Analogues of Zalcman's problem and their applications (Q2901759)

The present paper is devoted to the analogue of Zalcman's problem for regular triangle and tetrahedron. The case, when function has zero integral over all regular triangles (tetrahedrons), which is tangent to given one by inner way, is fully analyzed. It is proved that such functions are with necessity equal to zero almost everywhere. We also obtain some analogue of theorem of Thompson and Shconbek for discrete set of parameters \(\alpha\). With the help of proved results the new criterion of holomorphy of functions for regular triangle is obtained, the result about completeness of some system of functions in \(L_p\) and the analogue of Dzyadyk's theorem are proved, the new result about homeomorphisms with Lusin's \(N\)-property is obtained.