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Norms of embeddings of logarithmic Bessel potential spaces

Authors:	David E Edmunds Petr Gurka Bohumí r Opic

Institution:	Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, Brighton BN1 9QH, England ; Department of Mathematics, Czech University of Agriculture, 16521 Prague 6, Czech Republic ; Mathematical Institute, Academy of Sciences of the Czech Republic, Zitná 25, 11567 Prague 1, Czech Republic

Abstract:	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 . We show that the norm on the Orlicz space $L\sb {\Phi } (\Omega )$ is equivalent to $\begin{equation}\sup \sb {1<q<\infty } (e+\log q)\sp{-1/\nu } \\|f\\|\sb {L\sp{q}(\Omega )}. \end{equation}$ 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 provided that is large enough.

Keywords:	Generalized Lorentz-Zygmund spaces logarithmic Bessel potential spaces Orlicz spaces of double and single exponential types equivalent norms embeddings

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