Consider four-dimensional space $\mathbb{R}^2 \times \mathbb{C}$ with coordinates $w=x,y,z,\bar{z}$. Define $D^i = (\partial_x,\partial_y,4\partial_z)$, and consider the following equation for a Green's function:
\begin{equation} \frac{1}{\pi}(D^iD^k)\Delta_{k\ell}(w,w') = \,\delta^i_{\,\ell}\,\delta^{(4)}(w-w')\,. \end{equation} The solution is supposed to be \begin{equation} \label{PropagatoronM} \Delta_{ij}(w,w') = -\frac{c}{4\pi}\varepsilon_{ijk}D^k\bigg(\frac{1}{(x-x')^2+(y-y')^2+(z-z')(\bar{z}-\bar{z}')}\bigg) \,. \end{equation}
Away from $w=w'$, this is obvious from the antisymmetry of the Levi-Civita tensor. However, it is not clear to me why this is the case for $w=w'$. In fact, by computing $D^k\Delta_{kl}$ explicitly, I obtain zero. What am I missing?