= Cauchy-Green operator
{c}
For a smooth <function> $g$ compactly supported in a <disc> $D\subset\mathbb C$, the Cauchy-Green operator is
$$
Tg(z)=\frac1{2\pi i}\int_D\frac{g(w)}{w-z}\,dw\wedge d\bar w
=\frac1\pi\int_D\frac{g(w)}{z-w}\,dA(w).
$$
Here $w=x+iy$ and $dw\wedge d\bar w=-2i\,dx\wedge dy$, fixing the sign between the two forms of the integral. The kernel is locally integrable, and
$$
\partial_{\bar z}Tg=g.
$$
Indeed, the <distribution> $1/(\pi z)$ is a <fundamental solution> of $\partial_{\bar z}$, so differentiating its <convolution> with $g$ gives $\delta_0*g=g$. Smooth compactly supported input makes the convolution smooth and the identity classical.<Complex Analytic and Differential Geometry by Jean-Pierre Demailly>[Chapter I, Cauchy formula (3.2), Corollary (3.4) and Remark (3.5), printed p.21]{ref}
For local data without compact support, multiply by a smooth <cutoff function> equal to one on a smaller disc. The resulting operator solves the original equation there. Applying this construction to coefficients in one coordinate at a time gives the local primitives in the <Dolbeault-Poincaré lemma>. The solution need not itself have compact support.
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