Suppose $Pf=\sum_{|\alpha|\leq m}a_{\alpha}D^{\alpha}f$ is a partial differential operator with constant coefficient (at least one is different from 0). Prove that there are no function in Schwartz class other than 0 s.t $Pf=0$.
I want to use Fourier transform, but it seems doesn’t work.
Let $\phi\in\mathcal{S}$ be a solution of the equation. Then $$ \widehat{(P\,\phi)}(\xi)=Q(\xi)\,\widehat{\phi}(\xi)=0\quad\forall\xi\in\Bbb R^n, $$ where $Q$ is a polynomial. The only way this can happen is if $\widehat{\phi}(\xi)=0$ for all $\xi\in\Bbb R^n$.