Heisenberg群上与Schrödinger算子相关的Riesz变换在Hardy空间上的有界性

令L=?ΔHn+V是Heisenberg群Hn上Schr?dinger算子,其中非负位势V属于逆H?lder类.该文用分子刻画与L相关的Hardy型空间HL^p(H^n),进而得到了与L相关的Riesz变换的HL^p-有界性.     

Gelfand transforms of polyradial Schwartz functions on the Heisenberg group

We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.     

Strichartz estimates on the quaternion Heisenberg group

The aim of this article is to prove dispersive estimates and the Strichartz estimates on the quaternion Heisenberg group. In order to obtain these results, we first study the properties of Fourier transform for radial functions and the Besov spaces, Sobolev spaces on the quaternion Heisenberg group, then we give the proofs for the main results.     

Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.     

Uncertainty Principles for the Segal-Bargmann Transform

We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark''s uncertainty principle and Matolcsi-Sz\"ucs uncertainty principle.     

Octonion short-time linear canonical transform

We investigate the octonion short-time linear canonical transform (OCSTLCT) in this paper. First, we propose the new definition of the OCSTLCT, and then several important properties of newly defined OCSTLCT, such as bounded, shift, modulation, time-frequency shift, inversion formula, and orthogonality relation, are derived based on the spectral representation of the octonion linear canonical transform (OCLCT). Second, by the Heisenberg uncertainty principle for the OCLCT and the orthogonality relation property for the OCSTLCT, the Heisenberg uncertainty principle for the OCSTLCT is established. Finally, we give an example of the OCSTLCT.     

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ? ⁿ . Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform $\tilde f$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ? ⁿ , the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.     

SOME LIOUVILLE TYPE THEOREMS FOR THE P-SUB-LAPLACIAN ON THE GROUP OF HEISENBERG TYPE

In this paper we prove some Liouville type results for the p-sub-Laplacian on the group of Heisenberg type. A strong maximum principle and a Hopf type principle concerning p-sub-Laplacian are established.     

Heisenberg uncertainty inequality for Gabor transform on nilpotent Lie groups

Analysis Mathematica - In this paper, we define and prove an analog of the Heisenberg uncertainty inequality for Gabor transform in the setup of connected, simply connected nilpotent Lie groups....     

Hardy’s Inequalities for the Heisenberg Group

Potential Analysis - This article presents two types of Hardy’s inequalities for the Heisenberg group. The proofs are mainly based on estimates of the Fourier transform for atomic functions...     

Radar Waveform Design and the Heisenberg Group

This paper presents a number of constructions based on the Heisenberg group that are relevant to the problem of radar waveform design. All of the constructions are based on the modification of the Weil transform of a waveform.     

Real clifford windowed Fourier transform

We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform (WFT) and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.     

Quantitative uncertainty principles for the short time Fourier transform and the radar ambiguity function

Logarithmic uncertainty principle and Beckner’s uncertainty principle in terms of entropy are proved for the short time Fourier transform and the radar ambiguity function, also a Heisenberg inequality for generalized dispersion and Price’s local uncertainty principle are obtained.     

Note on the Fenchel transform in the Heisenberg group

Given a real-valued function defined on the Heisenberg group H, we provide a definition of abstract convexity and Fenchel transform in H, that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the classical case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

Weyl transforms associated with the Heisenberg group

In this paper, we define the Wigner transform and the corresponding Weyl transform associated with the Heisenberg group. We established some harmonic analysis results. Then we present that the Weyl transform with the Sp-valued symbol in Lp (p∈[1,2]) is not only bounded but also compacted, while when 2<p<+∞, the Weyl transform is not a bounded operator.     

Joint eigenfunctions of invariant differential operators on the quaternion Heisenberg group

Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N.In this article,we characterize the H c-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.     

Symplectic structures¶on Heisenberg-type nilmanifolds

We use cohomological methods to study the existence of symplectic structures on nilmanifolds associated to two-step nilpotent Lie groups. We construct a new family of symplectic nilmanifolds with building blocks the quaternionic analogue of the Heisenberg group, determining the dimension of the space of all left invariant symplectic structures. Such structures can not be K?hlerian. Also, we prove that the nilmanifolds associated to H type groups are not symplectic unless they correspond to the classical Heisenberg groups. Received: 26 May 1999 / Revised version: 10 April 2000     

The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups

In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.