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

### Daniel Spector

National Chiao Tung University, Hsinchu, Taiwan, National Taiwan University, Taipei, Taiwan and Washington University, S### Jean Van Schaftingen

Université Catholique de Louvain, Louvain-la-Neuve, Belgium

## Abstract

We prove a family of Sobolev inequalities of the form

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.

## Cite this article

Daniel Spector, Jean Van Schaftingen, Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 3, pp. 413–436

DOI 10.4171/RLM/854