Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

The heat kernel transform for the Heisenberg group

The heat kernel transform H_t is studied for the Heisenberg group in detail. The main result shows that the image of H_t is a direct sum of two weighted Bergman spaces, in contrast to the classical case of Rⁿ and compact symmetric spaces, and the weight functions are found to be (surprisingly) not non-negative.     

Strongly singular Radon transforms on the Heisenberg group and folding singularities

We prove sharp regularity results for classes of strongly singular Radon transforms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.

     

Singular Radon transforms along odd curves on the Heisenberg group

In this article we study the singular integral operators along the curve on the Heisenberg group. It is a variable coefficient extension of the singular integrals along the odd curves on the Euclidean space ℝ². The proof is based on the generalized Calderon-Zygmund theory on the space of homogeneous type.     

Spectral transform for the sub-Laplacian on the Heisenberg group

We continue our analysis of nilpotent groups related to quantum mechanical systems whose Hamiltonians have polynomial interactions. For the spinless particle in a constant external magnetic field, the associated nilpotent group is the Heisenberg group. We solve the heat equation for the Heisenberg group by diagonalizing the sub-Laplacian. The unitary map to the Hilbert space in which the sub-Laplacian is a multiplication operator with positive spectrum is given. The spectral multiplicity is shown to be related to the irreducible representations of SU(2). A Lax pair, generated from the Heisenberg sub-Laplacian, is used to find operators unitarily equivalent to the sub-Laplacian, but not arising from the SL(2,R) automorphisms of the Heisenberg group. Department of Mathematics, supported in part by NSF. Department of Physics and Astronomy, supported in part by DOE.     

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.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on Cn

Given a principal value convolution on the Heisenberg group Hn = Cn × R, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on Cn. We also calculate the Dirichlet kernel for the Laguerre expansion on the group Hn.     

Fourier transform of symmetric gauss measures on the Heisenberg group

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two symmetric Gauss measures to be a symmetric Gauss measure and for commutability of two symmetric Gauss measures. Moreover, necessary and sufficient conditions are presented for the convolution of two symmetric Gauss convolution semigroups to be a convolution semigroup.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H _n = ? ⁿ × ?, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ? ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n .     

Poisson transform for the Heisenberg group and eigenfunctions of the sublaplacian

We define an analogue of Poisson transform on the Heisenberg group and use it to characterise joint eigenfunctions of the sublaplacian and T=i∂_t in terms of certain analytic functionals.     

The Radon transform on hyperbolic space

The Radon transform that integrates a function in ⁿ , the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.     

The horocyclic Radon transform on non-homogeneous trees

This paper studies horocyles on trees and the corresponding Radon transformation. It is seen that a function can be reconstructed from the induced values on the horocycles. A formula is produced for the adjoint transformation and for the inverse. Research partially supported by the Consiglio Nazionale delle Ricerche and the Ministero dell’Università e della Ricerca Scientifica e Tecnologica.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ?n

Given a principal value convolution on the Heisenberg group H _n = ℂ ⁿ × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n . Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.