scholarly journals Complex tangencies to embeddings of Heisenberg groups and odd-dimensional spheres

2014 ◽  
Vol 25 (03) ◽  
pp. 1450028
Author(s):  
Ali M. Elgindi
Keyword(s):  
Heisenberg Group ◽  
Topological Structure ◽  
Dimensional Sphere ◽  
Heisenberg Groups ◽  
3 Dimensional ◽  
Complex Spaces ◽  
Higher Dimensional ◽  
Complex Tangent

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂn. The generic topological structure of the set complex tangents to such embeddings Mn ↪ ℂn takes the form of a (stratified) (n-2)-dimensional submanifold of Mn. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S2n-1 ↪ ℂ2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.

scholarly journals Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heisenberg Groups ◽  
Related Measure

AbstractIn the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

Dynamical Variation of Weierstrass-Mandelbrot Function in Higher Dimensional Space

2013 ◽  
Vol 470 ◽  
pp. 767-771
Author(s):  
L. Zhang ◽  
Shu Tang Liu
Keyword(s):  
Hurst Exponent ◽  
Dimensional Space ◽  
Dynamical Behavior ◽  
Fractal Dimensions ◽  
Statistical Characteristics ◽  
3 Dimensional ◽  
Higher Dimensional ◽  
Complex Phenomena ◽  
Real Complex

Many real complex phenomena are related with Weierstrass-Mandelbrot function (WMF). Most researches focus on the systems as parameters fixed, such as calculations of its different fractal dimensions or the statistical characteristics of its generalized form and so on. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these systems as parameters change is important. However, there is few results about this aim. In this paper, we propose simulated results for the effects of parameters changeably on the graph of WMF in higher dimensional space. In addition, the relationships between the Hurst exponent of WMF and its parameters dynamically in 2-and 3-dimensional spaces are also given.

scholarly journals Generalized Heisenberg Groups and Shtern's Question

2004 ◽  
Vol 11 (4) ◽  
pp. 775-782
Author(s):  
M. Megrelishvili
Keyword(s):  
Heisenberg Group ◽  
Hilbert Spaces ◽  
Normed Space ◽  
Unitary Representations ◽  
Heisenberg Groups ◽  
Minimal Subgroups ◽  
Weakly Continuous

Abstract Let 𝐻(𝑋) := (ℝ × 𝑋) ⋋ 𝑋* be the generalized Heisenberg group induced by a normed space 𝑋. We prove that 𝑋 and 𝑋* are relatively minimal subgroups of 𝐻(𝑋). We show that the group 𝐺 := 𝐻(𝐿4[0, 1]) is reflexively representable but weakly continuous unitary representations of 𝐺 in Hilbert spaces do not separate points of 𝐺. This answers the question of A. Shtern.

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

scholarly journals Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

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