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.



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



The division problem for tempered functions
Your space $T(\mathbb R^d)$ is what Laurent Schwartz introduced as $\mathscr O\_M$ because these *slowly increasing* smooth functions (each partial derivative is bounded by a polynomial whose degree depends on the derivative) act as *opérateurs de multiplication* on $\mathscr S'$ (L. Schwartz, Théorie des distributions (1966), page 246).


Surjectivity of linear partial differential operators with constant coefficients is discussed in the article *Surjectivity of differential operators and the division problem in certain function and distribution spaces* of Julian Larcher in *Journal of Mathematical Analysis and Applications* 409, p. 91-99. 


For example, it is proved there that every hypoelliptic operator is surjective (this not completely trivial: If $P(D)v=f\in \mathscr O\_M$ with $v\in\mathscr S'$ then $v$ is smooth and a tempered distribution but a priori need not be a smooth tempered function -- this is a good reason to avoid the term tempered function). Generalizing an example of M.S. Baouendi from 1965, Larcher shows that in $\mathbb R^2$ the operator $\partial\_x\partial\_y+c$ is surjective if and only if $c$ is real. There is also a conjecture about a *characterization* of surjectivity on $\mathscr O\_M$ but this is an open problem.


