*Preprint*

**Inserted:** 9 nov 2018

**Last Updated:** 9 nov 2018

**Pages:** 20

**Year:** 2018

**Abstract:**

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.