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

Abstract

We prove a family of Sobolev inequalities of the formis a vector first-order homogeneous linear differential operator with constant coefficients, u is a vector field on R n and L n n−1 ,1 (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.

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

“…Recently, a related (dual) notion of "ℓ-canceling" operators has been introduced in [30]. We conclude this introduction by remarking that the above results can be used to provide dimensional estimates and rectifiability results for measures whose decomposability bundle, defined in [2], has dimension at least ℓ. Namely, in this case the measure is absolutely continuous with respect to I ℓ and the set where the upper ℓ-dimensional density is positive, is rectifiable, compare with [11, Theorem 2.19] and with [3].…”

Endpoint $L^1$ estimates for Hodge systems

“…A program in this direction was pioneered in the seminal work of J. Bourgain and H. Brezis [5] (see also [6,25]) and received remarkable contributions from L. Lanzani and E. Stein [14] and J. Van Schaftingen [26][27][28], while endpoint fine parameter improvements on the Lorentz [11,23] and Besov-Lorentz [24] scales have only recently been obtained.…”

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

“…These operators have been recently considered in e.g. [17,46,47,37,10,57]. The deviator operator E D u := Eu − 1 n (div u) Id n is C-elliptic for n ≥ 3, but not for n = 2 (see Remark 2.5).…”