Strong Stein neighborhood bases

Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\in C^n:r(z)<a} is pseudoconvex for sufficiently small a>0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary.Comment: 14 pages, fixed same references, to appear in Complex Var. Elliptic Eq

Compactness of products of Hankel operators on the polydisk and some product domains in C2\mathbb{C}^2

Let $\mathbb{D}^n$ be the polydisk in $\mathbb{C}^n$ and the symbols $\phi,\psi\in C(\bar{\mathbb{D}^n})$ such that $\phi$ and $\psi$ are pluriharmonic on any $(n-1)$-dimensional polydisk in the boundary of $\mathbb{D}^{n}.$ Then $H^*_{\psi}H_{\phi}$ is compact on $A^2(\mathbb{D}^n)$ if and only if for every $1\leq j,k\leq n$ such that $j\neq k$ and any $(n-1)$-dimensional polydisk $D$, orthogonal to the $z_j$-axis in the boundary of $\mathbb{D}^n,$ either $\phi$ or $\psi$ is holomorphic in $z_k$ on $D.$ Furthermore, we prove a different sufficient condition for compactnes of the products of Hankel operators. In $\mathbb{C}^2,$ our techniques can be used to get a necessary condition on some product domains involving annuli.Comment: 9 pages. Fixed some typos, to appear in J. Math. Anal. App

Continuity of Plurisubharmonic Envelopes in C2\mathbb{C}^2

We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in $\mathbb{C}^{2}$ are $c$-regular.Comment: 13 page

Essential norm estimates for Hankel operators on convex domains in C2\mathbb{C}^2

Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi$ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary

On convergence of the Berezin transforms

We prove approximation results about sequences of Berezin transforms of finite sums of finite product of Toeplitz operators (and bounded linear maps, in general) in the spirit of Ramadanov and Skwarczynski Theorems that are about convergence of Bergman kernels

A sufficient condition for L-p regularity of the Berezin transform

We prove that the Berezin transform is (Formula presented.) regular on a large class of domains in (Formula presented.) and not (Formula presented.) regular on the Hartogs triangle

Schatten class Hankel and (partial derivative)over-bar-Neumann operators on pseudoconvex domains in C-n

Let similar to be a C-2-smooth bounded pseudoconvex domain in Cn for n = 2 and let. be a holomorphic function on similar to that is C-2-smooth on the closure of similar to. We prove that if H. is in Schatten p-class for p = 2n then. is a constant function. As a corollary, we show that the.-Neumann operator on similar to is not Hilbert-Schmidt

A local weighted Axler-Zheng theorem in C-n

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in C-n on which the (partial derivative) over bar -Neumann operator N is compact. In this work we remove the assumption on N, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights