Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

Citti, Giovanna; Manfredini, Maria; Pinamonti, Andrea; Cassano, Francesco Serra

doi:10.1007/s00526-013-0622-8

Cited by 34 publications

(63 citation statements)

References 34 publications

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“…dµ E ≤ C(n) e(1) ≤ C ′ (n) e(k 3 ), (5.20) where C(n) and C ′ (n) are dimensional constants. On the other hand, again by [8,…”

Section: Approximation Via Maximal Functionsmentioning

confidence: 91%

“…Let A ⊂⊂ W be fixed. Then, by the area formula (2.16), For the proof, see [8,Proposition 3.6]. Finally, the following result compares the distance d ϕ with the distance d ∞ in W .…”

Section: Approximation Via Maximal Functionsmentioning

confidence: 96%

“…Finally, we prove (5.15). Since D 1 = K ∪ J with disjoint union, we can split (5.19) On the one hand, by [8,Proposition 4.4] and by (5.2), we have K |∇ ϕ ϕ| 2 dL 2n = gr(ϕ| K )…”

Section: Approximation Via Maximal Functionsmentioning

confidence: 99%

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Improved Lipschitz approximation of H-perimeter minimizing boundaries

Monti

Stefani

2017

Journal de Mathématiques Pures et Appliquées

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We prove two new approximation results of H-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group H n with n ≥ 2. The first one is an improvement of [19] and is the natural reformulation in H n of the classical Lipschitz approximation in R n . The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro, [11]. 1 2 R. MONTI AND G. STEFANIHere, we prove two new intrinsic Lipschitz approximation theorems for H-perimeter minimizers in the setting of the Heisenberg group H n with n ≥ 2.The first result is an improvement of [19] and is the natural reformulation in H n of the classical Lipschitz approximation in R n , see [17, Theorem 23.7]. Let W = R × H n−1 be the hyperplane passing through the origin and orthogonal to the direction ν = −X 1 . The disk D r ⊂ W centered at the origin is defined using the natural box norm of H n and the cylinder C r (p), p ∈ H n , is defined as C r (p) = p * C r , where C r = D r * (−r, r). We denote by e(E, C r (p), ν) the excess of E in C r (p) with respect to the fixed direction ν, that is, the L 2 -averaged oscillation of ν E , the inner horizontal unit normal to E, from the direction ν in the cylinder. The 2n + 1-dimensional spherical Hausdorff measure S 2n+1 is defined by the natural distance of H n . Finally, ∇ ϕ ϕ is the intrinsic gradient of ϕ. We refer the reader to Section 2 for precise definitions.Theorem 1.1. Let n ≥ 2. There exist positive dimensional constants C 1 (n), ε 1 (n) and δ 1 (n) with the following property. If E ⊂ H n is an H-perimeter minimizer in the cylinder C 5124 with 0 ∈ ∂E and e(E, C 5124 , ν) ≤ ε 1 (n) then, letting

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“…dµ E ≤ C(n) e(1) ≤ C ′ (n) e(k 3 ), (5.20) where C(n) and C ′ (n) are dimensional constants. On the other hand, again by [8,…”

Section: Approximation Via Maximal Functionsmentioning

confidence: 91%

“…Let A ⊂⊂ W be fixed. Then, by the area formula (2.16), For the proof, see [8,Proposition 3.6]. Finally, the following result compares the distance d ϕ with the distance d ∞ in W .…”

Section: Approximation Via Maximal Functionsmentioning

confidence: 96%

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Improved Lipschitz approximation of H-perimeter minimizing boundaries

Monti

Stefani

2017

Journal de Mathématiques Pures et Appliquées

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“…The notion of intrinsic Lipschitz function in H is defined with this characterization in mind, with "graph" replaced by "intrinsic graph", and "cone" replaced by a natural H-analogue, see (2.2). Intrinsic Lipschitz functions were introduced by B. Franchi, R. Serapioni and F. Serra Cassano in [27], and they turned out to be very influential in the evolution of geometric analysis in Heisenberg groups, see for instance [1,3,5,9,14,29,28,39]. Curiously, the definition does not guarantee that an intrinsic Lipschitz function is (metrically) Lipschitz between the spaces W y,t and V x .…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group

Chousionis¹,

Fässler²,

Orponen³

2019

American Journal of Mathematics

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The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group H. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in H has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in R 3 .Our main object of study are the intrinsic Lipschitz graphs in H, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that these 3dimensional sets in H, if any, deserve to be called quantitatively 3-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical β-numbers. Conversely, extending a result of David and Semmes from R n , we prove that a 3-Ahlfors-David regular subset in H, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.

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“…More broadly, a great effort has been made to understand differentiability properties of Lipschitz functions in different settings, like stratified groups, Banach spaces and more general metric measure spaces. We mention only a few papers [2,4,7,8,9,10,16,21,25,29,33,34,35] to give a glimpse of the many works in this area.…”

Section: Introductionmentioning

confidence: 99%

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

Magnani

Pinamonti

Speight

2019

Annali di Matematica

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Let f be a Lipschitz map from a subset A of a stratified group to a Banach homogeneous group. We show that directional derivatives of f act as homogeneous homomorphisms at density points of A outside a σ-porous set. At all density points of A we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property.

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Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

Cited by 34 publications

References 34 publications

Improved Lipschitz approximation of H-perimeter minimizing boundaries

Improved Lipschitz approximation of H-perimeter minimizing boundaries

Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

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