Circular maximal functions on the Heisenberg group

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.