Samangi Munasinghe, Yunus E. Zeytuncu
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 $p\not =2$.
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Boundary behaviour of the Fefferman--Szegö metric in strictly pseudoconvex domains
本文研究严格伪凸域中 Fefferman--Szegö 度量及相关不变量的边界行为。
Let $(X, T^{1,0}X)$ be a compact strongly pseudoconvex CR manifold of dimension $2n+1$. Assume that $X$ admits a Torus action $T^d$. In this work, we study the behavior of torus equivariant Szegő kernels and prove that the weighted torus equivariant Szegő kernels admit asymptotic expansions.
The Szegö kernel for non-pseudoconvex tube domains in C^2
考虑C^2中非伪凸域的Szegö核，其中Ω= {（z，w）：Im w > b（Re z）}，其中b是具有正首项系数的非凸偶次多项式.这是一个扩展的结果，以前得到的作者的情况下，其中b的程度为4。我们证明了Szegö核在\bar Ω \times \bar Ω的边界的对角线上有奇点，在边界的对角线上也有奇点。
The Szegö kernel for certain non-pseudoconvex domains in C^2
我们考虑C^2中定义域Ω的Szegö核：Ω= {（z，w）：Im w > b（Re z）}，其中b是具有正首项系数的非凸四次多项式.这样的域不是伪凸的。我们描述了\bar Ω \times \bar Ω的子集，在该子集上核及其所有导数是有限的。特别是，我们表明，有点的对角线上的边界处的Szegö内核是infitie以及点的对角线上，它是有限的。
Logarithmic singularity of the Szegö kernel and a global invariant of strictly pseudoconvex domains
严格拟凸区域的Szego核在边界对角线上存在奇点，它由极点和对数型奇点组成。本文证明了对数奇性系数边界上的积分给出区域的双全纯不变量或边界的CR不变量。我们还表明，相同的不变量出现的系数的对数项的体积膨胀的域相对于Bergman体积元。
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。
In this paper, we consider bounded strictly pseudoconvex domains $D\subset \mathbb C^2$ with smooth boundary $M=M^3:=\partial D$. If we consider the asymptotic expansion of the Bergman kernel on the diagonal $$ K_B\sim \frac{ϕ_B}{ρ^{n+1}}+ψ_B\logρ, $$ where $ρ>0$ is a Fefferman defining equation for $D$, then it is well known that the trace of the log term $bψ_B:=(ψ_B)|_M$ on $M$ does not determine the CR geometry of $M$ locally; e.g., the vanishing of $bψ_B$ on an open subset of $M$ does not imply that $M$ is locally spherical there. Nevertheless, the main result in this paper is that if $D\s…
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 Invariant Szegő metric on strongly pseudoconvex domains
本文研究强伪凸域上的Fefferman-Szegő不变度量，建立其L²-Dolbeault上同调在中度以外的消失性，证明其具有C∞有界几何，并利用这些性质得到刚性结果，包括度量梯度Kähler-Ricci孤立子或常数量曲率时域双全纯于单位球，以及在二维情形下通过Kähler浸入条件得到边界局部球面性。
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.
A remark on irregularity of the dbar-Neumann problem on non-smooth domains
J. J. Kohn的一个观察是，对于$C^n$中的光滑有界伪凸域D，存在s>0，使得D上的dbar-Neumann算子连续地将$W^s_{(0,1)}(D)$（具有s阶$L^2$ -索伯列夫空间中的系数函数的$(0,1)$ -形式的空间）映射到其自身。通过在$C^2$中构造一个有界伪凸域D，证明了这个结论在没有光滑性假设的情况下是不成立的，该域D除了在一点处是光滑的，其dbar-Neumann算子对任何s>0在$W^s_{(0,1)}(D)$上都是有界的.
Let $(X, T^{1,0}X)$ be a compact connected orientable strongly pseudoconvex CR manifold of dimension $2n+1$, $n\geq1$. Assume that $X$ admits a connected compact Lie group $G$ action and a transversal CR $S^1$ action, we compute the coefficients of the first two lower order terms of the equivariant Szegő kernel asymptotic expansions with respect to the $S^1$ action.
Asymptotic expansion of the Bergman kernel for weakly pseudoconvex tube domains in C^2
本文给出了C^2中某些有限型弱伪凸管域的Bergman核的一个渐近展开.我们的渐近公式断言，伯格曼核在弱伪凸点处的奇异性本质上是用两个变量表示的;此外，某些实爆破是理解其奇异性所必需的。关于每个变量的渐近展开的形式类似于C的严格伪凸情形。弗利曼。在Szego核的情况下，我们也给出了类似的结果。
This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szegő kernel on $X$. In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sjöstrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline M$.
Boundary behaviour of the Bergman and Szegő kernels on generalized decoupled domains
本文在广义解耦伪凸域上证明了Bergman核、Szegő核及Bergman度量在边界附近的最优估计，该域允许复切方向非独立解耦且边界点可兼具有限与无限型方向。
Comparison of the Bergman and Szegö kernels
对于${\mathbb C}^n$中的光滑有界伪凸域，Szegö核和Bergman核的商对于任何$p>n$都由$δ|\logδ|^p$自上而下定界，其中$δ$是到边界的距离。对于一类包含D'Angelo有限型域和具有多重次调和定义函数的域的域，对于任何$p<-1$，其商也由$δ|\logδ|^p$从下有界.此外，对于凸区域，商从上到下由$δ$的常数倍界定。
On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains
本文计算了一类伪凸Reinhardt域的Szegö核在其对角线上的渐近展开中的首项和次首项，其中伪凸Reinhardt域的边界被赋予了一类一般的光滑测度.我们这样做是通过将它与射影空间上的Bergman核联系起来。
Singularities of the Bergman kernel for certain weakly pseudoconvex domains
考虑域$\ellip = \{z \in \Comp^n ; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$的Bergman核$K^B(z)$，其中$m=(m_1,\ldots,m_n) \in \Natl^n$和$m_n \neq 1$。设$z^0 \in \partial \ellip$是任意弱伪凸点，$k \in \Natl$是Levi型在$z^0$处的退化秩.给出了$K^B(z)$模解析函数在$z^0$周围极坐标$(t_1, \ldots, t_k, r)$下的显式表达式.该公式提供了关于$K^B(z)$奇异性的详细信息，改进了A. Bonami和N. Lohoué \cite{bol}。对于$\ellip$的Szegö核$K^S(z)$也建立了类似的结果。
Let $X$ be a compact strongly pseudoconvex CR manifold with a transversal CR $S^1$-action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components and by using the asymptotics, we show that $X$ can be equivariant CR embedded into some $\mathbb C^N$ equipped with a simple $S^1$-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.
Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation o…
Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szegő kernel function as $m\rightarrow\infty$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.
Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $γ$. A surprising consequence of the analysis is that, whenever $γ$ is irrational, the Bergman projection is bounded only for $p=2$.
In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ as in the recent result of Lanzani--Stein. We point out that in this setting the Cauchy type integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calderón--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calderón--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{blac…
Regularity equivalence of the Szegö projection and the complex Green operator
本文证明了在满足弱$Y(q)$条件的超曲面型CR流形上，复Green算子$G_q$是（整体）正正则的当且仅当Szegö投影$S_{q-1}, S_q$和第三正交投影$S'_{q+1}$是（整体）正正则的.投影$S'_{q+1}$与塞格投影$S_{q+1}$密切相关，并且如果调和$(0,q+1)$ -形式的空间是平凡的，则实际上与它重合。这个结果推广了H. Boas和E. Straube关于光滑有界伪凸域上的$\bar\partial$ -Neumann算子与Bergman投影的正则性的等价性。我们还证明了这一结果的推广的情况下，有界光滑域满足弱$Z(q)$条件的Stein流形。
We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an $\mathbb R$-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasa…
Sobolev regularity of the Bergman and Szegö projections in terms of $\overline{\partial}\oplus\overline{\partial}^{*}$ and $\overline{\partial}_{b}\oplus\overline{\partial}_{b}^{*}$
本文研究了光滑有界伪凸域上的 Bergman 与 Szegö 投影的 Sobolev 正则性，证明了嵌入映射与投影的连续性等价，并指出在 $\mathbb{C}^{2}$ 中 $j_{1}$ 始终正则。
Weighted endpoint bounds for the Bergman and Cauchy-Szegő projections on domains with near minimal smoothness
研究近最小光滑性域上Bergman和Cauchy-Szegő投影的加权端点界，证明了弱型$(1,1)$性质并推导出加权Kolmogorov和Zygmund不等式。
The Cauchy--Szegö Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory
针对最小光滑性（$C^2$ 边界）的有界强伪凸域，利用定量外推方法，证明了 Cauchy--Szegö 投影在 $A_p$-测度加权 $L^p$ 空间上的显式有界性，并识别了 $p=2$ 时可定义该投影的 Hardy 空间类。
The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity
针对$\boldsymbol{\text{ℂ}^n}$中边界为$C^2$类的有界强拟凸域，研究了加权柯西-塞格投影交换子$[b, S_ω]$在$L^p(bD, Ω_p)$中的紧性与有界性（含显式界），并刻画了$p=2$时交换子$[b,S_{Ω_2}]$在对应空间的相关性质。
Weighted Bergman spaces and the $\bar{\partial}-$equation
在任意具有$C^2-$边界的有界拟凸区域上，给出了$\bar{\partial}-$方程关于测度$δ_Ω^{-α}dV$的Hörmander型$L^2-$估计.给出了加权Bergman空间$A^2_α(Ω)$在函数论中的几个应用，包括一个冠型定理、一个Gleason型定理和一个密度定理.我们特别研究了$A^2_α(Ω)$中函数的边界表现，证明了$A^2_α(Ω)$中Levi问题的类似性，并给出了$A^2_α(Ω)$中函数的最优Gehring型估计.对任意有界区域建立了$A^2_1(Ω)$的消灭定理。讨论了加权Bergman核与Szegö核之间的关系。
Regularity of the $p-$Bergman kernel
本文证明了有界域上p-Bergman核的局部C^{1,1}正则性，并给出了全局非正则性例子及Levi形式上界估计和对数Lipschitz连续性结果。
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