Detlef Müller, Andreas Seeger:

Singular spherical maximal operators on a class of step two nilpotent Lie groups

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $\mu_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t > 0} ¦f*\mu_t ¦$. We prove for $n >= 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p > 2n/(2n-1)$. This improves an earlier result by Nevo and Thangavelu, and the range for $L^p$ boundedness is optimal. We also extend the result to a more general setting of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.

Electronic Print: Front for the Mathematics ArXiv

Mathematics Subject Classification (1991): 42B25, 22E25, 43A80

Bibliographical note: erscheint im Israel Journal

Keywords: spherical maximal operators, Heisenberg groups, step two nilpotent groups, oscillatory integral operators, fold singularities

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[Thu Feb 19 18:56:36 2009]

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