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51. CJM 1999 (vol 51 pp. 915)

Balogh, Zoltán M.; Leuenberger, Christoph
Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers
Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Kor\'anyi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in $\C^2$ is studied in details.

Categories:32E20, 30C65

52. CJM 1998 (vol 50 pp. 658)

Symesak, Frédéric
Hankel operators on pseudoconvex domains of finite type in ${\Bbb C}^2$
The aim of this paper is to study small Hankel operators $h$ on the Hardy space or on weighted Bergman spaces, where $\Omega$ is a finite type domain in ${\Bbbvii C}^2$ or a strictly pseudoconvex domain in ${\Bbbvii C}^n$. We give a sufficient condition on the symbol $f$ so that $h$ belongs to the Schatten class ${\cal S}_p$, $1\le p<+\infty$.

Categories:32A37, 47B35, 47B10, 46E22

53. CJM 1998 (vol 50 pp. 99)

Izuchi, Keiji; Matsugu, Yasuo
$A_\phi$-invariant subspaces on the torus
Generalizing the notion of invariant subspaces on the 2-dimensional torus $T^2$, we study the structure of $A_\phi$-invariant subspaces of $L^2(T^2)$. A complete description is given of $A_\phi$-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

Categories:32A35, 47A15

54. CJM 1997 (vol 49 pp. 1299)

Tie, Jingzhi
The explicit solution of the $\bar\partial$-Neumann problem in a non-isotropic Siegel domain
In this paper, we solve the $\dbar$-Neumann problem on $(0,q)$ forms, $0\leq q \leq n$, in the strictly pseudoconvex non-isotropic Siegel domain: \[ \cU=\left\{ \begin{array}{clc} &\bz=(z_1,\ldots,z_n) \in \C^{n},\\ (\bz,z_{n+1}):&&\Im (z_{n+1}) > \sum_{j=1}^{n}a_j |z_j|^2 \\ &z_{n+1}\in \C; \end{array} \right\}, \] where $a_j> 0$ for $j=1,2,\ldots, n$. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

Categories:32F15, 32F20, 35N15

55. 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

56. CJM 1997 (vol 49 pp. 916)

Brylinski, Ranee
Quantization of the $4$-dimensional nilpotent orbit of SL(3,$\mathbb{R}$)
We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit $O_\mathbb{R}$ of SL$(3,\mathbb{R})$. The space of quantization is the space of holomorphic functions on $\mathbb{C}^2- \{ 0 \}$ which are square integrable with respect to a signed measure defined by a Meijer $G$-function. We construct the quantization out a non-flat Kaehler structure on $\mathbb{C}^2 - \{ 0 \}$ (the universal cover of $O_\mathbb{R}$ ) with Kaehler potential $\rho=|z|^4$.

Categories:81S10, 32C17, 22E70

57. CJM 1997 (vol 49 pp. 653)

Cascante, Carme; Ortega, Joaquin M.
On $\lowercase{q}$-Carleson measures for spaces of ${\cal M}$-harmonic functions
In this paper we study the $q$-Carleson measures for a space $h_\alpha^p$ of ${\cal M}$-harmonic potentials in the unit ball of $\C^n$, when $q
Categories:32A35, 31C15
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