Spector, Daniel; van Schaftingen, Jean Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma. (English) Zbl 1442.46027 Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 30, No. 3, 413-436 (2019). Summary: 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 A. Alvino [Boll. Unione Mat. Ital., V. 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