$L^p - L^{p'}$ estimates for overdetermined Radon transforms

TL;DR

This paper extends bounds for Radon transforms involving convolutions with rotated measures, covering convex curves, hypersurfaces, and non-translation invariant families, advancing understanding of geometric averaging operators.

Contribution

It introduces new $L^p - L^{p'}$ estimates for Radon transforms over various geometric surfaces, generalizing previous results to higher dimensions and more complex surface families.

Findings

Bounds for convolution with rotations of measures supported on convex curves.

Estimates for averages over higher-dimensional convex hypersurfaces.

Results for non-translation invariant surface families.

Abstract

We prove several variations on the results of Ricci and Travaglini concerning bounds for convolution with all rotations of a measure supported by a fixed convex curve in the plane. Estimates are obtained for averages over higher-dimensional convex hypersurfaces, smooth k-dimensional surfaces and non-translation invariant families of surfaces.