= Solution
The <Malgrange–Ehrenpreis theorem> states that every nonzero constant-coefficient <linear partial differential operator> $P(D)$ on $\mathbb R^n$ has a <fundamental solution of a linear differential operator>: there is an $E\in\mathcal D'(\mathbb R^n)$ such that $P(D)E=\delta_0$.
Write $D=-i\partial$. After an <orthogonal matrix>[orthogonal change of coordinates] and multiplication by a nonzero constant, its polynomial symbol may be written as a monic polynomial in the last frequency,
$$
P(\xi',z)=z^M+\sum_{m=0}^{M-1}a_m(\xi')z^m.
$$
For each real $\mu'$, this polynomial has $M$ complex roots counted with multiplicity. Among a fixed finite collection of horizontal lines at bounded heights, one can choose a line that stays a positive distance from all those roots. Continuity of the roots preserves the choice on a neighborhood $N(\mu')$. Take a countable locally finite cover by such neighborhoods, refine it to a measurable disjoint partition $\mathbb R^{n-1}=\bigsqcup_j\Delta_j$, and let $c_j$ be the chosen height on $\Delta_j$. The resulting <Hörmander staircase>
$$
\Sigma=\bigcup_j\{(\xi',s+ic_j):\xi'\in\Delta_j,\ s\in\mathbb R\}
$$
has bounded heights and may be chosen so that $|P(\xi',s+ic_j)|\geq1$ on each step.
For a <test function> $\varphi$, define
$$
\langle E,\varphi\rangle
=\frac1{(2\pi)^n}\sum_j
\int_{\Delta_j}\int_{\mathbb R+ic_j}
\frac{\widehat\varphi(-\xi',-z)}{P(\xi',z)}\,dz\,d\xi'.
$$
The <Paley–Wiener–Schwartz theorem> gives rapid decay in the real frequency directions and at most a fixed exponential factor in the bounded imaginary direction. Together with $|P|\geq1$, this proves that the integral defines a continuous <distribution>. Applying $P(D)$ cancels the denominator. The remaining integrand is <entire function>[entire] in $z$, so the <Cauchy integral theorem> shifts every horizontal contour to the real axis; the partition then recombines into $\mathbb R^{n-1}$. The <Fourier inversion theorem> gives
$$
\langle P(D)E,\varphi\rangle=\varphi(0)=\langle\delta_0,\varphi\rangle,
$$
which proves the theorem.
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