Let $Ω\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $Ω$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partialΩ$ has a holomorphic extension to $Ω$. For unbounded domains this extension property may fail, for example if $Ω$ contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of $\mathbb{C}^n\backslash\overlineΩ$ is $\mathbb{C}^n$. It seems that it is a first result in the literature which gives a geometric characterization of unbounded domains in $\mathbb C^n$ for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z.~Słodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface and the boundary of the domain.
A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $Σ_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional gen…
In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new unexpected phenomena. In particular we prove that the Dolbeault isomorphism fails to hold for the Dolbeault cohomology with prescribed support
We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L_{2}$ functions on coverings of pseudoconvex manifolds.
The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is defined by $$ D_{n, m}(μ):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w \Vert^2<e^{-μ\Vert z \Vert^2}\},$$ where $μ>0.$ The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n, m}(μ)$ with respect to the weight $(-ρ)^α$, where $ρ(z,w):=\|w\|^2-e^{-μ\|z\|^2}$ is a defining function for $D_{n, m}(μ)$ and $α>-1$. Then, for $p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P…
We prove that on smooth bounded pseudoconvex Hartogs domains in $\mathbb{C}^2$ compactness of the $\bar{\partial}$-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain.
The present paper have been replaced by the paper: Hartogs-Bochner type theorem in Projective Space (math.CV/0011095) in which we prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.
In this paper we investigate a class of domains $Ω^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. we first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …
We prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.
It is shown that a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{C}^2$ has trivial Nebenhülle. The smoothness assumption is used to invoke a theorem of D. Catlin.
We show that Property $(P)$ of $\partialΩ$, compactness of the $\bar{\partial}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $Ω={\{(z, w_1, w_2,\dots, w_n) \in \mathbb{C}^{n+1} \mid\sum_{k=1}^{n} |w_k|^2 < e^{-2φ(z)}, z\in\mathit{D}\}}$ are equivalent, where $D$ is a smooth bounded connected open set in $\mathbb{C}$.
The Bergman theory of domains $\{ |{z_{1} |^γ} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $γ$, including all positive integers. For such $γ$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_γ$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_γ(z,z)$.
We study weighted Bergman projections on the Hartogs triangle in $\mathbb{C}^2$. We show that projections corresponding to exponentially vanishing weights have degenerate $L^p$ mapping properties.
We show that every holomorphic map $f\in\mathcal{H}(Ω\setminus K)$ ($K\subsetΩ\subset\mathbb{C}^n$, with $K$ compact, $Ω$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(Ω)$ matching $f$ on an open subset $C_{K,Ω}\subsetΩ\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps $f\in\mathc…
In this paper we investigate a two classes of domains in $\mathbb{C}^n$ generalizing the Hartogs triangle. We prove optimal estimates for the mapping properties of the Bergman projection on these domains.
A question of Poletsky was to know if there exists a thin Hartogs figure such that any of its neighborhoods cannot be imbedded in Stein spaces. In \cite{chirka}, Chirka and Ivashkovitch gave such an example arising in an open complex manifold. In this paper, we answer to the question of the existence of such a figure in compact surfaces by giving an example arising in $P_2(\mathbb{C})$. By smoothing it, we obtain a smooth (non analytic) disc with boundary $\bar{D} \subset P_2(\mathbb{C})$ having the same property. Consequently, this disc intersects all algebraic curves of $P_2(\mathbb{C})$. Mo…
In this paper we determine the automorphism group of the Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^n\times\mathbb{C}^m$ which is defined by the inequality ${\|ζ\|}^2<e^{-μ{\|z\|}^2}$.
A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.
Let $ϕ$ be a globally defined real Kähler potential on a domain $Ω\subset \mathbb{C}^d$, and $g_{F}$ be a Kähler metric on the Hartogs domain $ M=\{(z,w)\in Ω\times\mathbb{C}^{d_0}: \|w\|^2<e^{-ϕ(z)}\}$ associated with the Kähler potential $Φ_{F}(z,w)=ϕ(z)+F(ϕ(z)+\ln\|w\|^2)$. Firstly, we obtain explicit formulas of the coefficients $\mathbf{a}_j\;(j=1,2)$ of the Bergman function expansion for the Hartogs domain $( M,g_F)$ in a momentum profile $φ$. Secondly, using explicit expressions of $\mathbf{a}_j\;(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $\mathbf{a}_j\…
We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on $$\{(z_1,z_2,w)\in\mathbb C^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\}$$ and on $$\{(z_1,z_2,w)\in\mathbb C^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;{\rm and} \;|w|<1\}.$$ We use our formulas to determine th…
The purpose of this paper is twofold. Firstly, we will compute the explicit expression of the Rawnsley's $\varepsilon$-function $\varepsilon_{(α,g(μ;ν))}$ of $\big(\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ),g(μ;ν)\big)$, where $g(μ;ν)$ is a Kähler metric associated with the Kähler potential $-\sum_{j=1}^kν_j\ln N_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\ln(\prod_{j=1}^kN_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\|w\|^2)$ on the generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ and obtain necessary and sufficient conditions for $\varepsilon_{(α,g(μ;ν))}$ to become a poly…
In this paper, we obtain some $L^{p}$ mapping properties of the Bergman-Toeplitz operator \[ f\longrightarrow T_{K^{-α}}\left(f\right):=\intop_ΩK_Ω\left(\cdot,w\right)K^{-α}\left(w,w\right)f\left(w\right)dV(w) \] on fat Hartogs triangles $Ω_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\left|z_{1}\right|^{k}<\left|z_{2}\right|<1\right\} $, where $α\in\mathbb{R}$ and $k\in \mathbb Z^+$.
The Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper mainly consists of three parts. Firstly, we give the explicit expression of geodesics of $D_{n,1}$ in the sense of Kobayashi pseudometric; Secondly, using the formula of geodesics, we calculate explicitly the Kobayashi pseudometric on $D_{1,1}$; Lastly, we establish the Schwarz lemma at the boundary for holomorphic mappings between the nonequidimension…
We apply modern techniques of dyadic harmonic analysis to obtain sharp estimates for the Bergman projection in weighted Bergman spaces. Our main theorem focuses on the Bergman projection on the Hartogs triangle. The estimates of the operator norm are in terms of a Bekollé-Bonami type constant. As an application of the results obtained, we give, for example, an upper bound for the $L^p$ norm of the Bergman projection on the generalized Hartogs triangle $\mathbb H_{m/n}$ in $\mathbb C^2$.
Consider a domain $\varOmega$ in $\mathbb{C}^n$ with $n\geqslant 2$ and a compact subset $K\subset\varOmega$ such that $\varOmega\backslash K$ is connected. We address the problem whether a holomorphic line bundle defined on $\varOmega\backslash K$ extends to $\varOmega$. In 2013, Fornæss, Sibony and Wold gave a positive answer in dimension $n\geqslant 3$, when $\varOmega$ is pseudoconvex and $K$ is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for $K$ of general shape, we construct counterexamples in any dimension $n\geqslant 2$. The key is a certain gluing lemma…
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(μ)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(μ)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(μ)$ and $D_{N,1}(μ)$ with $N<2n$.
In this paper, we show that for each $k\in \mathbb Z^+, p>4$, there exists a solution operator $\mathcal T_k$ to the $\bar\partial$ problem on the Hartogs triangle that maintains the same $W^{k, p}$ regularity as that of the data. According to a Kerzman-type example, this operator provides solutions with the optimal Sobolev regularity.
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a Kähler metric $αg(μ;ν)$ $(α>0)$ on $D_{n,m}(μ)$, where $g(μ;ν)$ is the Kähler metric associated with the Kähler potential $Φ(z,w):=μν{\Vert z\Vert}^{2}-\ln(e^{-μ{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($ν>-1$) on $D_{n,m}(μ)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of th…
The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ is defined as the Hartogs type domain constructed over the product $\prod_{j=1}^kΩ_j$ of irreducible bounded symmetric domains $Ω_j$ $(1\leq j \le…
We provide characterizations of Carleson measures on a certain class of bounded pseudoconvex domains. An example of a vanishing Carleson measure whose Berezin transform does not vanish on the boundary is given in the class of the Hartogs triangles $$\mathbb{H}_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\left|z_{1}\right|^{k}<\left|z_{2}\right|<1\right\},\;k\in\mathbb{Z}^{+}.$$
We prove two density theorems for quadrature domains in $\mathbb{C}^n$, $n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D \times Ω$, where $D \subset \mathbb{C}^{n-1}$ is a smoothly bounded domain satisfying Bell's Condition R and $Ω\subset \mathbb{C}$ is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in $\mathbb{C}^2$.
We consider the family $\mathcal P$ of $n$-tuples $P$ consisting of polynomials $P_1, \ldots, P_n$ with nonnegative coefficients which satisfy $\partial_i P_j(0) = δ_{i, j},$ $i, j=1, \ldots, n.$ With any such $P,$ we associate a Reinhardt domain $\triangle^{\!n}_{_P}$ that we will call the generalized Hartogs triangle. We are particularly interested in the choices $P_a = (P_{1, a}, \ldots, P_{n, a}),$ $a \geq 0,$ where $P_{j, a}(z) = z_j + a \prod_{k=1}^n z_k,~ j=1, \ldots, n.$ The generalized Hartogs triangle associated with $P_a$ is given by \begin{equation} \triangle^{\!n}_a = \Big\{z \in …
We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$.
We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than known optimal estimates. When the weight function is radial, we prove that our estimate provides the $L^2$-minimum extension.
We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $\mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $\mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.
We use convex geometry tools, in particular John ellipsoids, to obtain a size estimate for the Szegő kernel on the boundary of a class of unbounded convex domains in $\mathbb{C}^n.$ Given a polynomial $b:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying a certain growth condition, we consider domains of the type $Ω_b = \{ z\in\mathbb{C}^{n+1}\,:\, {\rm Im}[z_{n+1}] > b({\rm Re}[z_1],\ldots,{\rm Re}[z_n]) \}.$
Let $Ω\subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial Ω$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial Ω\to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial Ω$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\barΩ$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial Ω$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\barΩ$ in $…
Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $Ω\subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial Ω$ has only nondegenerate CR singularities. We prove that if $f \…
The goal of this note is to explore the relationship between the Folland-Kohn basic estimate and the $Z(q)$-condition. In particular, on unbounded pseudoconvex (resp., pseudoconcave) domains, we prove that the Folland-Kohn basic estimate is equivalent to uniform strict pseudoconvexity (resp., pseudoconcavity). As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to a the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland-Kohn basic estimate fails. On unbounded non-pseudoconvex domains, we show that th…
In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$ and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in $\mathbb{C}^n$. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and even more critically, in $L^2$-Sobolev spaces.
Let $Ω\subset\mathbb{C}^n$ be a domain and $1 \leq q \leq n-1$ fixed. Our purpose in this article is to establish a general sufficient condition for the closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$-Sobolev spaces on $(0,q)$-forms. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and, even more critically, in $L^2$-Sobolev spaces.
We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of $L_h^2$-domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in $\mathbb C^2$. This allows easily to decide which pseudoconvex balanced domain in $\mathbb C^2$ has a positive Bergman kernel and which admits the Bergman metric…
We construct examples of twice differentiable functions in $\mathbb{R}^n$ with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability.
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a Levi flat unbounded domain of $\mathbb{C}^n$ with the complementary is also Levi flat and unbounded.
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
In this paper we study the global geometry of the Kobayashi metric on "convex" sets. We provide new examples of non-Gromov hyperbolic domains in $\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex, bounded and unbounded. Our first aim is to prove that if $Ω$ is a bounded weakly linearly convex domain in $\mathbb{C}^n,\,n\geq 2,$ and $S$ is an affine complex hyperplane intersecting $Ω,$ then the domain $Ω\setminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we sho…
We construct bounded pseudoconvex domains in $\mathbb{C}^2$ for which the Szegö projection operators are unbounded on $L^p$ spaces of the boundary for all $p\not =2$.
