Estimates of the spectrum of some differential operators (Q1274033)

Let \(\Omega\) be an open set in \({\mathbb{R}}^m\), and let \(d_x=\inf_{y\in \partial\Omega}| x-y| \) be the distance from \(x\) to the boundary of \(\Omega\) (if \(\Omega={\mathbb{R}}^m\), set \(d_x=1+| x| \)). Define the class \(W^l_{p,\lambda}\) as the closure of \(C_0^\infty\) with respect to the norm \[ \| f\| =\left(\int_\Omega \left(\sum_{k=l}| D^kf| ^p d_x^{-\lambda}+ | f| ^p d_x^{-pl-\lambda}\right) dx \right)^{1/p}, \] with \(l=1,2\), \(1\leq p<\infty\); \(\lambda\in (-\infty,\infty).\) The author provides necessary and sufficient conditions under which the embedding operator \(J:W^l_{p,\lambda}\to L_2(\Omega,\mu)\), where \(\mu\) is an arbitrary Borel measure in \(\Omega\), belongs to the class \(\sigma_\rho, \rho >0.\)