Sub-Riemannian limit of the differential form spectrum of contact manifolds

Rumin, Michel

doi:10.1007/s000390050013

Cited by 41 publications

(30 citation statements)

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“…It is not known whether the symmetry assumption is necessary for this inequality to hold [34,Open problem 1]. This result was used to obtain the Gagliardo-Nirenberg inequality for forms in the Rumin complex [78,79,80,81] and for forms on the Heisenberg groups H 1 and H 2 [10]. Since the Rumin complex contains higher-order differential operators, the higher-order estimates play a crucial role in the proof.…”

Section: Noncommutative Situationsmentioning

confidence: 95%

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: Noncommutative Situationsmentioning

confidence: 95%

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

Schaftingen

2014

J. Fixed Point Theory Appl.

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“…In [4,5], adiabatic limits were used to study the spectral geometry for Riemannian foliations. In [6], Rumin studied the adiabatic limits of some geometric objects for contact manifolds.…”

Section: Introductionmentioning

confidence: 99%

Adiabatic limits, vanishing theorems and the noncommutative residue

Liu

Wang

2009

Sci. China Ser. A-Math.

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“…Such approximations have been widely used since the mid-80's in a variety of contexts. As example we recall the work of Debiard [33], Koranyi [55,56], Ge [45], Rumin [77] as well as the references in [67] and [68]. More recently this technique has been used in the study of minimal surfaces and mean curvature flow in the Heisenberg group.…”

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confidence: 99%

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

Capogna

Citti

2015

Bull. Math. Sci.

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We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175-1198, 2013) and Manfredini (Anal Geom Metric Spaces 1:255-275, 2013) concerning stability of doubling properties, Poincare' inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hörmander vector fields.

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Sub-Riemannian limit of the differential form spectrum of contact manifolds

Cited by 41 publications

References 10 publications

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

Adiabatic limits, vanishing theorems and the noncommutative residue

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

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