Injectivity of spherical mean on Métivier Group

Abstract

In this article, we study the injectivity of the spherical mean for continuous functions on the Métivier group. The spherical mean is injective for $f(z,.)\in L^p({\mathbb{R}}^m),1\le p\le 2$ with tempered growth in $z$ variable. This result is also true for a class of functions in $L^p({\mathbb{C}}^n),1\le p\le \infty$ without tempered growth. Further, we obtain a two-radii theorem for functions on the Métivier group, which are tempered in $z$ variable and periodic in the centre variable.

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