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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
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

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