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 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.