Lacunary elliptic maximal operator on the Heisenberg group

In this paper, we prove $L^p$ boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these $L^p$ estimates from skew-symmetric matrices, which naturally arise in Heisenberg group operations, to arbitrary matrices $A$, investigating how the curvature induced by $A$ governs the $L^p$ boundedness of lacunary circular and elliptic maximal operators. Specifically, we provide necessary and sufficient conditions on $A$ that determine whether these operators are bounded or unbounded on $L^p$.