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

2017-08-27陈大钊

邵阳学院学报（自然科学版） 2017年4期

关键词：交换子特征函数邵阳

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

当m>1时,有

从而

Ⅰ=

≤

注定理2对非齐次Herz空间也成立。

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BoundednessformultilinearcommutatorofLittlewood-PaleyoperatoronHardyandHerz-Hardyspaces

CHEN Dazhao

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