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

    2017-04-12 14:31:39MOHuixiaYUDongyanSUIXin
    數(shù)學(xué)雜志 2017年2期
    關(guān)鍵詞:交換子型函數(shù)北京郵電大學(xué)

    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

    [1]Feff erman C.The uncertainty principle[J].Bull.Amer.Math.Soc.,1983,9:129–206.

    [2]Shen Z.Lpestimates for Schr¨odinger operators with certain potentials[J].Ann.Inst.Fourier,1995, 45(2):513–546.

    [3]Zhong J.Harmonic analysis for some Schr¨odinger type operators[D].New Jersey:Princeton Univ., 1993.

    [4]Guo Z,Li P,Peng L.Lpboundedness of commutators of Riesz transforms associated to Schr¨odinger operator[J].J.Math.Anal.Appl.,2008,341:421–32.

    [5]Liu Y.Weighted Lpboundedness of commutators of Schr¨odinger type operators[J].Acta Math. Sinica(Chinese Series),2009,6:1091–1100.

    [6]Bongioanni B,Harboure E,Salinas O.Commutators of Riesz transforms related to Schr¨odinger operators[J].J.Fourier Anal.Appl.,2011,17:115–134.

    [7]Chu T.Boundedness of commutators associated with schr¨odinger operator and fourier multiplier[D]. Changsha:Hunan Univ.,2006.

    [8]Mo H,Zhou H,Yu D.Boundedness of commutators generated by the Campanato-type functions and Riesz transform associated with Schr¨odinger operators[J].Comm.Math.Res.,accepted.

    [9]Lu S,Mo H.Toeplitz type operators on Lebesgue spaces[J].Acta Math.Scientia,2009,29:140–150.

    [10]Zhang L,Zheng Q.Boundedness of commutators for singular integral operators with oscillating kernels on weighted Morrey spaces[J].J.Math.,2014,34(4):684–690.

    [11]Huang C,Liu L.Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator[J].Publ.Inst.Math.,2012,92(106):165–176.

    [12]Jiang Y.Some properties of Riesz potential associated with Schr¨odinger operators[J].Appl.Math. J.Chineses Univ.,2012,7(1):59–68.

    [13]Yang D,Zhou Y.Endpoint properties of localized Riesz transforms and fractional integrals associated to Schr¨odinger operators[J].Potential Anal.,2009,30:271–300.

    [14]Chanillo S.A note on commutators[J].Indiana Univ.Math.J.,1982,31:7–16.

    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.

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