Intrinsic metrics and boundary values of functions of the Zygmund classes (Q1177540)

Zygmund spaces \(\Lambda^ k\) are also known as the limiting case of Nikol'skii-Besov spaces \(B^ k_{\infty,\infty}=H^ k_ \infty\), \(k\in N\). The authors consider the spaces \(\Lambda^ k(G)\), \(G\) being an arbitrary region in \(R^ n\), \(\Lambda^ k(G)\subseteq H^ k_ \infty(G)\) under the approximative definition of \(\Lambda^ k(G)\) the authors use. To characterize boundary values of functions in \(\Lambda^ k(G)\), the authors apply the approach developed by the second author [see e.g. Function Spaces Appl. Proc. US -- Swedish Seminar. Lund, Sweden, June 1986, 397-409 (1988; Zbl 0646.46028), or Sib. Mat. J. 30, No. 2, 29- 42 (1989; Zbl 0692.46025)]. This approach is based on the usage of the internal metrics \(d_ G(x,y)\), its definition depending on the functional space considered. The application of this approach to Zygmund spaces required special adaptation because of the second differences involved. The norm in \(\Lambda^ k(G)\) is introduced in terms of \(d_ G(x,y)\) which in this case is the infimum of length of rectifiable curves connecting \(x\) and \(y\) in \(G\). Let \(G_ 1=(G,d_ G)\) and \(\partial G_ 1=\tilde G_ 1\backslash G_ 1\) where \(\tilde G_ 1\) is the Hausdorff completion of \(G_ 1\). The main results are the following: Theorem 1 on boundary values (the trace operator Tr\(_ k:\Lambda^ k(G)\to\Lambda^ k(G_ 1)\) is bounded with the norm not exceeding 1); Theorem 2.1 (on the existence of the bounded extension operator Ext: \(\Lambda^ k(\partial G_ 1)\to\Lambda^ k(G)\) and Theorems 3.1 and 3.2 on the extension beyond the boundary.