Lipschitz continuity of holomorphic mappings with respect to Bergman metric

TL;DR

This paper establishes sharp Lipschitz continuity estimates for holomorphic mappings concerning the Bergman metric, improving and generalizing previous results in the field.

Contribution

It provides the most precise Lipschitz estimates with respect to the Bergman metric for holomorphic mappings, extending earlier work.

Findings

Sharp Lipschitz continuity estimates derived

Results improve previous bounds by Ghatage, Yan, and Zheng

Generalization to broader classes of holomorphic mappings

Abstract

n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer. Math. Soc., 129: 2039-2044, 2000).

Equations

$$B(z)=\frac{(1-|z{|}^2)I+A(z)}{(1-|z{|}^2{)}^2}$$

$$A(z)=\left(\begin{array}[]{cccc}\displaystyle z_{1}\overline{z}_{1}\;\cdots\;z_{1}\overline{z}_{n}\\[11.38109pt] \vdots\\[5.69054pt] \displaystyle z_{n}\overline{z}_{1}\;\cdots\;z_{n}\overline{z}_{n}\end{array}\right).$$

$$\ell (\gamma )={\int }_0^1⟨B(\gamma (t)){\gamma }^{\prime }(t),{\gamma }^{\prime }(t){⟩}^{1/2}dt.$$

$$\Vert f{\Vert }_{\mathcal{P}(n,\alpha )}=\underset{z\in {\mathbb{B}}^n}{\sup }{D}_f^{n,\alpha }(z),$$

$$D_f^{n,\alpha }(z)=(1-|z{|}^2{)}^{\frac{\alpha (n+1)}{2n}}|\det f^{\prime }(z){|}^{\frac{1}{n}}.$$

$$\rho (z,w)=\left|\frac{z-w}{1-\overline{w}z}\right|.$$

$$\big{|}D_{f}^{1,1}(z)-D_{f}^{1,1}(w)\big{|}\leq 3.31\|f\|_{\mathcal{P}(1,1)}\rho(z,w).$$

$$\rho (\phi (z_w),w)<r,\text{mbox}and\frac{1-|z_w{|}^2}{1-|\phi (z_w){|}^2}|{\phi }^{\prime }(z_w)|>\epsilon ,$$

$$\big{|}D^{n,1}_{f}(z_{2})-D^{n,1}_{f}(z_{1})\big{|}\leq M(n)\|f\|_{\mathcal{P}(n,1)}\big{[}\tanh\beta(z_{1},z_{2})\big{]}^{\frac{1}{n}},$$

$$(1-|z|^{2})\left(\Big{|}\frac{\partial}{\partial z}D^{1,1}_{f}(z)\Big{|}+\Big{|}\frac{\partial}{\partial\overline{z}}D^{1,1}_{f}(z)\Big{|}\right)\leq\frac{3\sqrt{3}}{2}\|f\|_{\mathcal{P}(1,1)}$$

$$|f^{\prime \prime }(z)|\le \frac{\left(2|z|+\frac{3\sqrt{3}}{2}\right)\Vert f{\Vert }_{\mathcal{P}(1,1)}}{(1-|z{|}^2{)}^2}.$$

$${\tau }_{\phi }(z_w)={\left(\frac{1-|z_w{|}^2}{1-|\phi (z_w){|}^2}\right)}^{\frac{n+1}{2n}}{\left|\det {\phi }^{\prime }(z_w)\right|}^{\frac{1}{n}}$$

$$\Vert C_{\phi }(f){\Vert }_{\mathcal{P}(n,1)}\ge k(n,r,\epsilon )\Vert f{\Vert }_{\mathcal{P}(n,1)}.$$

$$\phi (x)=x(1-x^2{)}^{\frac{\alpha (n+1)}{2}}\sqrt{\alpha (1+n)+1}{\left[\frac{\alpha (n+1)+1}{\alpha (n+1)}\right]}^{\frac{\alpha (n+1)}{2}}$$

$$a_0(\alpha )=\frac{1}{\sqrt{\alpha (1+n)+1}}.$$

$$|\det f^{\prime}(z)|\geq{\rm Re\,}\big{(}\det f^{\prime}(z)\big{)}\geq\frac{\lambda(m_{\alpha}(\lambda)-|z|)}{m_{\alpha}(\lambda)(1-m_{\alpha}(\lambda)|z|)^{\alpha(n+1)+1}},$$

$$|\det f^{\prime }(z)|\le \frac{\lambda (m_{\alpha }(\lambda )+|z|)}{m_{\alpha }(\lambda )(1+m_{\alpha }(\lambda )|z|{)}^{\alpha (n+1)+1}}.$$

$$\tanh\beta(z_{1},z_{2})=\tanh\beta\big{(}\phi^{-1}(z_{1}),\phi^{-1}(z_{2})\big{)}=\tanh\beta(0,w)=|w|.$$

$$|\det {\varphi }^{\prime }(z)|={\left(\frac{1-|\varphi (z){|}^2}{1-|z{|}^2}\right)}^{\frac{n+1}{2}},$$

$$|\det g^{\prime }(0){|}^{\frac{1}{n}}=|\det f^{\prime }(\varphi (0)){|}^{\frac{1}{n}}|\det {\varphi }^{\prime }(0){|}^{\frac{1}{n}}=|\det f^{\prime }(z_1){|}^{\frac{1}{n}}(1-|z_1{|}^2{)}^{\frac{n+1}{2n}}$$

$$(1-|w{|}^2{)}^{\frac{n+1}{2n}}|\det g^{\prime }(w){|}^{\frac{1}{n}}$$

$$\text{mbox}Re\left(e^{-i\theta }\det g^{\prime }(z)\right)\ge \frac{\lambda (m_1(\lambda )-|z|)}{m_1(\lambda )(1-m_1(\lambda )|z|{)}^{n+2}},$$

$$m_1(\lambda )\le \frac{a_0(1)+m_1(\lambda )}{1+a_0(1)m_1(\lambda )}.$$

$$(1-|w{|}^2{)}^{\frac{n+1}{2n}}\ge (1-|w|m_1(\lambda ){)}^{\frac{n+1}{2n}}\ge (1-|w|m_1(\lambda ){)}^{\frac{n+2}{n}}$$

$$m_{1}^{\frac{1}{n}}(\lambda)-|w|^{\frac{1}{n}}\leq\big{(}m_{1}(\lambda)-|w|\big{)}^{\frac{1}{n}},$$

$$m_1^{\frac{1}{n}}(\lambda )-\frac{(1-|w{|}^2{)}^{\frac{n+1}{2n}}(m_1(\lambda )-|w|{)}^{\frac{1}{n}}}{(1-m_1(\lambda )|w|{)}^{\frac{n+2}{n}}}\le |w{|}^{\frac{1}{n}}.$$

$$m_1(\lambda )(1-m_1^2(\lambda ))\sqrt{2+n}{\left(\frac{n+2}{n+1}\right)}^{\frac{n+1}{2}}=\lambda ,$$

$$D_f^{n,1}(z_1)-D_f^{n,1}(z_2)$$

$$f_{\lambda}(z)=\left(\begin{array}[]{cccc}\int_{0}^{z_{1}}\frac{\lambda(m_{1}(\lambda)-\xi)}{m_{1}(\lambda)(1-m_{1}(\lambda)\xi)^{n+2}}\,d\xi\\ z_{2}\\ \vdots\\ z_{n}\end{array}\right).$$

Full text

1. Introduction and main results

Let $\mathbb{C}^{n}$ denote the Euclidean space of complex dimension $n$. Throughout this paper, we write a point $z\in\mathbb{C}^{n}$ as a column vector in $n\times 1$ matrix form $z=(z_{1},\ldots,z_{n})^{T},$ where the symbol $T$ stands for the transpose of vectors or matrices. For $z\in\mathbb{C}^{n}$, the conjugate of $z$, denoted by $\overline{z}$, is defined by $\overline{z}=(\overline{z}_{1},\ldots,\overline{z}_{n})^{T}.$ For $z\in\mathbb{C}^{n}$ and $w=(w_{1},\ldots,w_{n})^{T}\in\mathbb{C}^{n}$, we write $\langle z,w\rangle:=\sum_{k=1}^{n}z_{k}\overline{w}_{k}$ and $|z|:={\langle z,z\rangle}^{1/2}=(|z_{1}|^{2}+\cdots+|z_{n}|^{2})^{1/2}.$ For $a=(a_{1},\ldots,a_{n})^{T}\in\mathbb{C}^{n}$, we set $\mathbb{B}^{n}(a,r)=\{z\in\mathbb{C}^{n}:\,|z-a|<r\}.$ Also, we use $\mathbb{B}^{n}$ to denote the unit ball $\mathbb{B}^{n}(0,1)$ and let $\mathbb{D}=\mathbb{B}^{1}$.

The class of all holomorphic functions from $\mathbb{B}^{n}$ into $\mathbb{C}^{n}$ is denoted by $H(\mathbb{B}^{n},\mathbb{C}^{n})$. Let ${\operatorname{Aut}}(\mathbb{B}^{n})$ be the automorphism group consisting of all biholomorphic self mappings of the unit ball $\mathbb{B}^{n}$.

For $z\in\mathbb{B}^{n}$, let

[TABLE]

be the Bergman matrix, where $I$ is the $n\times n$ identity matrix and

[TABLE]

For a smooth curve $\gamma:~{}[0,1]\rightarrow\mathbb{B}^{n}$, let

[TABLE]

For any two points $z$ and $w$ in $\mathbb{B}^{n}$, let $\beta(z,w)$ be the infimum of the set consisting of all $\ell(\gamma)$, where $\gamma$ is a piecewise smooth curve in $\mathbb{B}^{n}$ from $z$ to $w$. Then we call $\beta$ the Bergman metric in $\mathbb{B}^{n}$ (cf. [9]).

As in [4], the prenorm $\|f\|_{\mathcal{P}(n,\alpha)}$ of $f\in H(\mathbb{B}^{n},\mathbb{C}^{n})$ is given by

[TABLE]

where $\alpha>0$ and

[TABLE]

Let $\mathcal{B}_{\mathcal{P}(n,\alpha)}$ be the class of all holomorphic mappings $f\in H(\mathbb{B}^{n},\mathbb{C}^{n})$ satisfying $\|f\|_{\mathcal{P}(n,\alpha)}<\infty$. In particular, $\mathcal{B}_{\mathcal{P}(1,\alpha)}$ is the classical family of $\alpha$-Bloch functions (cf. [8, 9]).

For $z,w\in\mathbb{D}$, the pseudo-hyperbolic distance is defined as

[TABLE]

In particular, if $n=1$, then, for $z,w\in\mathbb{D}$, $\tanh\beta(z,w)=\rho(z,w).$ In [3], Ghatage, Yan and Zheng showed that $D_{f}^{1,1}(z)$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric, which is given in Theorem A below. For more details on this topic, see [1, 2, 5, 7].

Theorem A. ([3, Theorem 1])* Let $f\in\mathcal{B}_{\mathcal{P}(1,1)}$. Then, for all $z,w\in\mathbb{D}$,*

[TABLE]

Let $\varphi$ be a holomorphic mapping of $\mathbb{B}^{n}$ into $\mathbb{B}^{n}$. For all $f\in\mathcal{B}_{\mathcal{P}(n,\alpha)}$, let $C_{\varphi}:~{}f\mapsto f\circ\varphi$ be a pre-composition operator. As an application of Theorem A, they proved

Theorem B. ([3, Theorem 2])* Let $\varphi$ be a holomorphic mapping of $\mathbb{D}$ into $\mathbb{D}$. If for some constants $r\in(0,1/4)$, and $\varepsilon>0$, for each $w\in\mathbb{D}$, there is a point $z_{w}\in\mathbb{D}$ such that*

[TABLE]

*then $C_{\varphi}:~{}\mathcal{B}_{\mathcal{P}(1,1)}\rightarrow\mathcal{B}_{\mathcal{P}(1,1)}$ is bounded below. *

In this paper, by using a different method, we generalize Theorems A and  B to several dimensional case and obtain the sharp estimate of the Lipschitz constant with respect to the Bergman metric.

Theorem 1**.**

Let $f\in\mathcal{B}_{\mathcal{P}(n,1)}$. Then, for $z_{1},z_{2}\in\mathbb{B}^{n}$,

[TABLE]

where $M(n)=(2+n)^{\frac{1}{2n}}\left(\frac{n+2}{n+1}\right)^{\frac{n+1}{2n}}.$ Moreover, the constant $M(n)$ in (1) cannot be replaced by a smaller number.

The following result is an application of Theorem 1.

Theorem 2**.**

Let $f\in\mathcal{B}_{\mathcal{P}(1,1)}$. Then, for $z\in\mathbb{D}$,

[TABLE]

and

[TABLE]

Moreover, the extreme functions $f(z)=\pm 3\sqrt{3}z^{2}/4$ show that the estimates (2) and (3) are sharp.

Applying Theorem 1, we get the following result which is an improvement of Theorem B.

Theorem 3**.**

Let $\varphi$ be a holomorphic mapping of $\mathbb{B}^{n}$ into $\mathbb{B}^{n}$. Suppose that there is constants $0<r<\frac{1}{M(n)}\left(\frac{n+2}{1+n}\right)^{\frac{1}{n}}$ and $\varepsilon>0$ such that, for each $w\in\mathbb{B}^{n}$, there is a point $z_{w}\in\mathbb{B}^{n}$ satisfying $\tanh\beta(\varphi(z_{w}),w)<r^{n}$ and $|\tau_{\varphi}(z_{w})|>\varepsilon$, where

[TABLE]

and $M(n)$ is defined as in Theorem 1. Then, for all $f\in\mathcal{B}_{\mathcal{P}(n,1)}$, there is a constant $k(n,r,\varepsilon)>0$ depended only on $r$, $\varepsilon$ and $n$ such that

[TABLE]

The proofs of Theorems 1-3 will be presented in Section 2.

2. Proofs of the main results

We begin the section by recalling the following results which play an important role in the proofs of Theorem 1.

Lemma C. ([2, Lemma 1.1])* For $x\in[0,1]$, let*

[TABLE]

and

[TABLE]

*Then $\varphi$ is increasing in $[0,a_{0}(\alpha)]$, decreasing in $[a_{0}(\alpha),1]$ and $\varphi(a_{0}(\alpha))=1.$ *

Theorem D. ([2, Theorem 1.2])* Suppose that $f\in H(\mathbb{B}^{n},\mathbb{C}^{n})$ such that $\|f\|_{\mathcal{P}(n,\alpha)}=1$ and $\det f^{\prime}(0)=\lambda\in(0,1]$. Then, for all $z$ with $|z|\leq\frac{a_{0}(\alpha)+m_{\alpha}(\lambda)}{1+a_{0}(\alpha)m_{\alpha}(\lambda)}$, we have*

[TABLE]

where $m_{\alpha}(\lambda)$ is the unique real root of the equation $\varphi(x)=\lambda$ in the interval $[0,a_{0}(\alpha)]$ and, $\varphi$ and $a_{0}(\alpha)$ are defined as in Lemma C. Moreover, for all $z$ with $|z|\leq\frac{a_{0}(\alpha)-m_{\alpha}(\lambda)}{1-a_{0}(\alpha)m_{\alpha}(\lambda)}$, we have

[TABLE]

*Moreover, the estimates of $(\ref{eq1.07})$ and $(\ref{eq1.08})$ are sharp. *

Proof of Theorem 1

Without loss of generality, we assume that $\|f\|_{\mathcal{P}(n,1)}=1$ and $B^{n,1}_{f}(z_{2})\leq B^{n,1}_{f}(z_{1}).$ Let $\phi\in{\operatorname{Aut}}(\mathbb{B}^{n})$ such that $\phi(0)=z_{1}$ and $w=\phi^{-1}(z_{2})$. For $z\in\mathbb{B}^{n}$, set $g=f(\phi(z))$. By [9, Proposition 1.21], we have

[TABLE]

Since

[TABLE]

we see that

[TABLE]

and

[TABLE]

$\mathbf{Case~{}1.}$ If $|\det g^{\prime}(0)|=0$, then it is obvious.

$\mathbf{Case~{}2.}$ Let $\det g^{\prime}(0)=\lambda e^{i\theta}$, where $\lambda>0$ and $\theta\in[0,2\pi]$. Applying Theorem D (4) to $e^{-i\frac{\theta}{n}}g(z)$, for $|z|\leq\frac{a_{0}(1)+m_{1}(\lambda)}{1+a_{0}(1)m_{1}(\lambda)}$, we have

[TABLE]

where $m_{1}(\lambda)$ and $a_{0}(1)$ are defined as in Theorem  D.

$\mathbf{Subcase~{}2.1.}$ $|w|\leq m_{1}(\lambda)$. It is easy to know that

[TABLE]

On the other hand, by calculations, we have

[TABLE]

and

[TABLE]

which gives

[TABLE]

By Lemma C and Theorem  D, we have

[TABLE]

which, together with (6), (2), (8), (9) and (10), implies that

[TABLE]

$\mathbf{Subcase~{}2.2.}$ $|w|>m_{1}(\lambda)$. Then, by (6), (2) and (11),

[TABLE]

Now we prove the sharpness part. For $z\in\mathbb{B}^{n}$, let

[TABLE]

For any $\epsilon\in\left(0,M(n)\right]$, let

[TABLE]

Then, for $w^{\prime}=(0,0,\ldots,0)$ and $w^{\prime\prime}=(m_{1}(\lambda),0,\ldots,0)$, we have

[TABLE]

which shows that the constant $M(n)$ is sharp. The proof of this theorem is complete. $\Box$

Proof of Theorem 2

For $z=x+iy\in\mathbb{D}$, let $w=z+re^{i\theta}$. Then, by Theorem 1, we have

[TABLE]

where

[TABLE]

On the other hand,

[TABLE]

which, together with (12), gives that

[TABLE]

The proof of this theorem is complete. $\Box$

Proof of Theorem 3

Without loss of generality, we assume that $\|f\|_{\mathcal{P}(n,1)}=1$. For $z\in\mathbb{B}^{n}$, it follows from Theorem 1 that there is a point $w\in\mathbb{B}^{n}$ such that

[TABLE]

and

[TABLE]

where

[TABLE]

By the assumption, there is a point $z_{w}$ such that

[TABLE]

and $|\tau_{\varphi}(z_{w})|>\varepsilon$, which imply that

[TABLE]

By (13), we conclude that

[TABLE]

where

[TABLE]

The proof of this theorem is compete. $\Box$

Acknowledgement: This research of the first author was partly supported by the National Natural Science Foundation of China (No. 11401184 and No. 11571216), the Construct Program of the Key Discipline in Hunan Province, the Science and Technology Plan of Hunan Province (2016TP1020), the Fifty-ninth Batch of Post Doctoral Foundation of China (No. 2016M590492) and the Post Doctoral Foundation of Jiangsu Province (No. 1601182C).