Decomposition of approximable harmonic functions
Let \(\Omega\) be an open subset of \(\mathbb{R}^n\) \((n\geq 2)\) and let \(E\) be a relatively closed subset of \(\Omega\). Let \({\mathcal H}(\Omega)\) denote the collection of harmonic functions on \(\Omega\), and let \(A(E)\) denote the collection of real-valued functions on \(E\) that can be uniformly approximated on \(E\) by elements of \(H(\Omega)\). The paper's main result characterizes the elements of \(A(E)\) in terms of decompositions. A corresponding theorem for holomorphic functions, subject to some restrictions on the (plane) sets \(\Omega\) and \(E\), was proved by \textit{A. Stray} [Ann. Math., II. Ser. 120, 225-235 (1984; Zbl 0568.30035)] in response to a question of N. U. Arakelyan (1970). Statements of the main results require some more notation. Let \(C_e(E)\) denote the collection of real-valued functions on \(E\) that have a continuous extension to the closure of \(E\) in the one-point compactification of \(\mathbb{R}^n\). Let \(A_e(E)\) denote the collection of functions on \(E\) that can be uniformly approximated on \(E\) by functions in \({\mathcal H}(\Omega)\) whose restriction to \(E\) belong to \(C_e(E)\). Finally, let \(\widetilde{E}\) denote the union of \(E\) with all the components of \(\Omega\setminus E\) from which the Alexandroff point of \(\Omega\) is not accessible. The main theorem states that for a real-valued function on \(E\) the following are equivalent: (i) \(u\in A(E)\); (ii) \(u=v+h\) for some \(v\in A_e(\widetilde{E})\) and \(h\in{\mathcal H}(\Omega)\); (iii) \(u=v+h\) for some \(v\in C_e(\widetilde{E})\) which is finely harmonic on the fine interior of \(E\) and some \(h\in{\mathcal H}(\Omega)\). As a corollary, it is shown that in the case \(n=2\) we have \(u\in A(E)\) if and only if \(u=v+h\) for some \(v\in C_e(\widetilde{E})\cap{\mathcal H}((\widetilde{E})^0)\) and \(h\in{\mathcal H}(\Omega)\). The proof of the main theorem makes use of a new theorem of independent interest concerning the approximation of finely harmonic functions by harmonic functions.
scientific article; zbMATH DE number 1031506