Polyhomogeneous mapping properties of the Radon transform and backprojection operator on the unit ball

Abstract

This article covers polyhomogeneous mapping properties of the Radon transform $R$ of smooth functions on the open unit ball $\Omega\subset\mathbb{R}^n$ and the back-projection operator $R^*$ on $Z=(-1,1)\times S^{n-1}\subset\mathbb{R}\times S^{n-1}$. We construct a double $b$-fibration which desingularizes the point-hyperplane relation of $\overline{\Omega}$ as the total space of a fibration over $\overline{Z}$. We provide formulas for $R$ and $R^*$ in operations generated by the associated $b$-fibrations and sharper estimates on the polyhomogeneous mapping properties of $R$ and $R^*$ compared to classic estimates using classic Mellin functional techniques. We include a discussion of a one (complex) parameter family of normal operators associated to $R$ mapping $C^{\infty}(\overline{\Omega})$ to itself.