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# Rendiconti Lincei - Matematica e Applicazioni

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**Volume 30, Issue 3, 2019, pp. 413–436**

**DOI: 10.4171/RLM/854**

Published online: 2019-09-02

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

Daniel Spector^{[1]}and Jean Van Schaftingen

^{[2]}(1) National Chiao Tung University, Hsinchu, Taiwan, National Taiwan University, Taipei, Taiwan and Washington University, S

(2) Université Catholique de Louvain, Louvain-la-Neuve, Belgium

We prove a family of Sobolev inequalities of the form $$ \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} \le C \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} $$ where $A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-order homogeneous linear differential operator with constant coefficients, $u$ is a vector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ 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.

*Keywords: *Korn–Sobolev inequality, Lorentz spaces, Loomis–Whitney inequality

Spector Daniel, Van Schaftingen Jean: Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma. *Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.* 30 (2019), 413-436. doi: 10.4171/RLM/854