Spherical means on Métivier groups and support theorem

Abstract

Let $Z_{r,R}$ be the space of continuous functions on the annulus $B_{r,R}$ in ${\mathbb{C}}^n$ whose $\lambda$-twisted spherical mean, in the set up of the Métivier group, vanishes over the spheres $S_s(z)\subset B_{r,R}$ with ball $B_r(0)\subseteq B_s(z).$ We characterize the spherical harmonic coefficients of functions in $Z_{r,R},$ eventually, in terms of polynomial growth, by which we infer support theorem. Further, we prove that non-harmonic complex cone and the boundary of a bounded domain are sets of injectivity for the $\lambda$-twisted spherical means.

18 pages

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