No Arabic abstract
We study the complete K{a}hler-Einstein metric of a Hartogs domain $widetilde {Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $Omega $, using a power $N^{mu}$ of the generic norm of $Omega$. The generating function of the K{a}hler-Einstein metric satisfies a complex Monge-Amp`{e}re equation with boundary condition. The domain $widetilde {Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $Xinlbrack0,1[$. This allows to reduce the Monge-Amp`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $mu_{0}$ of $mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G_h/K_h. We describe the boundary orbits and relate them to the boundary orbits of G_h/K_h. We relate the crown and the split-holomorphic crown of G/K to the crown Xi_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.
In this paper we consider Hankel operators on domains with bounded intrinsic geometry. For these domains we characterize the $L^2$-symbols where the associated Hankel operator is compact (respectively bounded) on the space of square integrable holomorphic functions.
The main result is that for a connected hyperbolic complete Kahler manifold with bounded geometry of order two and exactly one end, either the first compactly supported cohomology with values in the structure sheaf vanishes or the manifold admits a proper holomorphic mapping onto a Riemann surface.
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two upper-half planes.