Mathematics

In this paper we discuss some recent results concerning the regularity and irregularity of the Bergman and Szegő projections on some weakly pseudoconvex domains that have the common feature to possess a nontrivial Nebenhülle.

We introduce and study a class of starlike functions defined by S ∗ ℘ :={f∈A: z f ′ (z) f(z) ≺1+z e z =:℘(z)}, where ℘ maps the unit disk onto a cardioid domain. We find the radius of convexity of ℘(z) and establish the inclusion relations between the class S ∗ ℘ and some well-known classes. Further we derive sharp radius constants and coefficient related results for the class S ∗ ℘ .

A Central Limit Theorem for linear combinations of iterates of an inner function is proved. The main technical tool is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures.

One constructs an example of a generic quadratic submanifold of codimension 5 in C 9 which admits a real analytic infinitesimal CR automorphism with homogeneous polynomial coefficients of degree 3.

We consider a bounded open Stein subset Ω of a complex Stein manifold X of dimension n . We prove that if f is a current on X of bidegree (p,q+1) , ∂ ¯ -closed on Ω , we can find a current u on X of bidegree (p,q) which is a solution of the equation ∂ ¯ u=f in Ω . In other words, we prove that the Dolbeault complex of temperate currents on Ω (i.e. currents on Ω which extend to currents on X ) is concentrated in degree 0 . Moreover if f is a current on X= C n of order k , then we can find a solution u which is a current on C n of order k+2n+1 .

We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for the Gromov hyperbolic metric under some families of Möbius transformations.

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in C 2 , which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

We generalize Rado's extension theorem to complex spaces.

We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), 105-191. We then provide an example using our approach and extend our results from the real field R to an arbitrary field F different from characteristic 2 . The new approach we take is only based on elementary results from the theory of determinants, the theory of Schur complements, and basic properties of polynomials.

Although the Bergman projection operator B Ω is defined on L 2 (Ω) , its behavior on other L p (Ω) spaces for p≠2 is an active research area. We survey some of the recent results on L p estimates on the Bergman projection.