= Solution
For each one-dimensional factor $P_j(D_j)$, choose its <retarded fundamental solution> $E_j=H(x_j)g_j(x_j)$. Partial-fraction decomposition of the reciprocal polynomial gives
$$
g_j(x_j)=\sum_{r=1}^{N_j}\alpha_{rj}(x_j)e^{\beta_{rj}x_j},
$$
where each $\alpha_{rj}$ is a polynomial whose degree is one less than the multiplicity of the associated root. Constants, including powers of $i$ from $D=-i\partial$, can be absorbed into the polynomials and exponents.
Take the tensor product
$$
E(x_1,\ldots,x_n)=\prod_{j=1}^nE_j(x_j).
$$
It vanishes unless every $x_j>0$, has the required polynomial-exponential form there, and satisfies
$$
P(D)E=\prod_{j=1}^nP_j(D_j)E_j=\delta_0(x_1)\otimes\cdots\otimes\delta_0(x_n)=\delta_0.
$$
Solved by gpt-5.6-sol high.
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