相关论文: Analytic discs, plurisubharmonic hulls, and non-co…

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in…

We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness…

The d-bar-Neumann operator on (0,q)-forms ($1\le q \le n$) on a bounded convex domain Omega in C^n is compact if and only if the boundary of Omega contains no complex analytic (equivalently: affine) variety of dimension greater than or…

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in $\mathbb{C}^n$ with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the…

In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact…

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (the space of $(0,1)$-forms with coefficient functions in…

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because…

For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit…

Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves…

Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the…

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial…

Let $\Omega$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $\varphi$ be a $\mathcal C^2$-weight function plurisubharmonic on $\Omega$. We show both necessary and sufficient conditions for existence and compactness of a weighted…

Let $S$ be a generic submanifold of $C^N$ of real codimension m. In this work we continue the study, carried over by various authors, of the set of analytic discs attached to S. Let $M$ be the set of analytic discs attached to $S.$ Given $q…

We construct a real analytic Levi-flat hypersurface M in a neighborhood of an ellipsoid B in C^2 such that the each leaf of the Levi foliation of M is a complex disc, M intersects the boundary of B transversely, and the intersection A of M…

We survey results arising from the study of domains in C^n with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even…

We derive a sufficient condition on a bounded pseudoconvex domain $\Omega\subset\mathbb{C}^2$ with smooth boundary such that $-(-\rho)^\eta$ is plurisubharmonic on $\Omega$ for $\eta>0$ arbitrarily close to $1$ (the supremum of $\eta$ is…

We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on C^n.$ For this purpose we use a description of relatively compact subsets of L^2- spaces. We also point out how to use this method to show that…

We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.

Let $\Omega$ be a bounded, weakly pseudoconvex domain in C^n, n > 1, with real-analytic boundary. A real-analytic submanifold $M \subset bd\Omega$ is called an analytic interpolation manifold if every real-analytic function on M extends to…

For smooth bounded pseudoconvex domains in $mathbb{C}^{2}$, we provide geometric conditions on (the points of infinite type in) the boundary which imply compactness of the $\bar{\partial}$-Neumann operator. It is noteworthy that the proof…