A shortcut to weighted representation formulas for holomorphic functions (Q1112212)

The method of explicit formulas for solving the \(\overline{\partial}\)- equation with estimates starts with the works of \textit{G. M. Khenkin}, Mat. Sb., n. Ser. 82(124), 300-308 (1970; Zbl 0206.091), and \textit{R. Ramirez de Arellano}, Math. Ann. 184, 172-187 (1970; Zbl 0189.097). In order to get better estimates the method was modified by \textit{B. Berndtsson} and the first author, Ann. Inst. Fourier 32, No.3, 91-110 (1982; Zbl 0466.32001) introducing weight factors. The purpose of this paper is to give a short proof of a generalization of representation formulas of B. Berndtsson. Let us give an impression. In Theorem 1 the following representation theorem is proved. Let \(\Omega \subseteq {\mathbb{C}}^ n\) be a domain and f be a holomorphic function on \(\Omega\), which is continuous up to the boundary. Assume that there exists appropriate functions G and \(q=(q^ 1,q^ 2,...,q^ p)\). Then for all \(z\in \Omega:\) \[ f(z)=\frac{1}{(2\pi i)^ n}\int_{\Omega}f(\zeta)\sum_{| \alpha | =n}\frac{D^{\alpha}G}{\alpha !}(\overline{\partial}q)^{\alpha}. \] Here for \(\alpha =(\alpha_ 1,\alpha_ 2,...,\alpha_ p)\), we have used the shorthand notation \(\alpha !=\prod^{p}_{j=1}\alpha_ j!\) and (\(\overline{\partial}q)^{\alpha}=(\overline{\partial}q^ 1)^{\alpha_ 1}\wedge...\wedge (\overline{\partial}q^ p)^{\alpha_ p}\).