A Busemann-Petty Type Problem for Dual Radon Transforms

TL;DR

This paper extends the Busemann-Petty problem to the setting of the $(n-k)$-dimensional dual Radon transform, studying when pointwise inequalities for $\mathcal{R}_{n-k}^*$ imply $L^p$-norm comparisons of the underlying data on the affine Grassmannian. A complete classification is given: for $p=1$ the comparison holds universally; for $p>1$ it holds precisely for data in the $(p,k)$-admissible class $\mathcal{A}_p^k$, with a sharp counterexample when the data falls outside an extended class $\mathcal{A}_{p,\infty}^k$. The authors develop an extended Fourier-Slice theory to measures, establish injectivity, and connect the problem to reverse $L^p$–$L^q$ estimates and a slicing inequality, including a mean-value property for the dual transform. The results yield reverse forms of Solmon-type estimates for dual Radon transforms and illuminate when isomorphic-type bounds can be guaranteed, advancing the understanding of Radon transform inequalities in high-dimensional geometric contexts.

Abstract

Inspired by resolution of the Busemann-Petty problem (1956), we consider the following comparison problem for dual Radon transforms: Given a pair of continuous functions defined on the affine Grassmannian whose dual Radon transforms satisfy a pointwise inequality, can their $L^p$ norms be compared in a meaningful way? We characterize the solution to this problem for each $p\geq 1$, and as a consequence of our investigation, we prove reverse $L^p$-$L^q$-estimates for dual Radon transforms. In particular, we reverse an inequality of Solmon (1979).

Theorems & Definitions (33)

• Theorem 1.3
• Theorem 1.4
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• proof : Proof of Theorem \ref{['t:mappingproperty']}
• Theorem 3.2