Nodal sets and horizontal singular sets of ℍ-harmonic functions on the Heisenberg group

2014 ◽  
Vol 16 (04) ◽  
pp. 1350049 ◽  
Author(s):  
Long Tian ◽  
Xiaoping Yang
Keyword(s):  
Heisenberg Group ◽  
Geometric Structure ◽  
Harmonic Functions ◽  
Nodal Sets ◽  
Singular Sets ◽  
Definition Of

In this paper, we give measure estimates of nodal sets of ℍ-harmonic functions on the Heisenberg group ℍn. We also introduce a definition of horizontal singular sets and show the geometric structure of the horizontal singular sets of ℍ-harmonic functions.

scholarly journals Superunitary Representations of Heisenberg Supergroups

2018 ◽  
Vol 2020 (19) ◽  
pp. 5926-6006 ◽  
Author(s):  
Axel de Goursac ◽  
Jean-Philippe Michel
Keyword(s):  
Representation Theory ◽  
Heisenberg Group ◽  
Fourier Transformation ◽  
Coadjoint Orbits ◽  
New Approach ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Lie Supergroups ◽  
Definition Of ◽  
Von Neumann Theorem

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

scholarly journals r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

Boundary Value Problems for Harmonic Functions on the Heisenberg Group

1986 ◽  
Vol 38 (2) ◽  
pp. 478-512 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Differential Operators ◽  
Special Functions ◽  
Continuous Functions ◽  
Group Representations ◽  
Hypoelliptic Operators ◽  
Analysis Group ◽  
Partial Differential Operators

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ, (γ ∊ C), associated to the Laplacian L0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ-harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

scholarly journals On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties

2014 ◽  
Vol 34 (7) ◽  
pp. 2779-2793 ◽  
Author(s):  
Fausto Ferrari ◽  
◽  
Qing Liu ◽  
Juan Manfredi ◽  
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Mean Value

Hamiltonian Systems, Pseudo-Poisson Brackets and Their Scattering Representation for Physical Systems

1999 ◽  
Author(s):  
B. M. J. Maschke ◽  
A. J. van der Schaft
Keyword(s):  
Hamiltonian Systems ◽  
Network Model ◽  
Geometric Structure ◽  
Symplectic Structure ◽  
Poisson Brackets ◽  
Physical Systems ◽  
Energy Conserving ◽  
Definition Of

Abstract This paper is concerned with the definition of the geometric structure of Hamiltonian systems associated with energy–conserving systems in relation with an interconnection topology of their network model. It is also presented how the symplectic structure of standard Hamiltonian systems has to be extended to pseudo–Poisson tensors in order to cope with invariants, equilibria and constraints. Finally a scattering representation of these pseudo–Poisson tensors is defined.

scholarly journals Suitable Spaces for Shape Optimization

2021 ◽  
Author(s):  
Kathrin Welker
Keyword(s):  
Shape Optimization ◽  
Explicit Expression ◽  
Geometric Structure ◽  
Covariant Derivative ◽  
Optimization Problems ◽  
Optimization Techniques ◽  
Poincaré Metric ◽  
Shape Hessian ◽  
New Space ◽  
Definition Of

AbstractThe differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov–Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to $$H^{1/2}$$ H 1 / 2 -shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of $$H^{1/2}$$ H 1 / 2 -shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.

On a Theorem of Privaloff

1967 ◽  
Vol 10 (3) ◽  
pp. 353-359
Author(s):  
P. S. Bullen
Keyword(s):  
Harmonic Functions ◽  
Not Given ◽  
Definition Of

It is the object of this note to extend to general harmonic structures a theorem due to Privaloff [2] concerning the definition of harmonic functions. The notation is that of [8, 9, 10], where many of the definitions not given here will be found.

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