Sharp Sobolev type embeddings on the entire Euclidean space

Angela Alberico; Andrea Cianchi; Luboš Pick; Lenka Slavíková

doi:10.3934/cpaa.2018096

Article Contents

2018, Volume 17, Issue 5: 2011-2037. Doi: 10.3934/cpaa.2018096

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Sharp Sobolev type embeddings on the entire Euclidean space

1.
Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy
2.
Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
3.
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
4.
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received: September 2017

Revised: January 2018

Published: April 2018

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Abstract

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space ${\mathbb R^n}$, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Keywords:

Sobolev embeddings on $ {\mathbb{R} ^n}$,
optimal target spaces,
rearrangementinvariant spaces,
Orlicz-Sobolev spaces,
Lorentz-Sobolev spaces.

Mathematics Subject Classification: 46E35, 46E30.

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References

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