  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-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.
