Function spaces of Lizorkin-Triebel type on an irregular domain (Q5901947)

The author introduces \(\sigma\)-domains \(G\) in \(\mathbb R^n\), \(\sigma \geq 1\), satisfying the flexible \(\sigma\)-cone condition. Such domains may have peaks of type \[ \{ x = (x', x_n): |x'| < x_n^\sigma \} \] near the origin. Here, \(\sigma =1\) corresponds to Lipschitz domains. In such domains, spaces \(L^{s(m)}_{pq} (G)\), \(1 \leq p <\infty\), \(1\leq q \leq \infty\), \(0<s<m\), in terms of polynomial approximations of order \(m\), are considered. If \(\sigma =1\), then \(L^{s(m)}_{pq} (G)\) coincides with a Lizorkin-Triebel space. Let \(W^m_{p} (G)\) be the usual Sobolev spaces. The paper deals with embeddings of type \(W^m_p (G) \subset L^{s(m)}_{pq} (G).\)