Define (i) the Fourier transform of a tempered distribution $T \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$, and (ii) the convolution $T * g$ of a tempered distribution $T \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ and a Schwartz function $g \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Give a formula for the Fourier transform of $T * g$ ("convolution theorem").

Let $t>0$. Compute the Fourier transform of the tempered distribution $A_{t} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ defined by

$$\left\langle A_{t}, \phi\right\rangle=\int_{\|y\|=t} \phi(y) d \Sigma(y), \quad \forall \phi \in \mathcal{S}\left(\mathbb{R}^{3}\right),$$

and deduce the Kirchhoff formula for the solution $u(t, x)$ of

$$\begin{gathered} \frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0, \\ u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=g(x), \quad g \in \mathcal{S}\left(\mathbb{R}^{3}\right) \end{gathered}$$

Prove, by consideration of the quantities $e=\frac{1}{2}\left(u_{t}^{2}+|\nabla u|^{2}\right)$ and $p=-u_{t} \nabla u$, that any $C^{2}$ solution is also given by the Kirchhoff formula (uniqueness).

Prove a corresponding uniqueness statement for the initial value problem

$$\begin{gathered} \frac{\partial^{2} w}{\partial t^{2}}-\Delta w+V(x) w=0 \\ w(0, x)=0, \quad \frac{\partial w}{\partial t}(0, x)=g(x), \quad g \in \mathcal{S}\left(\mathbb{R}^{3}\right) \end{gathered}$$

where $V$ is a smooth positive real-valued function of $x \in \mathbb{R}^{3}$ only.