用几何方法研究非辐射线性波的一个不等式
An inequality regarding non-radiative linear waves via a geometric method
摘要：
在这项工作中，我们考虑运营商\[ (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot ω, ω) dω, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). \]这是伴随算子的Radon变换。如果函数$G$是紧支集的，我们利用几何方法给出了$\mathbf{T} G$在无穷远处的最优$L^6$衰减估计.作为应用，我们给出了三维线性波动方程无辐射解在外部区域$\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$的衰减估计。这种衰减估计在波动方程的能量通道法中是有用的
英文摘要：
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
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