Dyadic sets, maximal functions and applications on$ax+b$,-groups

LIU Li-Guang; Maria Vallarino; YANG Da-Chun

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Dyadic sets, maximal functions and applications on$ax+b$,-groups

LIU Li-Guang 1,Maria Vallarino 2,YANG Da-Chun 3 *

1.Department of Mathematics, School of Information, Renmin University

2.Dipartimento di Matematica e Applicazioni, Universit`{a} diMilano-Bicocca, via R. Cozzi 53,endgraf20125 Milano, Italy

3.School of Mathematical Sciences, Beijing NormalUniversity,

*Correspondence author

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Funding: and the third author is supported by theNational Natural Science Foundation ofChina （No.Grant No. 10871025）, and the Research Funds of Renmin University of China （No.Grant No. 10XNF090）

Opened online:25 March 2011

Accepted by: none

Citation: LIU Li-Guang,Maria Vallarino,YANG Da-Chun.Dyadic sets, maximal functions and applications on$ax+b$,-groups[OL]. [25 March 2011] http://en.paper.edu.cn/en_releasepaper/content/4417246

Let $S$ be the Lie group${mathbb R}^nltimes {mathbb R}$, where ${mathbb R}$acts on ${mathbb R}^n$ by dilations,endowed with the left-invariantRiemannian symmetric space structure and the right Haar measure$ ho$, which is a Lie group of exponential growth. Hebisch andSteger in [Math. Z. 245(2003), 37-61] proved that any integrablefunction on $(S, ho)$ admits a Calder'on-Zygmund decompositionwhich involves a particular family of sets, calledCalder'on-Zygmund sets. In this paper, we show theexistence of a dyadic grid in the group $S$, which has {nice} propertiessimilar to the classical Euclidean dyadic cubes. Using theproperties of the dyadic grid, we prove aFefferman-Stein type inequality, involving the dyadic Hardy-Littlewoodmaximal function and the dyadic sharp function. As a consequence,we obtain a complex interpolationtheorem involving the Hardy space $H^1$ and the space${mathopmathrm{,BMO,}}$introduced in [Collect. Math. 60(2009), 277-295].

Keywords:Exponential growth group; Dyadic set; Complex interpolation; Hardy space; $BMO$

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