Whitney decomposition, embedding theorems, and interpolation in weighted spaces of analytic functions

F. A. Shamoyan^a, E. V. Tasoeva<sup>b

a</sup> Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia
^b Bryansk State University, 14 Bezhitskaya St., Bryansk 241036, Russia

Abstract: According to the classical Whitney theorem, each open set on the plane can be decomposed as a union of special squares whose interiors do not intersect. In the paper, using the properties of Whitney squares, a new concept is introduced. For each center $a_k$ of the Whitney square, there is a point $a_k^*\in \mathbb{C}\setminus G$ such that the distance to the boundary of the open set $G$ is between two constants, regardless of $k$. In particular, a necessary and sufficient condition for a sequence $(z_k)_1^{\infty}\subset G$ under which the operator $R(f)=(f(z_1),f(z_2),\ldots,f(z_n),\ldots)$ maps generalized Nevanlinna's flat classes in a domain $G$ of a complex plane in $l^p$.

Key words: Nevanlinna class, interpolation, Witny decomposition, Berman space.

UDC: 517.53

MSC: 30H15, 32A35

Received: 28.02.2018

DOI: 10.23671/VNC.2019.1.27735