Dyadic sets, maximal functions and applications on ax?+?b-groups
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- 作者：Liguang Liu (1) (2)
Maria Vallarino (3)
Dachun Yang (1) - 关键词：Exponential growth group ; Dyadic set ; Complex interpolation ; Hardy space ; BMO ; 22E30 ; 42B30 ; 46B70
- 刊名：Mathematische Zeitschrift
- 出版年：2012
- 出版时间：2 - February 2012
- 年：2012
- 卷：270
- 期：1
- 页码：515-529
- 全文大小：280KB
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26. Vallarino M.: Spaces / H 1 and BMO on / ax?+? / b-groups. Collect. Math. 60, 277-95 (2009) 10.1007/BF03191372">CrossRef - 作者单位：Liguang Liu (1) (2)
Maria Vallarino (3)
Dachun Yang (1)
1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China
2. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, People’s Republic of China
3. Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via R. Cozzi 53, 20125, Milan, Italy - ISSN：1432-1823
文摘
Let S be the Lie group ${{\mathbb R}^n\ltimes {\mathbb R}}$ , where ${{\mathbb R}}$ acts on ${{\mathbb R}^n}$ by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37-1, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H 1 and the space BMO introduced in [Collect. Math. 60: 277-95, 2009].