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Optimal function spaces in weighted Sobolev embeddings with α-homogeneous weights

Published online by Cambridge University Press:  19 June 2025

Ladislav Drážný Open the ORCID record for Ladislav Drážný [Opens in a new window]
Ladislav Drážný*
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 49/83, 186 75 Praha 8, Czech Republic Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic (draznyl2@seznam.cz)
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Abstract

We study weighted Sobolev inequalities on open convex cones endowed with α-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal rearrangement-invariant function spaces for these weighted Sobolev inequalities. Both optimal target and optimal domain spaces are characterized. Abstract results are accompanied by general yet concrete examples of optimal function spaces. For these examples, the class of so-called Lorentz–Karamata spaces, which contains in particular Lebesgue spaces, Lorentz spaces, and some Orlicz spaces, is used.

Keywords

Lorentz–Karamata spacesoptimal function spacesrearrangement-invariant function spacesreduction principleSobolev embeddingsweighted Sobolev spaces46E3046E35
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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