6 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 $f$ be a nonzero compactly supported distribution or continuous function on $\mathbb R^3$ whose relevant support boundary is real analytic, strictly convex, and closed, and let…

Let a Paley–Wiener family be a sufficiently large family of Laplace eigenfunctions, and let a common nodal surface be a surface on which every eigenfunction in the family vanishes.…

Range characterization conjecture. Such a function $G$ is the image of $F$ under the parabolic transform $\mathcal{P}$.

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…