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Embeddings of homogeneous Sobolev spaces on the entire space

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

Published online by Cambridge University Press:  09 March 2020

Zdeněk Mihula Open the ORCID record for Zdeněk Mihula [Opens in a new window]
Zdeněk Mihula*
Affiliation:
Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
*
mihulaz@karlin.mff.cuni.cz
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Abstract

We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$, where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

Keywords

Optimal function spacesRearrangement-invariant spacesReduction principleSobolev spaces

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 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 47B38: Operators on function spaces (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 296 - 328
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Copyright
Copyright © The Author(s), 2020. Published by The Royal Society of Edinburgh

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