A CRITERION FOR APPROXIMABILITY BY HARMONIC FUNCTIONS IN LIPSCHITZ SPACES

摘要

Let X be a compact subset of R~3, and let f be a function that is harmonic inside X and belongs to the Lipschitz space C~γ(X), 0 ＜ γ ＜ 1. A criterion for approximability of f on X in C~γ(X) by functions that are harmonic on neighborhoods of X is obtained in terms of the Hausdorff content of order 1+γ. The proof is completely constructive, and Vitushkin's method of singularities separation and approximation by parts is applied. Bibliography: 15 titles.