## 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(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

$\Sigma({n} ; {p})=\left\{\left(\zeta_{1}, \ldots, \zeta_{r}\right) \in \mathbb{C}^{n_{1}} \times \cdots \times \mathbb{C}^{n_{r}}: \sum_{k=1}^{r}\left\|{\zeta}_{k}\right\|^{2 p_{k}}<1\right\}$ 广义Fock-Bargmann-Hartogs域$D_{n_{0}}^{n, p}(\mu)$定义如下

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

• 无界非双曲域

• 非强拟凸域+边界非光滑

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

• $b_{0} D_{n_{0}}^{n, p}(\mu)$ 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

### 边界Schwarz Lemma

• 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.

### 刚度性质

• 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

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

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

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.

### 算子有界性

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

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 $\operatorname{maps} L^{p}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ into $L^{q}\left(\mathcal{H}_{\left\{k_{j}, \phi_{j}, b\right\}}^{n}\right)$ if and only if $t>\frac{1}{2 p}+\frac{(1-p)}{2 p} \frac{n+1+C_{b, k}}{n-1+C_{b, k}}$

## 文献列举

(包括：已读文献和计划研读的文献。需要列出文献来源杂志。注意不能少于10篇。)

• 已读文献7篇，其中重要文献3篇：
1. (重要文献) Bi, E., Su, G. & Tu, Z. The Kobayashi Pseudometric for the Fock-Bargmann-Hartogs Domain and Its Application. J Geom Anal 30, 86–106 (2020).
2. (重要文献) Bi E , Tu Z . Rigidity of proper holomorphic mappings between generalized Fock–Bargmann–Hartogs domains[J]. Pacific Journal of Mathematics, 2018, 297(2):277-297.
3. Tu Z , Wang L . Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains[J]. Journal of Mathematical Analysis and Applications, 2014, 419(2):703-714.
4. Yamamori A . The Bergman kernel of the Fock–Bargmann–Hartogs domain and the polylogarithm function[J]. Complex Variables, Theory and Application: An International Journal, 2013, 58(6):783-793.
5. (重要文献) Tang, Y., Tu, Z. Special Toeplitz operators on a class of bounded Hartogs domains. Arch. Math. (2019).
6. Chen, Liwei. The Lp boundedness of the Bergman projection for a class of bounded Hartogs domains[J]. Journal of Mathematical Analysis and Applications, 2017, 448(1):598-610.
7. Vu K T , Jiakun L , Trong T P . Bergman–Toeplitz operators on weakly pseudoconvex domains[J]. Mathematische Zeitschrift, 2018.
• 计划研读文献4篇：
• He L , Tang Y , Tu Z . $L^p$ regularity of weighted Bergman projection on Fock-Bargmann-Hartogs domain[J]. 2019.
• Tu Z , Wang L . Rigidity of proper holomorphic mappings between equidimensional Hua domains[J]. Mathematische Annalen, 2015, 363(1-2):1-34.
• Blocki Z . The Bergman Metric and the Pluricomplex Green Function[J]. Transactions of the American Mathematical Society, 2005, 357(7):2613-2625.
• Khanh T V , Liu J , Thuc P T . Bergman–Toeplitz operators on fat Hartogs triangles[J]. Proceedings of the American Mathematical Society, 2019, 147.

## 研究展望

(列出两个自己的创新点并概括的做描述性证明.每个创新点后面需要写出创新点来源.)

### $D_{n_{0}}^{n, p}(\mu)$的边界Schwarz Lemma

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

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

==猜想==

Let $F = ( f , h) : D_{1}^{1,1} \to D_{n_{0}}^{n, p}$ be a holomorphic mapping and holomorphic at $p \in \partial D_{1}^{1,1}$ with $F(p) = q \in \partial D_{n_{0}}^{n, p}$. There exists $\lambda \in \mathbb{R}$ such that $\overline{J_{F}(p)}^{T} q^{T}=\lambda p^{T}$ with $\lambda \geq|1-\overline{h_{1}(0)}|^{2} /\left(1-\left|h_{1}(0)\right|^{2}\right)>0$.

==技术路线==

### $D_{n_{0}}^{n, p}(\mu)$的 Bergman-Toeplitz operators有界性

==猜想==

Let $T_{K^{-t}}$ be the Toeplitz operator with the symbol $K^{-t}(z, z), t \geq 0$. Let $1 < p \leq q < \infty$. If $q \in\left[?, \infty\right),$ then the Toeplitz operator $T_{K^{-t}}$ continuously.

==技术路线==