Necessary and sufficient conditions of compactness of certain embeddings of Sobolev spaces

Necessary and sufficient conditions on an open set $\Omega\subset \mathbb{R}^n$ are obtained ensuring that for $l,m\in\mathbb{N}_0$, $m < l$ the embedding $\mathring{W}_\infty^l(\Omega)\subset W_\infty^m(\Omega)$ is compact, where $W_\infty^m(\Omega)$ is the Sobolev space and $\mathring{W}_\infty^l(\Omega)$ is the closure in $W_\infty^l(\Omega)$ of the space of all infinitely continuously differentiable functions on $\Omega$ with supports compact in $\Omega$.