Jian Chen, Sheng Rao
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.
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We solve the $\partial\bar{\partial}$-problem for extensible currents defined on a strongly pseudoconvex domain of a contractible manifold.
In this present paper, we solve the $\partial\bar{\partial}$ for extendable currents definite in a pseudoconvexe unbounded domain of $\mathbb{C}^n$ .
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
We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a strongly pseudoconvex contractible domain of a complex manifold.
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.
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.
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 $s\ge 0$.
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.
It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maxima…
$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains
研究有界严格伪凸域上的∂̄算子相关Sobolev型不等式，推导带迹Sobolev不等式推广，得到全纯函数最大模原理积分形式及∂̄的Hörmander L²估计改进结果。
We show that for any bounded domain $Ω\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in bΩ$, having a Stein neighborhood basis, there is a biholomorphic map $f:\barΩ\rightarrow \Cp ^n $ such that $f(p)$ is a global extreme point of type $2k$ for $f{(\barΩ)}$.
We consider the Szegö kernel for domains Ωin C^2 given by Ω= {(z,w): Im w > b(Re z)} where b is a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe the subset of \barΩ \times \barΩ on which the kernel and all its derivatives are finite. In particular, we show that there are points off the diagonal of the boundary at which the Szegö kernel is infitie as well as points on the diagonal at which it is finite.
We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.
This paper is concerned with proving superlogarithmic estimates for the operator $\Box_b$ on pseudoconvex CR manifolds and using them to establish hypoellipticity of $\Box_b$ and of the $\bar{\partial}$-Neumann problem. These estimates are established under the assumption that subellipticity degenerates in certain specified ways.
Boas and Straube proved a general sufficient condition for global regularity of the d-bar Neumann problem in terms of families of vector fields that commute approximately with d-bar. In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.
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 $bdΩ$.
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 …
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 $$ \begin{cases} \det\left[\dfrac{\partial^2(u)}{\partial z_i\partial\bar z_j}\right]=h\ge 0 & \text{in}\quadΩ,\\ u=ϕ& \text{on} \q…
We study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.
We study the equation $\dot{u}=\log\det (u_{α\barβ})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the Kähler-Ricci flow. In this paper, we consider the case of the boundary conditions are smooth and the initial conditions are bounded.
Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds
本文为拟凸域上的 $\bar\partial$-Neumann 问题建立了 $(p,q)$-形式全局正则性的一般充分条件，主要利用严格多重次调和函数或正全纯双截曲率，以及满足弱紧性估计的横截向量场族，并给出应用实例。
The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is defined by $$ 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…
Subelliptic and Maximal $L^p$ Estimates for the Complex Green Operator on non-pseudoconvex domains
该研究在满足交换子有限型条件的非拟凸CR流形上，证明了$\bar\partial_b$-复特定层$q$处复格林算子$K_q$的次椭圆估计，通过闭值域估计得到极大$L^p$估计，结果适用于含弱$Y(q)$流形的族，方法为微局部分解和Calderón-Zygmund理论。
Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex Levi corank one domain with defining function $r$, i.e., the Levi form $\partial \bar {\partial} r$ of the boundary $\partial D$ has at least $(n - 2)$ positive eigenvalues everywhere on $\partial D$. The main goal of this article is to obtain a lower bound for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points $p, q \in D$ in terms of parameters that reflect the Levi geometry of $\partial D$ and the distance of these points to the boundary. Applications include an understanding of Fridman's invari…
For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.
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…
We prove that the tangential Cauchy-Riemann operator has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold $X$. Our result generalizes the known case when $X$ is a Stein manifold.
Suppose that a smooth holomorphic curve $V$ has order of contact $η$ at a point $w_0$ in the boundary of a pseudoconvex domain $Ω$ in $\mathbb{C}^3.$ We show that the maximal gain in Hölder regularity for solutions of the $\bar{\partial}$-equation is at most $\frac{1}η.$
We prove local hypoellipticity of the complex Laplacian $\Box$ and of the Kohn Laplacian $\Box_b$ in a pseudoconvex boundary when, for a system of cut-off $η$, the gradient $\partial_bη$ and the Levi form $\partial_b\bar\partial_bη^2$ are subelliptic multipliers in the sense of Kohn.
A sufficient condition for $\bar{\partial}$ to have closed range is given for pseudoconvex, possibly unbounded domains in $\mathbb{C}^n$.
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.
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 $\mathbb{C}$.
We show, under an assumption on the weakly pseudoconvex points, the trivial Diederich--Fornæss index directly implies the global regularities of the $\bar{\partial}$-Neumann operators.
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.
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.
In this paper we study holomorphic approximation using boundary value problems for $\bar\partial$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\bar\partial$ are the mixed boundary problems on an annulus. We characterize pseudoconvexity and Runge type property of the domain by the vanishing of related $L^2$ cohomology groups.
An integral solution operator for $\bar\partial$ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy $L^p$ estimates for all $1\leq p <\infty$, though with non-standard -- relative to strongly pseudoconvex domains -- bounding term. These estimates imply $L^p$ estimates for $\bar\partial$ on the Hartogs triangle, with greater range of $p$ than the canonical solution satisfies.
We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C^n is an obstruction to compactness of the d-bar-Neumann operator on the domain, provided that at some point of M, the Levi form has the maximal possible rank n-1-dim(M) (i.e. the domain is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue. We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmon…
We study the $\bar\partial$-Neumann Laplacian from spectral theoretic perspectives. In particular, we show how pseudoconvexity of a bounded domain is characterized by positivity of the $\bar\partial$-Neumann Laplacian.
We establish compactness estimates for $\bar{\partial}_{M}$ on a compact pseudoconvex CR-submanifold $M$ of $\mathbb{C}^{n}$ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\bar{\partial}$-Neumann operator given by the authors earlier. These conditions are formulated in terms of certain short time flows in complex tangential directions.
An extension theorem for bundles with respect to strictly pseudoconvex extensions
该研究针对复流形上的全纯主G丛，在严格伪凸延拓非临界或拓扑丛可延拓的假设下，证明其可延拓，为Donaldson提出的延拓问题提供新证明与推广。
We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.
A smooth bounded pseudoconvex domain in two complex variables is of finite type if and only if the number of eigenvalues of the d-bar-Neumann Laplacian that are less than or equal to $λ$ has at most polynomial growth as $λ$ goes to infinity.
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 of compactness does \emph{not} proceed via verifying the known potential theoretic sufficient conditions.
Let $M$ be a strongly pseudoconvex complex $G$-manifold with compact quotient $M/G$. We provide a simple condition on forms $α$ sufficient for the regular solvability of the equation $\square u=α$ and other problems related to the $\bar\partial$-Neumann problem on $M$.
Analytic discs and compactness of the $\bar\partial$-Neumann operator
本研究构造了一个特殊的有界拟凸完全Reinhardt域，其边界含解析圆盘且不满足相关性质，但对应的$\bar\backslash$partial-Neumann算子仍是紧的，否定解决了该算子紧性的公开问题。
We give several versions of Siu's $\partial\bar{\partial}$-formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold $(M^{2m+1}, θ)$ into a Kähler manifold $(N^n, g)$. We also define and study the notion of pseudo-Hermitian harmonicity for maps from $M$ into $N$. In particular, we prove a CR version of Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.
The $\bar{\partial}$-Neumann problem is the fundamental boundary value problem in several complex variables. It features an elliptic operator coupled with non-coercive boundary conditions. The problem is globally regular on many, but not all, pseudoconvex domains. We discuss several recent developments in the $L^{2}$-Sobolev theory of the $\bar{\partial}$-Neumann problem that concern compactness and global regularity.
Let $Ω\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(Ω)$. The key feature of our results is a precise formula for $k$ in terms of the geometry of the boundary of $Ω$.
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 $Ω$ be a complex manifold, and let $X\subset Ω$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $Π$ be a differentiable principal $G$-bundle on $Ω$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= Π|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $Π$ to a neighborhood of $\bar X$ in $Ω$. This answers positively and generalizes a problem stated in the article "Bou…
We prove heat kernel estimates for the $\bar\partial$-Neumann Laplacian acting in spaces of differential forms over noncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.
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.
Let Y be a pure dimensional analytic variety in C^n with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of this paper is to present a technique which allows to determine obstructions to the solvability of the d-bar-equation in the $L^2$ respectively $L^\infty$ sense on Y^*=Y-{0} in terms of certain cohomology classes on X. More precisely, let D be a Stein domain, relatively compact in Y, containing the origin, D^*=D-{0}. We give a sufficient condition for the solvability of the d-bar-equation in the $L^2$-sense o…
We give a Hörmander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $δ_Ω^{-α}dV$, $α<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function theory of weighed Bergman spaces $A^2_α(Ω)$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $A^2_α(Ω)$ by proving an analogue of the Levi problem for $A^2_α(Ω)$ and giving an optimal Gehring type estimate for functions in $A^2_α(Ω)$. A vanishing theorem for $A^2_1…
Let $Ω$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $φ$ be a $\mathcal C^2$-weight function plurisubharmonic on $Ω$. We show both necessary and sufficient conditions for existence and compactness of a weighted $\bar\partial$-Neumann operator $N_φ$ on the space $L^2_{(0,1)}(Ω,e^{-φ})$ in terms of the eigenvalues of the complex Hessian $(\partial ^2φ/\partial z_j\partial\bar z_k)_{j,k}$ of the weight. We also give some applications to the unweighted $\bar\partial$-Neumann problem on unbounded domains.
Tower multitype and global regularity of the $\bar\partial$-Neumann operator
本文通过引入基于嵌套 $(1,0)$ 子丛的多重类型不变量、相对接触阶和超切向量场等新工具，为有限 D'Angelo $q$-型拟凸超曲面及更一般超曲面建立了性质 $(P_q)$ 的自包含证明，从而得到 $\bar\partial$-Neumann 算子的全局正则性。
We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$ which imply that the $\bar{\partial}$-Neumann operator is compact. These conditions are formulated in terms of certain short time flows in suitable complex tangential directions. It is noteworthy that compactness is \emph{not} established via the known potential theoretic sufficient conditions. Our results generalize to $\mathbb{C}^{n}$ the corresponding $\mathbb C^{2}$ results due to the second author.
We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $Ω$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar\partial$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partialΩ$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then $\dbar_b$ has closed range on $(0,q)$-forms on $\partialΩ$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$.
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