We prove a family of Sobolev inequalities of the form (Equation presented) where A(D): Cl c (Rn;V) → Cl c (Rn;E) is a vector first-order homogeneous linear differential operator with constant coefficients, u is a vector field on Rn and L n n-1; 1(Rn) 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.
|Number of pages||24|
|Journal||Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni|
|State||Published - 1 Jan 2019|
- Korn-Sobolev inequality
- Loomis-Whitney inequality
- Lorentz spaces