О задаче продолжения функции внутрь круга в пространствах с весом, имеющим особенность на границе

Abstract

It has been proven that any quadratically summable function with a power weight in the unit circle is uniquely represented as an orthogonal sum of its analytic and coanalytic components; therefore, it is natural to consider the coanalytic component of a function as a certain characteristic of its nonanalyticity. The article considers the problem of finding the extension of a function with the unit circle inside the circle that it would have the minimal deviation from the Sobolev weight subspace of analytic functions (the problem of minimizing the coanalytic deviation). Similar coanalytic problems were considered by other researchers in the weightless case for a unit circle, half-band, and an arbitrary bounded simply connected domain with a smooth boundary. The problem is formulated in spaces with a weight having a power singularity at the unit circle entire boundary, and the boundary values of the functions are taken from the corresponding Besov space. The known properties of weight spaces, including the direct and inverse theorems about the traces of functions from the considered classes are formulated. By using these properties, a theorem about the existence and uniqueness of the problem solution is proved in the framework of the ideas of the monotonic operators theory.