Rotational smoothing

TL;DR

This paper systematically studies rotational smoothing, a regularity gain from averaging over rotations, for operators like Riesz potentials and elliptic differential operators, with applications to inverse problems.

Contribution

It provides a comprehensive analysis of rotational smoothing effects for a broad class of operators, including Riesz potentials and solutions to elliptic equations, expanding understanding of directional regularization.

Findings

Rotational averaging enhances regularity for certain operators.

The study includes Riesz potentials and elliptic differential operators.

Applications to inverse problems under low-regularity conditions.

Abstract

Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $\mathbb{R}^n$ with $k < n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $\mathbb{R}^n$, or a solution operator for the transport equation in $\mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.