Alessandro Monguzzi, Marco M. Peloso
In this paper we discuss some recent results concerning the regularity and irregularity of the Bergman and Szegő projections on some weakly pseudoconvex domains that have the common feature to possess a nontrivial Nebenhülle.
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We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of dbar.
We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.
We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap\mbox{ker }\bar{\partial}\rightarrow W^{s}(Ω)$ for all $s\ge 0$.
For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25
A uniform upper bound for the Diederich--Fornaess index is given for weakly pseudoconvex domains whose Levi-form of the boundary vanishes in $\ell$-directions everywhere.
In this paper we give an asymptotic expansion of the Bergman kernel for certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic formula asserts that the singularity of the Bergman kernel at weakly pseudoconvex points is essentially expressed by using two variables; moreover certain real blowing-up is necessary to understand its singularity. The form of the asymptotic expansion with respect to each variable is similar to that in the strictly pseudoconvex case due to C. Fefferman. We also give an analogous result in the case of the Szego kernel.
We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained in case the boundary is of finite type.
Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex Kähler case under mixed positivity conditions.
Let $Ω$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation $$ \begin{cases} \det\left[\dfrac{\partial^2(u)}{\partial z_i\partial\bar z_j}\right]=h\ge 0 & \text{in}\quadΩ,\\ u=ϕ& \text{on} \q…
A new explicit construction of Cauchy-Fantappié kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains.
We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $Ω$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap{ker} \bar{\partial} \rightarrow W^{s}(Ω)$ for all $s\ge 0$ if there is a corresponding solution operator to the $\bar{\partial}_b$-problem with analogous regularity properties.
Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains
本文研究了弱伪凸域中Bergman核及其相关不变量在逼近h-可延展边界点序列时的渐进行为。
Let $Ω$ be a bounded, weakly pseudoconvex domain in C^n, n > 1, with real-analytic boundary. A real-analytic submanifold $M \subset bdΩ$ is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to $\Cal{O}(\barΩ)$. 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 $bdΩ$.
In this short note we improve some of recent results of Cuckovic and Sahutoglu concerning regularity of the Berezin transform for a class of smooth pseudoconvex domains.
We show here a "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The $f$-property is a potential-theoretical condition which holds for all pseudoconvex domains of finite type and many examples of infinite type.
We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case…
Global embeddings of weakly pseudoconvex complex spaces and refined Runge-type approximation theorems
本文通过典范奇点解消构造奇异Hermite度量，利用强开性控制乘子理想层，建立了适用于复空间的精细Runge型逼近定理，并证明了弱伪凸复空间正则点的全局全纯嵌入，进而解决了弱伪凸复流形的并集问题。
We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth strictly pseudoconvex domains.
We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n ; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and $m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point, $k \in \Natl$ the degenerate rank of the Levi form at $z^0$. An explicit formula for $K^B(z)$ modulo analytic functions is given in terms of the polar coordinates $(t_1, \ldots, t_k, r)$ around $z^0$. This formula provides detailed information about the singularities of $K^B(z)$, which improves the result of A. Bonami and N. Lohoué \cite{bol}. A similar result is e…
Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type
本文证明满足 Bell-Ligocka 条件 R 且具有良好各向异性伸缩的伪凸域上，Bergman 投影具备局部 $L^p$-Sobolev 与 Hölder 正则性，并说明 $h$-可延拓域适用该结论。
Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex Kähler manifolds
本文针对弱伪凸凯勒流形提出对数阿基祖-纳科诺消失定理，并推广了诺里马atsu在紧凯勒流形上的研究结果。
Constructing bounded pseudoconvex domains of finite D'Angelo type with prescribed weak loci
研究在\(\mathbb{C}^n\)中构造特定域，核心方法是利用规定闭集，贡献为得到相关有界伪凸或凸域。
We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which are powers of the defining functions of the two domains.
Bigness of adjoint linear subsystem and approximation theorems with ideal sheaves on weakly pseudoconvex manifolds
本文研究了弱伪凸流形上余线性子系统的大小及其与理想层相关的近似定理，推广了德马尔在复几何中的正性特征。
$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics
本文研究了弱伪凸复空间上 $L^2$-Dolbeault 解析与 $L^2$-估计，推广了 Nadel 消失定理。
In this paper, we consider the boundary M of a weakly pseudoconvex domain in a Stein manifold. We point out a striking difference between the local cohomology and the global cohomology of M, and illustrate this with an example. We also discuss the first and second Cousin problems, and the strong Poincare problem for CR meromorphic functions on the weakly pseudoconvex boundary M.
In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carathéodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of th…
We make two tiny corrections to our previous paper with the same title, and also obtain, as a bonus, something new.
We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the bisectional curvatures of the Bergman metric of a smooth bounded pseudoconvex domain near the boundary with constant Levi rank.
We study proper holomorphic mappings between strictly pseudoconvex domains with low boundary regularity.
We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.
The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ann. of Math. 102 (1975), 223-290]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR geometry for almost a quarter of a century because one can use the theory of isolated singularities to study the theory of CR manifolds and vice versa.
By using a variant Property $(P_q)$ of Catlin, we discuss the relation of small set of weakly pseudoconvex points on the boundary of pseudoconvex domain and compactness of the $\overline{\partial}$-Neumann operator. In particular, 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 compact on $(0,n-1)$-level $L^2$-integrable forms.
We prove that for certain classes of pseudoconvex domains of finite type, the Bergman-Toeplitz operator $T_ψ$ with symbol $ψ=K^{-α}$ maps from $L^{p}$ to $L^{q}$ continuously with $1< p\le q<\infty$ if and only if $α\ge\frac{1}{p}-\frac{1}{q}$, where $K$ is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains by Čučković and McNeal, and Abeta, Raissy and Saracco.
Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C^N (n less than or equal to N), of codimension one or more in C^N, and endowed with the induced CR structure. We show the tangential Cauchy-Riemann operator has closed range on such a manifold M, hence we get global existence and regularity results for the \bar\partial_b problem. We also show the middle (i.e. corresponding to (p,q) forms for q between 1 and n-2) \bar\partial_b cohomology groups of M with respect to L^2, Sobolev s, and smooth coefficients are finite and i…
In this article, we consider minimal $L^2$ integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex Kähler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal $L^2$ integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to the modules, a strong openness property of the modules and a twisted version, an effect…
Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds
本文为拟凸域上的 $\bar\partial$-Neumann 问题建立了 $(p,q)$-形式全局正则性的一般充分条件，主要利用严格多重次调和函数或正全纯双截曲率，以及满足弱紧性估计的横截向量场族，并给出应用实例。
We review some recent results on existence and regularity of Monge-Ampère exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.
Rarity of $\boldsymbol{\mathcal{C}^{1,1}}$ solutions to the complex Monge--Ampère equation on weakly pseudoconvex domains
研究显示，在弱伪凸B-正则域上，经典Dirichlet问题的$\mathcal{C}^\infty$光滑数据通常不具有$\mathcal{C}^{1,1}$光滑解，为潜在的势理论解扩展问题奠定基础。
We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $\mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $\mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the …
In this paper we study the Bergman-Toeplitz operator $T_ψ$ induced by $ψ(w) = K_Ω^{-α}(w,w)d_Ω^β(w)$ with $α, β\geq 0$ acting from a weighted $L^p$-space $L_a^p(Ω)$ to another one $L_a^q(Ω)$ on a large class of pseudoconvex domains of finite type. In the case $1 < p \leq q < \infty$, the following results are established: \\ - Necessary and sufficient conditions for boundedness, which generalize the recent results obtained by Khanh, Liu and Thuc.\\ - Upper and lower estimates for essential norm, in particular, a criterion for compactness.\\ - A characterization of Schatten class membership of …
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…
A Békollè-Bonami Class of Weights for Certain Pseudoconvex Domains
本文证明了特定伪凸域上普通 Bergman 投影在推广的 Békollè-Bonami 权重类下的加权 L^p 正则性，主要利用 Bergman 核估计和好-λ不等式。
Removable sets for pseudoconvexity for weakly smooth boundaries
研究 C^{1,1} 弱光滑边界上拟凸性可去集的判定条件，证明零测闭集外局部拟凸可推出全局拟凸，并以峰集型假设给出 Grauert–Remmert 结果的新证明。
Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C^N (N greater than or equal to 2), of codimension one or more in C^N, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator \bar\partial_b has closed range in L^2 in order to rule out the Rossi example, we push regularity up to show \bar\partial_b has closed range in all Sobolev spaces s for s greater than zero. We then use the Szegö projection to show there is a smooth solution to the \bar\partial_b problem given smooth data. The results are obtained via …
Sphericity and Analyticity of a strictly pdeusoconvex hypersurface in low regularity I
本文将严格伪凸超曲面的解析可正则化理论推广至有限光滑性情形，引入正则化 $(0,1)$ 截面工具，解决了 $\mathbb C^2$ 中低正则性下检验超曲面球面性的开放问题，并推导出 $C^6$ 正则性下 CR 曲率消失超曲面的球面性。
The purpose of this article is to study Lipschitz CR mappings from an $h$-extendible (or semi-regular) hypersurface in $\mbb C^n$. Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.
For a pseudoconvex domain in complex space, we prove the equivalence of the local hypoellipticity of the system (di-bar, di-bar*) with the system (di-bar_b,di-bar*_b) induced in the boundary. This develops a result of ours which used the theory of the "harmonic" extension by Kohn. This technique is inadequate for the purpose of the present paper and must be replaced by the "holomorphic" extension introduced by the authors in former work.
The Levi geometry at weakly pseudoconvex boundary points of domains in C^n, n \geq 3, is sufficiently complicated that there are no universal model domains with which to compare a general domain. Good models may be constructed by bumping outward a pseudoconvex, finite-type Ω\subset C^3 in such a way that: i) pseudoconvexity is preserved, ii) the (locally) larger domain has a simpler defining function, and iii) the lowest possible orders of contact of the bumped domain with \bdyΩ, at the site of the bumping, are realised. When Ω\subset C^n, n\geq 3, it is, in general, hard to meet the last two …
Weighted $L^p$ Estimates for the Bergman and Szegő Projections on Strongly Pseudoconvex Domains with Near Minimal Smoothness
本文证明近最小光滑强伪凸域上 Bergman 与 Cauchy-Szegő 投影的加权 $L^p$ 正则性，将 $B_p/A_p$ 条件推广至基于齐型空间拟度量的球上。
Sobolev regularity for the $\overline\partial$-Neumann operator and transverse vector fields
本文通过切向向量场提出新条件，研究$\overline\partial$-Neumann算子在伪凸域上的Sobolev正则性问题。
We give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.
We show that if the closure of a smooth pseudoconvex domain admits a "sufficiently nice" Stein neighborhood basis, then the d-bar Neumann problem on the domain is globally regular.
The Eigenvalue Problem for the Complex Hessian Operator on $m$-Pseudoconvex Manifolds
本文研究强$m$-伪凸流形上复Hessian算子的特征值问题，证明了第一特征函数的$C^{1,1}$正则性与唯一性，并给出第一特征值的变分公式，进而推导出分岔型定理和几何界。
Let $Ω$ be a $C^4$-smooth bounded pseudoconvex domain in $\mathbb{C}^2$. We show that if the $\overline{\partial}$-Neumann operator $N_1$ is compact on $L^2_{(0,1)}(Ω)$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial})\cap Dom(\overline{\partial}^*) \to L^2_{(0,1)}(Ω)$ is $L^p$-regular for all $2\leq p<\infty$.
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