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 $$extract_itex$$mathbb{H}^n$$/extract_itex$$ 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 $$extract_itex$$L^p$$/extract_itex$$ boundedness of the spherical maximal means on the Heisenberg group $$extract_itex$$mathbb{H}^n$$/extract_itex$$ for $$extract_itex$$nge 2$$/extract_itex$$.
In this talk, we discuss about the $$extract_itex$$L^p$$/extract_itex$$ boundedness of the circular maximal function on the Heisenberg group $$extract_itex$$mathbb{H}^1$$/extract_itex$$. 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.