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

Abstract

In this work we consider the operator \[ (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot ω, ω) dω, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). \] 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

arXiv comment35 pages, 17 figures