Nilpotent Lie Groups Of Step Research Articles

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

Local convergence of the Newton’s method in two step nilpotent Lie groups

Local convergence of the Newton’s method in two step nilpotent Lie groups

Coadjoint Orbits and Time-Optimal Problems for Step-$$2$$ Free Nilpotent Lie Groups

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step $$2$$ . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step $$2$$ for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.

The structure of abnormal extremals in a sub-Riemannian problem with growth vector

A left-invariant sub-Riemannian problem on a free nilpotent Lie group of step 4 with two generators is considered. The structure of abnormal extremals is described. These extremals are shown to define an abnormal foliation of the annihilator of the square of the distribution, which is given by the intersections of this annihilator with the leaves of the symplectic foliation of the Lie coalgebra. The question of abnormal trajectories being strictly/nonstrictly abnormal is investigated, their projection onto a plane of the distribution are described, estimates for the corank are given and examples of nonsmooth trajectories are constructed. Bibliography: 14 titles.

The Paneitz operator on the anisotropic quaternionic Heisenberg group

ABSTRACT We use the representation theory of nilpotent Lie groups of step two to find the fundamental solution for the Paneitz operator on the anisotropic quaternionic Heisenberg group. For the isotropic case, we also construct the closed-form of this solution. This result is the extension of the conclusion on the Heisenberg group.

Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

The Laguerre calculus on the nilpotent Lie groups of step two

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. We extend the Laguerre calculus for nilpotent groups of step two, and test it in the determining of the fundamental solution of the sub-Laplace operator. We also apply it to find the Szegö kernels of the projection operators to a kind of regular functions on the quaternion Heisenberg group.

The heat kernel of sub-Laplace operator on nilpotent Lie groups of step two

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: http://arxiv.org/abs/1901.06513], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis.

Around theorems of Ingham-type regarding decay of Fourier transform on [formula omitted], [formula omitted] and two step nilpotent Lie groups

Classical results due to Ingham, Levinson and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We prove some analogues of these results on the n-dimensional Euclidean space, the n-dimensional torus and connected, simply connected two step nilpotent Lie groups. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on the Euclidean space and a class of connected, simply connected two step nilpotent Lie groups.

Spectral zeta function on pseudo H-type nilmanifolds

We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

Fundamental solution of a higher step Grushin type operator

We examine a class of Grushin type operators Pk where k∈N0 defined in (1.1). The operators Pk are non-elliptic and degenerate on a sub-manifold of RN+ℓ. Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k+1. We explain the geometric framework and prove some analytic properties such as essential self-adjointness. The main purpose of the paper is to give an explicit expression of the fundamental solution of Pk. Our methods rely on an appropriate change of coordinates and involve the theory of Bessel and modified Bessel functions together with Weber's second exponential integral.

Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

AbstractLet F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;2 is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.

Harmonic analysis on two and three-step nilmanifolds

Let G be a connected, simply connected nilpotent Lie group of step 1, 2 or 3, which contains a discrete cocompact subgroup Γ. Let τ=indΓG1 be the quasi-regular representa-tion of G. Our main goals in this paper are twofold. The first is to give an orbital description of the decomposition of τ into irreducibles. The second main goal is to give an explicit intertwining operator between τ and its disintegration. As a straight application, we give a new multiplicity formula which is a partial answer to a question proposed by J. Brezin.

A restriction theorem for Métivier groups

In the spirit of an earlier result of D. Müller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of Müller also in the framework of the Heisenberg group.

Sub-semi-Riemannian geometry of general H-type groups

We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian general H-type groups.

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

AbstractOn quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^2$\end{document}, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^2$\end{document} cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.

Spectral zeta function of the sub-Laplacian on two step nilmanifolds

We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator ΔL∖Gsub on a two step compact nilmanifold L∖G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L∖G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L∖G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of ΔL∖Gsub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian ΔL∖Gsub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂G by using a decomposition of ΔL∖Gsub into a family of elliptic operators.

A characterization of the inverse Radon transform associated with the classical domain of type one

Let D(Ω, Φ) be the unbounded realization of the classical domain of type one. In general, its Šilov boundary is a nilpotent Lie group of step two. In this article, we characterize a subspace of (Schwartz space) on which the Radon transform is a bijection and give another characterization for this subspace . Also, we show that the two characterizations are equivalent. Finally, we give an inversion formula of the Radon transform on by using continuous wavelet transform.

On the best constant for the Friedrichs-Knapp-Stein inequality in free nilpotent Lie groups of step two and applications to subelliptic PDE

In this article we propose to find the best constant for the Friedrichs-Knapp-Stein inequality in F2n,2, that is the free nilpotent Lie group of step two on 2n generators, and to prove the second-order differentiability of subelliptic p-harmonic functions in an interval of p.

On stability of CR-mappings between nilpotent lie groups of step two

A quasi-CR-mapping from a nilpotent Lie group Open image in new window of step two to another such group satisfies a Beltrami-type system of partial differential equations which is usually not elliptic but subelliptic when the group Open image in new window is strongly 2-pseudoconcave. We derive an integral representation formula for CR-mappings from a strongly 2-pseudoconcave nilpotent Lie group of step two to another such group and establish the Holder continuity of e-quasi-CR-mappings and the stability of CR-mappings between such groups.