吳　瑛，束立生，程美芳

(安徽師范大學(xué)數(shù)學(xué)計算機科學(xué)學(xué)院, 安徽 蕪湖 241003)

吳瑛，束立生，程美芳

(安徽師范大學(xué)數(shù)學(xué)計算機科學(xué)學(xué)院, 安徽 蕪湖 241003)

摘要：利用Aspan權(quán)性質(zhì)及分析中的不等式，得到 Bochner-Riesz 算子及由BMO(Rn)函數(shù)b(x)和生成的交換子在加權(quán)共合空間(Rn)上的有界性,其中1

關(guān)鍵詞：Bochner-Riesz算子；交換子；加權(quán)共合空間；Ap權(quán)

0引言

自1975年Holland[2]研究了共合空間(Lq,Lp)(Rn)的一些性質(zhì)后,共合空間受到了廣泛關(guān)注[3].1988年,Fofana[4]引入了空間(Lq,Lp)α(Rn).對于1≤q,p,α≤∞,定義,其中是一個伸縮變換.B(x0,r)表示以x0為中心,r為半徑的球.χB(x0,r)表示其特征函數(shù).|B(x0,r)|表示B(x0,r)的Lebesgue測度.共合空間(Lq,Lp)α(Rn)定義如下：(Lq,Lp)α(}.Fofana還證明了當(dāng)且僅當(dāng)q≤α≤p時,該空間是非平凡的.

文中出現(xiàn)的C表示與主要變量無關(guān)的正常數(shù),并且在不同的地方可能取值不同.

1定義和引理

對于一個給定的權(quán)函數(shù)ω(x),記B的Lebesgue測度為|B|以及B的加權(quán)測度為ω(B),其中ω(B)=∫Bω(x)dx.

定義1[5]設(shè)10,使得對每個球B?Rn,有,則稱ω(x)為一個Ap權(quán),記作ω∈Ap.

定義2[6]設(shè)s>1,若存在一個常數(shù)C>0,使得對每個球B?Rn,有，則稱ω(x)滿足反向不等式,記作ω∈RHs.

文中主要結(jié)論的證明,還需要用到以下引理.

引理3[10-11]設(shè)b(x)∈BMO(Rn),則對任意的1≤p<∞,有.

在證明定理之前,先指出下面兩個事實.

2定理的證明

由引理2知

(1)

則有

(2)

(3)

(4)

(5)

把式(5)代入式(4)，有

(6)

另一方面，

(7)

記P(y)=ω(y)1-q′,因為ω∈Aq則有P(y)∈Aq′.按照式(5)的推導(dǎo)過程有

(8)

(9)

則有

(10)

參考文獻:

[1] Bochner S. Summation of multiple Fourier series by spherical means[J].Proc Natl Acad Sci U S A,1935,21(6):353-355.

[2] Holland F. Harmonic analysis on amalgams ofLpandLq[J]. London Math Soc,1975,10(3):295-305.

[3] Fourier J J F，Stewart J. Amalgams ofLpandLq[J]. Bull Amer Math Soc,1985,13:1-21.

[4] Fofana I. étude d′une class d′espaces de fonctions contenant les espaces de lorentz[J]. Afrika Mat,1988，1(2):29-50.

[5] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function[J]. Trans Amer Math Soc,1972,165:207-226.

[6] Garcia-Cuerva J,Rubio de Francia J L. Weighted norm inequalities and related topics[M]. Amsterdam:North-Holland,1985.

[7] Wei X M,Tao S P.The boundedness of littlewood-paley operatoes with rough kernels on weighted(Lq,Lp)α(Rn) spaces[J]. Anal Theory Appl,2013,29(2):135-148.

[8] Gundy R F，Wheedem R L. Weighted integral inequalities for the nontangential maximal function，Lusin and Walsh-Paley series[J]. Studia Math,1974,49(2):107-124.

[9] Shi X L,Sun Q Y. Weighted norm inequalities for Bochner-Riesz operators and singular integral operators[J]. Pron Amer Math Soc,1992,116:665-673.

[10] Duoandikoetxea J. Fourier analysis[M]. Providence:American Mathematical Society,2000.

[11] John F,Nirenberg L. On functions of bounded mean oscillation[J]. Comm Pure Appl Math, 1961,14(3):415-426.

[12] Lu S Z,Wang K Y.Bochner-Riesz means[M]. Beijing:Publish House of Beijing Normal University，1988.

[13] Stein E M, Weiss G. Introduction to fourier analysis on euclidean spaces[M]. Priceton: Princeton University Press,1971.

[14] Alvarez J,Bagby R J,Kurtz D S,etal. Weighted estimates for commutators of linear operators[J]. Studia Math,1993,104(2):195-209.

WU Ying, SHU Lisheng, CHENG Meifang

(College of Mathematics and Computer Science,Anhui Normal University,Wuhu 241003,China)

Abstract:To use the nature of Aspanweight and inequality, this paper obtains the boundedness of Bochner-Riesz operators and the commutator formed by a BMO(Rn) function b(x) and ) on the weighted ,Lp)α(Rn) spaces, where 1

Key words:Bochner-Riesz operators; commutator; weighted amalgam space; Apweight

文章編號:1674-232X(2015)03-0308-05

中圖分類號:O174.2MSC2010: 34K13

文獻標(biāo)志碼:A

doi:10.3969/j.issn.1674-232X.2015.03.014