I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if anyone knew about similar results in $H(\text{curl}, D)$ where D is some regular compact domain of $\mathbb{R}^3$. I'm especially interested in the subspace of zero divergence vector fields.
Thank you very much.
