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LetH  n ≅ℝ2n ⋉ℝ be the Heisenberg group and letμ  t  be the normalized surface measure for the sphere of radiust in ℝ2n . Consider the maximal function defined byM f=sup t>0|f*μ  t |. We prove forn≥2 thatM defines an operator bounded onL  p (H  n ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL  p  boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.

Singular spherical maximal operators on a class of two step nilpotent lie groups