The trace class property of embedding operators of Sobolev classes into weighted spaces (Q1128376)

Let \(\overset\circ W^\ell_{p,\lambda}(\Omega)\) be the weighted Sobolev space, \(\Omega \subset \mathbb{R}^m\), defined as the closure of \(C^\infty_0(\Omega)\) with respect to the norm \[ \| f\| _{\overset\circ W^\ell_{p,\lambda}(\Omega)}= \left(\int_\Omega \left(\sum_{| \beta| =\ell}| D^\beta f| ^p[\rho(x)]^{-\lambda} + | f| ^p [\rho(x)]^{-p\ell-\lambda}\right)dx\right)^\frac{1}{p} , \] where \(\rho(x)\) is the distance from \(x\) to the boundary of \(\Omega\) in the case \(\Omega \neq R^m\) and \( \rho(x) = 1+ | x| \) in the case when \(\Omega= R^m\); \(\ell = 1,2,3,\dots\), \(1\leq p < \infty\), with the usual modification in the case \(p=\infty\). The following result is announced, in which \(\mu\) is an arbitrary measure on \(\Omega\) and \[ \beta_n = r_n^\frac{(p\ell+\lambda-m)}{p}\bigg(\mu(B_n)\bigg)^ \frac{1}{q} , \] where \(r_n\) and \(B_n\) arise from the Whitney type covering, \(B_n\) being the ball of the radius \(r_n= \gamma \rho (x_n)\), centered at \(x_n\), \(0<\gamma < 1\). Theorem 2. Let \(J\) be the embedding operator from \(\overset\circ W^\ell_{p,\lambda}(\Omega)\) into \(L_q(\Omega,\mu)\) and let \(\ell > m\). Then (a) For \(1\leq q\leq p \leq \infty\), the operator \(J\) is nuclear if and only if \(\sum_{n}\beta_n < \infty\); (b) For \(1\leq p < q \leq \infty\), the operator \(J\) is nuclear if and only if \(\sum_{n}\beta_n^s < \infty\), where \(\frac{1}{s}=1-\frac{1}{p}+ \frac{1}{q}\). The author remarks that the proof is based essentially on his approach developed in [\textit{O. G. Parfenov}, Math. Nachr. 154, 105-115 (1991; Zbl 0761.47009)].