关于单位球Bergman空间上Luzin面积算子作用的一些注记

关于解析函数加权Bergman空间中极值问题的若干注记

Some remarks on extremal problems in weighted Bergman spaces of analytic functions

该研究针对解析函数的加权Bergman空间极值问题，分别在上半平面和带光滑边界的有界严格伪凸域场景下，证明了若干精确的极值距离结果。

关于Hartogs域的Bergman空间维数的一个注记

A remark on the dimension of the Bergman space of some Hartogs domains

设D是Hartogs域，其形式为D={（z，w）\in CxC^N：|w| < e^{-u（z）}}，其中u是C上的下调和函数.证明了D上全纯平方可积〔的〕函数的Bergman空间是平凡的或无穷维的。

A remark on weighted Bergman kernels on orbifolds

In this note, we explain that Ross-Thomas' result on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result. This result plays an important role in the companion paper to prove an orbifold version of Donaldson Theorem.

Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a Kähler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.

Remark on the off-diagonal expansion of the Bergman kernel on compact Kähler manifolds

In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkhäuser, 2007).

Remarks on one-component inner functions

A one-component inner function $Θ$ is an inner function whose level set $$Ω_Θ(\varepsilon)=\{z\in \mathbb{D}:|Θ(z)|<\varepsilon\}$$ is connected for some $\varepsilon\in (0,1)$. We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for $0<p<\infty$, the derivative of a one-component inner function $Θ$ is a member o…