel of(−∆+V)−α.The following results give the estimates on the kernel Kα(x,y).

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Lemma 2.4(see[4,12]).Suppose V ∈RHq1 with

(i)For every N>0,there exists a constant C such that

(ii)For every 0<δ<δ' there exists a constant C such that for every N>0,we have

We shall use the symbol AB to indicate that there exists a universal positive constant C,independent of all important parameters,such that A ≤CB.A ≈B means that A B and BA.

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Proposition 2.4(see[13]).Suppose that V ∈RHq1 with Let 0<β2 ≤β1 ≤1,1<andThen

Let β1=β2=α,by Proposition 2.4 and duality we get

Corollary 2.1.Suppose that V ∈RHq1 withLet 0<α≤1

(i)Foris bounded on Lp(Rn);

(ii)For is bounded on Lp(Rn).

3 The Lp-boundedness of[b,Tα]

To prove Theorem 1.1,we need the following Lemmas.

Lemma 3.1.Suppose V ∈RHq1 with and If then for all f ∈and every critical ball Q=B(x0,ρ(x0)),we have

where

Proof.Since

then

Proposition 2.3(see[9]).For 1k}k is a sequence of balls as in Proposition 2.2,then

where f=f1+f2 with f1=f χ2Q.

By the Ls-boundedness of(Corollary 2.1),we have

By Lemma 2.4,

For any y∈Q and z∈(2Q)c,we haveandSo,decomposing(2Q)c into annuli we get

Sincewe can choose t1 such thatBy Hoïder’s inequality and Lemma 2.2,we get

Then

Thus,taking N>l0α we get

The estimate for I2 can be proceeded in the same way of I1.The decomposition f=f1+f2 gives

LetBy Hoïder’s inequality,L>-boundedness of and Lemma 2.3,for some u>1 we have

The estimate can be obtained by the similar approach to ones of I12 and I21.Then we omit the details here.

Lemma 3.2.Let B=B(x0,r)with r≤γρ(x0)and let x∈B,then for any y,z∈B we have

Proof.Setting Q=B(x0,γρ(x0)),due to the factandthen by Lemma 2.4 we get

where

and

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Let j0 be the least integer such that Splitting into annuli,we have

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Bywe have We choose t1 and v>1 such that Then by inequality and Lemma 2.3,we have

Notewe get

For K2,splitting into annuli,

Sincetaking N ≥θ'+l0,we get

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Thus,we complete the proof.

Lemma 3.3.Letlet B=B(x0,r)with r≤γρ(x0)and let x∈B.Then

Proof.Write

where f=f1+f2 with f1=f χ2B.

Since r≤γρ(x0)and ρ(x)≈ρ(x0),by Hölder’s inequality and Lemma 2.3,we get

Select r0 so thatthen by Hoïder’s inequality and Lemma 2.3,

By Lemma 3.2,

So,we complete the proof.

We now come to prove Theorem 1.1.By Proposition 2.3,Lemma 3.1 and Lemma 3.3 we have

where we have used the finite overlapping property given by Proposition 2.2.

4 The -boundedness of[b,Tα]

We have the following atomic characterization of Hardy space.

Definition 4.1.LetA function a ∈L2(Rn)is called an -atom if r<ρ(x0)and the following conditions hold:

(i)supp a⊂B(x0,r),

(ii)

(iii)if r<ρ(x0)/4,then

Proposition 4.1(see[11]).LetThen if and only if f can be written aswhere aj are -atoms,and the sum converges in thequasi-norm.Moreover

where the infimum is taken over all atomic decompositions of f into -atoms.

Let us prove Theorems 1.2.Choose τ such that By Proposition 4.1,we only need to show that for any -atom a,

holds,where C is a constant independent of a.

Suppose supp a⊂B=B(x0,r)with r<ρ(x0).Then

Let 1/t=1/τ−β/n.By Corollary 1.1 and the size condition of atom a,we have

For A2,we consider two case,that are r<ρ(x0)/4 and ρ(x0)/4≤r<ρ(x0).

Case I:When r<ρ(x0)/4,by the vanishing condition of a,we have

Note that

By we choose s so that αq1.Then by inequality,Lemma 2.3 and Lemma 2.2 we have

When and y ∈B,by Lemma 2.4 and Lemma 2.1,we can take 0<β<δ<δ'such that

Noticeandthen we get

For x∈2k+1B2kB,y∈B,we have|x−y|≈2kr.Then by Lemma 2.4 and Lemma 2.1,

Choosing s such that αq1,we get

By inequality and Lemma 2.3 we get

Then,by Minkowiski’s inequality and taking N>l0α(k0+1),we get

Case II:When ρ(x0)/4≤r<ρ(x0),this means r ≈ρ(x0).The atom a does not satisfy the vanishing condition.By Minkowiski’s inequality,

Note r≈ρ(x0),then by(4.1),(4.2)and(4.3)we get

The estimate of is exactly the same as A22,we omit the detail of the proof.

Proof of Theorem 1.3. Let we write where each aj is an-atomand

Suppose that suppaj ⊂Bj=B(xj,rj)with rj<ρ(xj).Write

Note that

Choose t such thatBy Hoïder’s inequality and Corollary 1.1 we get

Note we get

Then

Since we haveand by Lemma 2.1 we get

Sincewe select s such that α1.Then

Note and Then

Then

When and y∈Bj,by Lemma 2.4 and Lemma 2.1,we have

Thus,by the vanishing condition of aj and 0<β<δ<δ'we have

Therefore

Note that

and

By Corollary 2.1,we know that Tα is bounded from L1(Rn)to WL1(Rn),then

Thus,

Thus,we complete the proof of Theorem 1.3

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