Detlef Müller, Andreas Seeger
Singular Spherical Maximal Operators on a Class of Two Step Nilpotent Lie Groups
Preprint series: ESI preprints

MSC:

42B25 Maximal functions, Littlewood-Paley theory

22E25 Nilpotent and solvable Lie groups

43A80 Analysis on other specific Lie groups, See also {22Exx}

Abstract: 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 $n\ge 2$ 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.

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