TY - JOUR

T1 - Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma

AU - Spector, Daniel

AU - Van Schaftingen, Jean

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Korn-Sobolev inequality

KW - Loomis-Whitney inequality

KW - Lorentz spaces

UR - http://www.scopus.com/inward/record.url?scp=85075206265&partnerID=8YFLogxK

U2 - 10.4171/RLM/854

DO - 10.4171/RLM/854

M3 - Article

AN - SCOPUS:85075206265

VL - 30

SP - 413

EP - 436

JO - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni

JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni

SN - 1120-6330

IS - 3

ER -