For a divergence free field $w\in [L^2(\mathbb{R}^d)]^d$, especially for $d = 3$, it is well known that there exists a skew-symmetric matrix $\psi \in [W^{1,2}_{loc}(\mathbb{R}^d)]^{d\times d}$ with $\nabla \psi_{ij} \in [L^2(\mathbb{R}^d)]^d$ such that $w = \text{div }\psi$, the divergence of the matrix $\psi$ is computed along its rows and the following estimates exists: $$ \|\nabla \psi_{ij}\|_{L^2(\mathbb{R}^d)} \leq C \|w\|_{L^2(\mathbb{R}^d)}. $$
On any bounded region $\omega$ and $\Omega$ with Lipschitz boundary or any nice properties and $\omega \subset \Omega \subset \mathbb{R}^d$, I wonder whether we still have the same type estimate for the above $\psi_{ij}$ and $w$: $$ \|\nabla \psi_{ij}\|_{L^2(\omega)} \leq C' \|w\|_{L^2(\omega)}, $$ where $C'$ is independent of $\omega$ but could depend on $\Omega$.