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
Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma cover
Download PDF

A subscription is required to access this article.

Abstract

We prove a family of Sobolev inequalities of the form

uLnn1,1(Rn,V)CA(D)uL1(Rn,E)\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):Cc(Rn,V)Cc(Rn,E)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, uu is a vector field on Rn\mathbb{R}^n and Lnn1,1(Rn)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