Let $P(D)$ be a nontrivial constant coefficient partial differential operator, where $D:=-i\partial \,.$
Show that the equation $P(D)u=0$ has no nonzero solutions $u\in M^\prime$, where $M^\prime$ means the set of distributions over the space of smooth functions.
I have come up with an idea, but there are some details which I can not fill in.
The story is using the Fourier Transform. Note that $\widehat{Du}=\xi \hat u$, where $\hat u$ means the Fourier Transform of $u$. So we have $0=\widehat {P(D)u}=P(\xi)\hat u$.
(True Statement) Assume $u\in M^\prime$, then $\hat u$ is analytic.
Note that $P$ is a polynomial, and the number of its zeros is finite, so the set of $P$ vanishing is closed. And so $\hat u$ vanishes on an open set. Since it is analytic, we conclude that $\hat u$ is identically zero. Since the Fourier Transform is bijective on the dual of the Schwartz space, which contains $M^\prime$, we conclude that $u$ is identically zero.
My problem now is how to prove the "Ture Statement" above. If it is proved, then we are done.
Any help will be appreciated.
I've found a proof. See Theorem $7.1.14$, The Analysis of Linear Partial Differential Operators I, Lars Hormander.