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

Authors:	Email author" target="_blank">E?K?Narayanan Email author S?Thangavelu

Institution:	(1) Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India;(2) Stat-Math Division, Indian Statistical Institute, 8th Mile, Mysore Road, 560 059 Bangalore, India

Abstract:	Let $\mathbb{I}^n = \mathbb{C}^n \times \mathbb{R}$ be the Heisenberg group and μ_r be the normalized surface measure on the sphere of radiusr in ℂⁿ. Let $Mf = \sup _{r > 0} \left\| {f * \mu _r } \right\|$ . We prove an optimalL ^p-boundedness result for the spherical maximal functionMf, namely we prove thatM is bounded onL ^p(Iⁿ) if and only ifp>2n/2n−1.

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