= Coordinate elimination for local Dolbeault primitives
For a $\bar\partial$-closed positive-antiholomorphic-degree <differential form of type (p, q)> on a <polydisc>, write $\varphi=d\bar z_j\wedge\alpha+\beta$ with no $d\bar z_j$ in $\alpha,\beta$. Apply the <Cauchy-Green operator> in coordinate $j$ to the coefficients of $\alpha$. Subtracting its <Dolbeault operator> derivative removes every $d\bar z_j$ term. Closedness then makes the remaining coefficients holomorphic in $z_j$. Eliminating successive coordinates preserves holomorphicity in earlier coordinates and produces a local primitive after finitely many steps.
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