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Proceedings of the American Mathematical Society

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Norms of embeddings
of logarithmic Bessel potential spaces

Authors: David E. Edmunds, Petr Gurka and Bohumír Opic
Journal: Proc. Amer. Math. Soc. 126 (1998), 2417-2425
MSC (1991): Primary 46E35, 46E30
MathSciNet review: 1451796
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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 $t$. 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 $q$ provided that $q$ is large enough.

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Additional Information

David E. Edmunds
Affiliation: Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, Brighton BN1 9QH, England

Petr Gurka
Affiliation: Department of Mathematics, Czech University of Agriculture, 16521 Prague 6, Czech Republic

Bohumír Opic
Affiliation: Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 11567 Prague 1, Czech Republic

Keywords: Generalized Lorentz-Zygmund spaces, logarithmic Bessel potential spaces, Orlicz spaces of double and single exponential types, equivalent norms, embeddings
Received by editor(s): January 23, 1997
Additional Notes: This research was partially supported by grant no. 201/94/1066 of the Grant Agency of the Czech Republic and by NATO Collaborative Research Grant no. CRG 930358; the research of the second author was also partially supported by EPSRC grant no. GR/L02937.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society