Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma

Daniel Spector; Jean Van Schaftingen

doi:10.4171/RLM/854

Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma

Daniel Spector, Jean Van Schaftingen

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

Abstract

We prove a family of Sobolev inequalities of the form (Equation presented) where A(D): Cl c (Rn;V) → Cl c (Rn;E) is a vector first-order homogeneous linear differential operator with constant coefficients, u is a vector field on Rn and L n n-1; 1(Rn) is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn-Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis-Whitney inequality and Gagliardo's lemma.

Original language	English
Pages (from-to)	413-436
Number of pages	24
Journal	Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
Volume	30
Issue number	3
DOIs	https://doi.org/10.4171/RLM/854
Publication status	Published - 2019

Keywords

Korn-Sobolev inequality
Loomis-Whitney inequality
Lorentz spaces

ASJC Scopus subject areas

General Mathematics

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10.4171/RLM/854

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Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma. / Spector, Daniel; Van Schaftingen, Jean.
In: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, Vol. 30, No. 3, 2019, p. 413-436.

Research output: Contribution to journal › Article › peer-review

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Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma

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