Norms of embeddings of logarithmic Bessel potential spaces

Abstract

Let Ω \Omega be a subset of R n \mathbb {R}^{n} with finite volume, let ν > 0 \nu >0 and let Φ \Phi be a Young function with Φ ( t ) = exp ⁡ ( exp ⁡ t ν ) \Phi (t) = \exp (\exp t^{\nu }) for large t t . We show that the norm on the Orlicz space L Φ ( Ω ) L_ {\Phi } (\Omega ) is equivalent to sup 1 > q > ∞ ( e + log ⁡ q ) − 1 / ν ‖ f ‖ L q ( Ω ) . \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 ( Ω ) L^{q}(\Omega ) which are sharp in their dependences on q q provided that q q is large enough.