An inequality regarding non-radiative linear waves via a geometric method

TL;DR

This paper establishes optimal decay estimates for a Radon transform adjoint operator using geometric methods, and applies these results to analyze decay of non-radiative solutions to the 3D linear wave equation.

Contribution

It introduces a geometric approach to obtain optimal decay estimates for the Radon transform's adjoint operator and applies this to wave equation solutions.

Findings

Optimal $L^6$ decay estimate for the operator near infinity

Decay estimates for non-radiative solutions in exterior regions

Application to the channel of energy method for wave equations

Abstract

In this work we consider the operator \[ (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot \omega, \omega) d\omega, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). \] This is the adjoint operator of the Radon transform. We manage to give an optimal $L^6$ decay estimate of $\mathbf{T} G$ near the infinity by a geometric method, if the function $G$ is compactly supported. As an application we give decay estimate of non-radiative solutions to the 3D linear wave equation in the exterior region $\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$. This kind of decay estimate is useful in the channel of energy method for wave equations