论文标题
$ p- $ bergman内核和公制的某些属性
Some properties of the $p-$Bergman kernel and metric
论文作者
论文摘要
$ p- $ bergman内核$ k_p（\ cdot）$显示为$ c^{1,1/2} $，$ 1 <p <\ p <\ infty $。 Off-Diagonal $ p-$ bergman内核$ k_p（\ cdot，z）$与某些加权$ l^2 $ bergman内核之间的意外关系，价格为$ 1 \ le P \ le 2 $。作为应用程序，我们表明，对于每个$ 1 \ le p \ le 2 $，$ k_p（\ cdot，z）\ in l^q（ω）$ in l^q（ω）$ for $ q <\ frac {2pn} {2pn} {2n-α（ω）} $和$ | k_s（z）-k_p（z）-k_p（z）-k_p（z）| \ Lessim | S-p || \ log | s-p || $每当hyperConvexity index $α（ω）$为正时。 $ 2 <p <\ infty $的反例分别给出。获得$ 2 \ le p <\ infty $时，$ p- $ bergman度量的全体形态截面曲率的最佳上限。对于有限的$ c^2 $域，这表明耐力空间和伯格曼空间满足$ h^p（ω）\ subset a^q（ω）$，其中$ q = p（1+ \ frac1n）$。引入了一个新概念所谓的$ p-$ schwarz内容。作为应用程序，给出了$ p-$ bergman空间之间的Banach-Mazur距离的上限，并且$ a^p（ω）$在$ l^p（ω）$中显示为$ 0 <p \ p \ le 1 $的非chebyshev。对于平面域，我们获得了$ p-$ bergman内核（在高维情况下无效）的刚性定理，并且通过Narasimhan-Simha Metric的完整性来表征非分离的边界点。
The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As applications, we show that for each $1\le p\le 2$, $K_p(\cdot,z)\in L^q(Ω)$ for $q< \frac{2pn}{2n-α(Ω)}$ and $|K_s(z)-K_p(z)| \lesssim |s-p||\log |s-p||$ whenever the hyperconvexity index $α(Ω)$ is positive. Counterexamples for $2<p<\infty$ are given respectively. An optimal upper bound for the holomorphic sectional curvature of the $p-$Bergman metric when $2\le p<\infty$ is obtained. For bounded $C^2$ domains, it is shown that the Hardy space and the Bergman space satisfy $H^p(Ω)\subset A^q(Ω)$ where $q=p(1+\frac1n)$. A new concept so-called the $p-$Schwarz content is introduced. As applications, upper bounds of the Banach-Mazur distance between $p-$Bergman spaces are given, and $A^p(Ω)$ is shown to be non-Chebyshev in $L^p(Ω)$ for $0<p\le 1$. For planar domains, we obtain a rigidity theorem for the $p-$Bergman kernel (which is not valid in high dimensional cases), and a characterization of non-isolated boundary points through completeness of the Narasimhan-Simha metric.
