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On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case

Author: Jan Vybíral
Journal: Proc. Amer. Math. Soc. 138 (2010), 141-146
MSC (2000): Primary 46E35, 46E30
Published electronically: September 2, 2009
MathSciNet review: 2550178
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Abstract: We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $ B^s_{p,q}(\mathbb{R}^n)$ and $ F^s_{p,q} (\mathbb{R}^n)$ in the limiting case $ s=\sigma_p:=n\max(\frac 1p-1,0)$. These results may also be reformulated as optimal embeddings into the scale of Lorentz spaces $ L_{p,q}(\mathbb{R}^n)$. We close several open problems outlined already in [H. Triebel, The structure of functions, Birkhäuser, Basel, 2001] and explicitly stated in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall/CRC, Boca Raton, FL, 2007].

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Jan Vybíral
Affiliation: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07740 Jena, Germany

Keywords: Besov spaces, Triebel-Lizorkin spaces, rearrangement invariant spaces, Lorentz spaces, growth envelopes
Received by editor(s): July 14, 2008
Published electronically: September 2, 2009
Communicated by: Andreas Seeger
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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