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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.

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

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