We prove L p → L q estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse domination for global maximal operators. We also consider lacunary variants, and extensions to Métivier groups.

Let Z(Ann(r, R)) be the class of all continuous functions f on the annulus Ann(r, R) in C n with twisted spherical mean f × µ s (z) = 0, whenever z ∈ C n and s > 0 satisfy the condition that the sphere S s (z) ⊆ Ann(r, R) and ball B r (0) ⊆ B s (z). In this paper, we give a characterization for functions in Z(Ann(r, R)) in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in C n which improve some of the earlier results.

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse domination for global maximal operators. We also consider lacunary variants and extensions to Métivier groups.

We prove L p → L q estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp L p → L q estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in [10] for the associated lacunary maximal operator.

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