# 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 Copyright of this article belongs to The Hebrew University Magnes Press Jerusalem. Division of Physical & Mathematical Sciences > Mathematics 27 Feb 2008 08 Feb 2012 05:11 http://eprints.iisc.ernet.in/id/eprint/12442