HARDY AND MIYACHI THEOREMS FOR HEISENBERG MOTION GROUPS

2016 ◽  
Vol 229 ◽  
pp. 1-20 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SUNDARAM THANGAVELU
Keyword(s):  
Heisenberg Group ◽  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Motion Group ◽  
Hardy’S Theorem ◽  
Motion Groups ◽  
Miyachi’S Theorem

Let $G=\mathbb{H}^{n}\rtimes K$ be the Heisenberg motion group, where $K=U(n)$ acts on the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$ by automorphisms. We formulate and prove two analogues of Hardy’s theorem on $G$. An analogue of Miyachi’s theorem for $G$ is also formulated and proved. This allows us to generalize and prove an analogue of the Cowling–Price uncertainty principle and prove the sharpness of the constant $1/4$ in all the settings.

scholarly journals Uncertainty principles like Hardy's theorem on some Lie groups

1998 ◽  
Vol 65 (3) ◽  
pp. 289-302 ◽  
Author(s):  
S. C. Bagchi ◽  
Swagato K. Ray
Keyword(s):  
Lie Groups ◽  
Uncertainty Principle ◽  
Real Line ◽  
Classical Result ◽  
Uncertainty Principles ◽  
Heisenberg Groups ◽  
Motion Group ◽  
Euclidean Motion Group ◽  
Hardy’S Theorem ◽  
Euclidean Motion

AbstractWe extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.

Local Price uncertainty principle and time-frequency localization operators for the Hankel–Stockwell transform

2020 ◽  
Vol 18 (06) ◽  
pp. 2050050
Author(s):  
Nadia Ben Hamadi ◽  
Zineb Hafirassou
Keyword(s):  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Localization Operators ◽  
Von Neumann ◽  
Time Frequency ◽  
Neumann Class ◽  
Time Frequency Localization ◽  
Stockwell Transform

For the Hankel–Stockwell transform, the Price uncertainty principle is proved, we define the Localization operators and we study their boundedness and compactness. We also show that these operators belong to the so-called Schatten–von Neumann class.

Uncertainty principle related to Flensted–Jensen partial differential operators

2019 ◽  
pp. 2150004
Author(s):  
Raoudha Laffi ◽  
Selma Negzaoui
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Differential Operators ◽  
Generalized Fourier Transform ◽  
Partial Differential ◽  
Hardy’S Theorem ◽  
Partial Differential Operators ◽  
Hörmander's Theorem

This paper deals with some formulations of the uncertainty principle associated to generalized Fourier transform [Formula: see text] related to Flensted–Jensen partial differential operators. The aim result is to prove the analogue of Bonami–Demange–Jaming’s theorem : A version of Beurling–Hörmander’s theorem which gives more precision in the form of nonzero functions verifying modified-Beurling’s condition. As application, we get analogous of Gelfand–Schilov’s theorem, Cowling–Price’s theorem and Hardy’s theorem for [Formula: see text].

Deforming discontinuous groups for Heisenberg motion groups

2019 ◽  
Vol 30 (09) ◽  
pp. 1950045
Author(s):  
Ali Baklouti ◽  
Souhail Bejar ◽  
Khaireddine Dhahri
Keyword(s):  
Homogeneous Space ◽  
Parameter Space ◽  
Closed Subgroup ◽  
Motion Group ◽  
Discontinuous Group ◽  
Natural Action ◽  
Local Rigidity ◽  
Deformation Parameters ◽  
Motion Groups ◽  
Discontinuous Groups

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group [Formula: see text] acting on a homogeneous space [Formula: see text], where [Formula: see text] stands for a closed subgroup of the Heisenberg motion group [Formula: see text]. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if [Formula: see text] is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.

Variants of stability of discontinuous groups for Euclidean motion groups

2017 ◽  
Vol 28 (06) ◽  
pp. 1750046 ◽  
Author(s):  
Ali Baklouti ◽  
Souhail Bejar
Keyword(s):  
Lie Group ◽  
Closed Subgroup ◽  
Deformation Parameter ◽  
Motion Group ◽  
Discontinuous Group ◽  
Euclidean Motion Group ◽  
Euclidean Motion ◽  
Motion Groups ◽  
Discontinuous Groups ◽  
Properly Discontinuous Action

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]. To understand this phenomenon in the case when [Formula: see text] stands for an Euclidean motion group [Formula: see text], we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of [Formula: see text] on [Formula: see text], Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when [Formula: see text] turns out to be a crystallographic subgroup of [Formula: see text].

Dispersion’s Uncertainty Principles Associated with the Directional Short-Time Fourier Transform

2020 ◽  
Vol 57 (4) ◽  
pp. 508-540
Author(s):  
Siwar Hkimi ◽  
Hatem Mejjaoli ◽  
Slim Omri
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Uncertainty Principles ◽  
Short Time Fourier Transform ◽  
Short Time

We introduce the directional short-time Fourier transform for which we prove a new Plancherel’s formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, logarithmic uncertainty principle, Faris–Price uncertainty principles and Donoho–Stark’s uncertainty principles.

scholarly journals An Lp Version of the Hardy Theorem for Motion Groups

2000 ◽  
Vol 68 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Masaaki Eguchi ◽  
Shin Koizumi ◽  
Keisaku Kimahara
Keyword(s):  
Measurable Function ◽  
Weight Functions ◽  
Motion Group ◽  
Almost Everywhere ◽  
Motion Groups

AbstractWe describe a generalization of the Hardy theorem on the motion group. We prove that for some weight functions νω growing very rapidly and a measurable function f, the finiteness of the Lp-norm of vf and the Lq-norm of ωf implies f=0 (almost everywhere).

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