A Note on the Sonar Transform and Related Radon Transforms

TL;DR

This paper provides sharp $L^p$-$L^q$ estimates and explicit inversion formulas for the sonar transform, revealing connections with Radon transforms over paraboloids and hyperplanes, advancing geometric tomography techniques.

Contribution

It introduces new sharp estimates and explicit inversion formulas for the sonar transform, linking it to Radon transforms over paraboloids and hyperplanes.

Findings

Established sharp $L^p$-$L^q$ estimates for the sonar transform.

Derived explicit inversion formulas under minimal assumptions.

Revealed connections between the sonar transform and Radon transforms over paraboloids and hyperplanes.

Abstract

The sonar transform in geometric tomography maps functions on the Euclidean half-space to integrals of those functions over hemispheres centered on the boundary hyperplane. We obtain sharp $L^p$-$L^q$ estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The main results follow from intriguing connection between the sonar transform, the Radon transform over paraboloids, and the transversal Radon transform, which integrates functions over hyperplanes, meeting the last coordinate axis.