, Volume 144, Issue 2, pp 211-219

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

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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 ℂ n . 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(I n ) if and only ifp>2n/2n−1.