Non-harmonic cones are Heisenberg uniqueness pairs for the Fourier transform on $\mathbb R^n$

R. K. Srivastava

Published 2015-07-08Version 1

A Heisenberg uniqueness pair is a pair $\left(\Gamma, \Lambda\right)$, where $\Gamma$ is a surface and $\Lambda\subset\mathbb R^n$ be such that any finite Borel measure $\mu$ which is supported on $\Gamma$ and absolutely continuous with respect to the surface measure, whose Fourier transform $\widehat\mu$ vanishes on $\Lambda,$ implies $\mu=0.$ In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to the unit sphere as long as it does not completely lay on the level surface of any homogeneous harmonic polynomial on $\mathbb R^n.$