Show that $E(t,x) = \frac{\textbf{1}_{(0,+\infty)}(t)}{t^{1/3}}\textbf{Ai}(\frac{x}{t^{1/3}})$ is the unique tempered fundamental solution with support in $\{ (t,x) \mid t\geq 0 \}$ of the partial differential operator $P(D) = \partial_t + \frac{1}{3}\partial_x^3$.
I am puzzled about an exercise problem from an undergraduate course about distributions and Fourier transfrom.
The exercise concerns about the constant-coefficients 1+1 dimensional differential operator $$P(D) = \partial_t + \frac{1}{3}\partial_x^3.$$ The first part of the exercise is to check that $$E(t,x) = \frac{\textbf{1}_{(0,+\infty)}(t)}{t^{1/3}}\textbf{Ai}(\frac{x}{t^{1/3}})$$ is a fundamental solution of $P(D)$, where $\textbf{Ai}(x) = \mathcal{F}^{-1}_{\xi \rightarrow x}(e^{i\xi^3/3})$ is the Airy function (it turns out that $\textbf{Ai}$ is $C^\infty$). This part is straightforward and I could work out.
But the second part of the exercise asks to show that $u(t,x) = E(t,x)$ is the unique solution in the sense that $$ u \in \mathcal{S}'(\mathbb{R}^2_{t,x}) \\ (\partial_t + \frac{1}{3}\partial_x^3)u = \delta_{(0,0)} \\ \textbf{supp}~u \subset \mathbb{R}_{t \geq 0} \times \mathbb{R}_x$$
I could not work out. I have consulted Hormander's book $$\textit{The Analysis of Linear Partial Differential Operators II}$$ and found some topics very close to it, mostly in Chap. XII "The Cauchy and Mixed Problems", but it seems that there is no theorem I could directly quote.
$\textbf{My Question}$
Is there any general theorem about the uniqueness of tempered fundamental solutions of linear differential operators with constant coefficients? Is there any necessary or sufficient conditions such that $P(D)u = \delta$ has a unique tempered solution $u$, maybe with some requirement of the support $\textbf{supp}~u$ (for instance $\textbf{supp}~u \subset \text{half space}$ like above.)
Does the requirement $\textbf{supp}~u \subset \mathbb{R}_{t \geq 0} \times \mathbb{R}_x$ take the most important role so that actually this exercise is very easy?