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Maximal noncompactness of limiting Sobolev embeddings

Part of: Linear function spaces and their duals Nonlinear operators and their properties Special classes of linear operators

Published online by Cambridge University Press:  18 November 2024

Jan Lang Open the ORCID record for Jan Lang [Opens in a new window] ,
Zdeněk Mihula Open the ORCID record for Zdeněk Mihula [Opens in a new window]  and
Luboš Pick Open the ORCID record for Luboš Pick [Opens in a new window]
Jan Lang
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174 and Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic ([email protected])
Zdeněk Mihula
Affiliation:
Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic ([email protected]) (corresponding author)
Luboš Pick
Affiliation:
Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic ([email protected])
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Abstract

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited—namely the absolute continuity of the target quasinorm.

We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis–Wainger type are so-called maximally noncompact, i.e. their ball measure of noncompactness is the worst possible.

Keywords

limiting Sobolev embeddingsmaximal noncompactnessmeasure of noncompactnessSobolev spacesweighted Lorentz spaces

MSC classification

Primary: 47B06: Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
Secondary: 47H08: Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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