Let \begin{equation} P(\xi)=\xi_{1}^2 \xi_{2}^2 + \xi_{3}^2 + i \xi_{4}, \end{equation} and consider the partial differential operator \begin{equation} P(D)=\left( -i \frac{\partial}{\partial x_1},\dots, - i \frac{\partial}{\partial x_4} \right). \end{equation} Hörmander (The Analysis of Linear Partial Differential Operators, Vol. II, Example 10.2.15) states that \begin{equation} E(x) = \begin{cases} (2 \pi)^{-3} \int \exp[i(x_1 \xi_1+x_2 \xi_2 + x_3 \xi_3)) - x_4 (\xi_{1}^2 \xi_{2}^2 + \xi_{3}^2)] d\xi_{1} d \xi_{2} d \xi_{3} & \textit{ if } x_4 > 0, \\ 0 & \textit{if } x_4 \leq 0 \end{cases} \end{equation} is the unique temperate fundamental solution of the operator $P(D)$.
There are two big flaws for me in this sentence.
(i) Hörmander's statement about the uniqueness is for sure false, since $P$ has real zeros. So for example, if $T$ is a temperate fundamental solution, e.g. $T + c_0 + c_1 x_1$ is too, for any $c_0, c_1 \in \mathbb{C}$.
(ii) The most relevant fact is that $E(x)$ is not even well-defined for $x_4 > 0$. Indeed, we have \begin{equation} \int_{\mathbb{R}^3} \exp \left( - x_4 \left(\xi_{1}^{2} \xi_{2}^{2} + \xi_{3}^{2} \right) \right) d \xi_\ d \xi_2 d \xi_3 = \int_{\mathbb{R}} \frac{\pi}{x_4 |\xi_1|} d \xi_1 = \infty. \end{equation}
It seems very strange to me that such a great mathematician like Hörmander made two big mistakes like these in the same sentence. What do you think about?
Thank you very much for your attention.
NOTE. Hörmander uses this example of partial differential operator to show that there are partial differential operators which do not admit a "regular" temperate fundamental solution (for the definition of "regular solution", see Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Definition 10.2.2). Actually, his proof that the differential operator $P(D)$ defined above does not admit a regular temperate fundamental solution is correct and has nothing to do with the statement above.