Littlewood-Paley多線性交換子在Hardy空間和Herz-Hardy的有界性

陳大釗

(邵陽學(xué)院 理學(xué)院，湖南 邵陽，422000)

Littlewood-Paley多線性交換子在Hardy空間和Herz-Hardy的有界性

陳大釗

(邵陽學(xué)院 理學(xué)院，湖南 邵陽，422000)

Littlewood-Paley算子;多線性交換子;BMO(Rn);Hardy空間;Herz型Hardy空間

(1)suppa∈B=B(x0,r);

(2)‖a‖L≤‖

定義2 令0

(2)非齊次Herz 空間定義為

定義3 令α∈R,1

定義4 令bj(j=1,…,m)為固定的局部可積函數(shù),0<δ0.ψ為滿足如下條件的函數(shù):

(1)∫Rnψ(x)dx=0;

y)f(y)dy,ψt(x)=t-n+δψ(x/t)當(dāng)t>0。

記Ft(f)=ψt*f。定義

此即為Littlewood-Paley算子(見文獻(xiàn)[15])。

1 定理和證明

對(duì)Ⅰ,取r,s>1,使得

對(duì)Ⅱ,記

λ=(λ1,…,λm),λi=(bi)B,1≤i≤m,

注意到當(dāng)

y∈B,x∈B(x0,2k+1l)B(x0,2kl),有

而且

定理2 令0

Ⅰ+Ⅱ。

對(duì)Ⅰ,令

故,當(dāng)m=1時(shí),

因此，

C‖b1‖BMO

C‖b1‖BMO

當(dāng)m>1時(shí),有

從而

Ⅰ=

≤

注 定理2對(duì)非齊次Herz空間也成立。

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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

湖南省教育廳科學(xué)研究一般項(xiàng)目(15C1236)

陳大釗(1979-)，男，湖南益陽人，副教授，碩士，從事調(diào)和分析方向的研究；E-mail:chendazhao27@sina.com

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