Some remarks on the Lipschitz regularity of Radon transforms

TL;DR

This paper constructs a specific set in the plane with a Radon transform that is Lipschitz continuous in all directions, and demonstrates that the Lipschitz constant function cannot be uniformly bounded under mild conditions.

Contribution

It provides a novel example of a set with a Radon transform that is Lipschitz in all directions and proves the unboundedness of the Lipschitz constant function in general.

Findings

Constructed a set with Lipschitz Radon transform in all directions

Showed the Lipschitz constant function cannot be bounded under mild hypotheses

Highlighted limitations in Lipschitz regularity of Radon transforms

Abstract

A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the corresponding Lipschitz constant cannot be bounded.