### 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 |
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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 | |

State | Published - 1 Jan 2019 |

### Keywords

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

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## Cite this

Spector, D., & Van Schaftingen, J. (2019). Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma.

*Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni*,*30*(3), 413-436. https://doi.org/10.4171/RLM/854