Nuclearity of imbedding operators of Sobolev classes into weighted spaces

Let $\Omega$ be an open set in $\mathbf R^m$. Denote by $d_x$ the distance from a point $x$ to the boundary of $\Omega$:
$$ d_x=\inf_{y\in\partial\Omega}|x-y|; $$
if $\Omega=\mathbf R^m$, then $d_x=1+|x|$. Define the class $\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)$ as the closure of $\mathbf C^\infty_0(\Omega)$ with respect to the norm
$$ \|f\|_{\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)}=\left(\int\limits_\Omega\left(\sum_{|\beta|=l}|D^\beta f|^p d^{-\lambda}_x+|f|^p d^{-pl-\lambda}_x\right)dx\right)^{1/p}; $$
here $l=1,2$; $1\le p<\infty$; $\lambda\in(-\infty,\infty)$. Let $\mu$ be a measure in $\Omega$ and $\mathbf L_q(\mu)$ the Lebesgue space. A criterion for the nuclearity of the imbedding of $\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)$ into $\mathbf L_q(\Omega)$ is given for $l>m$.