On the solvability of the Cauchy-Riemann equation with constraints of the growth of the functions and weighted approximation of analytic functions. (Russian) Zbl 0696.30041

Epifanov, O. V.

Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 2(333), 49-52 (1990).

Let \(\phi\) be a subharmonic function in a domain \(\Omega\) such that \(\phi =\sup \{\log | f|:\) f is holomorphic in \(\Omega\) and log\(| f| \leq \phi\) in \(\Omega\) \(\}\). Let g be a measurable function such that \(| g(z)| \leq e^{\phi (z)}/| z|\) in \(\Omega\). Then there exists a measurable function f such that \[ | f(z)| \leq c(1+| \log | z| |)e^{\phi (z)} \] on \(\Omega\) and \(\partial f/\partial \bar z=g\) on \(\Omega\) in the distributional sense, where c is a constant. The theorem is applied to the problem of approximation of analytic functions of a given growth by analytic functions of a smaller growth.

Reviewer: J.Siciak

MSC:

30E99	Miscellaneous topics of analysis in the complex plane
31A05	Harmonic, subharmonic, superharmonic functions in two dimensions

Keywords:

subharmonic function