An optimal theorem for the spherical maximal operator on the Heisenberg group

Narayanan, EK and Thangavelu, S (2004) An optimal theorem for the spherical maximal operator on the Heisenberg group. In: Israel Journal of Mathematics, 144 (2). pp. 211-219.

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Abstract

Let $H_n = C^n \times R$ be the Heisenberg group of dimension $2n+1$. Let $\sigma _r$ be the normalized surface measure on the sphere of radius $r$ in $C^n$ . For a function f on $H_n$ , let $\[ M_\sigma_f = sup_r>0 \ |f \ast \sigma_r| \]$. It had been shown in [A. Nevo and S. Thangavelu, Adv. Math. 127 (1997), no. 2, 307–334; MR1448717 (98f:22005)] that $M_\sigma$ is bounded on $L^p(H_n)$ for all $\[ p > \frac {2n−1}{2n−2}\]$ . In this paper the authors modify the arguments of that paper and combine them with the square function method used in [E. M. Stein and S. Wainger, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295; MR0508453 (80k:42023)] to prove the maximal theorem on R^n; they show that $M_\sigma$ is bounded on $ L^p(H_n)$ if and only if $\[p > \frac {2n}{2n−1} \]$. As a consequence of their result the authors obtain the result that for $n \geq 2$, the family $( \sigma_r)_r$ is pointwise ergodic in $\[ L^p$ for all $p > \frac {2n}{2n−1} \]$. The maximal theorem on $H_n$ has also been obtained in a more general setting by Fourier integral methods in [D. M¨uller and A. Seeger, Israel J. Math. 141 (2004), 315–340; MR2063040 (2005e:22005)].

Item Type:	Journal Article
Publication:	Israel Journal of Mathematics
Publisher:	The Hebrew University Magnes Press,Jerusalem
Additional Information:	Copyright of this article belongs to The Hebrew University Magnes Press Jerusalem.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	27 Feb 2008
Last Modified:	08 Feb 2012 05:11
URI:	http://eprints.iisc.ac.in/id/eprint/12442

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