Spherical maximal operators on rank one Riemannian symmetric spaces of noncompact type

The spherical maximal operator of order $\mu$ was introduced by El Kohen in the setting of real hyperbolic spaces, where its $L^p$-boundedness properties were studied. In this paper, we extend this notion to rank-one Riemannian symmetric spaces of noncompact type. We establish sufficient conditions on the parameter $\mu$ for the corresponding spherical maximal operator to be bounded on $L^p$ for $1<p\le \mathrm{\infty }$. We also obtain a necessary condition for $L^p$-boundedness when $\mu$ is real and $1<p<\mathrm{\infty }$. In particular, for $1<p\le 2$, the necessary condition agrees with the sufficient condition, yielding the sharp range of real $\mu$ in this regime.