On the Kor\'anyi Spherical maximal function on Heisenberg groups

We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the K\'oranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Kor\'anyi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.