Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces

In this article we investigate spherical maximal operators $\mathfrak m^α$ of real order $α$ acting on radial functions in the hyperbolic space $\mathbb H^n$. We show that for every $n\geq 2$, $1<p\leq\infty$ and $α>(1-n)/2$, $\mathfrak m^α$ is bounded on $L^p_{\mathrm{rad}}(\mathbb H^n)$ if and only if \begin{align*} \begin{cases} α>1-n+\frac np, & 1<p\leq 2, α>\frac1p-\frac{n-1}2, & 2<p\leq\frac{2n}{n-1}, α\geq\frac{1-n}p, & p>\frac{2n}{n-1}. \end{cases} \end{align*} Moreover, we show that for $1<p\leq 2n/(n-1)$ and \begin{align*} α= \begin{cases} 1-n+\frac np, & 1<p\leq 2, \frac1p-\frac{n-1}2, & 2<p\leq\frac{2n}{n-1}, \end{cases} \end{align*} $\mathfrak m^α$ is bounded from $L^{p,1}_{\mathrm{rad}}(\mathbb H^n)$ to $L^{p,\infty}_{\mathrm{rad}}(\mathbb H^n)$.