It is well known (see for example S Łojasiewicz, *Sur le problème de la division*, Studia Math. **8** (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective on $\mathcal{S}'(\mathbb{R}^n).$ (The space of tempered distributions.)

A function $f \in C_{\infty}(\mathbb{R}^n,\mathbb{C})$ is tempered if it can be controlled (as well as all its derivatives) by a polynomial when $|x|\to +\infty$. Of course, the distribution associated to a tempered function is a tempered distribution. Let us note $T(\mathbb{R}^n)$ the space of tempered functions.

Is every linear partial differential operator with constant coefficients surjective on $T(\mathbb{R}^n)$ ?

Of course if $f \in T(\mathbb{R}^n)$ and if $P$ is such an operator, I can find $v\in \mathcal{S}'(\mathbb{R}^n)$ such that $P.v = u_f$, but nothing tells me that $v$ is associated to a tempered function.

Thanks for any help.