1. The Bergman metric on unit disk

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1.1 Schwarz’s lemma

Lemma 1.1. If $f(z) \in Hol(\mathbb{D})$, and $f(0) = 0$, then

$\left| f(z) \right| \le |z|, and |f'(0)| \le 1.$

Equality holds at some point $z$ if and only if $f(z) = e^{i \gamma} z$, where $\gamma$ is a real constant.

We shall use the invariant form of Schwarz’s lemma. A $M\ddot{o}bius\ transformation$ is a conformal self-map of the unit disk. Every $\text M\ddot{\text{o}}\text{bius}$ transformation can be written as with $\gamma$ real and $|z_0| \le 0$.

Lemma 1.2. If $f \in Hol(\mathbb{D})$​, then

$\frac{|f(z) - f(z_0)|}{|1 - \overline{f(z_0)} f(z)|} \le \left|\frac{z - z_0}{1 - \overline{z_0}z}\right|, \quad z \neq z_0,$

and

$\frac{|f'(z)|}{1 - \left| f(z) \right|^2} \le \frac{1}{1 - |z|^2}.$

Equality holds at some point $z$ if and only if $f(z)$ is a $\text M\ddot{\text{o}}\text{bius}$ transformation.

Definition: If we define the standard $\text M\ddot{\text{o}}\text{bius}$ transform on $\mathbb{D}$​ by

$\varphi_{w} (z) = \frac{w-z}{1-z\overline{w}}.$

The pseudo-hyperbolic distance on $\mathbb{D}$​ is defined by

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

The Schwartz’s lemma says that analytic mappings from $\mathbb{D}$ to $\mathbb{D}$ are Lipschitz continuous in the pseudo-hyperbolic distance:

The lemma also says that the distance $\rho(z,w)$ is invariant under $\text M\ddot{\text{o}}\text{bius}$ transformations:

for all $z, w \in \mathbb{D}$.

A useful identity:

$1 - |\varphi_z(w)|^2 = 1 - \left|\frac{z-w}{1-\overline{z}w}\right| ^2 = \frac{(1-|z|^2)(1-|w|^2)}{|{1 - \overline{z}w}|}.$

For any $z \in \mathbb{D}$ and $r \in (0,1)$​, let

$\Delta(z,r) = \{w \in \mathbb{D}: \rho(w,z) < r\}$

be the pseudo-hyperbolic disk with “center” $z$ and “radius” $r$. $\Delta(z,r)$ is the image of the Euclidean disk $|\xi| < r$ under the $\text M\ddot{\text{o}}\text{bius}$ transformation $w = \varphi_{z}(\xi)$. It follows that $\Delta(z,r)$ is actually a Euclidean disk contained in $\mathbb{D}$. The Euclidean center and radius of $\Delta(z,r)$ are

1.2 The Bergman metric

The Bergman metric on $\mathbb{D}$, also called the hyperbolic metric or the $\text{Poincar\'e}$ metric, is given by $\sqrt{H(z)}ds$, where $ds$ is the Euclidean length element and

The induced distance on $\mathbb{D}$​ is given by

$\beta(z,w) = \frac{1}{2}\log \left(\frac{1+|\varphi_{z}(w)|}{1 - |\varphi_{z}(w)|}\right), \quad z, w \in \mathbb{D}.$

Remark: The arc tangent hyperbolic function has the form Hence, the Bergman metric can be written by $\beta(z,w) = \tanh^{-1}(|\varphi_z(w)|)$.

In particular,

The Bergman metric is also $\text M\ddot{\text{o}}\text{bius}$ invariant: for all $\varphi$ in $\text{Aut}({\mathbb{D}})$, and $z, w \in \mathbb{D}$.

Lemma 1.3. For any $z \in \mathbb{D}$ and $r > 0$, let be the Bergman metric disk with ’center’ $z$ and ’radius’ $r$. $D(z,r)$ is a Euclidean disk with Euclidean center and radius respectively, where

For any $r > 0$ and $z\in \mathbb{D}$, we have the following equalities:

1. $|D(z,r)| = \frac{(1 - |z|^2)^2 s^2}{(1 - |z|^2 s^2)^2}$;

2. $\inf \{|k_z(w)|^2: w \in D(z,r)\} = \frac{(1 - s|z|)^4}{(1-|z|^2)^2};$

3. $\sup \{|k_z(w)|^2: w \in D(z,r)\} = \frac{(1 + s|z|)^4}{(1-|z|^2)^2};$

where $s = \tanh r \in (0,1)$, and $|D(z,r)|$ is the (normalized) area of $D(z,r)$.

Proof. (1) follows from the fact that $D(z,r)$ is a Euclidean disk with Euclidean radius $R = s(1 - |z|^2) / (1 - |z|^2 s^2)$.

(2) We just calculate:

$\begin{aligned} \inf \{|k_z(w)|^2 : w \in D(z,r)\} &= \inf \{|k_{z|(w)}^2: w \in \varphi_z (D(0,r))\} \\ &= \inf \{|K_z(\varphi_z(w))|^2: w \in D(0,r)\} \\ &= \inf \left\{ \frac{|1 - z \overline{w|}^4}{(1 - |z|^2)^2}: w \in D(0,r) \right\} \\ &= \frac{(1 - s|z|^2)^4}{(1 - |z|^2)^2}. \end{aligned}$

The last equality follows from the fact that $D(0,r) = B(0,s)$, where $B(0,s)$ is the Euclidean disk with center $0$, and radius $s = \tanh r$​.

(3) is similar with (2). ◻

Corollary 1.1. For any $r_1, r_2, r_3 > 0$, there exists a constant $C> 0$​ such that

$\frac{1}{C} \le \frac{|D(z,r_1)|}{|D(w,r_2)|} \le C$

for all $\beta(z,w) \le r_3$.

Lemma 1.4. For any positive number $R$, there exists a constant $C > 0$ such that

$\frac{1}{C} \le \frac{|D(z,r)|}{|D(w,r)|} \le C$

for all $\beta(z, w) \le R$ and $r \le R$​.

Proof. By lemma 1.3, we have

$\frac{|D(z,r)|}{|D(w,r)|} = \frac{(1 - |z|^2)^2 s^2}{(1 - |z|^2s^2)^2} \cdot \frac{(1 - |w|^2s^2)^2}{(1 - |w|^2)^2 s^2} = \frac{(1 - |z|^2)^2}{(1 - |w|^2)^2} \cdot \frac{(1 - |w|^2s^2)^2}{(1 - |z|^2 s^2)^2}.$

Note that, $\tanh(r)$ is a monotone-increasing function. Hence, let $S = \tanh R$​, we have

$(1 - S^2)^2 \le \frac{(1 - |w|^2 s^2)^2}{(1 - |z|^2s^2)^2} \le \frac{1}{(1 - S^2)^2}.$

Thus, we only need to show there exists a constant $C > 0$ such that

$\frac{1}{C} \le \frac{1 - |z|^2}{1 - |w|^2} \le C$

for all $\beta(z, w) \le R$.

If $\beta(z, w) \le R$, then we have $\rho(z,w) = |\varphi_z(w)| \le S$, let $w = \varphi_z(a)$, we have $a = \varphi_z(w)$, which implies $|a| \le S$ and

$1 - |w|^2 = 1 - |\varphi_z(a)|^2 = \frac{(1 - |a|^2)(1 - |z|^2)}{|1 - z\overline{a|}^2}.$

Since, $1 - S^2 \le 1 - |a|^2 < 1$ and $1 - S \le |1 - \overline{a|z} \le 2$​, thus, we have

$\frac{1-S^2}{2} \le \frac{1 - |a|^2}{|1 - z\overline{a|}^2} = \frac{1 - |w|^2}{1 - |z|^2} \le \frac{1}{1 - S}.$

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By (1) of Lemma 1.3, $|D(z,r_1)|$ and $|D(z,r_2)|$ are comparable (with an absolute constant) if $r_1$ and $r_2$ are comparable and bounded above. Thus we have the following generalization of the above lemma.

Corollary 1.2. For any $R_1, R_2 > 0$, there exists a constant $C > 0$​ such that

$\frac{1}{C} \le \frac{|D(z,r_1)|}{|D(w,r_2)|} \le C$

for all $r_1, r_2, r_3 \le R_1$, $R_1^{-1} \le r_1 / r_2 \le R_2$, and $\beta(z,w) \le r_3$.

Lemma 1.5. (Covering lemma) There exists a positive integer $N$ such that any Bergman metric disk of radius $r(\le 1)$ can be covered by $N$ Bergman metric disk of radius $\frac{r}{2}$.

Proof. Fix $D(a,r)$ and let $D(z_1, \frac{r}{2}), \cdots, D(z_n, \frac{r}{2})$ be a cover of $D(a,r)$ such that $\beta(z_i, z_j) \ge \frac{r}{4}$ for all $i \neq j$ and $D(z_i, \frac{r}{8}) \subset D(a,r)$ for all $i$.

We claim that $D(z_i, \frac{r}{8})$ are pairwise disjoint since $\beta(z_i, z_j) \ge \frac{r}{4}$. Thus By the corollary (1.2), there is a constant $C > 0$ independent of $r(\le 1)$ such that $|D(a,r)| \le C |D(z_i, \frac{r}{8})|$ for all $1 \le i \le n$. This implies that $n \le C$ and hence taking $N = [C]+1$ will finish the proof. ◻

Remark: Note that the restriction $r \le 1$ in the statement of the above lemma can be replaced by $r \le R$ for any fixed positive number $R$. Mimicking the proof of the above lemma, we can prove the following generalization:

Corollary 1.3. Given a positive integer $n$, there is another positive integer $N$ such that any Bergman metric disk of radius $r(\le 1)$ can be covered by $N$ Bergman metric disk of radius $\frac{r}{n}$.

Lemma 1.6. There is a positive number $N$ such that for any $r \le 1$, there exists a sequence $\{\lambda_n\}$ in $\mathbb{D}$ satisfying the following conditions:

1. $\mathbb{D} = \bigcup_{n = 1}^{+\infty} D(\lambda_n,r)$;

2. $D(\lambda_n, \frac{r}{4}) \cap D(\lambda_m, \frac{r}{4}) = \emptyset$ if $m \neq n$;

3. Any point in $\mathbb{D}$ belongs to at most $N$ of the sets $D(\lambda_n, 2r)$.

Proof. We can find a sequence $\{\lambda_n\}$ such that $\mathbb{D} = \bigcup_{n = 1}^{+ \infty} D(\lambda_n, r)$ and $\beta(\lambda_n, \lambda_m) \ge \frac{r}{2}$ for all $m \neq n$. Now (2) follows immediately from the triangle inequality. We show that (3) has to hold.

Let $N$ be a positive integer such that any Bergman metric disk of radius $2r$ can be covered by $N$ Bergman metric disk of radius $\frac{r}{4}$. If $z \in \bigcap\{ D(\lambda_{n_i}, 2r): 1 \le i \le N+1\}$, then $\lambda_{n_i} \in D(z,2r)$ for all $1\le i \le N+1$. Let $\{D(w_j, \frac{r}{4}):1 \le j \le N\}$ be a cover of $D(z,2r)$, then at least one $D(w_j,\frac{r}{4})$ between these two $\lambda_{n_i}$’s is less than $\frac{r}{2}$, which is a contracdiction. ◻

Generally covering lemma: Let $r \le 1$ and $\{D(\lambda_n, r)\}$ be a cover of $\mathbb{D}$ satisfying the conditions of Lemma (1.6). For each $n$ we can find a measurable set $D_n$ with the following properties:

1. $D(\lambda_n,\frac{r}{4}) \subset D_n \subset D(\lambda_n,r)$ for all $n \ge 1$;

2. $D_m \cap D_n = \emptyset$ if $n \neq m$;

3. $\bigcup_{n = 1}^{+\infty} D_n = \mathbb{D}$.

This disjoint decomposition $\{D_n\}$ of $\mathbb{D}$ will play an important role in the atomic decomposition of functions in the Bergman spaces.

Proposition 1.1. There is a constant $C > 0$ such that

$\left| f(z) \right|^p \le \frac{C}{|D(z,r)|} \int_{D(z,r)} |f(w)|^p d A(w)$

for all $f$ analytic, $z \in \mathbb{D}, p > 0$ and $r \le 1$.

Proof. Since $D(0,r)$ is a Euclidean disk with Euclidean center $0$, and $\left| f(z) \right|^p$ is subharmonic for all analytic $f$ and $p > 0$, we have

$|f(0)|^p \le \frac{1}{|D(0,r)|} \int_{D(0,r)} |f(w)|^p dA(w) = \frac{1}{s^2} \int_{D(0,r)} |f(w)|^p dA(w),$

where $s = \tanh r \in (0,1)$. Replace $f$ by $f\circ \varphi_{z}$ and make a change of variable, we have

$\left| f(z) \right|^p \le \frac{1}{s^2} \int_{D(z,r)} |f(w)|^p |k_z(w)|^2 dA(w).$

By (3) of lemma (1.3), we have

$\left| f(z) \right|^p \le \frac{(1 - s^2 |z|^2)^4}{s^2(1 - |z|^2)^2} \int_{D(z,r)} |f(w)|^p dA(w).$

Since $r \le 1$ implies that $s \le \tanh 1 \in (0,1)$, (1) of lemma (1.3) implies that there is a constant $C$ such that

$\left| f(z) \right|^p \le \frac{C}{|D(z,r)|} \int_{D(z,r)} |f(w)|^p dA(w)$

for all $f$ analytic, $z \in \mathbb{D}$, $p > 0$, and $r \le 1$​​. ◻

Atomic decoposition

Recall that $k_z(w) = (1 - |z|^2)/(1 - \overline{z}w)^2$ are the normalized reproducing kernels of the Bergman space $L_a^2$. They are uint vectors in $L_a^2$. The purpose of this section is to show that these normalized reproducing kernels are the right building blocks for $L_a^2$, although they are clearly not mutually orthogonal. The same idea will be generalized to all the Bergman spaces $L_a^p$.

Total hypothesis: For any $0 < r \le 1$, we will fix a sequence $\{\lambda_n\}$ in $\mathbb{D}$ with the properties:

1. $\mathbb{D} = \bigcup_{n = 1}^{+\infty} D(\lambda_n,r)$;

2. $D(\lambda_n, \frac{r}{4}) \cap D(\lambda_m, \frac{r}{4}) = \emptyset$ if $m \neq n$;

3. Any point in $\mathbb{D}$ belongs to at most $N$ of the sets $D(\lambda_n, 2r)$.

The existence is supported by lemma (1.6). We also fix a disjoint decomposition of $\mathbb{D}$ satisfying:

1. $D(\lambda_n,\frac{r}{4}) \subset D_n \subset D(\lambda_n,r)$ for all $n \ge 1$;

2. $D_m \cap D_n = \emptyset$ if $n \neq m$;

3. $\bigcup_{n = 1}^{+\infty} D_n = \mathbb{D}$.

Lemma 1.7. For any $r > 0$, there is a constant $C > 0$ (depending on $r$) such that

$\sum_{n = 1}^{+ \infty} (1 - |\lambda_n|^2)^2 |f(\lambda_n)|^p \le C \int_{\mathbb{D}} \left| f(z) \right|^p dA(z)$

for all analytic $f$ and $p \ge 1$.

2. Dyadic structure of the unit disk

2.1 A dyadic structure of “kubes” on $\mathbb{D}$

If we fix the parameters $\theta_{0}, \lambda_0 > 0$. For $k \in \mathbb{N}$, let $\mathbb{S}_{k \theta_0} := \{z \in \mathbb{D} : \beta (0,z) = k\theta_0 \}$.

For each $k \in \mathbb{N}$, there exist points $\{w_j^k\}_{j = 1}^{J_k} \subset \mathbb{D}$ corresponding to Borel sets $Q_j^k \subset \mathbb{T}_{k\theta_0}$ containing $w_j^k$, and a constant $C > 0$ such that

1. $\mathbb{T}_{k\theta_0} = \bigcup_{j=1}^{J_k} Q_j^k,$

2. $Q_j^k \cap Q_{j'}^k = \emptyset$ whenever $j \neq j'$, and

3. $\mathbb{T}_{k\theta_0} \cap D(w_j^k, \lambda_0)\subset Q_j^k \subset \mathbb{T}_{k\theta_0}\cap D(w_j^k,C\lambda_0)$.

For $z \in \mathbb{D}$, let $P_{k\theta_0}z$ be the projection of $z$ onto $\mathbb{T}_{k\theta_0}$. Define and for $k \in \mathbb{N}$, $j \in \{1, 2, \cdots, J_k\}$

$K_j^k := \{z \in \mathbb{D} : k\theta_0 < \beta(0,z) \le (k+1)\theta_0 \text{ and } P_{k\theta_0} z \in Q_j^k\}.$

For each $k\in \mathbb{N}$, $j \in \{1, 2, \cdots, J_k\}$, $K_j^k$​ is called a kube. We can see these in the following figure:

The diagram of $\mathbb{T}_{k\theta_0}, Q_j^k, w_j^k$ and the kube $K_j^k$

We denote the collection of kubes $K_j^k$ obtained in this construction by $\mathcal{D}_0$. In what follows, we often drop the subscript and superscript labeling a kube if they are not necessary for clarity.

Fact: From the figure 1 and the definition of $K$, the kubes in $\mathcal{D}_0$ is a partition of $\mathbb{D}$.

Given a kube $K = K_j^k \in \mathcal{D}_0$, we define its center by $c_K = c_j^k := P_{(k+ \frac{1}{2})\theta_0}w_j^k$. Also, for $K = K_j^k$, we call $k$ the generation of $K$ and denote this by $d(K)$.

We define a tree structure on $\mathcal{D}_0$ in the following way.

1. For $k \ge 1$, we say a kube $K_{j_1}^{k+1}$ is a child of $K_{j_2}^k$ if $P_{k\theta_0}c_{j_1}^{k+1} \in Q_{j_2}^k$ and we declare every kube in $\{K_j^1\}_{j = 1}^{J_1}$ to be a child of $K_1^0$.

2. More generally, we say $K_{j_1}^{k_1}$ is a descendant of $K_{j_2}^{k_2}$ and write $K_{j_1}^{k_1} \prec K_{j_2}^{k_2}$ if $k_1 > k_2$ and $P_{k_2\theta_0}c_{j_1}^{k_1} \in Q_{j_2}^{k_2}$. We denote the collection of children of $K \in \mathcal{D}_0$ by $\text{ch}(K)$ and define $\mathcal{D}(K) = \{K\} \cup \{ K'\in \mathcal{D}_0 : K' \prec K\}$.

3. Given a fixed kube $K \in \mathcal{D}_0$, we define the dyadic tent over the kube, $\widehat{K}$, to be the following set:

Notice that by construction, given two tents $\widehat{K}$ and $\widehat{K'}$ associated to Kubes in $\mathcal{D}_0$​, either the two sets are disjoint or one is contained in the other. Here is an example:

The diagram of The relations of $K_j^k$.

Lemma 2.1. Let $t> -1$ and let $\mathcal{D}_0$ be a dyadic structure withe positive parameters $\lambda_0$ and $\theta_0$. Then it satisfies the following properties:

1. $\mathbb{D} = \bigcup_{K \in \mathcal{D}_0} K$ and the kubes $K_j^k$ are pairwise disjoint. Furthermore, there are constants, $C_1$ and $C_2$ depending on $\lambda_0$ and $\theta_0$ such that for all $K \in \mathcal{D}_0$ there holds:

2. $A_{t}(K) \asymp A_t(\widehat{K}) \asymp A_t(\mathcal{T}_z) \asymp (1 - |c_K|^2)^{1+t}$, where $dA_t(z) = (1 - |z|^2)^t dA(z)$.

3. Every element of $\mathcal{D}_0$ has at most $e^{2k\theta_0}$ children.

Lemma 2.2. We have for any $K \in \mathcal{D}_0$. We explicitly write where $\rho_0 \in (0,1)$, and therefore

Lemma 2.3. There exists $\alpha_0 > 1$ such that if $K \in \mathcal{D}_0$ and $K' \in \text{ch}(K)$, then

Proof. Note that by definition, for any kube $K \in \mathcal{D}_0$ with $d(K) = k$, we have

$\beta(0,c_K) = \frac{1}{2}\log \left( \frac{1+|c_K|}{1 - |c_K|}\right) = \left(k + \frac{1}{2}\right)\theta_0.$

We thus obtain

$|c_K| = \frac{e^{2(k + \frac{1}{2})\theta_0} - 1}{e^{2(k+\frac{1}{2})\theta_0} + 1}.$

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Lemma 2.4. If we define $\mathcal{T}_z = \{w \in \mathbb{D}: |1 - \overline{w}\frac{z}{|z|}| < 1 - |z| \}$. Then there exists a finite collection of dyadic structures $\{\mathcal{D}_{\ell}\}_{\ell = 1}^{M}$ such that for any $z \in \mathbb{D}$, there exists $K \in \bigcup_{\ell = 1}^{M} \mathcal{D}_{\ell}$ such that $\widehat{K} \supset \mathcal{T}_{z}$ and $|\widehat{K|} \asymp |\mathcal{T|_{z}}$. Moreover, for any $K' \in \bigcup_{\ell = 1}^{M} \mathcal{D}_{\ell}$, there exists $z' \in \mathbb{D}$ such that $\mathcal{T}_{z'} \supset \widehat{K'}$ and $|\mathcal{T|_{z'}} \asymp |\widehat{K'|}$.