The coarea formula for real‐valued Lipschitz maps on stratified groups

Magnani, Valentino

doi:10.1002/mana.200310334

Cited by 23 publications

(37 citation statements)

References 30 publications

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“…Due to Proposition 4.2, the differential equations (4.1) hold. Applying (4.1) for i = 2, [18] Towards differential calculus in stratified groups 93 hence we can writeγ 2 (t) aṡ…”

Section: Differentiability Of Curves In Graded Groupsmentioning

confidence: 99%

“…The intriguing feature of Theorem 1.1, along with its technical difficulty, stems from the fact that we do not assume any 'extrinsic' regularity of f . Here it is worth emphasizing the existence of continuously differentiable functions that are not C 1 in the Euclidean sense and in this sense they are not differentiable on sets of positive measure [18].…”

Section: Introductionmentioning

confidence: 99%

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Towards Differential Calculus in Stratified groups

Magnani¹

2013

J. Aust. Math. Soc.

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We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.2010 Mathematics subject classification: primary 22E30; secondary 26B10, 26B12.

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“…Due to Proposition 4.2, the differential equations (4.1) hold. Applying (4.1) for i = 2, [18] Towards differential calculus in stratified groups 93 hence we can writeγ 2 (t) aṡ…”

Section: Differentiability Of Curves In Graded Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Towards Differential Calculus in Stratified groups

Magnani¹

2013

J. Aust. Math. Soc.

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“…can be found in [28] and [49]. Due to H Q−1 -negligibility of characteristic points this formula holds for any set E with C 1 boundary and any measurable set of G. As a consequence, we can positively answer a conjecture raised by Danielli, Garofalo and Nhieu in [15].…”

Section: Introductionmentioning

confidence: 73%

“…They conjecture the validity of (27) for arbitrary stratified groups. Due to Theorem 2.5 of [49] (see also Theorem 3.5 of [28]) we have the formula…”

Section: Definition 33 (Rectifiability) We Say That a Subsetmentioning

confidence: 99%

Characteristic points, rectifiability and perimeter measure on stratified groups

Magnani

2006

J. Eur. Math. Soc.

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Abstract.We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q−1 restricted to ∂E, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q−1 (∂E) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The crucial ingredient of this result is the negligibility of "characteristic points" of the boundary. We introduce the notion of "horizontal point", which extends the notion of characteristic point to arbitrary submanifolds, and we prove that the set of horizontal points of a k-codimensional submanifold is S Q−k -negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, called (G, R k )-rectifiability, and we prove that Euclidean k-codimensional rectifiable sets are (G, R k )-rectifiable.

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“…There are several results concerning sub-Riemannian coarea formulae for realvalued functions; see, for instance, [28] and [27] for relevant discussions and references. Here we just point out that (5) extends previous coarea formulae obtained in [22], [26], [27] and [30].…”

Section: Introductionmentioning

confidence: 99%

Non-horizontal submanifolds and coarea formula

Magnani

2008

J Anal Math

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Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.

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The coarea formula for real‐valued Lipschitz maps on stratified groups

Cited by 23 publications

References 30 publications

Towards Differential Calculus in Stratified groups

Towards Differential Calculus in Stratified groups

Characteristic points, rectifiability and perimeter measure on stratified groups

Non-horizontal submanifolds and coarea formula

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