# The division problem for tempered functions

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.

• Do you mean any or every? – Jochen Wengenroth Apr 15 '18 at 18:11
• Every, I have modified accordingly. – Noether Apr 16 '18 at 13:32

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).
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.