Function spaces as Dirichlet spaces (about a paper by Maz'ya and Nagel) (Q558462)

The starting point of the paper is the definition of the space \(H^{\mu}(\mathbb R^ n)\), formulated as the closure of \(C^{\infty}_c(\mathbb R^ n)\) in the norm \[ \| u\| _{H^ {\mu}}^2:=\int_{\mathbb R^ n}| \hat u(\xi)| ^2\mu(\xi)\,d\xi + \int_{\mathbb R^ n}| \hat u(\xi)| ^ 2\,d\xi. \] \textit{W.~Maz'ya} and \textit{J.~Nagel} [Beitr.\ Anal.\ 12, 7--17 (1978; Zbl 0422.46029)] considered function spaces defined by integrals of differences of functions in the spirit of Slobodeckij spaces and found an equivalent norm, namely, \[ \int_{\mathbb R^ n}\left(\int_{\mathbb R^ n}| u(x+y)-u(x)| ^2N(y)\, dy\right)dx + \int_{\mathbb R^ n}| u(x)| ^2\,dx, \] where \(N(y)\) is a~weight function depending on an anisotropic measure. The authors notice that the latter norm describes a~Hilbert space invariant under contractions. Their main result states a~norm equivalence analogous to that of Maz'ya and Nagel for complete Bernstein functions. The paper contains several interesting auxiliary results and a~quite impressive table of complete Bernstein functions.