# 多复变近期进展

By pxxyyz

Fock-Bargmann-Hartogs域

## 研究方向

1. 借助度量推导边界Schwarz Lemma;
2. 利用Bergman核的精确形式来推导逆紧映射的刚度性质;
3. 判断Bergman算子和Toeplitz算子的有界性;

## 研究背景

• 介绍域: Fock-Bargmann-Hartogs域、广义复椭球、广义Fock-Bargmann-Hartogs域
• Schwarz引理: 域的性质、边界性质、函数性质
• 刚度性质: 域的Bergman核精确形式、域的自同构群、边界点的属性
• 算子有界性: Bergman算子的有界性和正则性、Toeplitz算子的有界性

Fock-Bargmann-Hartogs域(无界非双曲强拟凸+具有光滑实解析边界)

$D_{n, m}(\mu) = \left\{ (z, w) \in \mathbb{C}^{n} \times \mathbb{C}^{m}: \|w\|^{2} < e^{-\mu\|z\|^{2}} \right\}$

• 有界域的几何和解析性质不能直接推广至无界域甚至非双曲强拟凸域
• 与Fock-Bargmann域密切相关,利用加权Hilbert空间的高斯核计算其Bergman核

$\Sigma({n} ; {p})=\left\{\left(\zeta_{1}, \ldots, \zeta_{r}\right) \in \mathbb{C}^{n_{1}} \times \cdots \times \mathbb{C}^{n_{r}}: \sum_{k=1}^{r}\left\|{\zeta}_{k}\right\|^{2 p_{k}}<1\right\}$

$D_{n_{0}}^{n, p}(\mu)=\left\{\left(z, w_{(1)}, \ldots, w_{(\ell)}\right) \in \mathbb{C}^{n_{0}} \times \mathbb{C}^{n_{1}} \times \cdots \times \mathbb{C}^{n_{\ell}}: \right.\left.\sum_{j=1}^{\ell}\left\|w_{(j)}\right\|^{2 p_{j}} < e^{-\mu\|z\|^{2}}\right\}$

• 无界非双曲域

$b D_{n_{0}}^{n, p}(\mu)=b_{0} D_{n_{0}}^{n, p}(\mu) \cup b_{1} D_{n_{0}}^{n, p}(\mu) \cup b_{2} D_{n_{0}}^{n, p}(\mu)$

• 非强拟凸域+边界非光滑
• $b_{0} D_{n_{0}}^{n, p}(\mu)$ 实解析+强拟凸
• $b_{1} D_{n_{0}}^{n, p}(\mu)$ 弱拟凸但非强拟凸
• $b_{2} D_{n_{0}}^{n, p}(\mu)$ 非光滑

• 域的性质： 有(无)界域、凸性、度量、广义域
• 边界性质： 等维度、光滑性、不动点
• 函数性质： 全纯(调和)、(高阶)导、特征值

Let $F = ( f , h) : D_{1,1} \to D_{n,m}$ be a holomorphic mapping and holomorphic at $p \in \partial D_{1,1}$ with $F(p) = q \in \partial D_{n,m}$. Then we have the result as follows::

• There exists $\lambda \in \mathbb{R}$ such that $\overline{J_{F}(p)}^{T} q^{T}=\lambda p^{T}$ with $\lambda \geq|1-\overline{h_{1}(0)}|^{2} /\left(1-\left|h_{1}(0)\right|^{2}\right)>0$. Notice that $p^T$ and $q^T$ are the normal vectors to the boundary of $D_{1,1}$ at $p$ and $D_{n,m}$ at $q$, respectively.
• $J_F(p)$ can be regarded as a linear operator from $T_{p}^{1,0}\left(\partial D_{1,1}\right)$ to $T_{F(p)}^{1,0}\left(\partial D_{n,m}\right)$. Moreover, we have $\left\|J_{F}(p)\right\|_{o p} \leq \sqrt{\lambda}$ *where $\left\|\cdot\right\|_{o p}$ means the usual operator norm.

• 域的Bergman核精确形式
• 域的自同构群
• 边界点的属性

If $D_{n,m}(\mu)$ and $D_{n^{\prime}, m^{\prime}}\left(\mu^{\prime}\right)$ are two equidimensional Fock-Bargmann-Hartogs domains with $m \geq 2$ and $f$ is a proper holomorphic mapping of $D_{n,m}(\mu)$ into $D_{n^{\prime}, m^{\prime}}\left(\mu^{\prime}\right)$, then $f$ is a biholomorphism between $D_{n,m}(\mu)$ and $D_{n^{\prime}, m^{\prime}}\left(\mu^{\prime}\right)$.

Suppose $D_{n_{0}}^{n, p}(\mu)$ and $D_{m_{0}}^{m, q}(\nu)$ are two equidimensional generalized Fock-Bargmann-Hartogs domains with

$\min \left\{n_{1+\epsilon}, n_{2}, \ldots, n_{\ell}, n_{1}+\cdots+n_{\ell}\right\} \geq 2$

$\min \left\{m_{1+\delta}, m_{2}, \ldots, m_{\ell}, m_{1}+\cdots+m_{\ell}\right\} \geq 2$

Then any proper holomorphic mapping between $D_{n_{0}}^{n, p}(\mu)$ and $D_{m_{0}}^{m, q}(\nu)$ must be a biholomorphism; any proper holomorphic self-mapping of $D_{n_{0}}^{n, p}(\mu)$ must be an automorphism.

• Hartogs三角域

$\mathbb{H}=\left\{\left(z_{1}, z_{2}\right) \in \mathbb{C}^{2} :|z_{1}|<| z_{2} |<1\right\}$

• 广义Hartogs三角域

$\mathbb{H}_{\left\{k_{j}, \phi_{j}\right\}}^{n}=\left\{z \in \mathbb{C}^{n}:\max _{1 \leq j \leq l}| \phi_{j}\left(\tilde{z}_{j}\right)|<| z_{k+1}|<\cdots<| z_{n} |<1\right\}$

For $1 \leq p < \infty$ and $1 \leq k < n$, the Bergman projection $P_{\mathbb{H}_{\left\{k_{j}, \phi_{j}\right\}}^{n}}$ for $\mathbb{H}_{\left\{k_{j}, \phi_{j}\right\}}^{n}$ is bounded on $L^{p}\left(\mathbb{H}_{\left\{k_{j}, \phi_{j}\right\}}^{n}\right)$ if and only if $p$ is in the range $\left(\frac{2 n}{n+1}, \frac{2 n}{n-1}\right)$.

• Hartogs三角域

$\mathbb{H}=\left\{\left(z_{1}, z_{2}\right) \in \mathbb{C}^{2} :|z_{1}|<| z_{2} |<1\right\}$

• 广义Hartogs三角域

$\mathbb{H}_{\left\{k_{j}, \phi_{j}\right\}}^{n}=\left\{z \in \mathbb{C}^{n}:\max _{1 \leq j \leq l}| \phi_{j}\left(\tilde{z}_{j}\right)|<| z_{k+1}|<\cdots<| z_{n} |<1\right\}$

• 广义Hartogs三角域

$\mathcal{H}_{k}^{n+1}:=\left\{(z, w) \in \mathbb{C}^{n} \times \mathbb{C}:\|z\|<|w|^{k}<1\right\}$

• 广义Hartogs三角域

$\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n} = \left\{z \in \mathbb{C}^{n}: \max _{1 \leq j \leq l}\left\|\phi_{j}\left(\bar{z}_{j}\right)\right\|< \left|z_{k+1}\right|^{b} < \cdots< \left|z_{n} \right|^{b} < 1\right\}$

Let $T_{K^{-t}}$ be the Toeplitz operator with the symbol $K^{-t}(z, z), t \geq 0$. Let $1 < p \leq q < \infty$ and $C_{b, k}=k(b-1)$.\$
(1) If $q \in\left[\frac{2 n+2 C_{b, k}}{n-1+C_{b, k}}, \infty\right),$ then the Toeplitz operator $T_{K^{-t}}$ does not map $L^{p}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ into $L^{q}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ for any $t \geq 0$
(2) If $q \in\left(\frac{2(n-1)+2 C_{b, k}}{n+1+C_{b, k}-2 / p}, \frac{2 n+2 C_{b, k}}{n-1+C_{b, k}}\right),$ then the Toeplitz operator $T_{K^{-t}}$ continuously maps $L^{p}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ into $L^{q}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ if and only if $t \geq \frac{1}{p}-\frac{1}{q}$
(3) If $q \in\left[p, \frac{2(n-1)+2 C_{b, k}}{n+1+C_{b, k}-2 / p}\right],$ then the Toeplitz operator $T_{K^{-t}}$ continuously maps$L^{p}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ into $L^{q}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ if and only if $t>\frac{1}{2 p}+\frac{(1-p)}{2 p} \frac{n+1+C_{b, k}}{n-1+C_{b, k}}$

# Thanks

## U work so hard, but 干不过 write PPTs

nodeppt 助力你的人生逆袭之路！

* * *