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

Detlef Mueller·Andreas Seeger
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Abstract

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $μ_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*μ_t|$. We prove for $n\ge 2$ 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.

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Detlef Mueller, Andreas Seeger. 2003-07-03. Singular spherical maximal operators on a class of two step nilpotent Lie groups. https://arxiv.org/abs/math/0307042

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