On imbedding theorems for a natural extension of the sobolev class $W^l_p(\Omega)$

In this paper the class $W^l_{p,\varphi}(\Omega,g)$ of functions is considered which have generalized derivatives of order $l$ in the region $\Omega$ and finite norm
\begin{gather*} |f;W^l_{p,\varphi}(\Omega,g)|=|f;L_p(g)|+|f;L^l_{p,\varphi}(\Omega)| \\ (|f;L^l_{p,\varphi}(\Omega)|=\sum_{|r|=l}|\varphi D^rf;L_p(\Omega)|), \end{gather*}
where $g$ is a bounded interior subregion of the region $\Omega$, and $\varphi$ a weight that degenerates on the boundary $\partial\Omega$ or at infinity. Continuous and completely continuous imbeddings $W^l_{p,\varphi}(\Omega,g)\to L^k_{p,\varphi_r}(\Omega)$ $(0\leqslant k<l)$ are obtained.