If $v\in\ker\bar{\partial}\subset W^{1,2;\mu}$, then $v=fw$ for some real coefficient polynomial function $f$ on the upper half plane, such that $w$ is nonvanishing on $\mathbb{R}\setminus\{p_{1},\ldots p_{k}\}$, and vanishes minimally at $p_{i}$ (meaning $w$ is indivisible by $z-p_{i}$).