Injectivity of spherical mean on M\'{e}tivier Group

TL;DR

This paper investigates the injectivity of spherical means on the Métivier group, establishing conditions under which the spherical mean uniquely determines functions in certain L^p spaces and proving a two-radii theorem for specific function classes.

Contribution

It demonstrates the injectivity of spherical means for functions in L^p spaces on the Métivier group and introduces a two-radii theorem for tempered and periodic functions.

Findings

Injectivity holds for functions in L^p with 1 ≤ p ≤ 2 and tempered growth.

Injectivity extends to functions in L^p(ℂ^n) for 1 ≤ p ≤ ∞ without tempered growth.

A two-radii theorem is established for tempered and periodic functions on the group.

Abstract

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