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Abstract. Given a Gelfand pair where 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 . We prove local ergodic theorems for these measures when . 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.
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