MO Hui-xia,YU Dong-yan,SUI Xin

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

BOUNDEDNESS OF TOEPLITZ OPERATORS GENERATED BY THE CAMPANATO-TYPE FUNCTIONS AND RIESZ TRANSFORMS ASSOCIATED WITH SCHDINGER OPERATORS

MO Hui-xia,YU Dong-yan,SUI Xin

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

In the paper,we study the boundedness ofthe Toeplitz operators generated by the Campanato-type functions and Riesz transforms associated with the Schr¨odinge operators.Using the sharp maximal function estimate,we establish the boundedness of the Toeplitz operator Θbon the Lebesgue space,which extend the previous results about the comutators.

Commutator;Campanato-type functions;Riesz transform;Schr¨odinger operator

1 Introduction

Let L= ?△ +V be a Schr¨odinger operator on Rn(n ＞ 3),where △ is the Laplacian on Rnand V/=0 is a nonnegative locally integrable function.The problems about the Schr¨odinger operators L were wellstudied(see[1–3]for example).Especially,Fefferman[1], Shen[2]and Zhong[3]developed some basic results.

The commutators generated by the Riesz transforms associated with the Schr¨odinger operators and BMO functions or Lipschitz functions also attracted much attention(see[4–6] for example).Chu[7],consider the boundedness of commutators generalized by the BMOLfunctions and the Riesz transform ▽(?△ +V)?1/2on Lebesgue spaces.Mo et al.[8] established the boundedness of commutators generated by the Campanato-type functions and the Riesz transforms associated with Schr¨odinger operators.

First,let us introduce some notations.A nonnegative locally Lq(Rn)integrable function V is said to belong to Bq(1 ＜ q ＜ ∞)if there exists C=C(q,V) ＞ 0 such that the reverse H¨older’s inequalityholds for every ball B in Rn.

About the Toeplitz operator,there are some results.Mo et al.[9]established the boundedness of the Toeplitz operator generalized by the singular integral operator with nonsmooth kerneland the generalized fractional.Liu et al.[11]investigated the boundness of the Toeplitz operator related to the generalized fractionalintegraloperator.

The commutator[b,T](f)=bT(f)?T(bf)is a particular case ofthe Toeplitz operators. Inspired by[7–10],we willconsider the boundedness ofthe Toeplitz operators generated by the Campanato-type functions and Riesz transforms associated with Sch¨odinger operators.

Defi nition 1.1Let f ∈ Lloc(Rn),then the sharp maximal function associated with L= ?△ +V is defi ned by

where ρ is define by

Defi nition 1.2[12,13]Let L= ?△+V,p ∈ (0,∞)and β ∈ R.Afunction f ∈ Lploc(Rn) is said to be in ΛβL,p(Rn),if there exists a nonnegative constant C such that for all x ∈ Rnand 0 ＜ s ＜ ρ(x) ≤ r,then the kernel K(x,y)of operator ▽(?△ +V)?1/2satisfies the following estimates:there exists a constant δ＞ 0 such that for any nonnegative integrate i,

Hence, ▽(?△ +V)?1/2is boundedness on Lp(Rn)space for 1 ＜ p ≤ p0,where 1/p0= 1/q? 1/n.

Throughout this paper,the letter 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

Theorems 2.1Let V ∈ Bqsatisfy(1.2)for n/2 ≤ q ＜ n.Let 0 ≤ β ＜ 1,b ∈ ΛβL(Rn), 1 ＜ τ＜ ∞ and 1 ＜ s ＜ p0,where 1/p0=1/q ? 1/n.Suppose that Θ1f=0 for any f ∈ Lr(Rn)(1 ＜ r ＜ ∞),then there exists a constant C ＞ 0 such that

Theorems 2.2Let V ∈ Bqsatisfy(1.2)for n/2 ≤ q ＜ n and 1 ＜ p0＜ ∞ satisfy 1/p0=1/q ? 1/n.Suppose that Θ1f=0 for any f ∈ Lτ(Rn)(1 ＜ τ＜ ∞),0 ＜ β ＜ 1, and b ∈ ΛβL(Rn).Then for 1 ＜ r ＜ min{1/β,p0}and 1/p=1/r ? β,there exists a constant C ＞ 0,such that

Theorems 2.3Let V ∈ Bqsatisfy(1.2)for n/2 ≤ q ＜ n and 1 ＜ p0＜ ∞ satisfy 1/p0= 1/q ? 1/n.Suppose that Θ1f=0 for any f ∈ Lτ(Rn)(1 ＜ τ＜ ∞)and b ∈ B M OL(Rn), then for 1 ＜ p ＜ p0,there exists a constant C ＞ 0,such that

To prove the theorems,we need the following lemmas.

Lemma 2.1(see[7])Let 0 ＜ p0＜ ∞,p0≤ p ＜ ∞ and δ＞ 0.If f satisfies the condition M(|f|δ)1/δ∈ Lp0,then exists a constant C ＞ 0 such that

Lemma 2.2(see[14])For 1 ≤ γ ＜ ∞ and β ＞ 0,letSuppose that γ ＜ p ＜ n/β and 1/q=1/p ? β/n,then ‖Mγ,β(f)‖Lq≤ C‖f‖Lp.

Remark 2.1When β =0,we denote Mγ,β=Mr.And it is easy to see that Mris boundedness on Lp(Rn),for 1 ＜ r ＜ p.

Lemma 2.3(see[8])Let B=B(x,r)and 0 ＜ r ＜ ρ(x),then

3 Proofs of Theorems 2.1–2.2

First,let us prove Theorem 2.1.

Fix a ball B=B(x,r0)and let 2B=B(x,2r0).We need only to estimate

Case IWhen 0 ＜ r0＜ ρ(x),using the condition Θ1f=0,then we have

Let τand s be as in Theorem 2.1.Then using H¨older’s inequality and the Lsboundedness of Tj,1(Lemma 1.1),we have

Let’s estimate I2.From(1.4),it follows that

For H1,since δ＞ 0,by H¨older’s inequality,we have

From Lemma 2.3 and H¨older’s inequality,it follows that

Thus

So when Tj,1= ▽(?△ +V)?1/2,we conclude that

If Tj,1= ±I,it is obvious that

Thus using the above formula and H¨older’s inequality,we conclude

Thus for 0 ＜ r0＜ ρ(x),we conclude that

Case IIWhen r0＞ ρ(x),we have

If Tj,1= ▽(?△ +V)?1/2,then for 1 ＜ τ ＜ ∞ and 1 ＜ s ＜ p0are as in Theorem 2.1, we have

Thus

If Tj,1= ±I,then by H¨older’s inequality,we obtain

And

Thus for r0＞ ρ(x),

So whenever 0 ＜ r0＜ ρ(x)or r0＞ ρ(x),we have

Now,let us turn to prove Theorem 2.2.

Let s,τbe as in Theorem 2.1 and satisfy 1 ＜ sτ＜ p.Then applying Theorem 2.1, Lemma 2.1 and Lemma 2.2,we know that

Thus we complete the proof of Theorems 2.1–2.1.

4 Proof of Theorem 2.3

It is obvious that Λ0L=BMOL.Thus from the proof of Theorem 2.1,we have

Since Msτis boundedness on Lp(Rn),then

Therefore,we complete the proof of Theorem 2.3.

References

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由Campanato 型函數(shù)和與薛定諤算子相關(guān)的Riesz變換生成的Toeplitz算子的有界性

默會(huì)霞,余東艷,隋 鑫
(北京郵電大學(xué)理學(xué)院,北京 100876)

本文研究了由 Campanato 型函數(shù)及與 Schr¨odinger 算子相關(guān)的 Riesz 變換生成的 Toeplitz 算子的有界性. 利用 Sharp 極大函數(shù)估計(jì)得到了 Toeplitz 算子 Θb在 Lebesgue空間的有界性, 拓廣了已有交換子的結(jié)果.

交換子;Campanato 型函數(shù);Riesz 變換;Schr¨odinger 算子

:42B20;42B30;42B35

O174.2

tion:42B20;42B30;42B35

A < class="emphasis_bold">Article ID:0255-7797(2017)02-0239-08

0255-7797(2017)02-0239-08

?Received date:2014-09-28 Accepted date:2015-02-28

Foundation item:Supported by National Natural Science Foundation of China(11161042; 11471050;11601035).

Biography:Mo Huixia(1976–),female,born at Shijiazhung,Hebei,vice professor,major in harmonic analysis.