Let $\mathcal L$ be a differential operator with constant coefficients and $\mathcal{L} f=0$ for some $f\in C^{\infty}(\mathbb{R}^n).$
Under what conditions on $\mathcal {L}$ the function $f$ extends to an entire function in $\mathbb{C}^n$? 



Extension of solutions of PDEs with constant coefficients
If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to ellipticity, a sharp contrast with the $C^\infty$ case, where hypoellipticity was characterized by L. Hörmander and holds for instance for the heat equation (which is not elliptic).


As a reminder, the operator $$P=\sum\_{\vert \alpha\vert\le m}a\_\alpha D^\alpha$$ is elliptic means that
$
\sum\_{\vert \alpha\vert= m}a\_\alpha \xi^\alpha\not=0\text{ for $ \xi\in\mathbb R^n\backslash\{0\}$}.
$
The Laplace equation or $\overline{\partial}$ equation are examples of elliptic operators.


