Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

Abstract

Let G R d ⋉ R be a finite-dimensional two-step nilpotent group with the group multiplication (x, u) • (y, v) → (x + y, u + v + x T Jy) where J is a skew-symmetric matrix satisfying a degeneracy condition with 2 ≤ rank J < d. Consider the maximal function defined bywhere Σ is a smooth convex hypersurface and dµ is a compactly supported smooth density on Σ such that the Gaussian curvature of Σ is nonvanishing on supp dµ. In this paper we prove that when d ≥ 4, the maximal operator M is bounded on L p (G) for the range (d − 1)/(d − 2) < p ≤ ∞.