Littlewood-Paley算子的多線性交換子的Lipschitz估計

摘 要：在本文中，我們得到了由Littlewood-Paley算子和Lipschitz函數生成的多線性交換子在Triebel-Lizorkin空間、Hardy空間和Herz-Hardy空間的連續性．

關鍵詞：Littlewood-Paley算子；線性交換子；Triebel-Lizorkin空間；Herz-Hardy空間；Herz空間；Lipschitz空間

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中圖分類號：O174.3 文獻標識碼：A

Lipschitz Estimates for Multilinear Commutator 

of Littlewood-Paley Operator

GU Guang-ze1，CAI Ming-jie1，XIAO Qing-feng2

(1.College of Mathematics and Econometrics， Hunan Univ， Changsha， Hunan 410082， China; 

2. Dongguan Vocational College of Technology， Dongguan， Guangdong 523808， China)

Abstract: This paper studied the continuity of a multi-linear commutator generated by littlewood-Paley operator and Lipschitz functions on Triebel-Lizorkin space， Hardy space and Herz-Hardy space.

Key words: Littlewood-Paley operator; multi-linear commutator; Triebel-Lizorkin space; Herz-Hardy space; Herz space; Lipschitz space

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設T是Calderón-Zygmund算子，Coifman等［1］得到，交換子［b，T］(f)= bT(f) -T(bf)(b∈BMO(Rn))在Lp (Rn)的有界性(1<p<∞)．當T是分數次積分算子時，Chanillo［2］證明了類似的結果．Janson［3］和Paluszynski［4］在Triebel-Lizorkin空間和b屬于齊次Lipschitz空間時研究了這些結果．本文的主要目的是研究由Littlewood-Paley算子和b生成的極大多線性交換子在Triebel-Lizorkin空間，Hardy空間和Herz型Hardy空間的有界性，其中b= (b1，b2，…，bm)且函數bj∈Lipβ，1≤j≤m．

1 預備知識和定義

本文中，M(f)表示f的Hardy-Littlewood極大函數，且記Mp(f)=(M(fp))1/p對00和p>l，令Fβ，

SymboleB@ p為齊次Tribel-Lizorkin空間；用Lipβ(Rn)表示Lipschitz空間，函數f滿足

這就完成了定理3的證明．

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