 Cauchy-Green operator

For a smooth function $g$ compactly supported in a disc $D\subset \mathbb{C}$, the Cauchy-Green operator isHere $w=x+iy$ and $dw\wedge d\bar{w}=-2idx\wedge dy$, fixing the sign between the two forms of the integral. The kernel is locally integrable, andIndeed, the distribution $1/(\pi z)$ is a fundamental solution of ${\partial }_{\bar{z}}$, so differentiating its convolution with $g$ gives ${\delta }_0\ast g=g$. Smooth compactly supported input makes the convolution smooth and the identity classical.^[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.