Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Abstract

To Haïm Brezis who has taught me so much and has asked me many inspiring questions; on the occasion of his 70th birthday, with admiration and gratitude Abstract. J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L 1 of singular integral estimates and the critical Sobolev space W 1,n (R n ). This paper presents an overview of the results, further developments over the last ten years and challenging open problems. Mathematics Subject Classification. 35A23 26D15 46E35.Keywords. Critical Sobolev spaces, Gagliardo-Nirenberg-Sobolev inequality, canceling differential operator, overdetermined elliptic systems, underdetermined systems. Theme Limiting Hodge theory for Sobolev formsThe study of limiting estimates for systems of linear differential equations starts from the following problem: given a function g ∈ L n (R n ; R n ), find the best regularity that a vector field u : R n → R n can have such that div u = g in R n .(1.1)If n ≥ 2, equation (1.1) is strongly underdetermined. The standard way of finding a solution u consists in lifting the underdeterminacy by solving the system div u = g in R n , curl u = 0 in R n , (1.2)

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

“…These sparse directional Sobolev estimates into Lorentz space are analogous to L 2 estimates of de Figueiredo [12] and strengthen known results for Sobolev estimates into L n n−1 [7,Remark 16;33;41,Proposition 6.8].…”

Anisotropic Ornstein noninequalities

“…due to Gagliardo and Nirenberg had been generalized to the anisotropic case only in [16] and finally in [9]; if one deals with similar embeddings for vector fields, the isotropic case was successfully considered in [14] (see also the survey [15]), and there is almost no progress for anisotropic case (however, see [7,8]).…”

L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

“…In this paper, we prove (1−P oincaré) for h-forms of degree h < n in de Rham's complex (Ω • , d). We rely on Lanzani-Stein's observation (see [26]) that the duality estimate (emphasized by van Schaftingen [44]) underlying Bourgain-Brezis' result descends from (n−1)-forms to forms of lower degree, and the resulting Gagliardo-Nirenberg inequalities.…”