On the Nikol'skii Classes of Polyharmonic Functions

This paper is devoted to the study of the properties of polyharmonic functions defined on the unit ball $D^m$ of the Euclidean space $\mathbb R^m$, $D^m = \{x\in\mathbb R^m\mid |x|<1\}$. With the help of the well-known Almansi decomposition, the polyharmonic function is represented as a sum of components, each of which has a simple form. The main idea, developed in [1–3], is that, under a suitable choice of components, the behavior of these components near the boundary of the ball $D^m$ is no worse than that of the polyharmonic function itself. Here, in view of the smoothness at internal points, the boundary behavior of a polyharmonic function is naturally characterized by its membership in a certain functional class on $D^m$.