I need some help for the following statement
Let $P$ be a polynomial, and $P(\frac{d}{dx})u=f$, where $f$ is a distribution with compact support. Then it has a distributional solution $u$ with compact support if and only if $(f,\phi)=0$, for all $\phi$ satisfies $P(-\frac{d}{dx})\phi=0$
I want to know how to prove the sufficiency of the condition. I try to use the fundamental solution of $P$, but didn't solve it.
Besides, since this is the one-dimentional case, I also wondered if there are some similar cnclusions when treated in $\mathbb{R}^n$ ?
Thanks for your help.