The spherical average has been a source of many problems in harmonic analysis.
Since late 90's, the study of the maximal spherical means on the Heisenberg group $mathbb{H}^n$ has been started to show the pointwise ergodic theorems on the groups.
Later, it has turned out to be connected with the fold singularities of the Fourier integral operators, which leads to the $L^p$ boundedness of the spherical maximal means on the Heisenberg group $mathbb{H}^n$ for $nge 2$.
In this talk, we discuss about the $L^p$ boundedness of the circular maximal function on the Heisenberg group $mathbb{H}^1$. The proof is based on the the square sum estimate of the Fourier integral operators associated with the torus arising from the vector fields of the Heisenberg group algebra.
We compare this torus with the characteristic cone of the Euclidean space.
