Boundary theorems of the Gehring-Lohwater and Plessner type for polyanalytic functions
The theory of boundary properties of analytic functions is a beautiful and intensively elaborated chapter of complex analysis. There exist a lot of well-known results as the theorems of Fatou, F. and M. Riesz, Plessner, Luzin-Privalov a.o. It is impossible to transfer these results in their classical formulations to polyanalytic functions [see: \textit{M. B. Balk}, Polyanalytic functions (1991; Zbl 0764.30038)], these are the solutions of \(\partial^ n_{\overline z}u=0\). Nevertheless, in the present paper the authors describe a very large class of functions for which a lot of classical results remain true. The main idea is to assume, that the areolar derivative \(\partial_{\overline z}u\) belongs to \(L_ p(G)\), \(p>2\). Then every such function can be represented in the form \(u=h+g\), where \(h\) is analytic in \(G\) and \(g\) is continuous in \(\overline G\). Consequently, there is a close connection between the boundary properties of \(u\) and \(h\).
scientific article; zbMATH DE number 428725