On closed range for $\bar{\partial}$
A sufficient condition for $\bar{\partial}$ to have closed range is given for pseudoconvex, possibly unbounded domains in $\mathbb{C}^n$.
math.CV↗
arXiv subjects
A. -K. Herbig
Publications and source records attributed to A. -K. Herbig.
Integrating holomorphic $L^1$-functions
Let $Ω\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(Ω)$ may be represented as the integral of this function against a smooth function vanishing to order $k-1$ on $bΩ$. An application for a smoothing property of the Bergman projection for conjugate holomorphic functions is given.
math.CV↗
A note on a smoothing property of the harmonic Bergman projection
It is proved that on any smoothly bounded domain $D$ in $\mathbb{R}^n$, $n>1$, the output of the harmonic Bergman projection belongs to the Sobolev space of order $k$ whenever all tangential derivatives of order up to k of the input function belong to $L^2(D)$.
math.CV↗
Oka's lemma, convexity, and intermediate positivity conditions
A new proof of Oka's lemma is given for smoothly bounded, pseudoconvex domains $D\subset\mathbb{C}^n$. The method of proof is then also applied to other convexity-like hypotheses on the boundary of $D$.
math.CV↗
A smoothing property of the Bergman projection
Let B be the Bergman projection associated to a domain on which the dbar-Neumann operator is compact. We show that arbitrary L^2 derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions that are not smooth up to the boundary but are mapped by B to functions which are smooth up to the boundary are explicitly described.
math.CV↗
Convex defining functions for convex domains
We give three proofs of the fact that a smoothly bounded, convex domain in R^n has smooth defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.
math.CV↗
A note on plurisubharmonic defining functions in $\mathbb{C}^n$
Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
math.CV↗