Carnot Groups Research Articles

Isometries of Lipschitz‐free Banach spaces

AbstractWe describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3‐connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz‐free rigid space that has only three more points.

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

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

Constructing Hölder maps to Carnot groups

In this paper, we construct Hölder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group H \mathbb {H} . Pansu and Gromov [Carnot-Carathéodory spaces seen from within, Birkhäuser, Basel, 1996] observed that any surface embedded in H \mathbb {H} has Hausdorff dimension at least 3 3 , so there is no α \alpha -Hölder embedding of a surface into H \mathbb {H} when α > 2 3 \alpha >\frac {2}{3} . Züst [Anal. Geom. Metr. Spaces 3 (2015), pp. 73–92] improved this result to show that when α > 2 3 \alpha >\frac {2}{3} , any α \alpha -Hölder map from a simply-connected Riemannian manifold to H \mathbb {H} factors through a metric tree. In the present paper, we show that Züst’s result is sharp by constructing ( 2 3 − ϵ ) (\frac {2}{3}-\epsilon ) -Hölder maps from D 2 D^2 and D 3 D^3 to H \mathbb {H} that do not factor through a tree. We use these to show that if 0 > α > 2 3 0>\alpha > \frac {2}{3} , then the set of α \alpha -Hölder maps from a compact metric space to H \mathbb {H} is dense in the set of continuous maps and to construct proper degree-1 maps from R 3 \mathbb {R}^3 to H \mathbb {H} with Hölder exponents arbitrarily close to 2 3 \frac {2}{3} .

Besov Space via Heat Semigroup on Carnot Group and Its Capacity

Besov Space via Heat Semigroup on Carnot Group and Its Capacity

Escape from compact sets of normal curves in Carnot groups

In the setting of subFinsler Carnot groups, we consider curves that satisfy the normal equation coming from the Pontryagin Maximum Principle. We show that, unless it is constant, each such a curve leaves every compact set, quantitatively. Namely, the distance between the points at time 0 and time t grows at least of the order t^(1/s), where s denotes the step of the Carnot group. In particular, in subFinsler Carnot groups there are no periodic normal geodesics.

Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

In this paper we study Monge solutions to stationary Hamilton–Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

Some counterexamples to Alt–Caffarelli–Friedman monotonicity formulas in Carnot groups

Some counterexamples to Alt–Caffarelli–Friedman monotonicity formulas in Carnot groups

Loomis–Whitney inequalities on corank 1 Carnot groups

In this paper we provide another way to deduce the Loomis–Whitney inequality on higher dimensional Heisenberg groups \(\mathbb{H}^n\) based on the one on the first Heisenberg group \(\mathbb{H}^1\) and the known nonlinear Loomis–Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp–Lieb inequality and the subadditivity of the entropy developed in Carlen and Cordero-Erausquin (2009).

The set of intersections of several independent Brownian motions on Carnot group

In this paper the existence of intersections for functions of several Brownian motions on the Carnot group is studied. A condition is presented for the existence of such intersections with Probability 1, which is in the form of the asymptotics of a measure on a specific family of small balls. The measure is arbitrary but can be chosen as a surface measure on the manifold related to intersections, and the balls are constructed using the distances related to the processes. Additionally, if the same condition holds in a weaker form, it is shown that there is a Hausdorff measure, such that the value of this Hausdorff measure on the set of intersection points is finite with Probability 1.

Coercive inequalities on Carnot groups: taming singularities

In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson–Ornstein–Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators.

Maximal directional derivatives in Laakso space

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a [Formula: see text]-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a [Formula: see text]-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces, Carnot groups and Banach spaces. Hence, the techniques used in these spaces do not generalize to metric measure spaces.

A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and λ⩾0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\geqslant 0$$\\end{document}, we introduce a new notion of λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-subharmonicity for Lloc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1_\ extrm{loc}$$\\end{document}-functions, which we call localλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-shift defectivity, and which turns out to be equivalent to distributional λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-solutions of Δf⩽f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta f\\leqslant f$$\\end{document} for complete Riemannian manifolds.

Lower semicontinuity of distortion coefficients for homeomorphisms of bounded (1, \sigma )-weighted (q, p)-distortion on Carnot groups

In this paper we study the locally uniform convergence of homeomorphisms with bounded (1,σ)-weighted (q,p)-distortion to a limit homeomorphism. Under some additional conditions we prove that the limit homeomorphism is a mapping with bounded (1,σ)-weighted (q,p)-distortion. Moreover, we obtain the property of lower semicontinuity of distortion characteristics of homeomorphisms.

Sub-Riemannian Coarea Formula for Classes of Noncontact Mappings of Carnot Groups

Sub-Riemannian Coarea Formula for Classes of Noncontact Mappings of Carnot Groups

Lower Semicontinuity of Distortion Coefficients for Homeomorphisms of Bounded (1, σ)-Weighted (q, p)-Distortion on Carnot Groups

Lower Semicontinuity of Distortion Coefficients for Homeomorphisms of Bounded (1, σ)-Weighted (q, p)-Distortion on Carnot Groups

Symplectic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with sub-Riemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some Euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

AbstractIn this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range , being the homogeneous dimension of the group.

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

Topological Properties of Mappings with Finite Distortion on Carnot Groups

Topological Properties of Mappings with Finite Distortion on Carnot Groups

Schrödinger–Hardy system without the Ambrosetti–Rabinowitz condition on Carnot groups

Schrödinger–Hardy system without the Ambrosetti–Rabinowitz condition on Carnot groups