Yueshan Wangand Yuexiang He

Department of Mathematics,Jiaozuo University,Jiaozuo 454003,Henan,China

Abstract.Let L=??+V be a Schr?dinger operator on Rn(n ≥3),where the nonnegative potential V belongs to reverse Ho?der class RHq1 for Let be the Hardy space associated with L.In this paper,we consider the commutator[b,Tα],which associated with the Riesz transform with 0<α ≤1,and a locally integrable function b belongs to the new Campanato space We establish the boundedness of[b,Tα]from Lp(Rn)to Lq(Rn)for 1

Key Words:Riesz transform,Schr?dinger operator,commutator,Campanato space,Hardy space.

1 Introduction and results

Let L=??+V be a Schr?dinger operator on Rn,where n≥3.The function V is nonnegative,and belongs to a reverse H?lder class RHq1for some q1>n/2,that is to say,V satisfies the reverse H?lder 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)?αand its commutator have been obtained.Shen[1]established the Lp-boundedness of T1and T1/2,Liu and Tang[2]showed that T1and T1/2are bounded onforFor 0<α≤1,Sugano[3]studied the Lp-boundedness and Hu and Wang[4]obtained theboundedness.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 continuouslyinto weak L1(Rn).When b ∈BMOθ(ρ)and 0<α ≤1,the Lp-boundedness of[b,Tα]was investigated in[7]and the boundedness frominto 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 classLet us recall some concepts.

As in[1],for a given potential V∈RHq1with 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 classconsists of the locally integrable functions b such that

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

We recall the Hardy space associated with Schr?dinger operator L,which had been studied by Dziubański and Zienkiewicz in[10,11].Becausethe Schr?dinger 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 ofif the maximal function MLf 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 ∈RHq1with q1>n/2,and letIf 0<α ≤1 andthen

We immediately deduce the following result by duality.

Corollary 1.1.Let V ∈RHq1with q1>n/2,and letIf 0<α≤1 and 1

where 1/q=1/p?β/n.

Theorem 1.2.Let V ∈RHq1with q1>n/2,and let 0<α≤1.Supposeand 0<β<δ'.Ifandthen the commutator[b,Tα]is bounded frominto

Theorem 1.3.Let V ∈RHq1with q1>n/2,and let 0<α ≤1.Suppose0<β<δ'.

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

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 thatif|x?y|

Lemma 2.1.Let k∈N andThen 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].

Proposition 2.2.There exists a sequence of pointsin Rn,so that the family of critical ballsk≥1,satisfies

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

where Bρ,α={B(z,r):z∈Rnand r≤αρ(y)}.

We have the following Fefferman-Stein type inequality.

Proposition 2.3(see[9]).For 1

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 k0is the constant appearing in Proposition 2.1.

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

Lemma 2.4(see[4,12]).Suppose V ∈RHq1with

(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

where|y?z|≤|x?y|/4.

Proposition 2.4(see[13]).Suppose that V ∈RHq1withLet 0<β2≤β1≤1,1

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

Corollary 2.1.Suppose that V ∈RHq1withLet 0<α≤1

3 The Lp-boundedness of[b,Tα]

To prove Theorem 1.1,we need the following Lemmas.

Lemma 3.1.Suppose V ∈RHq1withandIfthen for all f ∈and every critical ball Q=B(x0,ρ(x0)),we have

where

Proof.Since

then

where f=f1+f2with 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)cinto annuliwe get

Then

Thus,taking N>l0α we get

The estimate for I2can be proceeded in the same way of I1.The decomposition f=f1+f2gives

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

Let j0be the least integer such thatSplitting into annuli,we have

For K2,splitting into annuli,

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+f2with f1=f χ2B.

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

Select r0so thatthen by Ho?der’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),

(iii)if r<ρ(x0)/4,then

Proposition 4.1(see[11]).LetThenif and only if f can be written aswhere ajare-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 thatBy 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

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

Choosing s such that αq

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

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

Note that

Then

Then

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

Therefore

Note that

and

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

Thus,

Thus,we complete the proof of Theorem 1.3