Pseudoconvex Points Research Articles

The Burns-Krantz rigidity with an interior fixed point

We prove a Burns-Krantz type boundary rigidity near strongly pseudoconvex points for holomorphic self-maps with an interior fixed point. This confirms a conjecture of Huang.

Sufficient conditions for compactness of the ∂-Neumann operator on high level forms

By establishing a unified estimate of the twisted Kohn-Morrey-H\"{o}rmander estimate and the $q$-pseudoconvex Ahn-Zampieri estimate, we discuss variants of Property $(P_q)$ of Catlin and Property $(\widetilde{P_q})$ of McNeal on the boundary of a smooth pseudoconvex domain in $\mathbb{C}^n$ for certain high level forms. These variant conditions on the one side, imply $L^2$-compactness of the $\overline{\partial}$-Neumann operator on the associated domain, on the other side, are different from the classical Property $(P_q)$ and Property $(\widetilde{P_q})$. As an application of our result, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is $L^2$-compact on $(0,n-1)$-level forms. This result generalizes Boas and Sibony's results on $(0,1)$-level forms.

On the boundary rigidity at strongly pseudoconvex points

We prove a localized version of Burns–Krantz type boundary rigidity theorem at strongly pseudoconvex points for holomorphic self-maps of some bounded taut domains in $${\mathbb {C}}^2$$ with an interior fixed point.

Hartogs domains and the Diederich–Fornæss index

We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich–Fornæ ss index. Using this property, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich–Fornæss index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis and show that, under the same hypotheses, if the Diederich–Fornæss index for a Hartogs domain is equal to one, then the domain admits a Stein neighborhood basis.

A local weighted Axler–Zheng theorem in ℂn

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 $\mathbb{C}^n$ on which the $\overline{\partial}$-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.

Levi problem in complex manifolds

Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive \({\partial \overline{\partial }}\)-closed currents T of bidimension (1, 1) and of mass 1 directed by the directions where all continuous psh functions in U have vanishing Levi-form. The extremal ones, are supported on the sets where all continuous psh functions are constant. We also construct under geometric conditions, bounded strictly psh exhaustion functions, and hence we obtain Donnelly–Fefferman weights. To any infinitesimally homogeneous manifold, we associate a foliation. The dynamics of the foliation determines the solution of the Levi-problem. Some of the results can be extended to the context of pseudoconvexity with respect to a Pfaff-system.

Almost strictly pseudo-convex domains. Examples and applications

In this work we introduce a class of smoothly bounded domains Ω in Cn with few non strictly pseudo-convex points in @ Ω with respect to a certain Minkowski dimension. We call them almost strictly pseudo-convex, aspc. For these domains we prove that a canonical measure associated to a separated se- quence of points in Ω which projects on the set of weakly pseudo-convex points is automatically a geometric Carleson measure. This class of aspc domains con- tains of course strictly pseudo-convex domains but also pseudo-convex domains of finite type in C2 , domains locally diagonalizable, convex domains of finite type 2 in Cn , domains with real analytic boundary and domains like |z1 | |z2 | 2 } < 1, which are not of finite type. + exp{1 As an application we study interpolating sequences for convex domains of finite type in Cn . After proving a Carleson-type embedding theorem, we get that if Ω is a convex domain of finite type in Cn and if S ⇢ Ω is a dual bounded sequence of points in H p (Ω), if p = 1 then for any q < 1, S is H q (Ω) interpolating with the linear extension property and if p < 1 then S is H q () interpolating with the linear extension property, provided that q < min( p, 2).

A Wong-Rosay type theorem for proper holomorphic self-maps

In this short paper, we show that the only proper holomorphic self-maps of bounded domains in ℂ k whose iterates approach a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type theorem for a sequence of maps whose degrees are a priori unbounded.

Analytic regularity of CR maps into spheres

Let M ⊂ CN be a connected real-analytic hypersurface and S2N−1 ⊂ C N′ the unit real sphere, N ′ > N ≥ 2. Assume that M does not contain any complex-analytic hypersurface of CN and that there exists at least one strongly pseudoconvex point on M . We show that any CR map f : M → S2N−1 of class CN−N+1 extends holomorphically to a neighborhood of M in CN .

Unbounded domains in ℂ2 with non-compact automorphisms group

Let D be a domain in ℂ2 such that the orbit of an internal point under the group Aut(D) of the holomorphic automorphisms of D accumulates to a smooth boundary point p. It is well known that if p is a strictly pseudoconvex point, such a local information forces D to be biholomorphic to the ball. We prove that if p is only weakly pseudoconvex of finite type, the domain D need not to be biholomorphic to a natural standard model. On the other end our results show that, under some convexity assumptions, D may be realized as a bounded domain with real analytic boundary except at most at one point.

On Analytic Interpolation Manifolds in Boundaries of Weakly Pseudoconvex Domains

Let Ω be a bounded, weakly pseudoconvex domain in C n , n ≤ 2, with real-analytic boundary. A real-analytic submanifold M ⊂ ∂Ω is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to (Ω¯). We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of ∂Ω.

A local version of Wong-Rosay’s theorem for proper holomorphic mappings

In the present paper, we generalize Wong-Rosay's theorem for proper holomorphic mappings with bounded multiplicity. As an application, we prove the non-existence of a proper holomorphic mapping from a bounded, homogenous domain in Cn onto a domain in Cn whose boundary contains strongly pseudoconvex points.

Failure of global regularity of ∂b on a convex domain with only one flat point

In this paper we exhibit a bounded domain in C 2 with real analytic boundary which is strictly convex except at one point and for which the @b operator is not analytic hypoelliptic modulo its kernel. The importance of such an example is twofold: First it shows that the theorem of Boas and Straube on global C 1 regularity for @b on convex domains cannot be extended to the analytic case; secondly it is the rst example of non analytic hypoellipticity of @b on a domain with isolated weakly pseudoconvex points in the boundary.

Hölder estimates for the $$\bar \partial - equation$$ in some domains of finite typein some domains of finite type

We obtain Holder estimates for the\(\bar \partial - equation\) on some domains of finite type in ℂn using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.

Approximation par des fonctions holomorphes à croissance contrôlée

Let $\Omega$ be a bounded pseuco-convex domain in $\Bbb C^n$ with a $\Cal C^{\infty}$ boundary, and let $S$ be the set of strictly pseudo-convex points of $\partial\Omega$. In this paper, we study the asymptotic of holomorphic functions along normals arising from points of $S$. We extend results obtained by M. Ortel and W. Schneider in the unit disc and those of A. Iordan and Y. Dupain in the unit ball of $\Bbb C^n$. We establish the existence of holomorphic functions of given growth having a prescribed behaviour on almost all normals arising from points of $S$.

A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications

A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications

Construction of P.S.H. functions on weakly pseudoconvex domains

Let Q c C be a smoothly bounded pseudoconvex domain, and let A(Q) denote the functions holomorphic on Q and continuous on Sl. A point p e aQ is a peak point if there is a function f e A(Q) such that f(p) = 1 and If(z)I 2) were obtained by Range [19], [20] and Hakim and Sibony [12]. These results do not extend automatically to arbitrary weakly pseudoconvex boundary points of Q, even if the boundary is real analytic. An example of Kohn and Nirenberg [17] shows that a p e aQ does not always have a separating function which is real analytic at p. Similarly, this example does not have a peak function which is e' at p (see Fornaess [6]). Here we show how peak functions and separating functions may be constructed for a large class of weakly pseudoconvex points on domains in C*. The results are most complete for domains in C2, so we state them here for that case.

Local analyticity for □band the ∂̸_-neumann probelm at certain weakly pseudoconvex points

On demontre l'hypoellipticite analytique reelle locale (en 0) dans le probleme ∂-Neumann et pour □ b pour des domaines pseudoconvexes qui sont localement de la forme Imw<h(|Z| 2 ) avec h(s) analytique reelle, h(0)=0

Finitely generated ideals in $A(\omega )$

The Gleason problem is solved on real analytic pseudoconvex domains in C 2 . In this case the weakly pseudoconvex points can be a two-dimensional subset of the boundary. To reduce the Gleason problem to a ∂ ¯ question it is shown that the set of Kohn-Nirenberg points is at most one-dimensional. In fact, except for a one-dimensional subset, the weakly pseudoconvex boundary points are R-points as studied by Range and therefore allow local sup-norm estimates for ∂ ¯.

Non-vanishing of the Bergman kernel function at boundary points of certain domains in ? n

Let f2 be a smooth bounded domain in (t TM whose associated Bergman kernel function K(w, z) extends continuously to O as a function ofw for each z ~ f~. We shall prove that for any fixed we bf2, K(w, z) cannot vanish identically in z for a wide class of domains O. Our theorem applies to any domain for which the L2(O) orthogonal projection P onto holomorphic functions is bounded in Sobolev norms ; in particular, if f2 is 1) smooth bounded strictly pseudoconvex, or 2) bounded pseudoconvex with smooth real analytic boundary (Kohn [7], Diederich and Fornaess [,2]), or 3) any smooth bounded domain for which subelliptic estimates hold for the ~-Neumann opcrator (Kohn [7]). Webster [8] has pointed out that the non-vanishing of K(w, z) for strongly pseudoconvex points of domains for which the c~-Neumann operator is pseudo-local follows easily from the asymptotic expansion for K(w, z) given by Fefferman [-3]. Our proof has no recourse to the asymptotic expansion, is elementary in spirit, and extends Webster's result to include certain weakly pseudoconvex domains. The non-vanishing question arises in connection with the problem of boundary behavior of biholomorphic mappings (see Webster [-8]). As a corollary to the non-vanishing theorem, we prove that if (2 is strictly pseudoconvex and Dis a set of determinacy for holomorphic functions in fL then the linear span of {K(., z) :ze ID} is dense in the subspace of holomorphic functions in w+(Q).