Lacunary elliptic maximal operator on the Heisenberg group

Abstract: 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 ).