An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in $\textbf {C}^{n}$

Abstract:

An integral representation formula for $(0,q)$ forms is constructed on a strictly pseudoconvex domain $D$ in $\mathbf {C}^n$ by using only the local geometry of the boundary of $D$. By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for $\bar {\partial }$ on $D$. The construction does not need any a priori knowledge of the solvability of $\bar {\partial }$ and thus allows us to establish fundamental global results by a direct and elementary method.