Solution formulas for the \(\partial {\bar\partial}\)-equation and weighted Nevanlinna classes in the polydisc. (English) Zbl 0598.32007

The interest in the Poincaré-Lelong equation \(i\partial {\bar \partial}u=\theta\) is because of its connection with the set of zeros of a holomorphic function of a certain Nevanlinna class. [See P. Lelong, Bull. Soc. Math. France 85, 239-262 (1957; Zbl 0079.309); G. M. Khenkin, Sov. Math., Dokl. 16(1975), 1310-1314 (1976); translation from Dokl. Akad. Nauk SSSR 224, 771-774 (1975; Zbl 0333.35056)]. Inspired by the theorem of Sh. A. Dautov and G. M. Khenkin in Math. USSR, Sb. 35, 449-459 (1979); translation from Mat. Sb., n. Ser. 107(149), 163-174 (1978; Zbl 0392.32001), Charpentier and Khenkin- Polyakov considered some weighted Nevanlinna classes in the polydisc \(D^ n=\{z\in {\mathbb{C}}^ n: \max \{| z_ j|: 1\leq j\leq n\}<1\}.\)
In this paper the author introduces some other weighted Nevanlinna classes \(N(d\lambda_{\alpha})\), corresponding to the product weights \(d\lambda_{\alpha}=\prod^{n}_{j=1}(1-| \zeta_ j|^ 2)^{\alpha_ j}d\lambda (\zeta_ j),\) where \(\alpha =(\alpha_ 1,...,\alpha_ n)\) and \(\alpha_ j>-1\). He gets necessary and sufficient conditions for the existence of solutions in \(N(d\lambda_{\alpha})\) of the Poincaré-Lelong equation for the polydisc in the complex n-space (Theorem 1). Explicit formulas for solutions of the Poincaré-Lelong equation in the polydisc allow to get estimates.
Reviewer: R.Braun


32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs