math.CV — Complex Variables

Supremum estimates for $\bar\partial$ on bounded convex domains in $\mathbb{C}^2$

Contributed by Yun-Heng Du

For every bounded convex domain $\Omega\subset\mathbb{C}^2$, we give a bounded distributional solution of $\bar\partial u=f$ for every distributionally $\bar\partial$-closed form $f\in L^\infty_{(0,1)}(\Omega)$. The solution satisfies $\|u\|_\infty\le C_\Omega\|f\|_\infty$, where $C_\Omega>0$ depends only on $\Omega$. The estimate follows from holomorphic localization of Leray rows and boundary cancellation.