The following apparently elementary question came out of a somewhat naive attempt to
prove that every distribution $u\in \mathscr D'(\mathbb R^2)$ with $\partial\_1 u=\partial\_2 u =0$ is a constant function (this can be reduced to $\mathscr C^1$-functions by convolution with an approximate identity and for $\mathscr C^1$-functions it is completely elementary). 



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> For which $\varphi \in \mathscr D(\mathbb R^2)$ does there exist $f,g \in \mathscr D(\mathbb R^2)$ such that $\varphi = \partial\_1 f + \partial\_2g$?
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The only necessary condition I see is $\int \varphi(x,y) d(x,y)=0$, and the conjecture is that this is sufficient.


However, all my ad hoc attempts failed.
On the side of Fourier transforms one would have to write $\hat{\varphi}(\xi,\eta) = \xi h(\xi,\eta) + \eta k(\xi,\eta)$ which is easy with smooth $h$ and $k$ (using $\hat{\varphi}(0)=0$) but I do not see how to do this with entire $h$ and $k$ satisfying
the Paley-Wiener conditions for $\hat{\mathscr D}$.


(If the conjecture is true one gets a solution of the above mentioned problem since then the kernel of $v(\varphi)=\int \varphi(x,y) d(x,y)$ is contained in the kernel of $u$ and therefore $u$ is a multiple of $v$.)



Which test functions are the divergence of a vector field?
If you use use the Euclidean metric on $\mathbb{R}^2$ and apply Hodge duality, your question is about $H^2\_c(\mathbb{R}^2)$, the De Rham cohomology with compact supports on $\mathbb{R}^2$.


Bott & Tu, *Differential forms in Algebraic Topology*, Theorem 4.7 and Corollary 4.7.1 give $H\_c^2(\mathbb{R}^2) = \mathbb{R}$, where a generator of the non trivial cohomology class can be represented by any smooth compactly support form $\alpha$ such that $\int\_{\mathbb{R}^2} \alpha \ne 1$. Their method of proof is by constructing an explicit cochain homotopy equivalence between the cochain complexes $\Omega\_c^{{\*}-1}(\mathbb{R}^{n-1}) \leftrightarrow \Omega\_c^{\*}(\mathbb{R}^n)$.


So, your condition $\int \phi(x,y) d(x,y) = 0$ is both necessary and sufficient.


