On the lacunary spherical maximal function on the Heisenberg group

In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $M_{\Ha}^{lac}$ associated to the spherical means $A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.