3 problems

Rubin's range characterization conjecture. A function $g\in C_c^\infty(\mathbb{S}^{n-1}\times(0,2))$ belongs to the range of the operator $f\mapsto\mathcal{M}f$ if and only if

Let $S\subset\mathbb R^d$. Call $S$ an injectivity set for the spherical mean transform if vanishing of all spherical means centered on $S$ forces the underlying function or distri…

Let $d$ be the ambient dimension and let $S\subset\mathbb{R}^d$. A set $S$ is a non-uniqueness set for the spherical mean transform in the space of compactly supported functions wh…