  $L^p$ boundedness of the circular maximal function $\mathcal
M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received
considerable attentions. While the problem still remains open, $L^p$
boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was
recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this
paper we extend their result considering the local maximal operator
$M_{\mathbb{H}^1}$ which is defined by taking supremum over $1<t<2$. We prove
$L^p-L^q$ estimates for $M_{\mathbb{H}^1}$ on Heisenberg radial functions on
the optimal range of $p,q$ modulo the borderline cases. Our argument also
provides a simpler proof of the aforementioned result due to Beltran et al.
