We prove a family of Sobolev inequalities of the form
where is a vector first-order homogeneous linear differential operator with constant coefficients, is a vector field on and 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. 413–436DOI 10.4171/RLM/854