Norms of embeddings of logarithmic Bessel potential spaces

Let $\Omega$ be a subset of $\mathbb{R}\sp{n}$ with finite volume, let $\nu >0$ and let $\Phi$ be a Young function with $\Phi (t) = \exp (\exp t\sp{\nu })$ for large $t$. We show that the norm on the Orlicz space $L\sb {\Phi } (\Omega )$ is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in $L\sp{q}(\Omega )$ which are sharp in their dependences on $q$ provided that $q$ is large enough.