Intrinsic Lipschitz graphs in Carnot groups of step 2

Donato, Daniela Di

doi:10.5186/aasfm.2020.4556

Cited by 6 publications

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“…The study of the relation between being a broad* solution to the system D ϕ ϕ = ω with a continuous ω and the intrinsic regularity of graph( ϕ) was first initiated, in H n for L one-dimensional in [ASCV06, Section 5], and then continued in [BSC10a,BSC10b,BCSC14]. For the case G = H n and L horizontal and k-dimensional see also [Cor19] and for the general case of Carnot groups of step 2 and L one-dimensional, see [DD18,DD19]. In Proposition 3.27 we give a sufficient condition for the map ϕ to be a broad* solution to the system D ϕ ϕ = ω, with ω continuous.…”

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 99%

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Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Antonelli,

Di Donato,

Don

et al. 2020

Preprint

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In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are C 1 H regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph.We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group.As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map. Contents 0. Notation 3 1. Introduction 3 1.1. An historical account of the notion of C 1 H -surface in Carnot groups 3 1.2. Main theorems 5 1.3. Structure of the paper 9 2. Preliminaries 10 2.1. Carnot groups 10 2.2. Little Hölder continuous functions 12 2.3. Intrinsic surfaces, Intrinsically Lipschitz and Intrinsically differentiable functions 13 3. Intrinsic projected vector fields on subgroups 19 3.1. Definition of D ϕ and main properties 20 3.2. Invariance properties of D ϕ 24 3.3. Metric properties of integral curves of D ϕ 27 3.4. Definition of broad* 34 4. Main Theorems in arbitrary Carnot groups 36 4.1. From regularity of ϕ along integral curves of D ϕ to regularity of ϕ 38 4.2. Area formula for codimension-one graphs in terms of intrinsic derivatives 45 4.3. Relations between intrinsic differentiability and local approximability 49

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Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 99%

“…For the expression of the projected vector fields in Example 3.6 we refer also to [DD18, Definition 5.2]. In the papers [DD18] and [DD19] the author started to generalize the results already proved in the Heisenberg groups H n (see Remark 3.5) to Carnot groups of step 2, in case L is one-dimensional.…”

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 99%

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Antonelli,

Di Donato,

Don

et al. 2020

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“…Let U ⊆ W be an open set, and let ϕ: U → L be a continuous function. Then the following conditions are equivalent Item (g) allows us to complete the chain of implications of Theorem 1.1 in the setting of step-2 Carnot groups generalizing the results scattered in [ASCV06,BSC10a,BSC10b] where the authors study the same problem in the Heisenberg groups, and [DD20a,DD20b] where partial generalizations of the results in [ASCV06,BSC10a,BSC10b] are obtained in the case of step-2 Carnot groups.…”

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Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

Antonelli¹,

Donato²,

Don³

2020

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We prove that in arbitrary Carnot groups G of step 2, with a splitting G = W•L with L one-dimensional, the graph of a continuous function ϕ: U ⊆ W → L is C 1 H -regular precisely when ϕ satisfies, in the distributional sense, a Burgers' type system D ϕ ϕ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group.As a tool for the proof we show that a continuous distributional solution ϕ to a Burgers' type system D ϕ ϕ = ω, with ω continuous, is actually a broad solution to D ϕ ϕ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to D ϕ ϕ = ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions.

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Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

2022

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We prove that in arbitrary Carnot groups $\mathbb {G}$ G of step 2, with a splitting $\mathbb {G}=\mathbb {W}\cdot \mathbb {L}$ G = W ⋅ L with $\mathbb {L}$ L one-dimensional, the intrinsic graph of a continuous function $\varphi \colon U\subseteq \mathbb {W}\to \mathbb {L}$ φ : U ⊆ W → L is $C^{1}_{\mathrm {H}}$ C H 1 -regular precisely when φ satisfies, in the distributional sense, a Burgers’ type system Dφφ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution φ to a Burgers’ type system Dφφ = ω, with ω continuous, is actually a broad solution to Dφφ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to Dφφ = ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions.

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Intrinsic Lipschitz graphs in Carnot groups of step 2

Cited by 6 publications

References 0 publications

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

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