It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz distributions.) I wonder if $$\frac{\partial}{\partial \overline{z}} : \mathcal{S}'(\mathbb{C}) \to \mathcal{S}'(\mathbb{C})$$ is surjective ? By a standard result of functional analysis, it is enough to show that $$\frac{\partial}{\partial \overline{z}} : \mathcal{S}(\mathbb{C}) \to \mathcal{S}(\mathbb{C})$$ is injective and has a sequencially closed image. The injectivity is clear since a rapid decaying entire function is $0$ by Liouville theorem. But the other property doesn't seem so clear. Is the result true ? If not, is there any easy counter-example ?
Thank you for any help.
(This question has also been asked on mathstackexchange without any answer :https://math.stackexchange.com/questions/2663083/neumann-dbar-problem-with-tempered-distributions)
