Local ergodic theorems for k-spherical averages on the Heisenberg group

Abstract

Given a Gelfand pair (Hⁿ, K) where Hⁿ is the Heisenberg group and K is a compact subgroup of the unitary group U(n) we consider the sphere and ball averages of certain K-invariant measures on Hⁿ. We prove local ergodic theorems for these measures when n ≥ 3. We also consider averages over annuli in the case of reduced Heisenberg group and show that when the functions have zero mean value the maximal function associated to the annulus averages behave better than the spherical maximal function. We use square function arguments which require several properties of the K-spherical functions.