Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on $${\mathbb {R}}^n$$ R n

Abstract

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on $${\mathbb {R}}^n$$ . We derive that $$\left( S^2, \text { paraboloid}\right) $$ and $$\left( S^2, \text { geodesic of } S_r(o)\right) $$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in $${\mathbb {R}}^3$$ . Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.