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

TL;DR

This paper investigates the detailed mathematical properties of the Radon transform and its backprojection operator on the unit ball, providing new formulas and sharper estimates using advanced geometric analysis techniques.

Contribution

It introduces a double $b$-fibration framework to desingularize the Radon transform's point-hyperplane relation and derives refined polyhomogeneous mapping properties with new formulas and estimates.

Findings

Constructed a double $b$-fibration for the Radon transform

Derived sharper estimates for the mapping properties of $R$ and $R^*$

Discussed a family of normal operators related to $R$

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.