Spherical maximal operators on Heisenberg groups: Restricted dilation sets

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_{E}f$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these maximal operators; the results depend on various notions of dimension of $E$.