Michael Roysdon
Case Western Reserve University
Title
Comparison problems and slicing inequalities for Radon transforms
Abstract
In 1956 Herbert Busemann and Clinton M. Petty created a list of 10 problems in Convex Geometry, among which only the first had been solved fully in 1999. A key feature of the solution to the first problem is that it (and in fact, the majority of the Busemann-Petty problems) can be reformulated into the language of Harmonic Analysis. Inspired by the Busemann-Petty problems and their connection to Harmonic Analysis, we consider the following natural question for various Radon transforms: Let p>1. Given a pair of nonnegative, even and continuous functions f,g such that the Radon transform of f is pointwise smaller than the Radon transform of g, is it necessarily true that the L^p-norm of f is smaller than the L^p-norm of g? As it turns out, this simple question has an immediate connection to the Busemann-Petty problem and the slicing problem of Bourgain. As a consequence of our investigation, we show that this implies reverse Oberlin-Stein type estimates for the spherical Radon transform when p >1, which is complementary to a recent work of Johnathan Bennett and Terence Tao in which similar reverse estimates were proven in the case 0 < p,q \leq 1 satisfying certain conditions. We will discuss a similar problem for the dual Radon transform.