A Radon Transform on the Cylinder

We define a parametric Radon transform $R$ that assigns to a Sobolev function on the cylinder $\mathbb{S}\times \mathbb{R}$ in ${\mathbb{R}}^3$ its mean values along sets $E_{\zeta }$ formed by the intersections of planes through the origin and the cylinder. We show that $R$ is a continuous operator, prove an inversion formula, provide a support theorem, as well as a characterization of its null space. We conclude by presenting a formula for the dual transform $R^{\ast }$. We show that $R$ and its dual $R^{\ast }$ are related to the right-sided and left-sided Chebyshev fractional integrals. Using this relationship, we characterize the null space of $R$ and $R^{\ast }$ and provide an inversion formula for $R^{\ast }$.