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

Daniel Spector, Jean Van Schaftingen

Research output: Contribution to journalArticle

2 Scopus citations

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 languageEnglish
Pages (from-to)413-436
Number of pages24
JournalAtti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni
Volume30
Issue number3
DOIs
StatePublished - 1 Jan 2019

Keywords

  • Korn-Sobolev inequality
  • Loomis-Whitney inequality
  • Lorentz spaces

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