Let $\Omega \subset {ℝ}^2$ be a bounded, convex and open set with real analytic boundary. Let $T_{\Omega }\subset {ℂ}^2$ be the tube with base $\Omega ,$ and let $ℬ$ be the Bergman kernel of $T_{\Omega }$. If $\Omega$ is strongly convex, then $ℬ$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of $T_{\Omega }$. Note that Trèves curves exist only...

In this paper we obtain results on approximation, in the multidimensional complex case, of functions from ${}^{\infty }\left(K\right)$ by complex polynomials. In particular, we generalize the results of Pawłucki and Pleśniak (1986) for the real case and of Siciak (1993) in the case of one complex variable. Furthermore, we extend the results of Baouendi and Goulaouic (1971) who obtained the order of approximation in the case of Gevrey classes over real compacts with smooth analytic boundary and we present the orders of approximation...

We give an equivalent condition for Bergman completeness of Zalcman type domains. This also solves a problem stated by Pflug.

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ 0,1. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves...

In strictly pseudoconvex domains with smooth boundary, we prove a commutator relationship between admissible integral operators, as introduced by Lieb and Range, and smooth vector fields which are tangential at boundary points. This makes it possible to gain estimates for admissible operators in function spaces which involve tangential derivatives. Examples are given under with circumstances these can be transformed into genuine Sobolev- and Ck-estimates.

Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br>    $R_{m,n}=Z\in M_{m,n}:I^{\left(m\right)}-ZZ*ispositivedefinite$. Here $I^{\left(m\right)}\in M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L_{\alpha }^p\left(R_{m,n}\right)$ is defined to be the space $L^p{R}_{m,n};{\left[det(I^{\left(m\right)}-ZZ*)\right]}^{\alpha }d{\mu }_{m,n}\left(Z\right)$, where ${\mu }_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H_{\alpha }^p\left(R_{m,n}\right)\subset L_{\alpha }^p\left(R_{m,n}\right)$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Re\beta >(\alpha +1)/p-1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then     $f\left(\right)=T_{m,n}^{\beta }\left(f\right)\left(\right),\in R_{m,n},$where $T_{m,n}^{\beta }$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...