= Distributional Cauchy-Riemann lemma
{title2=$\partial_{\bar z}u=0\Rightarrow u\text{ holomorphic}$}
A <distribution> $u$ on a plane <open set> with $\partial_{\bar z}u=0$ is represented by a <holomorphic function>. Indeed $\Delta u=4\partial_z\partial_{\bar z}u=0$, and <elliptic regularity> makes $u$ smooth. For a direct proof, mollify locally with a radial <mollifier> $\eta_\varepsilon$. The resulting <smooth functions> are harmonic. Their <mean value property> gives $(u*\eta_\varepsilon)*\eta_r=u*\eta_\varepsilon$ on a smaller ball for fixed small $r$. Passing to the <weak convergence of distributions> limit gives $u*\eta_r=u$ there. This proves smoothness without assuming <local integrability> beforehand. The ordinary <Cauchy-Riemann equations> now prove holomorphy.
The equation $\partial_zu=0$ similarly characterizes <antiholomorphic functions>. On the whole plane, the homogeneous solutions are respectively arbitrary <entire functions> $h(z)$ and the functions $h(\bar z)$ with $h$ entire. Consequently any two <fundamental solutions> for one of these <Wirtinger derivatives> differ by the corresponding homogeneous solution.
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