On the Nikol'skii Classes of Polyharmonic Functions

K. O. Besov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Investigations in the theory of differentiable functions of many variables and its applications. Part 18, Tome 227 (1999), pp. 43-55

Résumé

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 $D^m$.