Local convergence of Newton’s method on the Heisenberg group

2016

Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

Annales Academiae Scientiarum Fennicae Mathematica, 2017

In this manuscript, we prove uniqueness and existence results of viscosity solutions for a class of linear second-order equations in the Heisenberg group. We state uniqueness by proving a comparison result to our class of equations, and existence via an application of Perron's method adapted to our framework. We also provide the explicit construction of the appropriate sub-and supersolutions employed by Perron's method for a variety of domains in the Heisenberg group.

Communications on Pure and Applied Analysis, 2015

In this work, we propose and analyse approximation schemes for fully non-linear second order partial differential equations defined on the Heisenberg group. We prove that a consistent, stable and monotone scheme converges to a viscosity solution of a second order PDE on the Heisenberg group provided that comparison principles exists for the limiting equation. We also provide examples where this technique is applied.

Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

2009

General local convergence theorems with order of convergence r ≥ 1 are provided for iterative processes of the type x n+1 = Tx n , where T : D ⊂ X → X is an iteration function in a metric space X . The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Dočev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695-701; J.F. Traub, H. Woźniakowski, Convergence and complexity of Newton iteration for operator equations, J.

Advances in Applied Mathematics, 2014

It is well known that the system ∂ x f = a, ∂ y f = b on R 2 has a solution if and only if the closure condition ∂ x b = ∂ y a holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y). This paper deals with a similar problem, where the vector fields ∂ x , ∂ y are replaced by the Heisenberg vector fields X 1 , X 2. In this case the sub-Riemannian system X 1 f = a, X 2 f = b has a solution f if and only if the following integrability conditions hold X 2 1 b = (X 1 X 2 + [X 1 , X 2 ])a, X 2 2 a = (X 2 X 1 + [X 2 , X 1 ])b. The question addressed by this paper is whether we can provide a Poincaré-type Lemma for the Heisenberg distribution. The positive answer is given by Theorem 2, which provides a result similar to the Poincaré's Lemma in the integral form. The solution f in this case is the work done by the force vector field aX 1 + bX 2 along any horizontal curve from the origin to the current point.

We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of N and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.

Comptes Rendus Mathematique, 2014

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,2,3] of pseudo-differential calculi on graded groups. The relation between the Weyl quantisation and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in . Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group. © 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Dans cette note, nous présentons un calcul pseudo-différentiel symbolique sur le groupe de Heisenberg. Nous particularisons à ce groupe notre construction générale [4,2,3] des calculs pseudo-différentiels sur les groupes de Lie gradués. Le lien entre la quantification de Weyl et les représentations du groupe de Heisenberg nous permet de considérer, dans ce cas, des symboles à valeurs scalaires. Nous trouvons que les conditions qui définissent les classes de symboles sont similaires, mais différentes de celles dans [1]. Comme applications, nous obtenons des résultats d'hypoellipticité de type Schwartz ainsi que des estimations subelliptiques sur le groupe de Heisenberg. Dans [4], voir aussi [2,3], un calcul pseudo-différentiel est développé sur tout groupe gradué en utilisant les représentations de ces groupes. Ici, nous présentons nos résultats dans le cas particulier du groupe de Heisenberg H n . Il est bien connu que les représentations de H n sont intimement liées à la quantification de Weyl (voir [8] et la section 2 en anglais ci-dessous). Grâce à l'analogue de la quantification de Kohn-Nirenberg sur les groupes de Lie (voir par exemple [8, 6,4] et la section 4 en anglais ci-dessous), cela permet de développer un calcul pseudo-différentiel sur H n avec symboles à valeurs scalaires qui dépendent de paramètres. Cependant, une difficulté persiste : celle de trouver les conditions que l'on doit imposer sur ces symboles pour que la classe d'opérateurs qui en résulte ait bien les propriétés attendues d'un calcul d'opérateurs. Bien que M. Taylor ait expliqué ces grandes lignes d'idées dans [8], son choix fut de restreindre son analyse dans [8] principalement au cas d'opérateurs invariants (c'est-à-dire de convolution) sur H n avec des symboles définis par des développements asymptotiques. Jusqu'à ce jour, à la connaissance des auteurs de cette note, seule une étude porte sur des E-mail addresses: v.fischer@imperial.ac.uk (V. Fischer), m.ruzhansky@imperial.ac.uk (M. Ruzhansky).

Commentarii Mathematici Helvetici, 2000

Abstract: In this paper we prove some approximation results for biLipschitz maps in the Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz constant close to one can be pointwise approximated, quantitatively on any fixed ball, by an isometry. ...

Journal of Physics A: Mathematical and General, 1996

The differential calculus on the quantum Heisenberg group is constructed. The duality between quantum Heisenberg group and algebra is proved.