\(\overline\partial\)-closure of forms representable by a Koppelman integral on the basis of logarithmic residue (Q1022223)

The description of the integral characteristic properties of differential forms has been of big interest in multicomplex analysis for a long period, and it still is now. The paper under the review is clearly stimulated by the results of Kytmanov (1978) for the \((p, q)\)-forms with coefficients of the class \(C^2(\overline{\Omega})\), \(\overline{\Omega} \subset {\mathbb C}^n\), and also by the question of the \(\overline{\partial}\)-closure of \(CR\)-forms developped mostly by Khenkin (1985). As preliminary statements are obtained: an analogon of the Green formula in the basis of the Koppelman-Bochner-Martinelly kernel of Dolbeault cohomology, and also the Koppelman transform on the basis of logarithmic residue. Using these two statements, and other auxiliary lemmas, the author formulates and proves different theorems about \(\overline{\partial}\)-closure for the forms represented by the Koppelman integral. Of a main interest are {\parindent5mm \begin{itemize}\item[1.] the space of complex \((p, q)\)-forms \(F\) with coefficients of class \(C^2\) which satisfy the Hodge type condition \(\overline{\partial} * \overline{\partial} F =0\); \item[2.] the space of pluriharmonic \((p, q)\)-forms with coefficients of class \(C^2\) which satisfy the condition \(\overline{\partial} \overline{\partial} F =0\); \item[3.] the space of \(\overline{\partial}\)-closed \((p, q)\)-form in \(\Omega\); \item[4.] smooth type analogous of the above three spaces of differential forms. \end{itemize}} The paper is written very pedantically with long formulas, difficult to be interpreted in words.