Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier transform, thus $\tilde D$ is a polynomial on $\mathbb{R}^n$.

Let us assume that $\tilde D$ does not vanish. Does it tell something on solutions of the equation $Du=0$?

A basic example is $D=-\Delta +1$.

Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier transform, thus $\tilde D$ is a polynomial on $\mathbb{R}^n$.

Let us assume that $\tilde D$ does not vanish. Does it tell something on non-zero solutions of the equation $Du=0$?

A basic example is $D=-\Delta +1$.

Remark. As far as I understand it follows that $u$ does not belong to the Schwatz space of tempered growth distributions.