Littlewood-Paley多线性交换子在Hardy空间和Herz-Hardy的有界性

陈大钊

(邵阳学院 理学院，湖南 邵阳，422000)

Littlewood-Paley多线性交换子在Hardy空间和Herz-Hardy的有界性

陈大钊

(邵阳学院 理学院，湖南 邵阳，422000)

Littlewood-Paley算子;多线性交换子;BMO(Rn);Hardy空间;Herz型Hardy空间

(1)suppa∈B=B(x0,r);

(2)‖a‖L≤‖

定义2 令0

(2)非齐次Herz 空间定义为

定义3 令α∈R,1

定义4 令bj(j=1,…,m)为固定的局部可积函数,0<δ0.ψ为满足如下条件的函数:

(1)∫Rnψ(x)dx=0;

y)f(y)dy,ψt(x)=t-n+δψ(x/t)当t>0。

记Ft(f)=ψt*f。定义

此即为Littlewood-Paley算子(见文献[15])。

1 定理和证明

对Ⅰ,取r,s>1,使得

对Ⅱ,记

λ=(λ1,…,λm),λi=(bi)B,1≤i≤m,

注意到当

y∈B,x∈B(x0,2k+1l)B(x0,2kl),有

而且

定理2 令0

Ⅰ+Ⅱ。

对Ⅰ,令

故,当m=1时,

因此，

C‖b1‖BMO

C‖b1‖BMO

当m>1时,有

从而

Ⅰ=

≤

注 定理2对非齐次Herz空间也成立。

[1]LIU Lanzhe.Weighted weak type(H1,L1)estimates for commutators of Littlewood-Paley operators[J].Indian J of Math,2003,45(1):71-78.

[2]LIU Lanzhe.WeightedBlock-Hardy spaces estimates for commutators of Littlewood -Paley operators[J].Southeast Asian Bull of Math,2004,27:833-838.

[3]LIU Lanzhe.Weighted weak type estimates for commutators of Littlewood-Paley operator [J].Japanese J of Math,2003,29(1):1-13.

[4]LIU Lanzhe ,LU Shanzhen.Boundedness for commutators of Littlewood-Paley operator[J].Adv in Math(China),2003,32:473-480.

[5]LU Shanzhen.The local versions of Hp(Rn)spaces at the origin[J].Studia Math,1995,116(1):147-158.

[6]LU Shanzhen.The decomposition of the weighted Herz spaces and its applications[J].Sci in China(ser A),1995(1),38:147-158.

[7]LU Shanzhen.The weighted Herz type Hardy spaces and its applications[J].Sci in China(ser A),1995,38(5):662-673.

[8]LU Shanzhen.The continuity of commutat ors on Herz-type space[J].Michigan Math J,1997,44:255-281.

[9]CHEN Dazhao.Weighted Boundedness for Toeplitz Type Operators Related to Strongly Singular Integral Operators[J].Journal of Inequalities and Applications,2014,2014(1):42.

[10]CHEN Dazhao.Mean Oscillation and Boundedness of Toeplitz Type Operator Related to Singular Integral Operator with Variable Calderón-Zygmund Kernel[J].Integral Transforms And Special Functions,2014,25(7):588-596.

[11]CHEN Dazhao.Mk-type sharp estimates and boundedness on Morrey space for Toeplitz type operators associated to fractional integral and singular integral operator with general kernel[J].Journal of Pseudo-Differential Operators and Applications:2015,6(3):1-14.

[12]CHEN Dazhao.Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space[J].Open Mathematics,2015,13(1):678-690.

[13]LIU Lanzhe Sharp weighted estimates for multilinear commutators[J].J London Math Soc,2002,65(5):672-692

[14]STEIN EM.Harmonic analysis:real variable methods,orthogonality and oscillatory integrals[M].NJ Princeton:Princeton Univ Press,1993.

[15]Torchinsky A.Real variable methods in harmonic analysis[M].New York:Academic Press,1986.

BoundednessformultilinearcommutatorofLittlewood-PaleyoperatoronHardyandHerz-Hardyspaces

CHEN Dazhao

(School of Sciences,Shaoyang University，Shaoyang 422000，China)

Littlewood-Paleyoperator;multilinearcommutator;BMO(Rn);Hardy space;Herz-Hardyspace

1672-7010(2017)04-0019-06

2017-03-18

湖南省教育厅科学研究一般项目(15C1236)

陈大钊(1979-)，男，湖南益阳人，副教授，硕士，从事调和分析方向的研究；E-mail:chendazhao27@sina.com

O

A