Let $K\subset Q$ be compact, convex sets in ${ℝ}^n$ with $\stackrel{\circ }{K}\ne \varnothing$ and let $P\left(D\right)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P\left(D\right)$ in the space $ℰ\left(K\right)$ of all $C^{\infty }$-functions on $K$ extends to a zero solution in $ℰ\left(Q\right)$ resp. in $ℰ\left({ℝ}^n\right)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${ℂ}^n$ and in terms of fundamental solutions for $P\left(D\right)$ with lacunas.