Heisenberg Group Hn Research Articles

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space \( W^{{1,p}}_{{{\text{loc}}}} {\left( {\mathbb{H}^{n} } \right)} \) on the n-dimensional Heisenberg group ℍn = ℂn × ℝ, where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of ℍn. Suppose that the subelliptic gradient is gloablly Lp integrable, i.e., \( {\int_{\mathbb{H}^{n} } {{\left| {\nabla _{{\mathbb{H}^{n} }} f} \right|}} }^{p} du \) is finite. We prove a Poincare inequality for f on the entire space ℍn. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of \( C^{\infty }_{0} {\left( {\mathbb{H}^{n} } \right)} \) under the norm of $$ {\left( {{\int_{\mathbb{H}^{n} } {{\left| f \right|}} }^{{\frac{{Qp}} {{Q - p}}}} } \right)}^{{\frac{{Q - p}} {{Qp}}}} + {\left( {{\int_{\mathbb{H}^{n} } {{\left| {\nabla _{{\mathbb{H}^{n} }} f} \right|}} }^{p} } \right)}^{{\frac{1} {p}}} . $$ We will also prove that the best constants and extremals for such Poincare inequalities on ℍn are the same as those for Sobolev inequalities on ℍn. Using the results of Jerison and Lee on the sharp constant and extremals for L2 to \( L\frac{{2Q}} {{Q - 2}} \) Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincare inequality on ℍn when p = 2. We also derive the lower bound of the best constants for local Poincare inequalities over metric balls on the Heisenberg group ℍn.

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H n = ℂ n × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ n . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H n .

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn

The purpose of this paper is to prove existence of minimisers of the functional J(K,u):=∫f(Lu)dx+α∫‖u−g‖qdx+βDdQ−1(K∩Ω),where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in CH1(Ω/K),α,β>0,q≥1,g∈Lq(Ω)∩L∞(Ω) and f : R2n → R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the \(2n + 1\)-dimensional Heisenberg group Hn and the 4-manifolds \(Nil^4 \) and \(Nil^3 \times \mathbb{R}\) endowed with an arbitrary left-invariant metric admit no C3-regular immersions into Euclidean spaces \(\mathbb{R}^{2n + 2} \) and \(\mathbb{R}^5 \), respectively.

A Note on Hermite and Subelliptic Operators

In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈ℝ n . We also apply this result to obtain the fundamental solutions for the Grushin operator in ℝ2 and the sub-Laplacian in the Heisenberg group H n .

Approximate identities from Laguerre functions and singular integrals on the Heisenberg group

Suitably scaled Laguerre functions are an approximate identity for multiplicative convolution with test functions on the half line. As an application, we derive a precise connection between the Mikhlin-type expansion of a singular integral operator on a Heisenberg groupHn and its natural restriction toHn modulo the center.

Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group H n \mathbb {H}_n , of the form \[ P Λ = ∑ i , j = 1 n λ i j X i Y j = t X Λ Y , \mathcal {P}_\Lambda = \sum _{i,j=1}^{n} \lambda _{ij}X_i Y_j={\,}^t X\Lambda Y, \] where Λ = ( λ i j ) \Lambda =(\lambda _{ij}) is a complex n × n n\times n matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that P Λ \mathcal {P}_\Lambda cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that Re ⁡ Λ , \operatorname {Re}\Lambda , Im ⁡ Λ \operatorname {Im}\Lambda and their commutator are linearly independent, we show that P Λ \mathcal {P}_\Lambda is not locally solvable, even in the presence of lower-order terms, provided that n ≥ 7 n\ge 7 . In the case n = 3 n=3 we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group H 3 \mathbb {H}_3 a phenomenon first observed by Karadzhov and Müller in the case of H 2 . \mathbb {H}_2. It is interesting to notice that the analysis of the exceptional operators for the case n = 3 n=3 turns out to be more elementary than in the case n = 2. n=2. When 3 ≤ n ≤ 6 3\le n\le 6 the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

Generalized wavelets and inversion of the radon transform on the laguerre hypergroup

Let X=R+n×R denote the underlying manifold of polyradial functions on the Heisenberg group Hn. We construct a generalized translation on X=R+n×R, and establish the Plancherel formula on L2(X, dμ). Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.

Almost Everywhere Convergence of Bochner–Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group Hn and T r L a m b d a : = ( 1 − r L ) + λ the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclidean dimension of Hn. We prove convergence a.e. of the Bochner-Riesz means T r λ f as r → 0 for λ > 0 and for all f ∈ Lp(Hn), provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators ∂ ω a with a ∈ C, defined on the dual of Hn by ∂ ω a f ^ : = ω a f ^ , which may be of independent interest. Here ω is given by ω ( z , u ) : = | z | 2 − 4 i u for all (z,u) ∈ Hn. 2000 Mathematical Subject Classification: 22E30, 43A80.

Convolution kernels of (n + 1)-fold Marcinkiewicz multipliers on the Heisenberg group

We prove a characterisation, in terms of regularity and cancellation conditions, of the convolution kernels of Marcinkiewicz multiplier operators m (𝔏1,…,𝔏n, iT) on the Heisenberg group ℍn, where 𝔏1,…,𝔏n are the n partial sub-Laplacians. The necessity of these regularity and cancellation conditions was established by Fraser (2001); here, we prove their sufficiency.

Injectivity Sets for Spherical Means on the Heisenberg Group

In this paper we prove that cylinders of the form ΓR=SR×R, where SR is the sphere {z∈Cn: |z|=R}, are injectivity sets for the spherical mean value operator on the Heisenberg group Hn in Lp spaces. We prove this result as a consequence of a uniqueness theorem for the heat equation associated to the sub-Laplacian. A Hecke–Bochner type identity for the Weyl transform proved by D. Geller and spherical harmonic expansions are the main tools used.

Hardy's Theorem for simply connected nilpotent Lie groups

We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.

An identity related to the riesz transforms on the heisenberg group

Let be a real basis for the Heisenberg Lie algebra . In this paper, we calculate the explicit kernels of the Riesz transforms on the Heisenberg group Hn . Here is the sub-Laplacian operator. We show that for j = 1,2, … 2n where Rj 's are the regular Riesz transforms on . We also construct the “missing transform” such that in analogy of the classical result .

The Fourier transforms of Lipschitz functions on the Heisenberg group

We study the order of magnitude of the Fourier transforms of certain Lipschitz functions on the Heisenberg groupHn. We compare our conclusions with some previous results in the field.

ADMISSIBLE WAVELETS ON THE PRODUCT HEISENBERG GROUP Hn

Let Hn be the n-direct product of Heisenberg group H, P the affine group of Hn. Then P has a natural unitary representation U on L2(Hn). In this paper, the direct sum decomposition of irreducible invariant closed subspaces under unitary representation U for L2(Hn) is given. The restrictions of U on these subspaces are square-integrable. The characterization of admissible condition is obtained in terms of the Fourier transform. By selecting appropriately an orthogonal wavelet basis and the wavelet transform, the authors obtain the orthogonal direct sum decomposion of function space L2(P,dμl).

The Spherical Transform of a Schwartz Function on the Heisenberg Group

Suppose thatK⊂U(n) is a compact Lie group acting on the (2n+1)-dimensional Heisenberg groupHn. We say that (K,Hn) is a Gelfand pair if the convolution algebraL1K(Hn) of integrableK-invariant functions onHnis commutative. In this case, the Gelfand spaceΔ:(K,Hn) is equipped with the Godement–Plancherel measure, and the spherical transform∧:L2K(Hn)→L2(Δ(K,Hn)) is an isometry. The main result in this paper provides a complete characterization of the set SK(Hn)∧={f|f∈SK(Hn)} of spherical transforms ofK-invariant Schwartz functions onHn. We show that a functionFonΔ(K,Hn) belongs to SK(Hn)∧if and only if the functions obtained fromFvia application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions forK-invariant Schwartz functions onHnmodulo its center.

Pointwise Ergodic Theorems for Radial Averages on the Heisenberg Group

LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood–Paley–Stein square-functions and analytic interpolation.

Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane

LetP=NAMbe the minimal parabolic subgroup ofSU(n+1,1), which can be regarded as the affine automorphism group of the Siegel upper half-planeUn+1,Palso acts on the Heisenberg groupHn, the boundary ofUn+1. ThereforePhas a natural representationUonL2(Hn). We decomposeL2(Hn) into the direct sum of the irreducible invariant closed subspaces underU. The restrictions ofUon these subspaces are square-integrable. We give the characterization of the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform gives isometric operators from the irreducible invariant closed subspaces ofL2(Hn) toL2(P).

A geometric criterion for Gelfand pairs associated with the Heisenberg group

Let K be a closed subgroup of U(n) acting on the (2n+ 1)dimensional Heisenberg group Hn by automorphisms. One calls (K,Hn) a Gelfand pair when the integrable K-invariant functions on Hn form a commutative algebra under convolution. We prove that this is the case if and only if the coadjoint orbits for G := K n Hn which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. Equivalently, the representation of K on the polynomial algebra over C is multiplicity free if and only if the moment map from C to k∗ is one-to-one on K-orbits. It is also natural to conjecture that the spectrum of the quasi-regular representation of G on L(G/K) corresponds precisely to the integral coadjoint orbits that meet k⊥. We prove that the representations occurring in the quasi-regular representation are all given by integral coadjoint orbits that meet k⊥. Such orbits can, however, also give rise to representations that do not appear in L(G/K).

An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups

A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result to other Lie groups. (For example, see [18].) In sharp contrast to Schwartz’s theorem, is a result due to Hardy [5] which says that f and f cannot both be “very rapidly decreasing”. More precisely, if |f(x)| ≤ Ae−α|x|2 and |f(y)| ≤ Be−β|y| 2 and αβ > 1 4 , then f ≡ 0. (See [2], pp. 155-157.) However, as far as we are aware, until very recently no systematic attempt was made to generalize Hardy’s theorem to other Lie groups. In [12], [13], and [15], this result has been generalized to the Heisenberg groups Hn, the Euclidean motion groups M(n) and for certain eigenfunction expansions. In this paper we establish an analogue of Hardy’s theorem for a class of noncompact semisimple Lie groups and all symmetric spaces of the noncompact type. Hardy’s theorem can also be viewed as a sort of ‘Uncertainty Principle’. The results in [12] and [13] are presented from this point of view. (In [1], Cowling and Price have proved an “L − L” version of Hardy’s theorem on R. The theorem of Beurling in [9] is similar in spirit to Hardy’s theorem, although far more general, and indeed Hardy’s theorem, as well as the result of Cowling and Price, can be deduced from it as special cases.)