Norms of embeddings of logarithmic Bessel potential spaces

Abstract:

Let $\Omega$ be a subset of $\mathbb {R}^{n}$ with finite volume, let $\nu >0$ and let $\Phi$ be a Young function with $\Phi (t) = \exp (\exp t^{\nu })$ for large $t$. We show that the norm on the Orlicz space $L_ {\Phi } (\Omega )$ is equivalent to \begin{equation*}\sup _ {1<q<\infty } (e+\log q)^{-1/\nu } \|f\|_ {L^{q}(\Omega )}. \end{equation*} We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in $L^{q}(\Omega )$ which are sharp in their dependences on $q$ provided that $q$ is large enough.