2 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

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