Davide Vittone
Davide Vittone has a total of 20 co-authors. Here are the top co-authors by number of shared publications:
- Marco Di Marco3 shared publications
- Andrea Pinamonti
Dipartimento di Matematica, Universitá di Trento, Povo (TN), Italy
2 shared publications - Antoine Julia
Département de Mathématiques d’Orsay, Université Paris-Saclay, 91400 Orsay, France
2 shared publications - Francesco Boarotto
SISSA
2 shared publications - Francesco Rossi
Università Iuav di Venezia, Dipartimento di Culture del Progetto Ca’ Tron, S. Croce, 1957 30135, Venezia, Italy
2 shared publications - Kilian Zambanini
Dipartimento di Matematica Università degli Studi di Trento 38123 Povo (Trento), Italy
2 shared publications
Latest Publications
- 18 February 2026
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
Besicovitch’s \frac{1}2 problem and linear programming
We consider the following classical conjecture of Besicovitch: a 1 1 -dimensional Borel set in the plane with finite Hausdorff 1 1 -dimensional measure H 1 \mathcal {H}^1 which has lower density strictly larger than 1 2 \frac {1}{2} almost everywhere must be countably rectifiable. We improve the best known bound, due to Schechter, showing that the statement is indeed true if 1 2 \frac {1}{2} is replaced by 7 10 \frac {7}{10} . Concerning Schechter’s bound, which is in fact an improvement of an argument by Preiss and Tišer, and it is valid in any metric space, we give a simpler proof. More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.
- 18 February 2026
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
Besicovitch’s \frac{1}2 problem and linear programming
We consider the following classical conjecture of Besicovitch: a 1 1 -dimensional Borel set in the plane with finite Hausdorff 1 1 -dimensional measure H 1 \mathcal {H}^1 which has lower density strictly larger than 1 2 \frac {1}{2} almost everywhere must be countably rectifiable. We improve the best known bound, due to Schechter, showing that the statement is indeed true if 1 2 \frac {1}{2} is replaced by 7 10 \frac {7}{10} . Concerning Schechter’s bound, which is in fact an improvement of an argument by Preiss and Tišer, and it is valid in any metric space, we give a simpler proof. More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.
- Open Access
Renormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups
In this paper we obtain the well-posedness of the transport and continuity equations in the Heisenberg groups $\mathbb{H}^n$ for a class of contact vector fields $\mathbf b$, under natural assumptions on the regularity of $\mathbf b$ not covered by the, now classical, Euclidean theory [18]. It is the first example of well-posedness in a genuine sub-Riemannian setting, that we obtain adapting to the $\mathbb{H}^n$ geometry the mollification strategy of [18]. In the final part of the paper we illustrate why our result is not covered by the Euclidean $BV$ case solved by the first author in [1], and we compare it with the strategy of [7], based on the representation of the commutator by interpolation à la Bakry-Émery and an integral representation of the symmetrized derivative of $\mathbf b$.Open AccessRenormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups
In this paper we obtain the well-posedness of the transport and continuity equations in the Heisenberg groups $\mathbb{H}^n$ for a class of contact vector fields $\mathbf b$, under natural assumptions on the regularity of $\mathbf b$ not covered by the, now classical, Euclidean theory [18]. It is the first example of well-posedness in a genuine sub-Riemannian setting, that we obtain adapting to the $\mathbb{H}^n$ geometry the mollification strategy of [18]. In the final part of the paper we illustrate why our result is not covered by the Euclidean $BV$ case solved by the first author in [1], and we compare it with the strategy of [7], based on the representation of the commutator by interpolation à la Bakry-Émery and an integral representation of the symmetrized derivative of $\mathbf b$. - Open Access
Submanifolds with boundary in sub-Riemannian Heisenberg Groups
We discuss the notion of submanifolds with boundary with intrinsic $C^1$ regularity in sub-Riemannian Heisenberg groups and we provide some examples. Eventually, we present a Stokes' Theorem for such submanifolds involving the integration of Rumin's differential forms in Heisenberg groups.Open AccessSubmanifolds with boundary in sub-Riemannian Heisenberg Groups
We discuss the notion of submanifolds with boundary with intrinsic $C^1$ regularity in sub-Riemannian Heisenberg groups and we provide some examples. Eventually, we present a Stokes' Theorem for such submanifolds involving the integration of Rumin's differential forms in Heisenberg groups. - Open Access
- 17 November 2024
- Published by Walter de Gruyter GmbH in Advances in Calculus of Variations
- Vol. 18 (3) , 963-978
Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
- 17 November 2024
- Published by Walter de Gruyter GmbH in Advances in Calculus of Variations
- Vol. 18 (3) , 963-978
Open AccessStepanov differentiability theorem for intrinsic graphs in Heisenberg groups
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
- Open Access
Stepanov Differentiability Theorem for intrinsic graphs in Heisenberg groups
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
Open AccessStepanov Differentiability Theorem for intrinsic graphs in Heisenberg groups
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
- Open Access
Submanifolds with boundary and Stokes' Theorem in Heisenberg groups
We introduce and study the notion of $C^1_\mathbb{H}$-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for $C^1_\mathbb{H}$-regular submanifolds with boundary that takes into account Rumin's complex of differential forms in Heisenberg groups.
Open AccessSubmanifolds with boundary and Stokes' Theorem in Heisenberg groups
We introduce and study the notion of $C^1_\mathbb{H}$-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for $C^1_\mathbb{H}$-regular submanifolds with boundary that takes into account Rumin's complex of differential forms in Heisenberg groups.
- Open Access
- 22 December 2022
- Published by EDP Sciences in ESAIM: Control, Optimisation and Calculus of Variations
- Vol. 28, 75
The Sard problem in step 2 and in filiform Carnot groups,
We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in Le Donne et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1639–1666] and settles a question emerged in Ottazzi and Vittone [ESAIM: COCV 25 (2019) 18]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.
- 22 December 2022
- Published by EDP Sciences in ESAIM: Control, Optimisation and Calculus of Variations
- Vol. 28, 75
Open AccessThe Sard problem in step 2 and in filiform Carnot groups,
We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in Le Donne et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1639–1666] and settles a question emerged in Ottazzi and Vittone [ESAIM: COCV 25 (2019) 18]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.
- Top Cited PapersOpen Access
- 2 February 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10
Lipschitz graphs and currents in Heisenberg groups
The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.- 2 February 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10
Top Cited PapersOpen AccessLipschitz graphs and currents in Heisenberg groups
The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups. - Top Cited PapersOpen Access
- 18 January 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1369-1406
Area of intrinsic graphs and coarea formula in Carnot groups
We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ regularity ( $$C^1_H$$ ). Our first main result is an area formula for $$C^1_H$$ intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ submanifolds into level sets of a $$C^1_H$$ function.- 18 January 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1369-1406
Top Cited PapersOpen AccessArea of intrinsic graphs and coarea formula in Carnot groups
We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ regularity ( $$C^1_H$$ ). Our first main result is an area formula for $$C^1_H$$ intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ submanifolds into level sets of a $$C^1_H$$ function. - 17 November 2021
- Published by Elsevier in Systems & Control Letters
- Vol. 158, 105078
On the Lebesgue measure of the boundary of the evoluted set
- 17 November 2021
- Published by Elsevier in Systems & Control Letters
- Vol. 158, 105078
On the Lebesgue measure of the boundary of the evoluted set