A subelliptic Bourgain–Brezis inequality

Wang, Yi; Yung, Po-Lam

doi:10.4171/jems/443

Cited by 8 publications

(30 citation statements)

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“…This approximation result has been extended to some subscale of Triebel-Lizorkin spaces when rank T = 1 [20, Proposition 3.1] and to classical Sobolev spaces in the noncommutative setting of homogeneous groups [110,Lemma 1.7].…”

Section: About the Proofsmentioning

confidence: 99%

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

Schaftingen

2014

J. Fixed Point Theory Appl.

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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)

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Section: About the Proofsmentioning

confidence: 99%

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

Schaftingen

2014

J. Fixed Point Theory Appl.

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“…This seminal work has been followed by a number of approximation results of similar type [4], [5], [13], [6]. Our work is primarily motivated by two types of developments of the results in [13], [6] concerning functions in critical Sobolev spaces that barely fail the embedding in L 1 .…”

Section: Introductionmentioning

confidence: 99%

“…The rst of these results (Lemma 1.7 in [13]) deals with the extension of the approximation result given in [4] (Theorem 11) in the Euclidean case, to the more general case of strati ed homogeneous groups. Somewhat informally this reads (see Section 2 for de nitions): Theorem 1.1 Suppose G is a strati ed homogeneous group whose homogeneous dimension is Q and let X 1 ; :::; X n 1 be a minimal family of vector elds generating the Lie algebra of G. Then, for any Schwartz function f on G and any > 0 there exists a function F such that:…”

Section: Introductionmentioning

confidence: 99%

Approximation of critical regularity functions on stratified homogeneous groups

Curcă

2020

Commun. Contemp. Math.

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Let [Formula: see text] be a stratified homogeneous group with homogeneous dimension [Formula: see text] and whose Lie algebra is generated by the left-invariant vector fields [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text]. We prove that for any function [Formula: see text] there exists a function [Formula: see text] such that [Formula: see text] [Formula: see text] where [Formula: see text] is the largest integer smaller than [Formula: see text] and [Formula: see text] is a positive constant depending only on [Formula: see text]. Here, [Formula: see text] is a homogeneous Triebel–Lizorkin type space adapted to [Formula: see text]. This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9(2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal. 276(5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.

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“…Open problem 1.4. In [33], the conclusion of [5] is extended to a subelliptic context, namely the case of the Heisenberg groups endowed with a subelliptic Laplacian. The extension of Theorems 1.1 and 1.2 to the case of the Heisenberg group is an open problem.…”

Section: Let Us End Up This Introduction With Three Open Problemsmentioning

confidence: 99%

“…Step 2 Estimates on G m , H m and their derivatives. Let us collect the upper bounds for G m and H m which will be needed in the sequel (see [33,). In the lemmata below, the implicit constants may depend on |γ| but neither on m, σ, R nor on f .…”

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confidence: 99%

Approximation in higher-order Sobolev spaces and Hodge systems

Bousquet

Russ

Wang

et al. 2019

Journal of Functional Analysis

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Let d ≥ 2 be an integer, 1 ≤ l ≤ d − 1 and ϕ be a differential l-form on R d withẆ 1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on R d with coefficients in L ∞ ∩Ẇ 1,d such that dϕ = dψ. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order spaceẆ 2,d/2 or more generally in the fractional Sobolev spacesẆ s,p with sp = d. We give a positive answer to this question, provided that d − κ ≤ l ≤ d − 1, where κ is the largest positive integer such that κ < min(p, d). The proof relies on an approximation result (interesting in its own right) for functions inẆ s,p by functions inẆ s,p ∩L ∞ , even thoughẆ s,p does not embed into L ∞ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.

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A subelliptic Bourgain–Brezis inequality

Cited by 8 publications

References 1 publication

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

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

Approximation of critical regularity functions on stratified homogeneous groups

Approximation in higher-order Sobolev spaces and Hodge systems

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