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

Abstract

We prove a family of Sobolev inequalities of the form $$\Vert u{\Vert }_{L^{\frac{n}{n-1},1}({\mathbb{R}}^n,V)}\le \Vert A(D)u{\Vert }_{L^1({\mathbb{R}}^n,E)}$$ where $A(D):C_c^{\infty }({\mathbb{R}}^n,V)\to C_c^{\infty }({\mathbb{R}}^n,E)$ 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.

20 pages