Let $\Omega\subseteq\mathbb R^2$ be open and $w:\Omega\to\mathbb R$. I've read that there is a unique $v:\Omega\to\mathbb R^2$ with \begin{align}\nabla\cdot v&=0,\tag1\\\int v(x)\:{\rm d}x&=0,\tag2\end{align} and $$w=\nabla\wedge v:=\frac{\partial f_2}{\partial x_1}-\frac{\partial f_1}{\partial x_2}.\tag3$$
(a) How can we formulate this claim rigorously, i.e. which are the function spaces $W$ and $V$ we need to restrict $v$ and $w$ so that $(1)$ to $(3)$ are well-defined and the claim holds true? I'd be interested in this question for both ordinary (Fréchet) differentiability and distributional/weak differentiability.
(b) How can we prove the rigorous formulation of the claim?
Since I could imagine that this is related to the Helmholtz-Leray decomposition (and it's been awhile, since I've worked in this field), let me briefly recap what I already knew about it: $$G:=\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega,\mathbb R^2)\}$$ is a closed subspace of $L^2(\Omega,\mathbb R^2)$. Let $\operatorname P_H$ denote the orthogonal projection of $L^2(\Omega,\mathbb R^2)$ onto $$H:=G^\perp=\{u\in L^2(\Omega,\mathbb R^2):\langle\nabla p,u\rangle_{L^2(\Omega,\:\mathbb R^2)}=0\text{ for all }\nabla p\in G\}.$$ Moreover, $$E:=\left\{p\in H^1(\Omega):\int_\Omega p(x)\:{\rm d}x=0\right\}$$ is a closed subspace of $H^1(\Omega)$. If $\Omega$ is bounded and $\partial\Omega$ is Lipschitz, then $$G=\{\nabla p:p\in H^1(\Omega)\}.\tag4$$ If $u\in L^2(\Omega,\mathbb R^2)$ and $\Omega$ is bounded and connected and $\partial\Omega$ is Lipschitz, then there is a unique $p\in E$ with $$(\nabla p,u-\nabla p)\in G\times H,\tag5$$ i.e. $$\operatorname P_Hu=u-\nabla p.\tag6$$