Spaces of functions of fractional smoothness on an irregular domain (Q5895122)

A bounded domain \(G\) in \(\mathbb R^n\) is called irregular if it is not a cone domain. It is called a \(\sigma\)-domain, \(\sigma >1\), if (roughly speaking) any point of \(G\) admits an inward \(\sigma\)-cone which is roughly the local distortion of a peak of type \(x_n > |x'|^\sigma\), \(x' \in \mathbb R^{n-1}\), near \(0\). The paper deals with the extension of the Sobolev embeddings \[ W^m_p (G) \subset L_q (G), \;m \in\mathbb N, \;1<p,q< \infty, \;m - \tfrac{n}{p} + \tfrac{n}{q} \geq 0, \] for cone domains to \(\sigma\)-domains. For this purpose, the author introduces intrinsically defined spaces of type \(B^s_{pq} (G)\) and \(L^s_{pq} (G) = F^s_{pq} (G)\), adapted to the \(\sigma\)-domain \(G\), and proves sharp embeddings of type \[ W^m_p (G) \subset B^{s(m)}_{q \theta} (G) \quad \text{and} \quad W^m_p (G) \subset L^{s(m)}_{q \theta} (G). \] Personal comment: In my papers, I call the \(B^s_{pq}\) Besov spaces and leave \(L^s_{pq} = F^s_{pq}\) unnamed. Besov calls \(L^s_{pq} = F^s_{pq}\) Lizorkin-Triebel spaces and leaves \(B^s_{pq}\) unnamed.