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Search: MSC category 43A85 ( Analysis on homogeneous spaces )

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1. CJM 2012 (vol 66 pp. 700)

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

2. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator
Categories:43A85, 22D10, 39A70

3. CJM 2003 (vol 55 pp. 1134)

Casarino, Valentina
Norms of Complex Harmonic Projection Operators
In this paper we estimate the $(L^p-L^2)$-norm of the complex harmonic projectors $\pi_{\ell\ell'}$, $1\le p\le 2$, uniformly with respect to the indexes $\ell,\ell'$. We provide sharp estimates both for the projectors $\pi_{\ell\ell'}$, when $\ell,\ell'$ belong to a proper angular sector in $\mathbb{N} \times \mathbb{N}$, and for the projectors $\pi_{\ell 0}$ and $\pi_{0 \ell}$. The proof is based on an extension of a complex interpolation argument by C.~Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the $L^1$ norm on the unit sphere of $\mathbb{R}^{2n}$.

Categories:43A85, 33C55, 42B15

4. CJM 1999 (vol 51 pp. 952)

Deitmar, Anton; Hoffmann, Werner
On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.

Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus
Categories:11F72, 22E30, 22E40, 43A85, 58G25

5. CJM 1998 (vol 50 pp. 1090)

Lohoué, Noël; Mustapha, Sami
Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes
Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$ and let $\nabla$ be the associated gradient. In this paper we investigate the boundness of the Riesz transform $\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with exponential volume growth and on certain homogenous spaces.

Categories:22E30, 35H05, 43A80, 43A85

6. CJM 1997 (vol 49 pp. 1224)

Ørsted, Bent; Zhang, Genkai
Tensor products of analytic continuations of holomorphic discrete series
We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of $\SU(2, 2)$ with its conjugate.

Keywords:Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theorem
Categories:22E45, 43A85, 32M15, 33A65

7. CJM 1997 (vol 49 pp. 883)

Okounkov, Andrei
Proof of a conjecture of Goulden and Jackson
We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces.

Categories:05E05, 43A85, 57M15

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