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Title of Journal: Math Z

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Abbravation: Mathematische Zeitschrift

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Springer-Verlag

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DOI

10.1016/1044-5803(94)90135-x

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1432-1823

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On weighted critical imbeddings of Sobolev spaces

Authors: D E Edmunds H Hudzik M Krbec
Publish Date: 2010/02/16
Volume: 268, Issue: 1-2, Pages: 585-592
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Abstract

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces We characterize weights which do not change an exponential space up to equivalence of norms Specifically we first prove that L exp talphachi B=L exp talpharho if and only if rhoq in L q with some q  1 Second we consider the Sobolev space W1 NvarOmega where Ω is a bounded domain in mathbbRN with a sufficiently smooth boundary and its imbedding into a weighted exponential Orlicz space L exp tpvarOmegarho where ρ is a radial and nonincreasing weight function We show that there exists no effective weighted improvement of the standard target L exp tNvarOmega=L exp tNvarOmegachi varOmega in the sense that if W1 NvarOmega is imbedded into L exp tpvarOmegarho then L exp tpvarOmegarho and L exp tNvarOmega coincide up to equivalence of the norms that is we show that there exists no effective improvement of the standard target space The same holds for critical cases of higherorder Sobolev spaces and even Besov and Lizorkin–Triebel spacesH Hudzik and M Krbec appreciate the support of the grant No 1 PO3A 01127 of the State Committee for Scientific Research Poland M Krbec gratefully acknowledges the support of the Academy of Sciences of the Czech Republic Institutional Research Plan No AVOZ10190503 and of the grant No 201/06/0400 of GA ČR

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