Groups Of Heisenberg Type Research Articles

Overdetermined problems in groups of Heisenberg type: Conjectures and partial results

In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two problems. By exploiting some new invariances of these Lie groups, for domains having partial symmetry we solve these problems by converting them to known results for the classical p-Laplacian.

Quantum evolution and sub-Laplacian operators on groups of Heisenberg type

In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schrödinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits ) that are associated with this family. This allows us to prove an Egorov’s type Theorem describing the quantum evolution of a pseudodifferential semi-classical operator through the semi-group generated by a sub-Laplacian.

Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type

We give necessary and sufficient conditions for the controllability of a Schr\odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Oscillating spectral multipliers on groups of Heisenberg type

We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author [Springer INdAM Ser., vol. 45 (2021)], but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger [Anal. PDE 8 (2015)]. We highlight and develop the connection between sharp bounds for oscillating spectral multipliers and the problem of determining the minimal amount of smoothness required for Mihlin–Hörmander multipliers, a problem that has been solved for groups of Heisenberg type but remains open for other groups.

Almost everywhere convergence of Bochner–Riesz means on Heisenberg‐type groups

We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.

On Green functions for Dirichlet sub-Laplacians on H-type groups

We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc., as well as in a strip.

Three-spheres theorems for subelliptic quasilinear equations in Carnot groups of Heisenberg-type

We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p p -harmonic type in Carnot groups of Heisenberg type for 1 > p > ∞ 1>p>\infty . In the presentation we exhibit the special cases of sub-Laplace equations ( p = 2 p=2 ) and the case p p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.

Sharp Hardy–Littlewood–Sobolev inequalities on the octonionic Heisenberg group

This paper is the second one following Christ et al. (Nonlinear Anal 130:361–395, 2016) in a series, considering sharp Hardy–Littlewood–Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made in Frank et al. (Ann Math 176:349–381, 2012). In this paper, analogous results are obtained for the octonionic Heisenberg group.

Remainder terms for several inequalities on some groups of Heisenberg-type

We give some estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group, in analogy with the Euclidean case. By considering the variation of associated functionals, we give a stability of two dual forms: the fractional Sobolev (Folland-Stein) and Hardy-Littlewood-Sobolev inequality, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case s = Q (or $\lambda$ = 0), the remainder terms of Beckner-Onofri inequality and its dual Logarithmic Hardy-Littlewood-Sobolev inequality. Besides, we also list without proof some results for the other two cases of groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces by Chen, Frank, Weth [CFW13] and Dolbeault, Jankowiakin [DJ14] onto some groups of Heisenberg-type.

SharpLpbounds for the wave equation on groups of Heisenberg type

Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.

Strongly singular convolution operators on the Heisenberg-type groups

In this article, we obtain <italic>L</italic>² estimates for some strongly singular convolution operators on the Heisenberg-type groups. In particular, our results improve and generalize those obtained in the setting of Heisenberg groups by Laghi and Lyall. Furthermore, some simpler and more effective techniques are introduced in this article.

Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups

Let g = g 1 ⊕ g 2 , [ g , g ] = g 2 , \mathfrak {g}=\mathfrak {g}_1\oplus \mathfrak {g}_2,[\mathfrak {g},\mathfrak {g}] =\mathfrak {g}_2, a nilpotent Lie algebra of step 2, V 1 , ⋯ , V m V_1,\cdots , V_m a basis of g 1 \mathfrak {g}_1 and L = ∑ j , k m a j k V j V k L=\sum _{j,k}^{m}a_{jk}V_j V_k a left-invariant differential operator on G = e x p ( g ) G=\mathrm {exp} (\mathfrak {g}) , where ( a j k ) j k ∈ M n ( R ) (a_{jk})_{jk}\in M_n(\mathbb {R}) is symmetric. It is shown that if a solution w ( t , x ) w(t,x) to the Schrödinger equation ∂ t w ( t , g ) = i L w ( t , g ) , \partial _t w(t,g)=i Lw(t,g), g ∈ G , t ∈ R , w ( 0 , g ) = f ( g ) g\in G, t\in \mathbb {R}, w(0,g)=f(g) , satisfies a suitable Gaussian type estimate at time t = 0 t= 0 and at some time t = T ≠ 0 t=T\ne 0 , then w = 0 w=0 . The proof is based on Hardy’s uncertainty principle, on explicit computations within Howe’s oscillator semigroup and on methods developed by Fulvio Ricci and the second author. Our results extend work by Ben Saïd, Thangavelu and Dogga on the Schrödinger equation associated to the sub-Laplacian on Heisenberg type groups.

Invertible Carnot Groups

Abstract We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.

Sharp constants for weighted Moser–Trudinger inequalities on groups of Heisenberg type

Let G be a group of Heisenberg type, Q=m+2q be its homogeneous dimension, Qa=Q−a, Qa′=Q−aQ−a−1. For u∈G, we write u=(z(u),t(u))∈G, where t(u) is the coordinate of u corresponding to the center T of the Lie algebra G of G, z(u) is corresponding to the orthogonal complement of T. Let N(u)=(|z(u)|4+t(u)2)14 be the homogeneous norm of u∈G, W(u)=|z(u)|−a be a weight. The main purpose of this paper is to establish sharp constants for weighted Moser–Trudinger inequalities on domains of finite measure in G (Theorem 2.1) and on unbounded domains (Theorem 2.2). We also establish the weighted inequalities of Adachi–Tanaka type on the entire G (Theorem 2.3). Our results extend the sharp Moser–Trudinger inequalities on domains of finite measure in Cohn and Lu (2001, 2002) [13,14] and on unbounded domains in Lam et al. (2012) [19] to the weighted case and improve the sharp weighted Moser–Trudinger inequality proved in Tyson (2006) [16] on domains of finite measure on G. The usual symmetrization method (i.e., rearrangement argument) is not available on such groups and therefore our argument is a rearrangement-free argument recently developed in Lam and Lu (2012) [17,18]. Our weighted Adachi–Tanaka type inequalities extend the nonweighted results in Lam et al. (2012) [20].

Notes on non-archimedean topological groups

We show that the Heisenberg type group HX=(Z2⊕V)⋋V⁎, with the discrete Boolean group V:=C(X,Z2), canonically defined by any Stone space X, is always minimal. That is, HX does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean G there exists a (resp., locally compact) non-archimedean minimal group M such that G is a group retract of M. For discrete groups G the latter was proved by S. Dierolf and U. Schwanengel (1979) [6]. We unify some old and new characterization results for non-archimedean groups.

Riesz potentials and p -superharmonic functions in Lie groups of Heisenberg type

We prove a superposition principle for Riesz potentials of nonnegative continuous functions on Lie groups of Heisenberg type. More precisely, we show that the Riesz potential $$ R_\alpha(\rho)(g) = \int_{\G} N(g^{-1} g')^{\alpha-Q} \rho(g') dg', \qquad 0<\alpha<Q, $$ of a nonnegative function $\rho\in C_0(\G)$ on a group $\G$ of Heisenberg type is necessarily either $p$-subharmonic or $p$-superharmonic, depending on $p$ and $\alpha$. Here $N$ denotes the non-isotropic homogeneous norm on such groups, as introduced by Kaplan. This result extends to a wide class of nonabelian stratified Lie groups a recent remarkable superposition result of Lindqvist and Manfredi.

Improved Gagliardo-Nirenberg inequalities on Heisenberg type groups

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn's sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.

An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups

This note extends the classical theorem of Minkowski on lattice points and convex bodies in ℝ^n to 2-step simply connected nilpotent Lie groups with a ℚ -structure. This includes all groups of Heisenberg type. More generally (and more naturally), it works for any simply connected nilpotent Lie group with a ℚ -structure whose Lie algebra admits a grading of length 2. Here a new invariant associated with the grading occurs which we call the degree . It explains why some directions are more equal than others.

Gradient estimates for the heat kernels in higher dimensional Heisenberg groups

The author obtains sharp gradient estimates for the heat kernels in two kinds of higher dimensional Heisenberg groups — the non-isotropic Heisenberg group and the Heisenberg type group ℍ n,m . The method used here relies on the positive property of the Bakry-Emery curvature Γ2 on the radial functions and some associated semigroup technics.

Cohomology rings and formality properties of nilpotent groups

We analyze k -stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k -stage formal, the resonance varieties are trivial up to degree k . We also show that the cohomology ring, truncated up to degree k + 1 , of a finitely generated nilpotent, k -stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k -stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.