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Centered Hardy-Littlewood maximal functions on Heisenberg type groups


Authors: Hong-Quan Li and Bin Qian
Journal: Trans. Amer. Math. Soc. 366 (2014), 1497-1524
MSC (2010): Primary 42B25, 43A80
DOI: https://doi.org/10.1090/S0002-9947-2013-05965-X
Published electronically: September 26, 2013
MathSciNet review: 3145740
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Abstract: In this paper, by establishing uniform lower bounds for the Poisson kernel and $ (-\Delta )^{-\frac 12}$ on the Heisenberg type group $ \mathbb{H}(2n,m)$ with $ m \geq 2$, which follow from the various properties of Bessel functions and Legendre functions, we prove that there exists a constant $ A>0$ such that, for all $ f\in L^1(\mathbb{H}(2n,m))$ and all $ n,m \in \mathbb{N}^*$ satisfying $ 4 \leq m^2\ll \log n$, we have $ \Vert M_{K}f\Vert _{L^{1,\infty }}\le An\Vert f\Vert _1$, where $ M_{K}$ denotes the centered Hardy-Littlewood maximal function defined by the Korányi norm. For the centered Hardy-Littlewood maximal function $ M_{CC}$ defined by the Carnot-Carathédory distance, we prove $ \Vert M_{CC}f\Vert _{L^{1,\infty }}\le A(m)n\Vert f\Vert _1$ holds for some constant $ A(m)$ independent of $ n$.


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Hong-Quan Li
Affiliation: School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, People’s Republic of China
Email: hongquan{\textunderscore}li@fudan.edu.cn, hong{\textunderscore}quanli@yahoo.fr

Bin Qian
Affiliation: School of Mathematics and Statistics, Changshu Institute of Technology 215500, Changshu, People’s Republic of China
Email: binqiancn@yahoo.com.cn, binqiancn@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2013-05965-X
Received by editor(s): February 10, 2012
Published electronically: September 26, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.