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We prove a family of Sobolev inequalities of the form $$  \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)}  \le  \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} $$ where $A (D) : C^\infty_c(\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-orderhomogeneous linear differential operator with constant coefficients, $u$ is avector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ isa Lorentz space. These new inequalities imply in particular the extension ofthe classical Gagliardo-Nirenberg inequality to Lorentz spaces originally dueto Alvino and a sharpening of an inequality in terms of the deformationoperator by Strauss (Korn-Sobolev inequality) on the Lorentz scale. The proofrelies on a nonorthogonal application of the Loomis--Whitney inequality andGagliardo's lemma.

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