  We prove the $L^p$ boundedness of the circular maximal function on the
Heisenberg group $\mathbb{H}^1$ for $2<p\le \infty$. The proof is based on the
square sum estimate associated with the $2\times 2$ cone $|(\xi_1',\xi_2')|=
|(\xi_3',\xi_4')| $ of the phase space arising from the vector fields
$X_1,X_2,tX_3,\partial/\partial t$ on the Heisenberg group, rather than the
$2\times 1$ cone $ |(\xi_1,\xi_2)|= |\xi_3|$ of the frequency space arising
from $\partial/\partial x_1, \partial/\partial x_2, \partial/\partial t$ on the
Euclidean space.
