Heisenberg Group Research Articles

Instantaneous blow-up solutions for nonlinear Sobolev-type equations on the Heisenberg groups

Instantaneous blow-up solutions for nonlinear Sobolev-type equations on the Heisenberg groups

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

Existence results for a class of critical fractional elliptic system on Heisenberg group

Existence results for a class of critical fractional elliptic system on Heisenberg group

A quantum fluctuation description of charge qubits

We consider a specific instance of a superconducting circuit, the so-called charge-qubit, consisting of a capacitor and a Josephson junction that we describe by means of the BCS microscopic model in terms of two tunnelling superconducting systems in the strong-coupling quasi-spin formulation. Then, by means of collective observables we derive the Hamiltonian governing the quantum behaviour of the circuit in the limit of a large number N of quasi-spins. Our approach relies on suitable quantum fluctuations, i.e. on collective quasi-spin operators, different from mean-field observables, that retain a quantum character in the large-N limit. These collective operators generate the Heisenberg algebra on the circle and we show that their dynamics reproduces the phenomenological one generated by the charge qubit Hamiltonian obtained by quantizing the macroscopic classical Hamiltonian of the circuit. The microscopic derivation of the emergent, large-N behaviour provides a rigorous setting to investigate more in detail both general quantum circuits and quantum macroscopic scenarios; in particular, in the specific case of charge-qubits, it allows to explicitly obtain the temperature dependence of the critical Josephson current in the strong coupling regime, a result not accessible using standard approximation techniques.

Sub-Riemannian cut time and cut locus in Reiter–Heisenberg groups

We study the sub-Riemannian cut time and cut locus of a given point in a class of step-2 Carnot groups of Reiter–Heisenberg type. Following the Hamiltonian point of view, we write and analyze extremal curves, getting the cut time of any of them, and a precise description of the set of cut points.

A sharp multiplier theorem for solvable extensions of Heisenberg and related groups

A sharp multiplier theorem for solvable extensions of Heisenberg and related groups

Solutions for nonhomogeneous Kohn–Spencer Laplacian on Heisenberg group

In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form − di v H n ( ν ( ξ , u ) | D H n u | H n p − 2 D H n u ) = f ( ξ , u , D H n u ) in a bounded domain Ω ⊂ H n . We show the set of solutions is uniformly bounded by a special Moser's iteration.

Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group

Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group

Systems of left translates and oblique duals on the Heisenberg group

In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\lambda$. Here, $\mathbb{H}$ denotes the Heisenberg group and $g^\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ on $\mathbb{H}$. This result is also illustrated with an example.

The Cheeger cut and Cheeger problem in metric measure spaces

In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterization of the Cheeger constant: a variational one and another one through the eigenvalue of the 1-Laplacian. We obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by Cheeger in (In: Gunning RC (ed) Problems in analysis. Princeton University Press, Princeton, pp 195–199, 1970). We also study the Cheeger problem. Through a variational characterization of the Cheeger sets we prove the existence of Cheeger sets and obtain a characterization of the calibrable sets and a version of the Max Flow Min Cut Theorem.

Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

In this paper, we study low-dimensional Filiform Lie algebras. Specifically, three-dimensional standard Filiform Lie algebras and five-dimensional non-standard Filiform Lie algebras. The classification method was given in the following stage. For given a low-dimensional Filiform Lie algebra, we compute its second centre. We showed that three-dimensional Filiform Lie algebra-called Heisenberg Lie algebra-is type 𝐴 and 𝐴+ as well. On the other hand, for 𝑛≥3, the standard Filiform Lie algebras are type 𝐴 but not type 𝐴+. In this case, we give a concrete example of case five-dimensional Heisenberg Lie algebra. Moreover, we proved that five-dimensional non-standard Filiform Lie algebra is type 𝐴 but not type 𝐴+. It is still an open problem to classify types 𝐴 and 𝐴+ for the general case of non-standard Filiform Lie algebra of dimension ≥6.

Generalised Hardy type and Rellich type inequalities on the Heisenberg group

Generalised Hardy type and Rellich type inequalities on the Heisenberg group

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

Localization operator and Weyl transform on reduced Heisenberg group with multidimensional center

In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional centre G. The group G is a quotient group of non-isotropic Heisenberg group with multidimensional centre Hm by its centre subgroup. Firstly, we define the localization operator using a wavelet transform on G and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on G with the operator-valued symbol. Finally, we have shown that the Weyl transform is not only a bounded operator but also a compact operator when the operator-valued symbol is in Lp, 1 ≤ p ≤ 2, and it is an unbounded operator when p > 2.

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group

Abstract In this article, we establish some reverse weighted Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. We then show the existence of extremal functions for the above inequalities by combining the subcritical approach and the renormalization method.

On singular solutions of Lane-Emden equation on the Heisenberg group

Abstract By applying the gluing method, we construct infinitely many axial symmetric singular positive solutions to the Lane-Emden equation: Δ H u + u p = 0 , in H n \ { 0 } {\Delta }_{{\mathbb{H}}}u+{u}^{p}=0\left,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{H}}}^{n}\backslash \left\{0\right\} on the Heisenberg group H n {{\mathbb{H}}}^{n} , where n > 1 , Q ⁄ ( Q − 4 ) < p < p J L ( Q − 2 ) n\gt 1,Q/\left(Q-4)\lt p\lt {p}_{JL}\left(Q-2) , and Q = 2 n + 2 Q=2n+2 is the homogeneous dimension of H n {{\mathbb{H}}}^{n} .

Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group

Abstract In this article, the Liouville theorems for the Kirchhoff-type parabolic equations on the Heisenberg group were established. The main technique for proving the result relies on the method of test functions.

Kakeya maximal inequality in the Heisenberg group

Kakeya maximal inequality in the Heisenberg group

On Bohr compactifications and profinite completions of group extensions

AbstractLet $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.