TY - JOUR
T1 - Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma
AU - Spector, Daniel
AU - Van Schaftingen, Jean
N1 - Funding Information:
Acknowledgments. The authors would like to thank Wen-Wei Lin, the S. T. Yau Center at National Chiao Tung University, and the National Center for Theoretical Sciences of Taiwan for their support in the conference where this collaboration was initiated. The first author is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2, 107-2918-I-009-003 and 107-2115-M-009-002-MY2.
Publisher Copyright:
© 2019 European Mathematical Society Publishing House. All rights reserved.
PY - 2019
Y1 - 2019
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
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U2 - 10.4171/RLM/854
DO - 10.4171/RLM/854
M3 - Article
AN - SCOPUS:85075206265
SN - 1120-6330
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
IS - 3
ER -