In this paper we deal with lacunary and full versions of the spherical maximal function on the Heisenberg group Hⁿ, for n ≥ 2. By suitable adaptation of an approach developed by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In particular, we deduce the L^p boundedness, for 1 < p < ∞, of the lacunary maximal function associated to the spherical means on the Heisenberg group. In order to prove the sparse bounds, we establish L^p - L^q estimates for local (single scale) variants of the spherical means.

The first-named author was partially supported by Inspire Faculty Fellowship [DST/INSPIRE/04/2016/000776]. The first, second, and third authors were supported by 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundation. The fourth author was visiting Basque Center for Applied Mathematics through a Visiting Fellow programme. The third and fourth-named author were also supported by the Basque Government through BERC 2018--2021 program, by Spanish Ministry of Science, Innovation and Universities through BCAM Severo Ochoa accreditation SEV-2017-2018 and the project MTM2017-82160-C2-1-P funded by AEI/FEDER, UE. The third author also acknowledges the RyC project RYC2018-025477-I and IKERBASQUE.