Sobolev embedding into BMO and weak-L∞ for 1-dimensional probability measure

We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensional probability μ (e.g. log-concave measure) such that the Sobolev embedding‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ)) holds for any function u∈L1(R,μ), whose real-valued weakly derivative u′ belongs to X. Here BMO(R,μ) is th...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2015-02, Vol.422 (1), p.478-495
Main Authors: Feo, Filomena, Martin, Joaquim, Posteraro, M. Rosaria
Format: Article
Language:English
Subjects:
1-dimensional log-concave probability measure
BMO space
Embedding
Rearrangement invariant space
Weak-[formula omitted] space
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Summary:We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensional probability μ (e.g. log-concave measure) such that the Sobolev embedding‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ)) holds for any function u∈L1(R,μ), whose real-valued weakly derivative u′ belongs to X. Here BMO(R,μ) is the space of functions with bounded mean oscillation with respect to μ. We investigate the embedding in weak-L∞(R,μ), too.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2014.08.045