Hankel Operators on Weighted Bergman Spaces and Norm Ideals

Abstract

Consider Hankel operators $H_f$ on the weighted Bergman space $L^2_a(\mathbf{B}, dv_\alpha )$. In this paper we characterize the membership of $\left( H^*_fH_f\right) ^{s/2} = |H_f|^s$ in the norm ideal ${\mathcal C}_\Phi$, where $0 < s \le 1$ and the symmetric gauge function $\Phi$ is allowed to be arbitrary.

1 Introduction

Let B denote the open unit ball $\{z \in$ ${\mathbf{C}}^n$ : $|z| < 1\}$ in ${\mathbf{C}}^n$. Write dv for the volume measure on B with the normalization $v(\mathbf{B}) = 1$. For each $-1< \alpha < \infty$, we define the weighted measure

$$\begin{aligned} dv_\alpha (z) = c_\alpha (1-|z|^2)^\alpha dv(z) \end{aligned}$$

on B, where the coefficient $c_\alpha$ is chosen so that $v_\alpha (\mathbf{B}) = 1$. Recall that the weighted Bergman space $L^2_a(\text{ B },dv_\alpha )$ is defined to be the subspace

$$\begin{aligned} \{h \in L^2(\text{ B },dv_\alpha ) : h \ \text {is analytic on} \ \mathbf{B}\} \end{aligned}$$

of $L^2(\text{ B },dv_\alpha )$. The orthogonal projection from $L^2(\text{ B },dv_\alpha )$ onto $L^2_a(\text{ B },dv_\alpha )$ is given by

$$\begin{aligned} (Pf)(z) = \int {f(w)\over (1 - \langle z,w\rangle )^{n+1+\alpha }}dv_\alpha (w), \quad f \in L^2(\text{ B },dv_\alpha ). \end{aligned}$$

Note that this integral formula defines Pf as a function even for $f \in L^1(\text{ B },dv_\alpha )$. Although P is obviously $\alpha$ dependent, for the sake of simplicity we intentionally omit the weight of the space in the notation for this projection.

Given an appropriate symbol function f, the Hankel operator $H_f : L_a^2(\text{ B },dv_\alpha ) \rightarrow L^2(\text{ B },dv_\alpha )\ominus L^2_a(\text{ B },dv_\alpha )$ is defined by the formula

$$\begin{aligned} H_fh = fh - P(fh), \end{aligned}$$

$h \in L^2_a(\text{ B },dv_\alpha )$. A subject of intense research interest, the theory of Hankel operators can be conveniently divided into two natural components. Because of the relation

$$\begin{aligned} {[}M_f,P] = H_f - H_{\bar{f}}^*, \end{aligned}$$

the simultaneous study of the pair of Hankel operators $H_f$ and $H_{\bar{f}}$ is equivalent to the study of the commutator $[M_f,P]$. Results that simultaneous concern the pair $H_f$, $H_{\bar{f}}$ are often called the “two-sided” theory of Hankel operators, of which we cite [1, 9, 11, 17, 20] as typical examples.

By contrast, the study of $H_f$ alone is often called the “one-sided” theory of Hankel operators, which presents its unique challenges. As examples of “one-sided” theory in the Bergman space case, let us cite [13,14,15,16]. Recall that in these papers, Li and Luecking characterized the boundedness, compactness and Schatten-class membership of $H_f$. Building on these results, in this paper we will take the logical next step. Namely, we will determine exactly when the operator $|H_f|^s = (H_f^*H_f)^{s/2}$ belongs to the norm ideal ${\mathcal C}_\Phi$, where $0 < s \le 1$ and the symmetric gauge function $\Phi$ is allowed to be arbitrary.

Before going any further, a brief review of “symmetric gauge functions” and the associated “norm ideals” will be beneficial. Throughout the paper [10], will be our standard reference in this connection. Following [10], let $\hat{c}$ denote the linear space of sequences $\{a_j\}_{j\in {\mathbf{N}}}$, where $a_j \in$ R and for every sequence the set $\{j \in \mathbf{N} : a_j \ne 0\}$ is finite. A symmetric gauge function (also called symmetric norming function) is a map

$$\begin{aligned} \Phi : \hat{c} \rightarrow [0,\infty ) \end{aligned}$$

that has the following properties:

• (a) $\Phi$ is a norm on $\hat{c}$.

• (b) $\Phi (\{1$, 0, ..., 0, $\dots \}) = 1$.

• (c) $\Phi (\{a_j\}_{j\in {\mathbf{N}}}) = \Phi (\{|a_{\pi (j)}|\}_{j\in {\mathbf{N}}})$ for every bijection $\pi : {\mathbf{N}} \rightarrow {\mathbf{N}}$.

See [10, page 71]. Each symmetric gauge function $\Phi$ gives rise to the symmetric norm

$$\begin{aligned} \Vert A\Vert _\Phi = \sup _{j\ge 1}\Phi (\{s_1(A), \dots , s_j(A), 0, \dots , 0, \dots \}) \end{aligned}$$

(1.1)

for bounded operators. On any separable Hilbert space ${\mathcal H}$, the set of operators

$$\begin{aligned} {\mathcal C}_\Phi = \{A \in {\mathcal B}({\mathcal H}): \Vert A\Vert _\Phi < \infty \} \end{aligned}$$

(1.2)

is a norm ideal [10, page 68]. This term refers to the following properties of ${\mathcal C}_\Phi$:

• For any B, $C \in {\mathcal B}({\mathcal H})$ and $A \in {\mathcal C}_\Phi$, $BAC \in {\mathcal C}_\Phi$ and $\Vert BAC\Vert _\Phi \le \Vert B\Vert \Vert A\Vert _\Phi \Vert C\Vert$.

• If $A \in {\mathcal C}_\Phi$, then $A^*\in {\mathcal C}_\Phi$ and $\Vert A^*\Vert _\Phi = \Vert A\Vert _\Phi$.

• For any $A \in {\mathcal C}_\Phi$, $\Vert A\Vert \le \Vert A\Vert _\Phi$, and the equality holds when rank$(A) = 1$.

• ${\mathcal C}_\Phi$ is complete with respect to $\Vert .\Vert _\Phi$.

There are many familiar examples of symmetric gauge functions. For each $1 \le p < \infty$, the formula $\Phi _p(\{a_j\}_{j\in {\mathbf{N}}}) = (\sum _{j=1}^\infty |a_j|^p)^{1/p}$ defines a symmetric gauge function on $\hat{c}$, and the corresponding ideal ${\mathcal C}_{\Phi _p}$ defined by (1.2) is just the Schatten class ${\mathcal C}_p$. As another family of examples, let us mention the symmetric gauge function $\Phi _p^-$ defined by the formula

$$\begin{aligned} \Phi _p^-(\{a_j\}_{j\in \mathbf{N}}) = \sum _{j=1}^\infty {|a_{\pi (j)}|\over j^{(p-1)/p}}, \quad \{a_j\}_{j\in \mathbf{N}} \in \hat{c}, \end{aligned}$$

where $\pi : \mathbf{N} \rightarrow \mathbf{N}$ is any bijection such that $|a_{\pi (1)}| \ge |a_{\pi (2)}| \ge \cdots \ge |a_{\pi (j)}| \ge \cdots$, which exists because each $\{a_j\}_{j\in \mathbf{N}} \in \hat{c}$ only has a finite number of nonzero terms. In this case, the ideal ${\mathcal C}_{\Phi _p^-}$ defined by (1.2) is called a Lorentz ideal and often simply denoted by the symbol ${\mathcal C}_p^-$. When $p = 1$, ${\mathcal C}_1^-$ is just the trace class ${\mathcal C}_1$. But when $1< p < \infty$, ${\mathcal C}_p^-$ is strictly smaller than the Schatten class ${\mathcal C}_p$. Moreover, when $1< p < \infty$, the dual ${\mathcal C}_{p/(p-1)}^+$ of ${\mathcal C}_p^-$ is a norm ideal with interesting properties of its own [10].

Given a symmetric gauge $\Phi$, it is a common practice to extend its domain of definition beyond the space $\hat{c}$. Suppose that $\{b_j\}_{j\in \mathbf{N}}$ is an arbitrary sequence of real numbers, i.e., the set $\{j \in \mathbf{N} : b_j \ne 0\}$ is not necessarily finite. Then we define

$$\begin{aligned} \Phi (\{b_j\}_{j\in \mathbf{N}}) = \sup _{k\ge 1}\Phi (\{b_1, \dots , b_k, 0, \dots , 0, \dots \}). \end{aligned}$$

(1.3)

Thus if A is a bounded operator, then $\Vert A\Vert _\Phi = \Phi (\{s_j(A)\}_{j\in \mathbf{N}})$. For each $0< p < \infty$, the singular numbers of $|A|^p = (A^*A)^{p/2}$ are $\{(s_1(A))^p,\dots ,(s_j(A))^p,\dots \}$, and therefore

$$\begin{aligned} \Vert |A|^p\Vert _\Phi = \Phi (\{(s_j(A))^p\}_{j\in \mathbf{N}}). \end{aligned}$$

(1.4)

For an unbounded operator X, it is consistent with [10, Theorem II.7.1] to interpret all its singular numbers as infinity. Therefore it is consistent with (1.4) to adopt the convention that $\Vert |X|^p\Vert _\Phi = \infty$ for all $0< p < \infty$ whenever the operator X is unbounded.

For our purpose we also need to deal with sequences indexed by sets other than N. If W is a countable, infinite set, then we define

$$\begin{aligned} \Phi (\{b_\alpha \}_{\alpha \in W}) = \Phi (\{b_{\pi (j)}\}_{j \in \mathbf{N}}), \end{aligned}$$

where $\pi : \mathbf{N} \rightarrow W$ is any bijection. The definition of symmetric gauge functions guarantees that the value of $\Phi (\{b_\alpha \}_{\alpha \in W})$ is independent of the choice of the bijection $\pi$. For a finite index set $F = \{x_1, \dots ,x_\ell \}$, we simply define $\Phi (\{b_x\}_{x\in F}) = \Phi (\{b_{x_1}, \dots , b_{x_\ell }, 0, \dots , 0, \dots \})$.

Recall that the membership of the commutator $[M_f,P] = H_f - H_{\bar{f}}^*$ in ${\mathcal C}_\Phi$ was characterized in [20] for arbitrary symmetric gauge functions $\Phi$, although in [20] the weight of the Bergman space was set at $\alpha = 0$. This paper deals with the corresponding “one-sided” problem for arbitrary weight $-1< \alpha < \infty$, and we will introduce the power $0 < s \le 1$ mentioned earlier.

The statement of our result involves modified kernel functions and the Bergman metric, which we will now review. First of all, the formula

$$\begin{aligned} k_z(\zeta ) = {(1-|z|^2)^{(n+1+\alpha )/2}\over (1 - \langle \zeta ,z\rangle )^{n+1+\alpha }}, \quad z, \zeta \in \mathbf{B}, \end{aligned}$$

(1.5)

gives us the normalized reproducing kernel for $L^2_a(\text{ B },dv_\alpha )$. For each integer $i \ge 0$, we define the modified kernel function

$$\begin{aligned} \psi _{z,i}(\zeta ) = {(1 - |z|^2)^{\{(n+1+\alpha )/2\}+i} \over (1 - \langle \zeta ,z\rangle )^{n+1+\alpha +i}}, \quad z, \zeta \in \mathbf{B}. \end{aligned}$$

(1.6)

If we introduce the multiplier

$$\begin{aligned} m_z(\zeta ) = {1-|z|^2\over 1 - \langle \zeta ,z\rangle } \end{aligned}$$

(1.7)

for each $z \in$ B, then we have the relation $\psi _{z,i} = m_z^ik_z$. Similar to the analogous situations in the Hardy space and the Drury-Arveson space [6,7,8], this modification gives $\psi _{z,i}$ a faster “decaying rate” than $k_z$, which will allow us to establish certain crucial bounds.

Let $\beta$ denote the Bergman metric on B. That is,

$$\begin{aligned} \beta (z,w) = {1\over 2}\log {1 + |\varphi _z(w)|\over 1 - |\varphi _z(w)|}, \quad z, w \in \mathbf{B}, \end{aligned}$$

where $\varphi _z$ is the Möbius transform of B [18, Section 2.2]. For each $z \in$ B and each $a > 0$, we define the corresponding $\beta$-ball $D(z,a) = \{w \in \mathbf{B} : \beta (z,w) < a\}$.

Definition 1.1

[20, Definition 1.1]

• (i) Let a be a positive number. A subset $\Gamma$ of B is said to be a-separated if $D(z,a)\cap D(w,a) = \emptyset$ for all distinct elements z, w in $\Gamma$.

• (ii) Let $0< a< b < \infty$. A subset $\Gamma$ of B is said to be an a, b-lattice if it is a-separated and has the property $\cup _{z\in \Gamma }D(z,b) =$ B.

Given an operator A, for example a Toeplitz operator or a Hankel operator, one is always interested in formulas for its set of singular numbers. But as a practical matter, a formula that is both explicit and exact, is usually not available. Thus one is frequently forced to search for alternatives: are there quantities given by simple formulas that are equivalent to $\{s_1(A),s_2(A),\dots ,s_j(A),\dots \}$ in some clearly-defined sense?

In this general context, our investigation stems from the following intuition: if i is suitably large, i.e., if $\psi _{z,i}$ “decays fast enough”, then for an a, b-lattice $\Gamma$ in B, the set of scalar quantities

$$\begin{aligned} \{\Vert H_f\psi _{z,i}\Vert \}_{z\in \Gamma } \end{aligned}$$

should be equivalent to the set of singular numbers $\{s_1(H_f),s_2(H_f),\dots ,s_j(H_f),\dots \}$ of the Hankel operator $H_f$. The main result of this paper confirms our intuition in a very specific way: if one allows a constant multiple, then the s-powers of these two sets of numbers are not distinguishable by the application of symmetric functions.

Theorem 1.2

Let $0 < s \le 1$ be given, and let $i \in \mathbf{Z}_+$ satisfy the condition $s(n+1+\alpha +2i) > 2n$. Let $0< a< b < \infty$ be positive numbers such that $b \ge 2a$. Then there exist constants $0< c \le C < \infty$ which depend only on the given s, i, a, b, the complex dimension n and the weight $\alpha$ such that the inequality

$$\begin{aligned} c\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le \Vert |H_f|^s\Vert _\Phi \le C\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \end{aligned}$$

holds for every $f \in L^2(\mathbf{B},dv_\alpha )$, every symmetric gauge function $\Phi$ and every a, b-lattice $\Gamma$ in B.

The reader may wonder, why does Theorem 1.2 only cover the powers $0 < s \le 1$? The simple answer is, we could consider all $0< s < \infty$, but that would not add anything. The point is this: if $\Phi$ is a symmetric gauge function, then for each $1< p < \infty$ the formula

$$\begin{aligned} \{a_j\}_{j\in {\mathbf{N}}} \mapsto \left( \Phi (\{|a_j|^p\}_{j\in {\mathbf{N}}})\right) ^{1/p} \end{aligned}$$

defines just another symmetric gauge function on $\hat{c}$, which Theorem 1.2 already covers. That is why we only need to consider $0 < s \le 1$.

The proof of Theorem 1.2 involves a somewhat complicated scheme. To conclude the Introduction, let us outline the main steps in the proof.

For both directions in Theorem 1.2, it is necessary to control the projection $1 - P$ by certain differential operators. This will be achieved in terms of the inequality

$$\begin{aligned} \Vert f - Pf\Vert \le C(\Vert \rho \bar{\partial }f\Vert + \Vert \rho ^{1/2}\bar{\partial }f \wedge \bar{\partial }\rho \Vert ) \end{aligned}$$

(1.8)

for $f \in C^\infty (\mathbf{B})\cap L^2(\mathbf{B},dv_\alpha )$, which will be the main content of Sect. 2.

As one would expect, the proof of Theorem 1.2 uses properties of symmetric gauge functions and symmetric norms extensively. For that reason we begin Sect. 3 with a review of these properties. Another key ingredient in the proof is a workable decomposition system for the unit ball. For this we adopt the decomposition system from [20], which gives us the sets $T_{k,j}$ and $Q_{k,j}$, $(k,j) \in I$. Accordingly, we define the quantities $A(f;Q_{k,j})$, $(k,j) \in I$, for $f \in L^2(\mathbf{B},dv_\alpha )$. With this decomposition system we have

$$\begin{aligned} \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

(1.9)

if $\Gamma$ is a-separated for some $a > 0$. In (1.9), the integer $i \in \mathbf{Z}_+$ must satisfy the condition $s(n+1+\alpha + 2i) > 2n$, and that is why there is such a requirement in Theorem 1.2.

Section 4 is one of the two major steps, which shows that

$$\begin{aligned} \left\| \left| M_f \sum _{z\in \Gamma }\psi _{z,i'}\otimes e_z\right| ^s\right\| _\Phi \le C\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}), \end{aligned}$$

(1.10)

where $i'$ is appropriately large and $\{e_z : z \in \Gamma \}$ is an orthonormal set. Then, by using the atomic decomposition for $L_a^2(\mathbf{B},dv_\alpha )$, in Sect. 5 we show that (1.10) implies

$$\begin{aligned} \Vert |M_fP|^s\Vert _\Phi \le C\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

(1.11)

In Sect. 6, we adopt ideas from [15, 16] and introduce the local projections $P_{k,j}$, which have certain amazing properties. With the local projections $P_{k,j}$ we can define “analytic oscillations” M(f; k, j) for a given symbol function f. Then, using Luecking’s ideas in [16], we show that f admits a decomposition $f = f^{(1)} + f^{(2)}$ such that

$$\begin{aligned} \left\{ \begin{matrix} A(f^{(1)};Q_{k,j}), \ \ A(\rho |\bar{\partial }f^{(2)}|;Q_{k,j}), \ \ A(\rho ^{1/2}|\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho |;Q_{k,j}) \\ \text {can be controlled by} \ \ \{M(f;k,j) : (k,j) \in I\} \end{matrix} \right. . \end{aligned}$$

(1.12)

It is then easy to deduce from (1.8), (1.11) and (1.12) that

$$\begin{aligned} \Vert |H_f|^s\Vert _\Phi \le C\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}). \end{aligned}$$

This essentially proves the upper bound in Theorem 1.2, for it is routine to show that

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le C\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \end{aligned}$$

if $\Gamma$ has the property that $\cup _{z\in \Gamma }D(z,b) =$ B for some $0< b < \infty$.

For the proof of the lower bound in Theorem 1.2, the most crucial step is Proposition 6.8, which establishes the inequality

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le C\Vert |H_f|^s\Vert _\Phi . \end{aligned}$$

(1.13)

Then, using (1.12), (1.9) and (1.8), we can show that

$$\begin{aligned} \Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}). \end{aligned}$$

(1.14)

Obviously, the lower bound in Theorem 1.2 follows from (1.13) and (1.14).

To summarize, Sects. 2–6 contain the technical steps outlined above, and the proof of Theorem 1.2 itself is formally completed in Sect. 7. Finally, the Appendix at the end of the paper contains technical proofs that are judged to be either similar to what can be found in the literature, or too elementary for the main text.

2 Projection and D-bar Operators

We begin by recalling a particular integral estimate on B. As in [4], define

$$\begin{aligned} \Delta (\zeta ,z) = |1 - \langle \zeta ,z\rangle |^2 - (1-|\zeta |^2)(1 - |z|^2), \quad \zeta ,z \in \mathbf{B}. \end{aligned}$$

Lemma 2.1

[4, Lemma 24] Let $a, b, c, t \in$ R. If $c > -2n$ and $-2a< t + 1 < 2b + 2$, then the operator

$$\begin{aligned} (Tf)(z) = \int {(1-|z|^2)^a(1-|\zeta |^2)^b\Delta ^{c/2}(\zeta ,z)\over |1 - \langle \zeta ,z\rangle |^{n+1+a+b+c}}f(\zeta )dv(\zeta ) \end{aligned}$$

is bounded on $L^2(\mathbf{B},dv_t)$.

For any $f \in C^\infty (\mathbf{B})$, let $\bar{\partial }f$ denote the (0, 1)-form $\sum _{j=1}^n(\bar{\partial }_jf)(\zeta )d\bar{\zeta }_j$ as usual. Write

$$\begin{aligned} |(\bar{\partial }f)(\zeta )| = \{ |(\bar{\partial }_1f)(\zeta )|^2 + \cdots + |(\bar{\partial }_nf)(\zeta )|^2\}^{1/2} \end{aligned}$$

for $\zeta \in$ B. If $\varphi$ is a scalar function on B, then by $\Vert \varphi \bar{\partial }f\Vert$ we mean the norm of the scalar function $\varphi |\bar{\partial }f|$ in $L^2(\mathbf{B}, dv_\alpha )$, allowing the possibility that $\Vert \varphi \bar{\partial }f\Vert = \infty$. For any (p, q)-form F on B, $|F(\zeta )|$ and $\Vert \varphi F\Vert$ are similarly defined.

Let us write

$$\begin{aligned} \rho (\zeta ) = 1 - |\zeta |^2 \quad \text {for} \ \ \zeta \in \mathbf{B}, \end{aligned}$$

and this notation will be fixed for the rest of the paper.

The following proposition is a classic estimate of the minimum-norm solution of a $\bar{\partial }$-problem, which can be obtained using Lemma 2.1.

Proposition 2.2

[2, Theorem I.4] There is a constant $C_{2.2}$ which depends only on n and $\alpha$ such that

$$\begin{aligned} \Vert f - Pf\Vert \le C_{2.2}(\Vert \rho \bar{\partial }f\Vert + \Vert \rho ^{1/2}\bar{\partial }f \wedge \bar{\partial }\rho \Vert ) \end{aligned}$$

(2.1)

for every $f \in C^\infty (\mathbf{B})\cap L^2(\mathbf{B},dv_\alpha )$.

Recall that for each pair of $i \ne j$ in $\{1,\dots ,n\}$, one has the tangential derivatives

$$\begin{aligned} L_{i,j} = \bar{\zeta }_j\partial _i - \bar{\zeta }_i\partial _j \quad \text {and} \quad \bar{L}_{i,j} = \zeta _j\bar{\partial }_i - \zeta _i\bar{\partial }_j. \end{aligned}$$

Thus $|(\bar{\partial }f\wedge \bar{\partial }\rho )(\zeta )|^2$ is simply the sum of all $|(\bar{L}_{i,j}f)(\zeta )|^2$, $i < j$. Recall that $\varphi _z$ is the Möbius transform of B [18, Section 2.2]:

$$\begin{aligned} \varphi _z(\zeta ) = {1\over 1 -\langle \zeta ,z \rangle } \left\{ z - {\langle \zeta ,z\rangle \over |z|^2}z - (1-|z|^2)^{1/2} \left( \zeta - {\langle \zeta ,z\rangle \over |z|^2}z\right) \right\} . \end{aligned}$$

Note that $\varphi _z$ is an involution, i.e., $\varphi _z\circ \varphi _z =$ id. We end this section with an elementary estimate on derivatives that will be needed in Sect. 6.

Lemma 2.3

There is a constant $C_{2.3}$ such that for every $z \in$ B, we have $\Vert \rho \partial _i\varphi _z\Vert _\infty \le C_{2.3}$ for every $i \in \{1,\dots ,n\}$ and $\Vert \rho ^{1/2}L_{i,j}\varphi _z\Vert _\infty \le C_{2.3}$ for all $i \ne j$ in $\{1,\dots ,n\}$.

Proof

Write $D_z(\zeta )$ for $1 - \langle \zeta ,z\rangle$ and $N_z(\zeta )$ for the vector $\{\cdots \}$ above. In other words, we have $\varphi _z = D_z^{-1}N_z$. Note that $\Vert \rho /D_z\Vert _\infty \le 2$ and that $\Vert \partial _iN_z\Vert _\infty \le 3$. Since

$$\begin{aligned} (\partial _i\varphi _z)(\zeta ) = {\bar{z}_i\over D_z(\zeta )}\varphi _z(\zeta ) + {1\over D_z(\zeta )}(\partial _iN_z)(\zeta ), \end{aligned}$$

we have $\Vert \rho \partial _i\varphi _z\Vert _\infty \le 2 + 2\cdot 3 = 8$. For the tangential derivatives, we have

$$\begin{aligned} (L_{i,j}\varphi _z)(\zeta )= & {} {\bar{\zeta }_j\bar{z}_i - \bar{\zeta }_i\bar{z}_j\over D_z(\zeta )}\varphi _z(\zeta ) \nonumber \\&+ {1\over D_z(\zeta )}\left\{ ((1-|z|^2)^{1/2}-1){\bar{\zeta }_j\bar{z}_i - \bar{\zeta }_i\bar{z}_j\over |z|^2}z - (1-|z|^2)^{1/2}L_{i,j}\zeta \right\} . \nonumber \\ \end{aligned}$$

(2.2)

Note that $|\bar{\zeta }_i\bar{z}_j - \bar{\zeta }_j\bar{z}_i| = |(\bar{\zeta }_i - \bar{z}_i)\bar{z}_j - (\bar{\zeta }_j-\bar{z}_j)\bar{z}_i| \le 2|\zeta -z|$. On the other hand, $|\zeta - z|^2 = |\zeta |^2 - 2\text {Re}\langle \zeta ,z\rangle + |z|^2 \le 2(1-\text {Re}\langle \zeta ,z\rangle )$. Therefore $|\bar{\zeta }_j\bar{z}_i - \bar{\zeta }_i\bar{z}_j| \le 2\sqrt{2}|1-\langle \zeta ,z\rangle |^{1/2}$, which leads to

$$\begin{aligned} \rho ^{1/2}(\zeta )\left| {\bar{\zeta }_j\bar{z}_i - \bar{\zeta }_i\bar{z}_j\over D_z(\zeta )}\right| \le 4. \end{aligned}$$

(2.3)

Also, we have $\rho ^{1/2}(\zeta )(1-|z|^2)^{1/2}/|D_z(\zeta )| = (1 - |\varphi _z(\zeta )|^2)^{1/2} \le 1$ [18, Theorem 2.2.2]. Combining this with (2.2) and (2.3), we find that $\Vert \rho ^{1/2}L_{i,j}\varphi _z\Vert _\infty \le 4 + 4+1 = 9$. $\square$

3 Other Preliminaries

The proof of Theorem 1.2 requires a familiarity with symmetric norms.

Lemma 3.1

[20, Lemma 2.2] Suppose that X and Y are countable sets and that N is a natural number. Suppose that $T : X \rightarrow Y$ is a map that is at most N-to-1. That is, for every $y \in Y$, $\text {card}\{x \in X : T(x) = y\} \le N$. Then for every set of real numbers $\{b_y\}_{y\in Y}$ and every symmetric gauge function $\Phi$, we have $\Phi (\{b_{T(x)}\}_{x\in X}) \le N\Phi (\{b_y\}_{y\in Y})$.

Recall from [10, page 125] that given a symmetric gauge function $\Phi$, the formula

$$\begin{aligned} \Phi ^*(\{b_j\}_{j\in \mathbf{N}}) = \sup \left\{ \left| \sum _{j=1}^\infty a_jb_j\right| : \{a_j\}_{j\in \mathbf{N}} \in \hat{c}, \Phi (\{a_j\}_{j\in \mathbf{N}}) \le 1\right\} , \quad \{b_j\}_{j\in \mathbf{N}} \in \hat{c}, \end{aligned}$$

defines the symmetric gauge function that is dual to $\Phi$. Moreover, we have the relation $\Phi ^{**} = \Phi$ [10, page 125]. This relation implies that for every $\{a_j\}_{j\in \mathbf{N}} \in \hat{c}$, we have

$$\begin{aligned} \Phi (\{a_j\}_{j\in \mathbf{N}}) = \sup \left\{ \left| \sum _{j=1}^\infty a_jb_j\right| : \{b_j\}_{j\in \mathbf{N}} \in \hat{c}, \Phi ^*(\{b_j\}_{j\in \mathbf{N}}) \le 1\right\} . \end{aligned}$$

(3.1)

Lemma 3.2

[20, Lemma 5.1] Let $\{A_k\}$ be a sequence of bounded operators on a separable Hilbert space ${\mathcal H}$. If $\{A_k\}$ weakly converges to an operator A, then the inequality

$$\begin{aligned} \Vert A\Vert _\Phi \le \sup _k\Vert A_k\Vert _\Phi \end{aligned}$$

holds for every symmetric gauge function $\Phi$.

Lemma 3.3

Let A and B be two bounded operators. Then the inequalities

$$\begin{aligned} \Vert |AB|^s\Vert _\Phi \le \Vert B\Vert ^s\Vert |A|^s\Vert _\Phi \quad \text {and} \quad \Vert |BA|^s\Vert _\Phi \le \Vert B\Vert ^s\Vert |A|^s\Vert _\Phi \end{aligned}$$

hold for every symmetric gauge function $\Phi$ and every $0 < s \le 1$.

Proof

For the singular numbers of the operators involved, it is well known that

$$\begin{aligned} s_j(AB) \le s_j(A)\Vert B\Vert \quad \text {and} \quad s_j(BA) \le \Vert B\Vert s_j(A) \end{aligned}$$

for every $j \in$ N [10, page 61]. Therefore for any gauge function $\Phi$ and any $0 < s \le 1$,

$$\begin{aligned} \Vert |AB|^s\Vert _\Phi = \Phi (\{(s_j(AB))^s\}_{j\in \mathbf{N}}) \le \Vert B\Vert ^s\Phi (\{(s_j(A))^s\}_{j\in \mathbf{N}}) = \Vert B\Vert ^s\Vert |A|^s\Vert _\Phi . \end{aligned}$$

The other inequality is similarly proved. $\square$

Lemma 3.4

[20, Lemma 3.1] Suppose that $A_1, \dots , A_m$ are finite-rank operators on a Hilbert space ${\mathcal H}$ and let $A = A_1 + \cdots + A_m$. Then for every symmetric gauge function $\Phi$ and every $0 < s \le 1$, we have

$$\begin{aligned} \Vert |A|^s\Vert _\Phi \le 2^{1-s}(\Vert |A_1|^s\Vert _\Phi + \cdots + \Vert |A_m|^s\Vert _\Phi ). \end{aligned}$$

(3.2)

Remark 3.5

Although (3.2) was only proved for finite-rank operators $A_1, \dots , A_m$ in [20], it actually hold for all bounded operators $A_1, \dots , A_m$ and $A = A_1 + \cdots + A_m$ on any separable Hilbert space ${\mathcal H}$. Indeed let $A_1, \dots , A_m \in {\mathcal B}({\mathcal H})$ and $A = A_1 + \cdots + A_m$, and let E and F be finite-rank orthogonal projections on ${\mathcal H}$. Then by (3.2) and Lemma 3.3,

$$\begin{aligned} \begin{aligned} \Vert E|FA|^s\Vert _\Phi \le \Vert |FA|^s\Vert _\Phi&\le 2^{1-s}(\Vert |FA_1|^s\Vert _\Phi + \cdots + \Vert |FA_m|^s\Vert _\Phi ) \\&\le 2^{1-s}(\Vert |A_1|^s\Vert _\Phi + \cdots + \Vert |A_m|^s\Vert _\Phi ). \end{aligned} \end{aligned}$$

Since rank$(E) < \infty$, the supremum of $\Vert E|FA|^s\Vert _\Phi$ over all finite-rank orthogonal projections F dominates $\Vert E|A|^s\Vert _\Phi$. Then observe that, by (1.1), if we take the supremum of $\Vert E|A|^s\Vert _\Phi$ over all finite-rank orthogonal projections E, we obtain $\Vert |A|^s\Vert _\Phi$. Hence (3.2) holds for all $A_1, \dots , A_m \in {\mathcal B}({\mathcal H})$ and $A = A_1 + \cdots + A_m$.

As one would expect, the proof of Theorem 1.2 also requires a suitable decomposition of the ball and the sphere. We will adopt the decomposition system in [20], for that paper showed that the system, however complicated it may appear, actually works. Next let us review the decomposition system in [20] and estimates related to it.

Let S denote the unit sphere $\{\xi \in {\mathbf{C}}^n : |\xi | = 1\}$. Recall that the formula

$$\begin{aligned} d(u,\xi ) = |1 - \langle u,\xi \rangle |^{1/2}, \ \ \ \ u, \xi \in S, \end{aligned}$$

defines a metric on S [18, page 66]. Throughout the paper, we denote

$$\begin{aligned} B(u,r) = \{\xi \in S : |1 - \langle u,\xi \rangle |^{1/2} < r\} \end{aligned}$$

for $u \in S$ and $r > 0$. Let $\sigma$ be the positive, regular Borel measure on S which is invariant under the orthogonal group O(2n), i.e., the group of isometries on ${\mathbf{C}}^n \cong {\mathbf{R}}^{2n}$ which fix 0. We take the usual normalization $\sigma (S) = 1$. There is a constant $A_0 \in (2^{-n},\infty )$ such that

$$\begin{aligned} 2^{-n}r^{2n} \le \sigma (B(u,r)) \le A_0r^{2n} \end{aligned}$$

(3.3)

for all $u \in S$ and $0 < r \le \sqrt{2}$ [18, Proposition 5.1.4]. Note that the upper bound actually holds when $r>\sqrt{2}$.

For each integer $k \ge 0$, let $\{u_{k,1}, \dots , u_{k,m(k)}\}$ be a subset of S which is maximal with respect to the property

$$\begin{aligned} B(u_{k,j},2^{-k-1})\cap B(u_{k,j'},2^{-k-1}) = \emptyset \quad \text {for all} \ \ 1 \le j < j' \le m(k). \end{aligned}$$

(3.4)

The maximality of $\{u_{k,1}, \dots , u_{k,m(k)}\}$ implies that

$$\begin{aligned} \cup _{j=1}^{m(k)}B(u_{k,j},2^{-k}) = S. \end{aligned}$$

(3.5)

For each pair of $k \ge 0$ and $1 \le j \le m(k)$, define the subsets

$$\begin{aligned} T_{k,j}&= \{ru : 1 - 2^{-2k} \le r < 1 - 2^{-2(k+1)}, u \in B(u_{k,j},2^{-k})\} \quad \text {and} \end{aligned}$$

(3.6)

$$\begin{aligned} Q_{k,j}&= \{ru : 1 - 2^{-2k} \le r < 1 - 2^{-2(k+2)}, u \in B(u_{k,j},9\cdot 2^{-k})\} \end{aligned}$$

(3.7)

of B. Let us also introduce the index set

$$\begin{aligned} I = \{(k,j) : k \ge 0, 1 \le j \le m(k)\}. \end{aligned}$$

(3.8)

Lemma 3.6

[20, Lemma 2.4] Given any $0< a < \infty$, there exists a natural number K which depends only on a and the complex dimension n such that the following holds true: Suppose that $\Gamma$ is an a-separated subset of B. Then there exist pairwise disjoint subsets $\Gamma _1, \dots , \Gamma _K$ of $\Gamma$ such that $\cup _{\mu =1}^K\Gamma _\mu = \Gamma$ and such that $\text {card}(\Gamma _\mu \cap T_{k,j}) \le 1$ for all $\mu \in \{1, \dots , K\}$ and $(k,j) \in I$.

Let E be a Borel set in B with $v_\alpha (E) > 0$. For any $f \in L^2(\mathbf{B},dv_\alpha )$, we define

$$\begin{aligned} A(f;E) = \left( {1\over v_\alpha (E)}\int _E|f|^2dv_\alpha \right) ^{1/2}. \end{aligned}$$

Although we use the same decomposition system as that in [20], there is a major difference between [20] and this paper: Whereas most of the estimates in [20] were carried out in terms of the various mean oscillations introduced there, quantities of the form A(f; E) and $\Vert f\psi _{z,i}\Vert$ will be much more prominent in this paper.

Proposition 3.7

Let $0 < s \le 1$ be given, and let $i \in \mathbf{Z}_+$ satisfy the condition $s(n+1+\alpha +2i) > 2n$. Let $0< a < \infty$ also be given. Then there exists a constant $0< C_{3.7} < \infty$ which depends only on n, $\alpha$, s, i and a such that the inequality

$$\begin{aligned} \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C_{3.7}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

holds for every $f \in L^2(\mathbf{B},dv)$, every symmetric gauge function $\Phi$, and every a-separated subset $\Gamma$ of B.

The proof of this proposition is essentially a combination of a part of the work for the proof of [20, Lemma 6.4] and a part of the work in [20, Section 2]. For this reason the proof of Proposition 3.7 is relegated to the Appendix at the end of the paper.

Next we recall some elementary facts related to the Bergman metric.

Lemma 3.8

[21, Lemma 2.3] For all $u, v, x, y \in$ B we have

$$\begin{aligned} {(1- |\varphi _u(x)|^2)^{1/2}(1- |\varphi _v(y)|^2)^{1/2}\over |1 - \langle \varphi _u(x),\varphi _v(y)\rangle |} \le 2e^{\beta (x,0)+\beta (y,0)}{(1-|u|^2)^{1/2}(1-|v|^2)^{1/2}\over |1 - \langle u,v\rangle |}. \end{aligned}$$

Lemma 3.9

[22, Lemma 1.24] Given any $r > 0$, there are $0< c(r) \le C(r) < \infty$ such that

$$\begin{aligned} c(r)(1 - |z|^2)^{n+1+\alpha } \le v_\alpha (D(z,r)) \le C(r)(1 - |z|^2)^{n+1+\alpha } \end{aligned}$$

for every $z \in$ B.

Lemma 3.10

[22, Lemma 2.20] Given any $r > 0$, there is a $\delta (r) > 0$ such that $|m_z(w)| \ge \delta (r)$ for all $z, w \in$ B satisfying the condition $\beta (z,w) < r$.

The proof of Theorem 1.2 involves a familiar counting lemma:

Lemma 3.11

[19, Lemma 4.1] Let X be a set and let E be a subset of $X\times X$. Suppose that m is a natural number such that

$$\begin{aligned} \text {card}\{y \in X : (x,y) \in E\} \le m \ \ \ \ \text { and} \ \ \ \ \text {card}\{y \in X : (y,x) \in E\} \le m \end{aligned}$$

for every $x \in X$. Then there exist pairwise disjoint subsets $E_1$, $E_2$, ..., $E_{2m}$ of E such that

$$\begin{aligned} E = E_1\cup E_2 \cup ... \cup E_{2m} \end{aligned}$$

and such that for each $1 \le j \le 2m$, the conditions $(x,y), (x',y') \in E_j$ and $(x,y) \ne (x',y')$ imply both $x \ne x'$ and $y \ne y'$.

We end the preliminaries with an elementary operator-theoretical fact.

Lemma 3.12

Let $A : {\mathcal H} \rightarrow {\mathcal H}'$ and $B: {\mathcal H} \rightarrow {\mathcal H}''$ be bounded operators, where ${\mathcal H}$, ${\mathcal H}'$, ${\mathcal H}''$ are Hilbert spaces. Suppose that there is a positive number C such that $\Vert Ax\Vert \le C\Vert Bx\Vert$ for every $x \in {\mathcal H}$. Then there is an operator $T : {\mathcal H}'' \rightarrow {\mathcal H}'$ with $\Vert T\Vert \le C$ such that $A = TB$.

Proof

Let ${\mathcal R}_0$ denote the linear subspace $\{Bx : x \in {\mathcal H}\}$ of ${\mathcal H}''$, and let ${\mathcal R}$ be the closure of ${\mathcal R}_0$ in ${\mathcal H}''$. Since $\Vert Ax\Vert \le C\Vert Bx\Vert$ for every $x \in {\mathcal H}$, the formula

$$\begin{aligned} TBx = Ax, \quad \ \ x \in {\mathcal H}, \end{aligned}$$

(3.9)

gives us a well-defined linear operator T from ${\mathcal R}_0$ into ${\mathcal H}'$. Moreover, we have $\Vert Ty\Vert \le C\Vert y\Vert$ for every $y \in {\mathcal R}_0$. By the density of ${\mathcal R}_0$ in ${\mathcal R}$, T extends to a bounded operator $T : {\mathcal R} \rightarrow {\mathcal H}'$ with $\Vert T\Vert \le C$. It is then trivial to extend T to an operator on from ${\mathcal H}''$ to ${\mathcal H}'$ with the same norm. Finally, (3.9) implies the operator identity $A = TB$. $\square$

4 Estimates Involving the Modified Kernel

We begin with inner products involving $\psi _{z,i}$. First of all, there is a $\delta \in \mathbf{Z}_+$ such that

$$\begin{aligned} 0 \le \delta - \alpha < 1. \end{aligned}$$

(4.1)

Lemma 4.1

Given any $i \in \mathbf{Z}_+$, there is a constant $C_{4.1}$ which depends only on n, $\alpha$ and i such that if $z = |z|\xi$ and $w = |w|\eta$ with $\xi ,\eta \in S$, and if $0 \le |z| \le |w| < 1$, then

$$\begin{aligned} |\langle f\psi _{z,3i + n+1+\delta },f\psi _{w,3i + n+1+\delta }\rangle |\le & {} C_{4.1}\left( {1-|w|^2\over 1-|z|^2}\right) ^{(n+1+\alpha )/2}\left( {1-|z|^2\over d^2(\xi ,\eta )}\right) ^i\\&\times \,\Vert f\psi _{w,i}\Vert ^2 \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$.

Proof

By (1.7), $\Vert m_z\Vert _\infty \le 1 + |z| < 2$ for every $z \in$ B. Thus

$$\begin{aligned} \begin{aligned} |\psi _{z,3i+n+1+\delta }\psi _{w,3i+n+1+\delta }|&= |\psi _{w,i}|^2\left( {1-|w|^2\over 1 - |z|^2}\right) ^{(n+1+\alpha )/2}\\&\quad \,|m_w|^{i+\delta -\alpha } |m_z|^{3i+2n + 2+\alpha +\delta } \\&\le 2^{\delta -\alpha +2i+2n+2+\alpha +\delta }\left( {1-|w|^2\over 1 - |z|^2}\right) ^{(n+1+\alpha )/2}\\&\quad \,\left| m_wm_z\right| ^i|\psi _{w,i}|^2 \end{aligned} \end{aligned}$$

for all $z, w \in$ B. Thus if we write $C = 2^{2i+2n+2+2\delta }$, then

$$\begin{aligned} |\langle f\psi _{z,3i + n+1+\delta },f\psi _{w,3i + n+1+\delta }\rangle | {\le } C\left( {1-|w|^2\over 1-|z|^2}\right) ^{(n+1+\alpha )/2}\Vert (m_zm_w)^i\Vert _\infty \Vert f\psi _{w,i}\Vert ^2 \nonumber \\ \end{aligned}$$

(4.2)

for all $z, w \in$ B and $f \in L^2(\mathbf{B},dv_\alpha )$. Hence the proof will be complete if we can show that

$$\begin{aligned} \Vert m_zm_w\Vert _\infty \le 16{1-|z|^2\over d^2(\xi ,\eta )} \end{aligned}$$

(4.3)

for all $z, w \in$ B satisfying the conditions $z = |z|\xi$, $w = |w|\eta$, $\xi ,\eta \in S$ and $|z| \le |w|$. For this, consider any $\zeta \in$ B. Then $\zeta = |\zeta |x$ for some $x \in S$. We have

$$\begin{aligned} 2|1 {-} \langle \zeta ,z\rangle | \ge |1 - \langle x,\xi \rangle | {=} d^2(x,\xi ) \quad \text {and} \quad 2|1 - \langle \zeta ,w\rangle | \ge |1 - \langle x,\eta \rangle | = d^2(x,\eta ). \end{aligned}$$

Hence we have either $|1 - \langle \zeta ,z\rangle | \ge (1/8)d^2(\xi ,\eta )$ or $|1 - \langle \zeta ,w\rangle | \ge (1/8)d^2(\xi ,\eta )$. Since $1 - |w|^2 \le 1 - |z|^2$, $\Vert m_z\Vert _\infty \le 2$ and $\Vert m_w\Vert _\infty \le 2$, (4.3) follows. $\square$

Lemma 4.2

Suppose that $\{e_x : x \in X\}$ is an orthonormal set in a Hilbert space ${\mathcal H}$, where X is a countable index set. Furthermore, suppose that $\{g_x : x \in X\}$ are vectors in ${\mathcal H}$ satisfying the following two conditions:

• (1) There is an $N \in$ N such that card$\{y \in X : \langle g_x,g_y \rangle \ne 0\} \le N$ for every $x \in X$.

• (2) $g_x = 0$ for all but a finite number of $x \in X$.

Let $A = \sum _{x\in X}g_x\otimes e_x$. Then for every symmetric gauge function $\Phi$ and every $0 < s \le 1$, we have $\Vert |A|^s\Vert _\Phi \le 2N\Phi (\{\Vert g_x\Vert ^s\}_{x\in X})$.

Proof

By (1) and a standard maximality argument, there is a partition $X = X_1\cup \cdots \cup X_N$ such that for every $r \in \{1,\dots ,N\}$, the conditions $x, y \in X_r$ and $x \ne y$ imply $\langle g_x,g_y \rangle = 0$. Thus if we define $A_r = \sum _{x\in X_r}g_x\otimes e_x$, $r \in \{1,\dots ,N\}$, then

$$\begin{aligned} A_r^*A_r = \sum _{x\in X_r}\Vert g_x\Vert ^2e_x\otimes e_x. \end{aligned}$$

Thus for every $0 < s \le 1$ and every symmetric gauge function $\Phi$,

$$\begin{aligned} \Vert |A_r|^s\Vert _\Phi = \left\| \left( A_r^*A_r\right) ^{s/2}\right\| _\Phi = \Phi (\{\Vert g_x\Vert ^s\}_{x\in X_r}) \le \Phi (\{\Vert g_x\Vert ^s\}_{x\in X}). \end{aligned}$$

Since $A = A_1 + \cdots + A_N$, the conclusion of the lemma follows from this inequality and Lemma 3.4. $\square$

Lemma 4.3

Let $0 < s \le 1$ be given, and let $i \in$ N satisfy the condition $si > 4n$. Write $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Then there is a constant $C_{4.3}$ which depends only on n, $\alpha$, s and i such that the following holds for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$: Let $\{e_{k,j} : (k,j) \in I\}$ be an orthonormal set. Let $z_{k,j} \in T_{k,j}$ for every $(k,j) \in I$. For each $(k,j) \in I$, let $c_{k,j}$ be either 1 or 0, and suppose that $c_{k,j} = 0$ for all but a finite number of $(k,j) \in I$. Then the operator

$$\begin{aligned} F = M_f \sum _{(k,j)\in I}c_{k,j}\psi _{z_{k,j},i'}\otimes e_{k,j} = \sum _{(k,j)\in I}c_{k,j}(f\psi _{z_{k,j},i'})\otimes e_{k,j} \end{aligned}$$

satisfies the estimate $\Vert |F|^s\Vert _\Phi \le C_{4.3}\Phi (\{c_{k,j}\Vert f\psi _{z_{k,j},i}\Vert ^s\}_{(k,j)\in I})$.

Proof

By (3.4) and (3.3), there is an $N \in$ N such that for every $(k,j) \in I$,

$$\begin{aligned} \text {card}\{j' \in \{1,\dots ,m(k)\} : B(u_{k,j},2^{-k})\cap B(u_{k,j'},2^{-k}) \ne \emptyset \} \le N. \end{aligned}$$

(4.4)

This N will be fixed for the rest of the proof. To simplify the notation, let us write

$$\begin{aligned} \left\{ \begin{matrix} r(k,j) = c_{k,j}\Vert f\psi _{z_{k,j},i}\Vert &{}\text {for all} &{}(k,j) \in I \\ \ \ \\ a(k,j;t,h) = c_{t,h}c_{k,j}\langle f\psi _{z_{k,j},i'},f\psi _{z_{t,h},i'}\rangle &{}\text {for all} &{}(k,j), (t,h) \in I \end{matrix} \right. . \end{aligned}$$

(4.5)

Then

$$\begin{aligned} F^*F = \sum _{(k,j), (t,h) \in I}a(k,j;t,h)e_{t,h}\otimes e_{k,j} = B_0 + \sum _{\ell = 1}^\infty (B_\ell + B_\ell ^*), \end{aligned}$$

where

$$\begin{aligned} B_\ell = \sum _{(k,j), (k+\ell ,h) \in I}a(k,j;k+\ell ,h)e_{k+\ell ,h}\otimes e_{k,j}, \end{aligned}$$

$\ell \ge 0$. It follows from Lemma 3.4 that

$$\begin{aligned} \Vert |F|^s\Vert _\Phi = \Vert (F^*F)^{s/2}\Vert _\Phi \le 2^{1-(s/2)}\Vert |B_0|^{s/2}\Vert _\Phi + 2^{2-(s/2)}\sum _{\ell = 1}^\infty \Vert |B_\ell |^{s/2}\Vert _\Phi . \nonumber \\ \end{aligned}$$

(4.6)

To estimate each $\Vert |B_\ell |^{s/2}\Vert _\Phi$, we need to group the terms in $B_\ell$ is a specific way.

By the assumption $z_{k,j} \in T_{k,j}$, $(k,j) \in I$, we can write each $z_{k,j}$ in the form $z_{k,j} = |z_{k,j}|\xi _{k,j}$, where $\xi _{k,j} \in B(u_{k,j},2^{-k})$. By (3.5), we can rewrite each $B_\ell$ in the form

$$\begin{aligned} B_\ell = \sum _{k=0}^\infty \sum _{1\le j,j'\le m(k)} \sum _{\xi _{k+\ell ,h}\in B(u_{k,j'},2^{-k})}\epsilon (k,j';k+\ell ,h)a(k,j;k+\ell ,h)e_{k+\ell ,h}\otimes e_{k,j}, \nonumber \\ \end{aligned}$$

(4.7)

where each $\epsilon (k,j';k+\ell ,h)$ is either 1 or 0. Define the vector

$$\begin{aligned} g^{(\ell )}_{k,j;k,j'} = \sum _{\xi _{k+\ell ,h}\in B(u_{k,j'},2^{-k})}\epsilon (k,j';k+\ell ,h)a(k,j;k+\ell ,h)e_{k+\ell ,h} \end{aligned}$$

(4.8)

for such $\ell$, k and $j, j'$. Note that for all $j, j', q, q' \in \{1,\dots m(k)\}$, we have

$$\begin{aligned} \left\langle g^{(\ell )}_{k,j;k,j'},g^{(\ell )}_{k,q;k,q'}\right\rangle = 0 \quad \text {whenever} \ \ B(u_{k,j'},2^{-k})\cap B(u_{k,q'},2^{-k}) = \emptyset . \end{aligned}$$

(4.9)

Also, it is obvious that

$$\begin{aligned} \left\langle g^{(\ell )}_{k,j;k,j'},g^{(\ell )}_{k',q;k',q'}\right\rangle = 0 \quad \text {whenever} \ \ k \ne k'. \end{aligned}$$

(4.10)

Let us introduce the index sets

$$\begin{aligned} \begin{aligned} E^{(0)}&= \{((k,j),(k,j')) : d(u_{k,j},u_{k,j'})<2^{-k+2}\} \quad \text {and} \\ E^{(m)}&= \{((k,j),(k,j')) : 2^{-k+m+1} \le d(u_{k,j},u_{k,j'})<2^{-k+m+2}\}, \quad m \ge 1. \end{aligned} \end{aligned}$$

Then by (4.7) and (4.8), we have

$$\begin{aligned} \begin{aligned} B_\ell&= \sum _{k=0}^\infty \sum _{1\le j,j'\le m(k)}g^{(\ell )}_{k,j;k,j'}\otimes e_{k,j} = \sum _{m=0}^\infty B_\ell ^{(m)}, \quad \text {where} \\ B_\ell ^{(m)}&= \sum _{((k,j),(k,j'))\in E^{(m)}} g^{(\ell )}_{k,j;k,j'}\otimes e_{k,j} \quad \text {for each} \ \ m \ge 0. \end{aligned} \end{aligned}$$

But each $B_\ell ^{(m)}$ needs to be further decomposed. By (3.4) and (3.3), there is a natural number $C_1$ such that for each $(k,j) \in I$ and each $m \ge 0$, we have

$$\begin{aligned} \text {card}\{j' \in \{1,\dots ,m(k)\} : d(u_{k,j},u_{k,j'})<2^{-k+m+2}\} \le C_12^{2nm}. \end{aligned}$$

(4.11)

By (4.11) and Lemma 3.11, for each $m \ge 0$ we have a partition

$$\begin{aligned} E^{(m)} = E^{(m)}_1\cup \dots \cup E^{(m)}_{2C_12^{2nm}} \end{aligned}$$

(4.12)

such that for each $1 \le \nu \le 2C_12^{2nm}$, if $((k_1,j_1),(k_1,j_1'))$ and $((k_2,j_2),(k_2,j_2'))$ are two distinct elements in $E^{(m)}_\nu$, then we have both $(k_1,j_1) \ne (k_2,j_2)$ and $(k_1,j_1') \ne (k_2,j_2')$. Define

$$\begin{aligned} B_\ell ^{(m,\nu )} = \sum _{((k,j),(k,j'))\in E^{(m)}_\nu }g^{(\ell )}_{k,j;k,j'}\otimes e_{k,j} \end{aligned}$$

(4.13)

for $m \ge 0$ and $1 \le \nu \le 2C_12^{2nm}$. The above-mentioned property of $E^{(m)}_\nu$ implies that the projections $((k,j),(k,j')) \mapsto (k,j)$ and $((k,j),(k,j'))$ $\mapsto$ $(k,j')$ are both injective on $E^{(m)}_\nu$. It follows from the injectivity of this second projection and (4.9), (4.4) and (4.10) that for each $((k,j),(k,j'))\in E^{(m)}_\nu$, we have

$$\begin{aligned} \text {card}\left\{ ((k',q),(k',q'))\in E^{(m)}_\nu : \left\langle g^{(\ell )}_{k,j;k,j'},g^{(\ell )}_{k',q;k',q'}\right\rangle \ne 0\right\} \le N. \end{aligned}$$

Since $\{e_{k,j} : (k,j) \in I\}$ is an orthonormal set and since the projection $((k,j),(k,j')) \mapsto (k,j)$ is injective on $E^{(m)}_\nu$, we can now apply Lemma 4.2 to obtain

$$\begin{aligned} \left\| \left| B_\ell ^{(m,\nu )}\right| ^{s/2}\right\| _\Phi \le 2N\Phi \left( \left\{ \left\| g^{(\ell )}_{k,j;k,j'}\right\| ^{s/2}\right\} _{((k,j),(k,j'))\in E^{(m)}_\nu }\right) . \end{aligned}$$

(4.14)

Next we estimate the right-hand side of (4.14).

For each triple of $\ell \ge 0$, $(k,j) \in I$ and $m \ge 0$, there is an $h(\ell ;k,j;m) \in \{1,\dots ,m(k+\ell )\}$ such that $d(u_{k,j},u_{k+\ell ,h(\ell ;k,j;m)}) < 2^{-k+m+3}$ and

$$\begin{aligned} r(k+\ell ,h(\ell ;k,j;m)) \ge r(k+\ell ,h) \quad \text {whenever} \ \ d(u_{k,j},u_{k+\ell ,h}) < 2^{-k+m+3}. \end{aligned}$$

Claim: there is a $C_0$ such that if $((k,j),(k,j')) \in E^{(m)}$ and $\xi _{k+\ell ,h} \in B(u_{k,j'},2^{-k})$, then

$$\begin{aligned} |a(k,j;k+\ell ,h)| \le C_02^{-\ell (n+1+\alpha )}2^{-2im}r^2(k+\ell ,h(\ell ;k,j;m)). \end{aligned}$$

(4.15)

Using (4.5) and Lemma 4.1, let us verify it according to the following three cases.

• (1) Suppose that $\ell = 0$ and that $m = 0$. Since $z_{k,h} = |z_{k,h}|\xi _{k,h}$ and $\xi _{k,h} \in B(u_{k,h},2^{-k})$, if $((k,j),(k,j')) \in E^{(0)}$ and $\xi _{k,h} \in B(u_{k,j'},2^{-k})$, then $d(u_{k,j},u_{k,h}) \le d(u_{k,j},u_{k,j'}) + d(u _{k,j'},u_{k,h})$ $< 2^{-k+2} + 2^{-k+1}$ $< 2^{-k+3}$. In this case, recalling (4.5), it follows from (4.2) and the definition of $h(\cdot ;\cdot ,\cdot ;\cdot )$ that $|a(k,j;k,h)| \le 4^iCr^2(k,h(0;k,j;0))$.

• (2) Suppose that $\ell = 0$ and that $m \ge 1$. If $((k,j),(k,j')) \in E^{(m)}$ and $\xi _{k,h} \in B(u_{k,j'},2^{-k})$, then $d(u_{k,j},u_{k,h}) \le d(u_{k,j},u_{k,j'}) + d(u _{k,j'},u_{k,h})$ $< 2^{-k+m+3}$ in this case. Hence, recalling (4.5), it follows from Lemma 4.1 and the definition of $h(\cdot ;\cdot ,\cdot ;\cdot )$ that

  $$\begin{aligned} |a(k,j;k,h)| \le C_{4.1}\left( {2^{-2k+1}\over d^2(\xi _{k,j},\xi _{k,h})}\right) ^ir^2(k,h(0;k,j;m)). \end{aligned}$$

  (4.16)

  Since $((k,j),(k,j')) \in E^{(m)}$ and $m \ge 1$, it follows from the definition of $E^{(m)}$ that $d(u_{k,j},u_{k,j'}) \ge 2^{-k+m+1}$ $\ge 4d(u_{k,j},\xi _{k,j})$. Similarly, $d(u_{k,j},u_{k,j'}) \ge 4d(u_{k,j'},\xi _{k,h})$ since $\xi _{k,h} \in B(u_{k,j'},2^{-k})$. By the triangle inequality, we have $d(\xi _{k,j},\xi _{k,h}) \ge (1/2)d(u_{k,j},u_{k,j'}) \ge 2^{-k+m}$. Substituting this in (4.16), we obtain

  $$\begin{aligned} |a(k,j;k,h)| \le 2^iC_{4.1}2^{-2im}r^2(k,h(0;k,j;m)) \end{aligned}$$

  (4.17)

  if $\xi _{k,h} \in B(u_{k,j'},2^{-k})$ and $((k,j),(k,j')) \in E^{(m)}$.

• (3) Suppose that $\ell \ge 1$. Let $((k,j),(k,j')) \in E^{(m)}$ and $\xi _{k+\ell ,h} \in B(u_{k,j'},2^{-k})$. Then $d(u_{k,j},u_{k+\ell ,h})< 2^{-k+m+2} + 2^{-k} + 2^{-k-\ell } < 2^{-k+m+3}$. Applying Lemma 4.1, we have

  $$\begin{aligned} |a(k,j;k+\ell ,h)|\le & {} C_{4.1}\left( {1-|z_{k+\ell ,h}|^2\over 1 - |z_{k,j}|^2}\right) ^{(n+1+\alpha )/2} \left( {1 -|z_{k,j}|^2\over d^2(\xi _{k,j},\xi _{k+\ell ,h})}\right) ^i\nonumber \\&\times \,r^2(k+\ell ,h) \nonumber \\\le & {} C_{4.1}\left( {2^{-2(k+\ell )+1}\over 2^{-2(k+1)}}\right) ^{(n+1+\alpha )/2} \left( {2^{-2k+1}\over d^2(\xi _{k,j},\xi _{k+\ell ,h})}\right) ^i\nonumber \\&\, \times r^2(k+\ell ,h(\ell ;k,j;m)). \end{aligned}$$

  (4.18)

By (4.2), we can also replace the factor $(\cdots )^i$ above by $4^i$, which covers the case $m = 0$. For the case $m \ge 1$, we can repeat the triangle inequality-argument between (4.16) and (4.17) to obtain $d(\xi _{k,j},\xi _{k+\ell ,h}) \ge (1/2)d(u_{k,j},u_{k,j'}) \ge 2^{-k+m}$. Substituting this in (4.18), we see that (4.15) also holds in the case $\ell \ge 1$. This completes the verification of (4.15).

For each pair of $\ell \ge 0$ and $(k,j') \in I$, define

$$\begin{aligned} {\mathcal N}(\ell ;k,j') = \text {card}\{h : \xi _{k+\ell ,h} \in B(u_{k,j'},2^{-k})\}. \end{aligned}$$

Since $\xi _{k+\ell ,h} \in B(u_{k+\ell ,h},2^{-k-\ell })$, if $\xi _{k+\ell ,h} \in B(u_{k,j'},2^{-k})$, then $d(u_{k,j'},u_{k+\ell ,h}) < 2^{-k+1}$. Hence it follows from (3.4) and (3.3) that there is a $C_2$ such that

$$\begin{aligned} {\mathcal N}(\ell ;k,j') \le C_22^{2n\ell } \end{aligned}$$

for all $\ell \ge 0$ and $(k,j') \in I$. The fact that $\{e_{k,j} : (k,j) \in I\}$ is an orthonormal set now produces a quantitative effect: by (4.8), (4.15) and this orthonormality, we have

$$\begin{aligned} \left\| g^{(m)}_{k,j;k,j'}\right\|\le & {} C_02^{-\ell (n+1+\alpha )}2^{-2im}r^2(k+\ell ,h(\ell ;k,j;m))\sqrt{{\mathcal N}(\ell ;k,j')} \nonumber \\\le & {} C_02^{-\ell (n+1+\alpha )}2^{-2im}r^2(k+\ell ,h(\ell ;k,j;m))\cdot C_2^{1/2}2^{\ell n} \nonumber \\= & {} C_32^{-\ell (1+\alpha )}2^{-2im}r^2(k+\ell ,h(\ell ;k,j;m)) \end{aligned}$$

(4.19)

for every $((k,j),(k,j')) \in E^{(m)}$, where $C_3 = C_0C_2^{1/2}$. Thus

$$\begin{aligned} \left\| g^{(m)}_{k,j;k,j'}\right\| ^{s/2} \le C_3^{s/2}2^{-\ell (1+\alpha )(s/2)}2^{-sim}r^s(k+\ell ,h(\ell ;k,j;m)). \end{aligned}$$

Since the projection $((k,j),(k,j')) \mapsto (k,j)$ is injective on $E^{(m)}_\nu$, (4.14) now leads to

$$\begin{aligned} \left\| \left| B_\ell ^{(m,\nu )}\right| ^{s/2}\right\| _\Phi\le & {} 2N\Phi \left( \left\{ \left\| g^{(\ell )}_{k,j;k,j'}\right\| ^{s/2}\right\} _{((k,j),(k,j'))\in E^{(m)}_\nu }\right) \nonumber \\\le & {} C_42^{-\ell (1+\alpha )(s/2)}2^{-sim}\Phi \left( \{r^s(k+\ell ,h(\ell ;k,j;m))\}_{(k,j)\in I}\right) ,\nonumber \\ \end{aligned}$$

(4.20)

where $C_4 = 2NC_3^{s/2}$. If $h(\ell ;k,j;m) = h(\ell ;k,j';m)$, then $d(u_{k,j},u_{k,j'}) < 2^{-k+m+4}$. By (3.4) and (3.3), there is an $N_1 \in$ N such that for every pair of $\ell \ge 0$ and $m \ge 0$, the map

$$\begin{aligned} (k,j) \mapsto (k+\ell ,h(\ell ;k,j;m)) \end{aligned}$$

is at most $N_12^{2nm}$-to-1 on I. Applying Lemma 3.1 in (4.20), we obtain

$$\begin{aligned} \left\| \left| B_\ell ^{(m,\nu )}\right| ^{s/2}\right\| _\Phi \le N_1C_42^{-\ell (1+\alpha )(s/2)}2^{-(si-2n)m}\Phi \left( \{r^s(k,j)\}_{(k,j)\in I}\right) . \end{aligned}$$

By (4.12) and (4.13), $B_\ell ^{(m)} = B_\ell ^{(m,1)} + \cdots + B_\ell ^{(m,2C_12^{2nm})}$. Thus Lemma 3.4 leads to

$$\begin{aligned} \begin{aligned} \left\| \left| B_\ell ^{(m)}\right| ^{s/2}\right\| _\Phi&\le 2\sum _{\nu =1}^{2C_12^{2nm}}\left\| \left| B_\ell ^{(m,\nu )}\right| ^{s/2}\right\| _\Phi \\&\le 4C_1N_1C_42^{-\ell (1+\alpha )(s/2)}2^{-(si-4n)m}\Phi \left( \{r^s(k,j)\}_{(k,j)\in I}\right) . \end{aligned} \end{aligned}$$

Since $si > 4n$, another application of Lemma 3.4 gives us

$$\begin{aligned} \left\| \left| B_\ell \right| ^{s/2}\right\| _\Phi \le 2\sum _{m=0}^\infty \left\| \left| B_\ell ^{(m)}\right| ^{s/2}\right\| _\Phi \le C_52^{-\ell (1+\alpha )(s/2)}\Phi \left( \left\{ c_{k,j}\left\| f\psi _{z_{k,j},i}\right\| ^s\right\} _{(k,j)\in I}\right) . \end{aligned}$$

Finally, substituting this in (4.6), we see that the lemma holds for the constant

$$\begin{aligned} C_{4.3} = 2^{1-(s/2)}C_5 + 2^{2-(s/2)}C_5\sum _{\ell = 1}^\infty 2^{-\ell (1+\alpha )(s/2)}, \end{aligned}$$

which is finite because $\alpha > -1$. This completes the proof. $\square$

Proposition 4.4

Let $0 < s \le 1$ be given, and let $i \in$ N satisfy the condition $si > 4n$. Set $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Let $a > 0$ also be given. Then there is a constant $C_{4.4}$ which depends only on n, $\alpha$, s, i and a such that the following holds for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$: Let $\Gamma$ be an a-separated set in B, and let $\{e_z : z \in \Gamma \}$ be an orthonormal set. Then the operator

$$\begin{aligned} Y = M_f \sum _{z\in \Gamma }\psi _{z,i'}\otimes e_z = \sum _{z\in \Gamma }(f\psi _{z,i'})\otimes e_z \end{aligned}$$

satisfies the estimate $\Vert |Y|^s\Vert _\Phi \le C_{4.4}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I})$.

Proof

Given $a > 0$, let K denote the natural number provided by Lemma 3.6. According to that lemma, any a-separated set $\Gamma$ admits a partition $\Gamma = \Gamma _1\cup \cdots \cup \Gamma _K$ such that for each $\mu \in \{1,\dots ,K\}$, we have card$(\Gamma _\mu \cap T_{k,j}) \le 1$ for every $(k,j) \in I$. We can write $\Gamma$ as the union of an increasing sequence of finite subsets $G_1 \subset G_2 \subset \cdots \subset G_m \subset \cdots$.

Consider any $f \in L^2(\mathbf{B},dv_\alpha )$ and any symmetric gauge function $\Phi$. The condition $si > 4n$ certainly implies $s(n+1+\alpha +2i) > 2n$. Thus by Proposition 3.7,

$$\begin{aligned} \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C_{3.7}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

(4.21)

For every pair of $\mu \in \{1,\dots ,K\}$, and $m \ge 1$, define

$$\begin{aligned} Y_\mu ^{(m)} = M_f \sum _{z\in \Gamma _\mu \cap G_m}\psi _{z,i'}\otimes e_z = \sum _{z\in \Gamma _\mu \cap G_m}(f\psi _{z,i'})\otimes e_z. \end{aligned}$$

Since the finite set $\Gamma _\mu \cap G_m$ has the property card$(\Gamma _\mu \cap G_m\cap T_{k,j}) \le 1$ for every $(k,j) \in I$, it follows from Lemma 4.3 and (4.21) that

$$\begin{aligned} \left\| \left| Y_\mu ^{(m)}\right| ^s\right\| _\Phi \le C_{4.3}\Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma _\mu \cap G_m}) \le C_{4.3}C_{3.7}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

Set $C_{4.4} = 2^{1-s}KC_{4.3}C_{3.7}$. By the partition $\Gamma = \Gamma _1\cup \cdots \cup \Gamma _K$ and Lemma 3.4, we have

$$\begin{aligned} \Vert |Y^{(m)}|^s\Vert _\Phi {\le } 2^{1-s}\left( \left\| \left| Y_1^{(m)}\right| ^s\right\| _\Phi + \cdots + \left\| \left| Y_K^{(m)}\right| ^s\right\| _\Phi \right) {\le } C_{4.4}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}), \end{aligned}$$

where

$$\begin{aligned} Y^{(m)} = M_f \sum _{z\in G_m}\psi _{z,i'}\otimes e_z = \sum _{z\in G_m}(f\psi _{z,i'})\otimes e_z, \end{aligned}$$

$m \ge 1$. Thus for every $m \ge 1$ we have

$$\begin{aligned} \Vert (Y^{(m)}Y^{(m)*})^{s/2}\Vert _\Phi = \Vert |Y^{(m)*}|^s\Vert _\Phi = \Vert |Y^{(m)}|^s\Vert _\Phi \le C_{4.4}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

If $\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) < \infty$, then this bound guarantees that the increasing operator sequence $\{Y^{(m)}Y^{(m)*}\}$ converges to $YY^*$ strongly. Hence the sequence $\{(Y^{(m)}Y^{(m)*})^{s/2}\}$ strongly converges to $(YY^*)^{s/2}$. Thus it follows from Lemma 3.2 that

$$\begin{aligned} \Vert (YY^*)^{s/2}\Vert _\Phi = \sup _{m\ge 1}\Vert (Y^{(m)}Y^{(m)*})^{s/2}\Vert _\Phi \le C_{4.4}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

But if $\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) = \infty$, then this inequality holds trivially. Finally, since $(YY^*)^{s/2}$ $=$ $|Y^*|^s$ and $\Vert |Y^*|^s\Vert _\Phi = \Vert |Y|^s\Vert _\Phi$, the proposition follows. $\square$

Corollary 4.5

Let $i \in$ N satisfy the condition $i > 4n$. Set $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Let $a > 0$ also be given. Then there is a constant $C_{4.5}$ which depends only on n, $\alpha$, i and a such that if $\Gamma$ is an a-separated set in B and if $\{e_z : z \in \Gamma \}$ is an orthonormal set, then

$$\begin{aligned} \left\| \sum _{z\in \Gamma }\psi _{z,i'}\otimes e_z \right\| \le C_{4.5}. \end{aligned}$$

Proof

This follows from Proposition 4.4 by applying it to the specific symmetric gauge function

$$\begin{aligned} \Phi _\infty (\{a_j\}_{j\in \mathbf{N}}) = \sup \{|a_1|,\dots ,|a_j|,\dots \}, \quad \{a_j\}_{j\in \mathbf{N}} \in \hat{c}, \end{aligned}$$

with $s = 1$ and f being the constant function 1 on B. $\square$

5 Discrete Sums and the Bergman Projection

Next we will show that operators of the form $M_fP$ can be dominated by the kind of discrete sums Y in Proposition 4.4. This will reduce the main estimate in the proof of the upper bound in Theorem 1.2 to the estimate provided by Proposition 4.4. What is involved here is the familiar atomic decomposition for the weighted Bergman space [3, 5, 22].

Lemma 5.1

[21, Lemma 2.2] Let $\Gamma$ be an a-separated set in B for some $a > 0$.

(a) For each $0< R < \infty$, there is a natural number $N = N(\Gamma ,R)$ such that card$\{v \in \Gamma : \beta (u,v) \le R\} \le N$ for every $u \in \Gamma$.

(b) For every pair of $z \in$ B and $r > 0$, there is a finite partition $\Gamma = \Gamma _1\cup \cdots \cup \Gamma _m$ such that for every $\nu \in \{1,\dots ,m\}$, the conditions $u, v \in \Gamma _\nu$ and $u \ne v$ imply $\beta (\varphi _u(z),\varphi _v(z)) > r$.

Let ${\mathcal \Gamma }$ be an a-separated set in B. For each pair of $i \in \mathbf{Z}_+$ and $z \in$ B, denote

$$\begin{aligned} E_{\Gamma ,z,i} = \sum _{u\in \Gamma }\psi _{\varphi _u(z),i}\otimes \psi _{\varphi _u(z),i}. \end{aligned}$$

Lemma 5.2

Let ${\mathcal \Gamma }$ be an a-separated set in B for some $a > 0$. Given $0 < s \le 1$, let $i \in$ N satisfy the condition $si > 4n$. Set $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Then for every $z \in$ B, there is a constant $C_{5.2}(z)$ which depends only on n, $\alpha$, $\Gamma$, s, i, and z such that

$$\begin{aligned} \Vert |M_fE_{\Gamma ,z,i'}|^s\Vert _\Phi \le C_{5.2}(z)\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$.

Proof

For each $z \in$ B, Lemma 5.1(b) provides an $m = m(\Gamma ,z) \in$ N and a partition $\Gamma$ $=$ $\Gamma _1\cup \cdots \cup \Gamma _m$ such that for each $\nu \in \{1,\dots ,m\}$, the conditions $u, v \in \Gamma _\nu$ and $u \ne v$ imply $\beta (\varphi _u(z),\varphi _v(z)) > 2$. In other words, each $\{\varphi _u (z) : u \in \Gamma _\nu \}$ is a 1-separated set. Thus we can pick an orthonormal set $\{e_u : u \in \Gamma \}$ and decompose $E_{\Gamma ,z,i'}$ in the form

$$\begin{aligned} E_{\Gamma ,z,i'} = F_1F_1^*+ \cdots + F_mF_m^*, \quad \text {where} \quad F_\nu = \sum _{u\in \Gamma _\nu }\psi _{\varphi _u(z),i'}\otimes e_u, \end{aligned}$$

$1 \le \nu \le m$. Since each $\{\varphi _u (z) : u \in \Gamma _\nu \}$ is 1-separated, Corollary 4.5 guarantees that $F_\nu$ is bounded. For each $\nu \in \{1,\dots ,m\}$, we can apply Proposition 4.4 with $a = 1$ to obtain

$$\begin{aligned} \Vert |M_fF_\nu |^s\Vert _\Phi \le C_{4.4}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

(5.1)

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$. On the other hand, applying Lemma 3.4, Remark 3.5 and Lemma 3.3, we have

$$\begin{aligned} \begin{aligned} \Vert |M_fE_{\Gamma ,z,i'}|^s\Vert _\Phi&\le 2^{1-s}\left( \left\| \left| M_fF_1F_1^*\right| ^s\right\| _\Phi + \cdots + \left\| \left| M_fF_mF_m^*\right| ^s\right\| _\Phi \right) \\&\le 2^{1-s}(\Vert |M_fF_1|^s\Vert _\Phi \Vert F_1^*\Vert ^s + \cdots + \Vert |M_fF_m|^s\Vert _\Phi \Vert F_m^*\Vert ^s). \end{aligned} \end{aligned}$$

Combining this with (5.1), we see that the constant $C_{5.2}(z) = 2^{1-s}C_{4.4}(\Vert F_1\Vert ^s + \cdots + \Vert F_m\Vert ^s)$ will do for the lemma. $\square$

Let us recall the well-known atomic decomposition for $L_a^2(\mathbf{B},dv_\alpha )$:

Proposition 5.3

[22, pages 69–72] Let $i \in \mathbf{Z}_+$ be given. Then there exist an a-separated set $\Gamma$ in B for some $a > 0$ and a finite set $\{z_1,\dots ,z_q\}$ in B such that every $h \in L_a^2(\mathbf{B},dv_\alpha )$ admits the representation

$$\begin{aligned} h = \sum _{u\in \Gamma }\sum _{1\le j \le q}c_{u,j}\psi _{\varphi _u(z_j),i}, \end{aligned}$$

where the coefficients $c_{u,j}$ satisfy the condition $\sum _{u\in \Gamma }\sum _{1\le j \le q}|c_{u,j}|^2 < \infty$.

Lemma 5.4

Let $i \in$ N satisfy the condition $i > 4n$. Set $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Then there exist an a-separated set $\Gamma$ in B for some $a > 0$, a finite set $\{z_1,\dots ,z_q\}$ in B, and a bounded operator T on $L^2(\mathbf{B},dv_\alpha )$ such that

$$\begin{aligned} P = E_{\Gamma ,z_1,i'}T + \cdots + E_{\Gamma ,z_q,i'}T. \end{aligned}$$

(5.2)

Proof

We apply Propositions 5.3 to this integer $i'$: there is an a-separated set $\Gamma$ for some $a > 0$ and $\{z_1,\dots ,z_q\} \subset$ B such that every $h \in L_a^2(\mathbf{B},dv_\alpha )$ admits the representation

$$\begin{aligned} h = \sum _{u\in \Gamma }\sum _{1\le j \le q}c_{u,j}\psi _{\varphi _u(z_j),i'} \quad \text {with} \ \ \sum _{u\in \Gamma }\sum _{1\le j \le q}|c_{u,j}|^2 < \infty . \end{aligned}$$

(5.3)

Let $\{e_{u,j} : u \in \Gamma , 1 \le j \le q\}$ be an orthonormal set and define the operator

$$\begin{aligned} A = \sum _{u\in \Gamma }\sum _{1\le j \le q}\psi _{\varphi _u(z_j),i'}\otimes e_{u,j}. \end{aligned}$$

By Lemma 5.1(b) and Corollary 4.5, A is a bounded operator. By (5.3), the range of A equals $L_a^2(\mathbf{B},dv_\alpha )$. Thus a standard argument gives us a $c > 0$ such that $\Vert A^*h\Vert \ge c\Vert h\Vert$ for every $h \in L_a^2(\mathbf{B},dv_\alpha )$. This lower bound implies that $AA^*$, which we regard as an operator on the whole of $L^2(\mathbf{B},dv_\alpha )$, is invertible on the subspace $L_a^2(\mathbf{B},dv_\alpha )$. In other words, there is a bounded operator X on $L_a^2(\mathbf{B},dv_\alpha )$ such that $AA^*Xh = h$ for every $h \in L_a^2(\mathbf{B},dv_\alpha )$. Now define the operator T by the formula $T(h+g) = Xh$ for $h \in L_a^2(\mathbf{B},dv_\alpha )$ and $g \in L^2(\mathbf{B},dv_\alpha )\ominus L_a^2(\mathbf{B},dv_\alpha )$. Then $\Vert T\Vert = \Vert X\Vert < \infty$ and $P = AA^*T$. To complete the proof, simply observe that $AA^*= E_{\Gamma ,z_1,i'} + \cdots + E_{\Gamma ,z_q,i'}$. $\square$

Proposition 5.5

Let $0 < s \le 1$ be given. Then there is a constant $C_{5.5}$ which depends only on n, $\alpha$ and s such that

$$\begin{aligned} \Vert |M_fP|^s\Vert _\Phi \le C_{5.5}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and very symmetric gauge function $\Phi$.

Proof

Given any $0 < s \le 1$, we pick an $i \in$ N such that $si > 4n$. Then set $i' = 3i+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). For this $i'$, Lemma 5.4 provides an a-separated set $\Gamma$ in B for some $a > 0$, a finite set $\{z_1,\dots ,z_q\}$ in B and a bounded operator T such that (5.2) holds. Since $si > 4n$ and $i' = 3i+n+1+\delta$, by Lemma 5.2, for every $j \in \{1,\dots ,q\}$ we have

$$\begin{aligned} \Vert |M_fE_{\Gamma ,z_j,i'}|^s\Vert _\Phi \le C_{5.2}(z_j)\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

(5.4)

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$, where $C_{5.2}(z_j)$ depends only on n, $\alpha$, s, i, $\Gamma$ and $z_j$. By (5.2), we have $M_fP = M_fE_{\Gamma ,z_1,i'}T + \cdots + M_fE_{\Gamma ,z_q,i'}T$. Applying Lemma 3.4, Remark 3.5 and Lemma 3.3 to this sum, we obtain

$$\begin{aligned} \begin{aligned} \Vert |M_fP|^s\Vert _\Phi&\le 2(\Vert |M_fE_{\Gamma ,z_1,i'}T|^s\Vert _\Phi + \cdots + \Vert |M_fE_{\Gamma ,z_q,i'}T|^s\Vert _\Phi ) \\&\le 2\Vert T\Vert ^s(\Vert |M_fE_{\Gamma ,z_1,i'}|^s\Vert _\Phi + \cdots + \Vert |M_fE_{\Gamma ,z_q,i'}|^s\Vert _\Phi ). \\ \end{aligned} \end{aligned}$$

Combining this with (5.4), we have

$$\begin{aligned} \Vert |M_fP|^s\Vert _\Phi \le 2\Vert T\Vert ^s(C_{5.2}(z_1) + \cdots + C_{5.2}(z_q))\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$. $\square$

6 Bergman Balls and Local Projections

The cumbersome decomposition system adopted in Sect. 3 was designed to accommodate a disparity between the radial direction and the spherical direction of the ball. The best place to see this disparity is (4.19): the factor $2^{-\ell (1+\alpha )}$ is the best decaying rate that one can hope for in the radial direction. In contrast, the factor $2^{-2im}$ in (4.19), which is the decaying rate in the spherical direction, represents artificial improvement: one can pencil in as large an i as one pleases. But once we have proved Proposition 5.5, we no longer need to be concerned the disparity between the two directions. For the rest of the paper, it will simplify matters considerably if we adopt a new decomposition system in terms of balls in the Bergman metric.

For the rest of the paper the paper we fix the point

$$\begin{aligned} w_{k,j} = (1 - 2^{-2k-1})u_{k,j} \end{aligned}$$

for each $(k,j) \in I$, recalling that for each $k \ge 0$, the set $\{u_{k,j}\}$ is a subset of S which is maximal with respect to the property in (3.4) . Recalling (3.6) and (3.7), we have $w_{k,j} \in T_{k,j} \subset Q_{k,j}$ for every $(k,j) \in I$, and we think of $w_{k,j}$ as the “center” for $T_{k,j}$.

Lemma 6.1

• (1) There is a $\tau _0 > 0$ such that $D(w_{k,j},\tau _0)\cap D(w_{t,h},\tau _0) = \emptyset$ for all $(k,j) \ne (t,h)$ in I.

• (2) There is a $\tau _0< \tau < \infty$ such that $D(w_{k,j},\tau ) \supset Q_{k,j}$ for every $(k,j) \in I$.

• (3) There is an $N_0 \in$ N such that card$\{(t,h) \in I : D(w_{k,j},\tau +1)\cap D(w_{t,h},\tau +1) \ne \emptyset \} \le N_0$ for every $(k,j) \in I$.

Since the proof of Lemma 6.1 is completely elementary, it is omitted here.

Definition 6.2

For each $(k,j) \in I$, we denote

$$\begin{aligned} D_{k,j} = D(w_{k,j},\tau ), \quad G_{k,j} = D(w_{k,j},\tau + 1), \quad U_{k,j} = D(w_{k,j},3\tau + 3) \end{aligned}$$

and $I_{k,j} = \{(t,h) \in I : G_{k,j}\cap G_{t,h} \ne \emptyset \}$.

Note that

$$\begin{aligned} \text {if} \ \ (t,h) \in I_{k,j}, \ \ \text {then} \ \ U_{t,h} \supset G_{k,j} \supset Q_{k,j}. \end{aligned}$$

(6.1)

Also note that

$$\begin{aligned} \overline{D(0,\tau )}= & {} \left\{ w \in \mathbf{B} : |w| \le {e^{2\tau }-1\over e^{2\tau }+1}\right\} \quad \text {and} \\ D(0,\tau +1)= & {} \left\{ w \in \mathbf{B} : |w| < {e^{2\tau +2}-1\over e^{2\tau +2}+1}\right\} . \end{aligned}$$

We now fix a $C^\infty$ function $\eta$ on $[0,\infty )$ with the following properties:

• (i) $0 \le \eta (x) \le 1$ for every $x \in [0,\infty )$;

• (ii) $\eta (x^2) = 1$ if $0 \le x \le (e^{2\tau }-1)/(e^{2\tau }+1)$;

• (iii) $\eta (x^2) = 0$ if $x \ge (e^{2\tau +2}-1)/(e^{2\tau +2}+1)$.

For each $(k,j) \in I$, define

$$\begin{aligned} \eta _{k,j}(\zeta ) = \eta (|\varphi _{w_{k,j}}(\zeta )|^2), \quad \zeta \in \mathbf{B}. \end{aligned}$$

Then each $\eta _{k,j}$ is a $C^\infty$ function on B. Furthermore, because $\varphi _{w_{k,j}}(\overline{D_{k,j}}) = \overline{D(0,\tau )}$ and $\varphi _{w_{k,j}}(G_{k,j}) = D(0,\tau +1)$, we have

$$\begin{aligned} \eta _{k,j} = 1 \ \ \text {on} \ \ \overline{D_{k,j}} \ \ \text {and} \ \ \eta _{k,j} = 0 \ \ \text {on} \ \ \mathbf{B}\backslash G_{k,j}. \end{aligned}$$

By Lemma 6.1(3), we have $\sum _{(k,j)\in I}\eta _{k,j} \le N_0$ on B. On the other hand, since $\cup _{(k,j)\in I}T_{k,j}$ $=$ B, we have $\sum _{(k,j)\in I}\eta _{k,j} \ge 1$ on B. Now, for every $(k,j) \in I$ define

$$\begin{aligned} \gamma _{k,j} = {\eta _{k,j}\over \sum _{(t,h)\in I}\eta _{t,h}}. \end{aligned}$$

This gives us a family of $C^\infty$-partition of unity on B. More specifically, we have

• (A) $\sum _{(k,j)\in I}\gamma _{k,j} = 1$ on B;

• (B) for each $(k,j) \in I$, $\gamma _{k,j} = 0$ on $\mathbf{B}\backslash G_{k,j}$.

Lemma 6.3

There is a constant $C_{6.3}$ such that $\Vert \rho \bar{\partial }_\nu \gamma _{k,j}\Vert _\infty \le C_{6.3}$ and $\Vert \rho ^{1/2}\bar{L}_{\nu ,\mu }\gamma _{k,j}\Vert _\infty \le C_{6.3}$ for all $(k,j) \in I$, $\nu \in \{1,\dots ,n\}$ and $\mu \ne \nu$ in $\{1,\dots ,n\}$.

Proof

Write $H = \sum _{(t,h)\in I}\eta _{t,h}$. Then $H \ge 1$ on B. Straightforward differentiation yields

$$\begin{aligned} \begin{aligned} \bar{\partial }_\nu \gamma _{k,j}&= H^{-1}\bar{\partial }_\nu \eta _{k,j} - H^{-2}\eta _{k,j}\bar{\partial }_\nu H = H^{-1}\bar{\partial }_\nu \eta _{k,j} - H^{-2}\eta _{k,j}\sum _{(t,h)\in I_{k,j}}\bar{\partial }_\nu \eta _{t,h} \\&= H^{-1}\eta '(|\varphi _{w_{k,j}}|^2)\langle \varphi _{w_{k,j}},\partial _\nu \varphi _{w_{k,j}}\rangle \\&\quad - H^{-2}\eta _{k,j}\sum _{(t,h)\in I_{k,j}}\eta '(|\varphi _{w_{t,h}}|^2)\langle \varphi _{w_{t,h}},\partial _\nu \varphi _{w_{t,h}}\rangle , \end{aligned} \end{aligned}$$

where the $\langle \cdot ,\cdot \rangle$ is the inner product in $\mathbf{C}^n$. Similarly, for $\mu \ne \nu$ in $\{1,\dots ,n\}$ we have

$$\begin{aligned}&\bar{L}_{\nu ,\mu }\gamma _{k,j}\\&\quad = H^{-1}\eta '(|\varphi _{w_{k,j}}|^2)\langle \varphi _{w_{k,j}},L_{\nu ,\mu }\varphi _{w_{k,j}}\rangle \\&\quad -\, {\eta _{k,j}\over H^2}\sum _{(t,h)\in I_{k,j}}\eta '(|\varphi _{w_{t,h}}|^2)\langle \varphi _{w_{t,h}},L_{\nu ,\mu }\varphi _{w_{t,h}}\rangle . \end{aligned}$$

Obviously, $\eta '$ is bounded on $[0,\infty )$. Thus, combining the bounds provided by Lemma 2.3 with Lemma 6.1(3), the conclusion of the lemma follows. $\square$

Let E be any Borel set in B. Then by $L^2(E,dv_\alpha )$ we mean the collection of functions g in $L^2(\mathbf{B},dv_\alpha )$ satisfying the condition $g = 0$ on $\mathbf{B}\backslash E$. The point is that we consider each element in $L^2(E,dv_\alpha )$ as a function on the whole of the unit ball $\mathbf{B}$.

For each $(k,j) \in I$, let ${\mathcal B}_{k,j}$ be the collection of functions h in $L^2(U_{k,j},dv_\alpha )$ that are analytic on $U_{k,j}$. That is, ${\mathcal B}_{k,j}$ consists of functions h in $L^2(\mathbf{B},dv_\alpha )$ that are analytic on $U_{k,j}$ and identically zero on $\mathbf{B}\backslash U_{k,j}$. Obviously, ${\mathcal B}_{k,j}$ is a closed linear subspace of $L^2(\mathbf{B},dv_\alpha )$. One may think of ${\mathcal B}_{k,j}$ as a kind of “Bergman space”, but keep in mind that the measure in question is the restriction of the weighted volume measure $dv_\alpha$ to $U_{k,j}$. For each $(k,j) \in I$, let

$$\begin{aligned} P_{k,j} : L^2(\mathbf{B},dv_\alpha ) \rightarrow {\mathcal B}_{k,j} \end{aligned}$$

be the orthogonal projection. We consider each $P_{k,j}$ as a local projection (used in [15, 16]), and it performs a little magic:

Lemma 6.4

For all $f, g \in L^2(\mathbf{B},dv_\alpha )$ and $(k,j) \in I$, we have

$$\begin{aligned} \langle f - Pf,\chi _{U_{k,j}}g - P_{k,j}g\rangle = \langle \chi _{U_{k,j}}f - P_{k,j}f,\chi _{U_{k,j}}g - P_{k,j}g\rangle . \end{aligned}$$

Proof

Note that $\langle h,\chi _{U_{k,j}}g - P_{k,j}g\rangle = 0$ for every $h \in L^2(\mathbf{B},dv_\alpha )$ that is analytic on $U_{k,j}$. Therefore

$$\begin{aligned} \begin{aligned} \langle f - Pf,\chi _{U_{k,j}}g - P_{k,j}g\rangle&= \langle f,\chi _{U_{k,j}}g - P_{k,j}g\rangle \\&= \langle \chi _{U_{k,j}}f,\chi _{U_{k,j}}g - P_{k,j}g\rangle \\&= \langle \chi _{U_{k,j}}f - P_{k,j}f,\chi _{U_{k,j}}g - P_{k,j}g\rangle \end{aligned} \end{aligned}$$

as promised. $\square$

For all $f \in L^2(\mathbf{B},dv_\alpha )$ and $(k,j) \in I$, we define

$$\begin{aligned} M(f;k,j) = \left( {1\over v_\alpha (U_{k,j})}\int _{U_{k,j}} |f - P_{k,j}f|^2dv_\alpha \right) ^{1/2}. \end{aligned}$$

Proposition 6.5

There is a constant $C_{6.5}$ such that the following estimates hold: Every $f \in L^2(\mathbf{B},dv_\alpha )$ admits a decomposition

$$\begin{aligned} f = f^{(1)} + f^{(2)} \end{aligned}$$

with $f^{(2)} \in C^\infty (\mathbf{B})$ such that for every $(k,j) \in I$, we have

$$\begin{aligned} \begin{aligned} A^2(f^{(1)};Q_{k,j})&\le C_{6.5}\sum _{(t,h)\in I_{k,j}}M^2(f;t,h), \\ A^2(\rho |\bar{\partial }f^{(2)}|;Q_{k,j})&\le C_{6.5}\sum _{(t,h)\in I_{k,j}}M^2(f;t,h) \quad \text {and} \\ A^2(\rho ^{1/2}|\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho |;Q_{k,j})&\le C_{6.5}\sum _{(t,h)\in I_{k,j}}M^2(f;t,h). \end{aligned} \end{aligned}$$

Proof

If $(t,h) \in I_{k,j}$, then $U_{t,h} \subset D(w_{k,j},5\tau + 5)$. By Lemma 3.9, there is a $C_1$ such that

$$\begin{aligned} v_\alpha (U_{t,h}) \le C_1v_\alpha (Q_{k,j}) \quad \text {whenever} \ \ (t,h) \in I_{k,j}. \end{aligned}$$

(6.2)

Using the partition of unit $\{\gamma _{k,j} : (k,j) \in I\}$, for a given $f \in L^2(\mathbf{B},dv_\alpha )$ we define

$$\begin{aligned} f^{(2)} = \sum _{(k,j)\in I}\gamma _{k,j}P_{k,j}f \quad \text {and} \quad f^{(1)} = f - f^{(2)} = \sum _{(k,j)\in I}(f - P_{k,j}f)\gamma _{k,j}. \end{aligned}$$

If $(t,h) \notin I_{k,j}$, then $\gamma _{t,h} = 0$ on $G_{k,j} \supset Q_{k,j}$. Therefore for every $(k,j) \in I$ we have

$$\begin{aligned} \int _{Q_{k,j}}|f^{(1)}|^2dv_\alpha= & {} \int _{Q_{k,j}}\left| \sum _{(t,h)\in I_{k,j}}(f-P_{t,h}f)\gamma _{t,h}\right| ^2dv_\alpha \\\le & {} N_0\sum _{(t,h)\in I_{k,j}} \int _{Q_{k,j}}|f-P_{t,h}f|^2dv_\alpha , \end{aligned}$$

where the second $\le$ follows from the Cauchy-Schwarz inequality and $N_0$ is given by Lemma 6.1. Recalling (6.1), we have

$$\begin{aligned} \int _{Q_{k,j}}|f^{(1)}|^2dv_\alpha \le N_0\sum _{(t,h)\in I_{k,j}}\int _{U_{t,h}}|f-P_{t,h}f|^2dv_\alpha . \end{aligned}$$

Dividing both sides by $v_\alpha (Q_{k,j})$ and using (6.2), we find that

$$\begin{aligned} A^2(f^{(1)};Q_{k,j}) \le N_0C_1\sum _{(t,h)\in I_{k,j}}M^2(f;t,h), \end{aligned}$$

proving the first inequality.

Since each $\gamma _{k,j}$ vanishes on $\mathbf{B}\backslash G_{k,j}$, by Lemma 6.1(3) we have $f^{(2)} \in C^\infty (\mathbf{B})$. Moreover, since $P_{k,j}f$ is analytic on $U_{k,j}$, for each $\nu \in \{1,\dots ,n\}$ we have

$$\begin{aligned} \bar{\partial }_\nu f^{(2)} = \sum _{(k,j) \in I}P_{k,j}f\cdot \bar{\partial }_\nu \gamma _{k,j}. \end{aligned}$$

Thus if $\zeta \in G_{k,j}$, then

$$\begin{aligned} (\bar{\partial }_\nu f^{(2)})(\zeta )= & {} \sum _{(t,h)\in I_{k,j}}(P_{t,h}f)(\zeta )(\bar{\partial }_\nu \gamma _{t,h})(\zeta )\\= & {} \sum _{(t,h)\in I_{k,j}}\{(P_{t,h}f)(\zeta ) - (P_{k,j}f)(\zeta )\}(\bar{\partial }_\nu \gamma _{t,h})(\zeta ), \end{aligned}$$

where the second $=$ is due to the fact that $\sum _{(t,h)\in I}\bar{\partial }_\nu \gamma _{t,h} = \bar{\partial }_\nu 1 = 0$. Combining this with Lemma 6.3, we obtain

$$\begin{aligned} \rho (\zeta )|(\bar{\partial }_\nu f^{(2)})(\zeta ) | \le C_{6.3}\sum _{(t,h)\in I_{k,j}}|(P_{t,h}f)(\zeta ) - (P_{k,j}f)(\zeta )| \quad \text {if} \ \ \zeta \in G_{k,j}. \end{aligned}$$

Using the Cauchy-Schwarz inequality, Lemma 6.1(3) and (6.1) again, we have

$$\begin{aligned} \begin{aligned} \int _{Q_{k,j}} |\rho \bar{\partial }_\nu f^{(2)}|^2dv_\alpha&\le N_0C_{6.3}^2\sum _{(t,h)\in I_{k,j}}\int _{Q_{k,j}}|P_{t,h}f - P_{k,j}f|^2dv_\alpha \\&\le N_0C_{6.3}^2\sum _{(t,h)\in I_{k,j}}2\left( \int _{U_{t,h}}|P_{t,h}f - f|^2dv_\alpha \right. \\&\quad \left. + \int _{U_{k,j}}|f - P_{k,j}f|^2dv_\alpha \right) . \end{aligned} \end{aligned}$$

Again, dividing both sides by $v_\alpha (Q_{k,j})$ and using (6.2), we have

$$\begin{aligned} A^2(\rho \bar{\partial }_\nu f^{(2)};Q_{k,j}) \le 2\left( N_0 + N_0^2\right) C_{6.3}^2C_1\sum _{(t,h)\in I_{k,j}}M^2(f;t,h). \end{aligned}$$

Since this holds for every $\nu \in \{1,\dots ,n\}$, we obtain the second inequality. The proof of the third inequality is similar and will be omitted. $\square$

Lemma 6.6

Let $0 < s \le 1$, and suppose that $i \in$ N satisfies the condition $si > n$. Then for any given $\epsilon > 0$, there is an $0< R < \infty$ such that

$$\begin{aligned} \sup _{(k,j)\in I}v_\alpha ^{s/2}(U_{k,j}) \sum \limits _{\mathop {(t,h)\in I}\limits _{\beta (w_{k,j},w_{t,h})\ge R}} \sup _{\zeta \in U_{k,j}}|\psi _{w_{t,h},i}(\zeta )|^s \le \epsilon . \end{aligned}$$

This lemma is in fact a discrete variant of the familiar Forelli-Rudin estimates [12, 17, 18, 21]. The proof will be omitted.

Lemma 6.7

Let $0< p < \infty$. Then for every pair of finite-rank operators A and B,

$$\begin{aligned} \sum _{\nu =1}^\infty \left( s_\nu (AB)\right) ^p \le 2\sum _{\nu =1}^\infty \left( s_\nu (A)\right) ^p\left( s_\nu (B)\right) ^p. \end{aligned}$$

Proof

It is well known that $s_{\mu +\nu -1}(AB) \le s_\mu (A)s_\nu (B)$ for all $\mu ,\nu \in$ N [10, page 30]. In particular, we have $s_{2\nu -1}(AB) \le s_\nu (A)s_\nu (B)$ and $s_{2\nu }(AB) \le s_{\nu +1}(A)s_\nu (B) \le s_\nu (A)s_\nu (B)$ for every $\nu \in$ N. Hence for any $0< p < \infty$, we have

$$\begin{aligned} \left( s_{2\nu -1}(AB)\right) ^p \le \left( s_\nu (A)s_\nu (B)\right) ^p \quad \text {and} \quad \left( s_{2\nu }(AB)\right) ^p \le \left( s_\nu (A)s_\nu (B)\right) ^p \end{aligned}$$

for every $\nu \in$ N. The lemma obviously follows from these inequalities. $\square$

Proposition 6.8

Let $0 < s \le 1$ be given. Then there is a constant $C_{6.8}$ which depends only on n, $\alpha$ and s such that

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le C_{6.8}\Vert |H_f|^s\Vert _\Phi \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$.

Proof

We begin by fixing certain constants. Given $0 < s \le 1$, pick an $i_0 \in$ N such that $si_0 > 4n$. Then set $i = 3i_0+n+1+\delta$, where $\delta \in \mathbf{Z}_+$ satisfies (4.1). Let $\{e_{k,j} : (k,j) \in I\}$ be an orthonormal set. By Lemma 6.1(1) and Corollary 4.5, there is a $C_1$ such that

$$\begin{aligned} \left\| \sum _{(k,j)\in J}\psi _{w_{k,j},i}\otimes e_{k,j}\right\| \le C_1 \end{aligned}$$

(6.3)

for every subset J of I. Also, once this i is so fixed, by Lemmas 3.9 and 3.10, there is a $c > 0$ which depends only on n, $\alpha$ and i such that

$$\begin{aligned} v_\alpha ^{1/2}(U_{k,j})\inf _{\zeta \in U_{k,j}}|\psi _{w_{k,j},i}(\zeta )| \ge c \end{aligned}$$

(6.4)

for every $(k,j) \in I$. For $R > 0$, write

$$\begin{aligned} \epsilon (R) = \sup _{(k,j)\in I} v_\alpha ^{s/2}(U_{k,j})\sum \limits _{\mathop {(t,h)\in I}\limits _{\beta (w_{k,j},w_{t,h})\ge R}} \sup _{\zeta \in U_{k,j}}|\psi _{w_{t,h},i}(\zeta )|^s. \end{aligned}$$

For this i, Lemma 6.6 allows us to pick an $R > 6\tau +7$ such that

$$\begin{aligned} 2\epsilon (R) \le c^s/2, \end{aligned}$$

(6.5)

and this R is so fixed for the rest of the proof.

By Lemmas 6.1(1) and 5.1(a), there is an $M \in$ N such that

$$\begin{aligned} \text {card}\{(t,h) \in I : \beta (w_{k,j},w_{t,h}) < R\} \le M \end{aligned}$$

for every $(k,j) \in I$. By a standard maximality argument, there is a partition $I = E_1\cup \cdots \cup E_M$ such that for every $m \in \{1,\dots ,M\}$, we have $\beta (w_{k,j},w_{t,h})\ge R$ whenever $(k,j), (t,h) \in E_m$ and $(k,j) \ne (t,h)$. We will show that $C_{6.8} = 8M(C_1^s/c^s)$ suffices for the proposition.

Let a symmetric gauge function $\Phi$ be given, and let $\Phi ^*$ be its dual. Fix an $m \in \{1,\dots ,M\}$ for the moment. Given an $f \in L^2(\mathbf{B},dv_\alpha )$, consider any

$$\begin{aligned} J_m \subset \{(k,j) \in E_m : M(f;k,j) \ne 0\} \quad \text {with} \quad \text {card}(J_m) < \infty . \end{aligned}$$

(6.6)

For each $(k,j) \in J_m$, define the unit vector

$$\begin{aligned} g_{k,j} = {\chi _{U_{k,j}}f\psi _{w_{k,j},i} - P_{k,j}(f\psi _{w_{k,j},i})\over \Vert \chi _{U_{k,j}}f\psi _{w_{k,j},i} - P_{k,j}(f\psi _{w_{k,j},i})\Vert } \end{aligned}$$

(6.7)

in $L^2(U_{k,j},dv_\alpha )$. Let $\{b_{k,j} : (k,j) \in J_m\}$ be a family of non-negative numbers. We define the finite-rank operator

$$\begin{aligned} A = \sum _{(k,j)\in J_m}b_{k,j}e_{k,j}\otimes g_{k,j}. \end{aligned}$$

Note that the choice $R > 6\tau +7$ ensures that for $(k,j) \ne (t,h)$ in $E_m$, we have $U_{k,j}\cap U_{t,h} = \emptyset$. Hence $\langle g_{k,j},g_{t,h}\rangle = 0$ for $(k,j) \ne (t,h)$ in $E_m$. Consequently,

$$\begin{aligned} \Phi ^*(\{\left( s_\nu (A)\right) ^s\}_{\nu \in \mathbf{N}}) = \Phi ^*\left( \left\{ b^s_{k,j}\right\} _{(k,j)\in J_m}\right) . \end{aligned}$$

(6.8)

Also, define the operator

$$\begin{aligned} T = \sum _{(k,j)\in J_m}\psi _{w_{k,j},i}\otimes e_{k,j}. \end{aligned}$$

Then $\Vert T\Vert \le C_1$ by (6.3).

By straightforward multiplication,

$$\begin{aligned} AH_fT = \sum _{(k,j),(t,h)\in J_m}b_{k,j}\langle H_f\psi _{w_{t,h},i},g_{k,j}\rangle e_{k,j}\otimes e_{t,h} = Y + Z, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} Y&= \sum _{(k,j)\in J_m}b_{k,j}\langle H_f\psi _{w_{k,j},i},g_{k,j}\rangle e_{k,j}\otimes e_{k,j} \quad \text {and} \\ Z&= \sum \limits _{(k,j)\in J_m}\sum \limits _{\mathop {(t,h)\ne (k,j)}\limits _{(t,h)\in J_m }} b_{k,j}\langle H_f\psi _{w_{t,h},i},g_{k,j}\rangle e_{k,j}\otimes e_{t,h}. \end{aligned} \end{aligned}$$

Since $Y = AH_fT - Z$, an application of Lemma 3.4 to the symmetric gauge function for the trace class ${\mathcal C}_1$ yields

$$\begin{aligned} \Vert |Y|^s\Vert _1 \le 2\Vert |AH_fT|^s\Vert _1 + 2\Vert |Z|^s\Vert _1. \end{aligned}$$

(6.9)

By (6.7) and Lemma 6.4, we have

$$\begin{aligned} \begin{aligned} \langle H_f\psi _{w_{k,j},i},g_{k,j}\rangle&= \Vert \chi _{U_{k,j}}f\psi _{w_{k,j},i} - P_{k,j}(f\psi _{w_{k,j},i})\Vert \\&= \left\| \chi _{U_{k,j}}\psi _{w_{k,j},i}\left( f - \psi _{w_{k,j},i}^{-1}P_{k,j}(f\psi _{w_{k,j},i})\right) \right\| . \end{aligned} \end{aligned}$$

Recalling (6.4), we have

$$\begin{aligned} \langle H_f\psi _{w_{k,j},i},g_{k,j}\rangle\ge & {} c{\left\| \chi _{U_{k,j}}f - \psi _{w_{k,j},i}^{-1}P_{k,j}(f\psi _{w_{k,j},i})\right\| \over v_\alpha ^{1/2}(U_{k,j})} \ge c{\Vert \chi _{U_{k,j}}f - P_{k,j}f\Vert \over v_\alpha ^{1/2}(U_{k,j})} \\= & {} cM(f;k,j), \end{aligned}$$

where the second $\ge$ follows from the facts the that $\psi _{w_{k,j},i}^{-1}P_{k,j}(f\psi _{w_{k,j},i}) \in {\mathcal B}_{k,j}$ and that $P_{k,j}f$ is the element in ${\mathcal B}_{k,j}$ that minimizes the norm $\Vert \chi _{U_{k,j}}f - h\Vert$, $h \in {\mathcal B}_{k,j}$. Thus

$$\begin{aligned} \Vert |Y|^s\Vert _1 = \sum _{(k,j)\in J_m}\{b_{k,j}\langle H_f\psi _{w_{k,j},i},g_{k,j}\rangle \}^s \ge c^s\sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j). \nonumber \\ \end{aligned}$$

(6.10)

On the other hand, since $0 < s \le 1$, the orthonormality of $\{e_{k,j} : (k,j) \in I\}$ leads to

$$\begin{aligned} \Vert |Z|^s\Vert _1 \le \sum _{(k,j),(t,h)\in J_m}|\langle Ze_{t,h},e_{k,j}\rangle |^s = \sum \limits _{(k,j)\in J_m}\sum \limits _{\mathop {(t,h)\ne (k,j)}\limits _{(t,h)\in J_m }} b^s_{k,j}|\langle H_f\psi _{w_{t,h},i},g_{k,j}\rangle |^s. \nonumber \\ \end{aligned}$$

(6.11)

Using Lemma 6.4 and the norm-minimizing property of $P_{k,j}$ again, we have

$$\begin{aligned} \begin{aligned}&|\langle H_f\psi _{w_{t,h},i},g_{k,j}\rangle | \\&\quad = {|\langle \chi _{U_{k,j}}f\psi _{w_{t,h},i} - P_{k,j}(f\psi _{w_{t,h},i}), \chi _{U_{k,j}}f\psi _{w_{k,j},i} - P_{k,j}(f\psi _{w_{k,j},i})\rangle | \over \Vert \chi _{U_{k,j}}f\psi _{w_{k,j},i} - P_{k,j}(f\psi _{w_{k,j},i})\Vert } \\&\quad \le \Vert \chi _{U_{k,j}}f\psi _{w_{t,h},i} - P_{k,j}(f\psi _{w_{t,h},i})\Vert \le \Vert \chi _{U_{k,j}}f\psi _{w_{t,h},i} - \psi _{w_{t,h},i}P_{k,j}f\Vert \\&\quad \le v_\alpha ^{1/2}(U_{k,j})\sup _{\zeta \in U_{k,j}}|\psi _{w_{t,h},i}(\zeta )|M(f;k,j). \end{aligned} \end{aligned}$$

Substituting this in (6.11), since $\beta (w_{k,j},w_{t,h})\ge R$ for $(k,j) \ne (t,h)$ in $E_m$, we obtain

$$\begin{aligned} \begin{aligned} \Vert |Z|^s\Vert _1&\le \sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j) v_\alpha ^{s/2}(U_{k,j})\sum \limits _{\mathop {(t,h)\ne (k,j)}\limits _{(t,h)\in J_m }} \sup \limits _{\zeta \in U_{k,j}}|\psi _{w_{t,h},i}(\zeta )|^s \\&\le \epsilon (R)\sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j). \end{aligned} \end{aligned}$$

Combining this with (6.9) and (6.10), we find that

$$\begin{aligned} c^s\sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j) \le 2\Vert |AH_fT|^s\Vert _1 + 2\epsilon (R)\sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j). \end{aligned}$$

Since $J_m$ is a finite set, the sum $\sum _{(k,j)\in J_m}\cdots$ above is finite. By (6.5), $2\epsilon (R) \le c^s/2$. Thus the obvious cancellation leads to

$$\begin{aligned} (c^s/2)\sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j) \le 2\Vert |AH_fT|^s\Vert _1. \end{aligned}$$

(6.12)

To estimate $\Vert |AH_fT|^s\Vert _1$, we apply Lemma 6.7, which gives us

$$\begin{aligned} \Vert |AH_fT|^s\Vert _1 = \sum _{\nu =1}^\infty \left( s_\nu (AH_fT)\right) ^s \le 2\sum _{\nu =1}^\infty \left( s_\nu (A)\right) ^s\left( s_\nu (H_fT)\right) ^s. \end{aligned}$$

Applying (3.1) and (6.8) to the right-hand side, we obtain

$$\begin{aligned} \begin{aligned} \Vert |AH_fT|^s\Vert _1&\le 2\Phi ^*(\{\left( s_\nu (A)\right) ^s\}_{\nu \in \mathbf{N}})\Phi (\{\left( s_\nu (H_fT)\right) ^s\}_{\nu \in \mathbf{N}}) \\&= 2\Phi ^*\left( \left\{ b^s_{k,j}\right\} _{(k,j)\in J_m}\right) \Vert |H_fT|^s\Vert _\Phi \\&\le 2C_1^s\Phi ^*\left( \left\{ b^s_{k,j}\right\} _{(k,j)\in J_m}\right) \Vert |H_f|^s\Vert _\Phi , \end{aligned} \end{aligned}$$

where the second $\le$ follows from Lemma 3.3 and the fact that $\Vert T\Vert \le C_1$. Substituting this in (6.12) and simplifying, we find that

$$\begin{aligned} \sum _{(k,j)\in J_m}b^s_{k,j}M^s(f;k,j) \le 8(C_1^s/c^s)\Phi ^*\left( \left\{ b^s_{k,j}\right\} _{(k,j)\in J_m}\right) \Vert |H_f|^s\Vert _\Phi . \end{aligned}$$

Since the non-negative numbers $\{b^s_{k,j} : (k,j) \in J_m\}$ are arbitrary, the duality between $\Phi$ and $\Phi ^*$ (see (3.1)) implies

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in J_m}) \le 8\left( C_1^s/c^s\right) \Vert |H_f|^s\Vert _\Phi . \end{aligned}$$

Since the above holds for every $J_m$ given by (6.6), recalling (1.3), we conclude that

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in E_m}) \le 8\left( C_1^s/c^s\right) \Vert |H_f|^s\Vert _\Phi . \end{aligned}$$

Finally, since this holds for every $m \in \{1,\dots ,M\}$ and since $I = E_1\cup \dots \cup E_M$, we have

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le \sum _{m=1}^M\Phi (\{M^s(f;k,j)\}_{(k,j)\in E_m}) \le 8M\left( C_1^s/c^s\right) \Vert |H_f|^s\Vert _\Phi . \end{aligned}$$

This completes the proof. $\square$

Lemma 6.9

There is a constant $C_{6.9}$ such that

$$\begin{aligned} \Phi \left( \left\{ \sum _{(t,h)\in I_{k,j}}a_{t,h}\right\} _{(k,j)\in I}\right) \le C_{6.9}\Phi (\{a_{k,j}\}_{(k,j)\in I}) \end{aligned}$$

for every set of non-negative numbers $\{a_{k,j}\}_{(k,j)\in I}$ and every symmetric gauge function $\Phi$.

Proof

First of all, by Lemmas 6.1(1) and 5.1(a), there is an $N_1 \in$ N such that

$$\begin{aligned} \text {card}\{(t,h) \in I : \beta (w_{k,j},w_{t,h}) < 4\tau + 4\} \le N_1 \end{aligned}$$

(6.13)

for every $(k,j) \in I$. Let non-negative numbers $\{a_{k,j}\}_{(k,j)\in I}$ be given. For every $(k,j) \in I$, there is a $\pi (k,j) \in I_{k,j}$ such that $a_{\pi (k,j)} \ge a_{t,h}$ for every $(t,h) \in I_{k,j}$. Thus $\sum _{t,h}a_{t,h} \le \text {card}(I_{k,j})a_{\pi (k,j)} \le N_0a_{\pi (k,j)}$, where the second $\le$ follows from Lemma 6.1(3). Hence

$$\begin{aligned} \Phi \left( \left\{ \sum _{(t,h)\in I_{k,j}}a_{t,h}\right\} _{(k,j)\in I}\right) \le N_0\Phi (\{a_{\pi (k,j)}\}_{(k,j)\in I}). \end{aligned}$$

(6.14)

Obviously, $\beta (w_{k,j},w_{\pi (k,j)}) < 2\tau + 2$ for every $(k,j) \in I$. Thus for any pair of $(k,j), (k',j') \in I$, if $\pi (k,j) = \pi (k',j')$, then $\beta (w_{k,j},w_{k',j'}) < 4\tau + 4$ by the triangle inequality. By (6.13), the map $\pi : I \mapsto I$ is at most $N_1$-to-1. Applying Lemma 3.1, we obtain $\Phi (\{a_{\pi (k,j)}\}_{(k,j)\in I}) \le N_1\Phi (\{a_{k,j}\}_{(k,j)\in I})$. Recalling (6.14), the lemma holds for the constant $C_{6.9} = N_0N_1$. $\square$

Proposition 6.10

Let $0 < s \le 1$ be given, and let $i \in \mathbf{Z}_+$ satisfy the condition $s(n+1+\alpha +2i) > 2n$. Let $a > 0$ also be given. Then there is a constant $C_{6.10}$ which depends only on n, $\alpha$, s, i and a such that

$$\begin{aligned} \Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C_{6.10}\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$, every symmetric gauge function $\Phi$, and every a-separated set $\Gamma$ in B.

Proof

Given any $f \in L^2(\mathbf{B},dv_\alpha )$, let $f = f^{(1)} + f^{(2)}$ be the decomposition provided by Proposition 6.5. Applying Proposition 2.2 to $f^{(2)}\psi _{z,i} - P( f^{(2)}\psi _{z,i})$, $z \in$ B, we have

$$\begin{aligned} \begin{aligned} \Vert H_f\psi _{z,i}\Vert&\le \Vert H_{f^{(1)}}\psi _{z,i}\Vert + \Vert H_{f^{(2)}}\psi _{z,i}\Vert \le \Vert f^{(1)}\psi _{z,i}\Vert + \Vert H_{f^{(2)}}\psi _{z,i}\Vert \\&\le \Vert f^{(1)}\psi _{z,i}\Vert + C_{2.2}\Vert \rho \bar{\partial }(f^{(2)}\psi _{z,i})\Vert + C_{2.2}\Vert \rho ^{1/2}\bar{\partial }(f^{(2)}\psi _{z,i})\wedge \bar{\partial }\rho \Vert \\&= \Vert f^{(1)}\psi _{z,i}\Vert + C_{2.2}\Vert \rho \psi _{z,i}\bar{\partial }f^{(2)}\Vert + C_{2.2}\Vert \rho ^{1/2}\psi _{z,i}\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho \Vert . \end{aligned} \end{aligned}$$

For $0 < s \le 1$, the above implies

$$\begin{aligned} \Vert H_f\psi _{z,i}\Vert ^s \le \Vert f^{(1)}\psi _{z,i}\Vert ^s + C_{2.2}^s\Vert \rho \psi _{z,i}\bar{\partial }f^{(2)}\Vert ^s + C_{2.2}^s\Vert \rho ^{1/2}\psi _{z,i}\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho \Vert ^s. \end{aligned}$$

Thus it suffices to find a C that depends only on n, $\alpha$, s, i and a such that

$$\begin{aligned} \begin{aligned} \Phi (\{\Vert f^{(1)}\psi _{z,i}\Vert ^s\}_{z\in \Gamma })&\le C\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}), \\ \Phi (\{\Vert \rho \psi _{z,i}\bar{\partial }f^{(2)}\Vert ^s\}_{z\in \Gamma })&\le C\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \quad \text {and} \\ \Phi (\{\Vert \rho ^{1/2}\psi _{z,i}\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho \Vert ^s\}_{z\in \Gamma })&\le C\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \end{aligned} \end{aligned}$$

for every symmetric gauge function $\Phi$ and every a-separated set $\Gamma$ in B.

Since $s(n+1+\alpha +2i) > 2n$ and $\Gamma$ is a-separated, by Propositions 3.7 and 6.5,

$$\begin{aligned} \begin{aligned} \Phi (\{\Vert f^{(1)}\psi _{z,i}\Vert ^s\}_{z\in \Gamma })&\le C_{3.7}\Phi (\{A^s(f^{(1)};Q_{k,j})\}_{(k,j)\in I}) \\&\le C_{3.7}C_{6.5}^{s/2}\Phi \left( \left\{ \sum _{(t,h)\in I_{k,j}}M^s(f;t,h)\right\} _{(k,j)\in I}\right) . \end{aligned} \end{aligned}$$

Applying Lemma 6.9, we obtain

$$\begin{aligned} \Phi (\{\Vert f^{(1)}\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C_{3.7}C_{6.5}^{s/2}C_{6.9}\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}). \end{aligned}$$

That is, the first inequality holds for the constant $C = C_{3.7}C_{6.5}^{s/2}C_{6.9}$. By the same argument, the other two inequalities also hold for the same C. $\square$

Lemma 6.11

Let $i \in \mathbf{Z}_+$ and $b > 0$ be given. Then there is a constant $C_{6.11}$ which depends only on n, $\alpha$, i and b such that

$$\begin{aligned} M(f;k,j) \le C_{6.11} \Vert H_f\psi _{z,i}\Vert \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every pair of $(k,j) \in I$ and $z \in$ B satisfying the condition $\beta (w_{k,j},z) < b$.

Proof

Let $b > 0$ be given. By Lemma 3.9, there is a $C_1$ such that

$$\begin{aligned} v_\alpha (D(w,2b+3\tau + 3)) \le C_1v_\alpha (D(w,3\tau +3)) \quad \text {for every} \ \ w \in \mathbf{B}. \end{aligned}$$

(6.15)

Let $i \in \mathbf{Z}_+$. By Lemmas 3.9 and 3.10, there is a $c_0 > 0$ such that for every $z \in$ B,

$$\begin{aligned} |\psi _{z,i}(\zeta )| \ge c_0v_\alpha ^{-1/2}(D(z,b+3\tau + 3)) \quad \text {whenever} \ \ \zeta \in D(z,b+3\tau + 3). \nonumber \\ \end{aligned}$$

(6.16)

Let $(k,j) \in I$ and $z \in$ B be such that $\beta (w_{k,j},z) < b$. Then $D(z,b+3\tau + 3) \subset D(w_{k,j},2b+3\tau + 3)$. By (6.15), we have $v_\alpha (D(z,b+3\tau + 3)) \le C_1v_\alpha (D(w_{k,j},3\tau +3))$, and consequently

$$\begin{aligned} v^{-1/2}_\alpha (D(z,b+3\tau + 3)) \ge C_1^{-1/2}v^{-1/2}_\alpha (D(w_{k,j},3\tau +3)) = C_1^{-1/2}v^{-1/2}_\alpha (U_{k,j}). \nonumber \\ \end{aligned}$$

(6.17)

Since $U_{k,j} = D(w_{k,j},3\tau + 3)$, we have $U_{k,j} \subset D(z,b+3\tau + 3)$. Writing $c_1 = c_0C^{-1/2}_1$, from (6.16) and (6.17) we obtain

$$\begin{aligned} |\psi _{z,i}(\zeta )| \ge c_1v^{-1/2}_\alpha (U_{k,j}) \quad \text {for every} \ \ \zeta \in U_{k,j}. \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} \Vert H_f\psi _{z,i}\Vert&= \Vert f\psi _{z,i} - P(f\psi _{z,i})\Vert \ge \Vert \chi _{U_{k,j}}\psi _{z,i}(f - \psi ^{-1}_{z,i}P(f\psi _{z,i}))\Vert \\&\ge c_1v^{-1/2}_\alpha (U_{k,j})\Vert \chi _{U_{k,j}}f - \chi _{U_{k,j}}\psi ^{-1}_{z,i}P(f\psi _{z,i})\Vert \\&\ge c_1v^{-1/2}_\alpha (U_{k,j})\Vert \chi _{U_{k,j}}f - P_{k,j}f\Vert = c_1M(f;k,j), \end{aligned} \end{aligned}$$

where the last $\ge$ again follows from the norm-minimizing property of $P_{k,j}f$. $\square$

Proposition 6.12

Let $i \in \mathbf{Z}_+$ and $b > 0$ be given. Then there is a constant $C_{6.12}$ which depends only on n, $\alpha$, i and b such that

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le C_{6.12} \Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$, every $0 < s \le 1$, every symmetric gauge function $\Phi$, and every countable subset $\Gamma$ of B with the property $\cup _{z\in \Gamma }D(z,b) =$ B.

Proof

Let $b > 0$ be given. Then by Lemmas 6.1 and 5.1, there is an $N \in$ N such that

$$\begin{aligned} \text {card}\{(k',j') \in I : \beta (w_{k,j},w_{k',j'}) < 2b\} \le N \quad \text {for every} \ \ (k,j) \in I. \end{aligned}$$

(6.18)

Let $\Gamma$ be a countable subset of B with the property $\cup _{z\in \Gamma }D(z,b) =$ B. Then for every $(k,j) \in I$, there is a $z_{k,j} \in \Gamma$ such that $\beta (w_{k,j},z_{k,j})$ < b. Let $i \in \mathbf{Z}_+$ also be given. By Lemma 6.11, we have

$$\begin{aligned} M(f;k,j) \le C_{6.11} \Vert H_f\psi _{z_{k,j},i}\Vert \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every $(k,j) \in I$, where $C_{6.11}$ depends only on n, $\alpha$, i and b. Hence for every $0 < s \le 1$ and every symmetric gauge function $\Phi$ we have

$$\begin{aligned} \Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le \max \{C_{6.11},1\}\Phi (\{\Vert H_f\psi _{z_{k,j},i}\Vert ^s\}_{(k,j) \in I}). \end{aligned}$$

(6.19)

If $(k,j), (k',j') \in I$ are such that $z_{k,j} = z_{k',j'}$, then

$$\begin{aligned} \beta (w_{k,j},w_{k',j'}) \le \beta (w_{k,j},z_{k,j}) + \beta (z_{k,j},w_{k',j'})= & {} \beta (w_{k,j},z_{k,j}) + \beta (z_{k',j'},w_{k',j'}) \\< & {} 2b. \end{aligned}$$

Thus, by (6.18), the map $(k,j) \mapsto z_{k,j}$ is at most N-to-1. Applying Lemma 3.1, we have

$$\begin{aligned} \Phi (\{\Vert H_f\psi _{z_{k,j},i}\Vert ^s\}_{(k,j) \in I}) \le N\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }). \end{aligned}$$

The combination of this with (6.19) proves the proposition. $\square$

7 Proof of Theorem 1.2

We need one more proposition for the proof of the upper bound in Theorem 1.2.

Proposition 7.1

Set $C_{7.1} = 2(1+\sqrt{2}C_{2.2})$, where $C_{2.2}$ is the constant in Proposition 2.2. Then for every $f \in C^\infty (\mathbf{B})\cap L^2(\mathbf{B},dv_\alpha )$, every $0 < s \le 1$ and every symmetric symmetric gauge function $\Phi$ we have

$$\begin{aligned} \Vert |H_f|^s\Vert _\Phi \le C_{7.1}(\Vert |M_{\rho |\bar{\partial }f|}P|^s\Vert _\Phi + \Vert |M_{\rho ^{1/2}|\bar{\partial }f\wedge \bar{\partial }\rho |}P|^s\Vert _\Phi ). \end{aligned}$$

(7.1)

Proof

Given f, s and $\Phi$ as above, it suffices to consider the case where the right-hand side of (7.1) is finite, for otherwise the inequality holds trivially. This finiteness implies that every $M_{\rho \bar{\partial }_if}P$ and every $M_{\rho ^{1/2}\bar{L}_{i,j}f}P$ is a bounded operator on $L^2(\mathbf{B},dv_\alpha )$. Let ${\mathcal H}$ be the orthogonal sum of $n + (1/2)n(n-1)$ copies of $L^2(\mathbf{B},dv_\alpha )$. We now define an operator

$$\begin{aligned} X : L_a^2(\mathbf{B},dv_\alpha ) \rightarrow {\mathcal H} \end{aligned}$$

as follows: for each $h \in L_a^2(\mathbf{B},dv_\alpha )$, the first n components of Xh are $(\rho \bar{\partial }_1f)h, \dots ,(\rho \bar{\partial }_nf)h$, while the other $(1/2)n(n-1)$ components of Xh are $(\rho ^{1/2}\bar{L}_{i,j}f)h$, arranged according to a fixed enumeration of the pairs $i < j$ in $\{1,\dots ,n\}$. Then obviously we have

$$\begin{aligned} \Vert Xh\Vert ^2 = \langle X^*Xh,h\rangle = \Vert M_{\rho |\bar{\partial }f|}h\Vert ^2 + \Vert M_{\rho ^{1/2}|\bar{\partial }f\wedge \bar{\partial }\rho |}h\Vert ^2, \end{aligned}$$

$h \in L_a^2(\mathbf{B},dv_\alpha )$. For $h \in L_a^2(\mathbf{B},dv_\alpha )$, its analyticity leads to $\bar{\partial }(fh) = h\bar{\partial }f$. Hence

$$\begin{aligned} \Vert Xh\Vert ^2 = \Vert \rho \bar{\partial }(fh)\Vert ^2 + \Vert \rho ^{1/2}\bar{\partial }(fh)\wedge \bar{\partial }\rho \Vert ^2 \ge {1\over 2}(\Vert \rho \bar{\partial }(fh)\Vert {+} \Vert \rho ^{1/2}\bar{\partial }(fh)\wedge \bar{\partial }\rho \Vert )^2 \end{aligned}$$

for every $h \in L_a^2(\mathbf{B},dv_\alpha )$. Applying Proposition 2.2, for every $g \in H^\infty (\mathbf{B})$ we have

$$\begin{aligned} \Vert H_fg\Vert = \Vert fg - P(fg)\Vert \le C_{2.2}(\Vert \rho \bar{\partial }(fg)\Vert + \Vert \rho ^{1/2}\bar{\partial }(fg)\wedge \bar{\partial }\rho \Vert ) \le \sqrt{2}C_{2.2}\Vert Xg\Vert . \end{aligned}$$

For $h \in L_a^2(\mathbf{B},dv_\alpha )$ and $0< r < 1$, the function $h_r$ defined by the formula $h_r(z) = h(rz)$ belongs to $H^\infty (\mathbf{B})$. Thus an obvious application of Fatou’s lemma in the above gives us

$$\begin{aligned} \Vert H_fh\Vert = \Vert fh - P(fh)\Vert \le \sqrt{2}C_{2.2}\Vert Xh\Vert \quad \text {for every} \ \ h \in L_a^2(\mathbf{B},dv_\alpha ). \end{aligned}$$

By Lemma 3.12, there is an operator $T : {\mathcal H} \rightarrow L^2(\mathbf{B},dv_\alpha )$ with $\Vert T\Vert \le \sqrt{2}C_{2.2}$ such that

$$\begin{aligned} H_f = TX. \end{aligned}$$

Thus it follows from Lemma 3.3 that

$$\begin{aligned} \Vert |H_f|^s\Vert _\Phi \le \Vert T\Vert ^s\Vert |X|^s\Vert _\Phi \le (\sqrt{2}C_{2.2})^s\Vert |X|^s\Vert _\Phi \le (1 + \sqrt{2}C_{2.2})\Vert |X|^s\Vert _\Phi . \nonumber \\ \end{aligned}$$

(7.2)

To estimate $\Vert |X|^s\Vert _\Phi$, write $F = \rho |\bar{\partial }f|$ and $G = \rho ^{1/2}|\bar{\partial }f\wedge \bar{\partial }\rho |$. Then note that

$$\begin{aligned} X^*X = PM_{F^2}P + PM_{G^2}P = (M_FP)^*M_FP + (M_GP)^*M_GP. \end{aligned}$$

By Lemma 3.4 and Remark 3.5, we have

$$\begin{aligned} \begin{aligned} \Vert |X|^s\Vert _\Phi&= \Vert (X^*X)^{s/2}\Vert _\Phi \le 2\Vert ((M_FP)^*M_FP)^{s/2}\Vert _\Phi + 2\Vert ((M_GP)^*M_GP)^{s/2}\Vert _\Phi \\&= 2\Vert |M_FP|^s\Vert _\Phi + 2\Vert |M_GP|^s\Vert _\Phi \\&= 2(\Vert |M_{\rho |\bar{\partial }f|}P|^s\Vert _\Phi + \Vert |M_{\rho ^{1/2}|\bar{\partial }f\wedge \bar{\partial }\rho |}P|^s\Vert _\Phi ). \end{aligned} \end{aligned}$$

Combining this with (7.2), the proposition follows. $\square$

At this point, we are finally ready to assemble the previous steps and present

Proof of Theorem 1.2

Let s, i, $\Gamma$, f and $\Phi$ be given as in the statement of the theorem. Applying Propositions 6.10 and 6.8, we obtain

$$\begin{aligned} \Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le C_{6.10}\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \le C_{6.10}C_{6.8}\Vert |H_f|^s\Vert _\Phi , \end{aligned}$$

which establishes the lower bound in Theorem 1.2.

To prove the upper bound, let $f = f^{(1)} + f^{(2)}$ be the decomposition provided by Proposition 6.5. Then by Lemma 3.4 and Remark 3.5, we have

$$\begin{aligned} \Vert |H_f|^s\Vert _\Phi \le 2^{1-s}(\Vert |H_{f^{(1)}}|^s\Vert _\Phi + \Vert |H_{f^{(2)}}|^s\Vert _\Phi ). \end{aligned}$$

(7.3)

Since $H_{f^{(1)}} = (1-P)M_{f^{(1)}}P$, it follows from Lemma 3.3 and Proposition 5.5 that

$$\begin{aligned} \Vert |H_{f^{(1)}}|^s\Vert _\Phi \le \Vert |M_{f^{(1)}}P|^s\Vert _\Phi \le C_{5.5}\Phi (\{A^s(f^{(1)};Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

(7.4)

Since $0< s/2 < 1$, it follows from Propositions 6.5 that

$$\begin{aligned} A^s(f^{(1)};Q_{k,j}) \le C_{6.5}^{s/2}\sum _{(t,h)\in I_{k,j}}M^s(f;t,h) \end{aligned}$$

for every $(k,j) \in I$. Substituting this in (7.4) and then applying Lemma 6.9 and Proposition 6.12, we obtain

$$\begin{aligned} \begin{aligned} \Vert |H_{f^{(1)}}|^s\Vert _\Phi \le \Vert |M_{f^{(1)}}P|^s\Vert _\Phi&\le C_{5.5}C_{6.5}^{s/2}C_{6.9}\Phi (\{M^s(f;k,j)\}_{(k,j)\in I}) \\&\le C_{5.5}C_{6.5}^{s/2}C_{6.9}C_{6.12}\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }). \end{aligned} \end{aligned}$$

(7.5)

To bound $\Vert |H_{f^{(2)}}|^s\Vert _\Phi$, we first apply Proposition 7.1, which gives us

$$\begin{aligned} \Vert |H_{f^{(2)}}|^s\Vert _\Phi \le C_{7.1}(\Vert |M_{\rho |\bar{\partial }f^{(2)}|}P|^s\Vert _\Phi + \Vert |M_{\rho ^{1/2}|\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho |}P|^s\Vert _\Phi ). \end{aligned}$$

Then, applying Propositions 5.5 and 6.5, Lemma 6.9 and Proposition 6.12 in the same manner as above, we obtain

$$\begin{aligned} \begin{aligned} \Vert |M_{\rho |\bar{\partial }f^{(2)}|}P|^s\Vert _\Phi&\le C_{5.5}C_{6.5}^{s/2}C_{6.9}C_{6.12}\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \quad \text {and} \\\ \Vert |M_{\rho ^{1/2}|\bar{\partial }f^{(2)}\wedge \bar{\partial }\rho |}P|^s\Vert _\Phi&\le C_{5.5}C_{6.5}^{s/2}C_{6.9}C_{6.12}\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }). \end{aligned} \end{aligned}$$

That is,

$$\begin{aligned} \Vert |H_{f^{(2)}}|^s\Vert _\Phi \le 2C_{7.1}C_{5.5}C_{6.5}^{s/2}C_{6.9}C_{6.12}\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }). \end{aligned}$$

Finally, combining this with (7.5) and (7.3), we find that

$$\begin{aligned} \Vert |H_f|^s\Vert _\Phi \le 2^{1-s}(1 + 2C_{7.1})C_{5.5}C_{6.5}^{s/2}C_{6.9}C_{6.12}\Phi (\{\Vert H_f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }). \end{aligned}$$

This proves the upper bound in Theorem 1.2 and completes the proof. $\square$

Appendix

We present the proof of Prop. 3.7 in this Appendix. Note that the proofs of the lemmas are elementary hence the proofs will be omitted.

For each $(k,j) \in I$, we define the subset

$$\begin{aligned} F_{k,j} = \{(\ell ,i) : \ell > k, 1 \le i \le m(\ell ), B(u_{\ell ,i},2^{-\ell })\cap B(u_{k,j},3\cdot 2^{-k}) \ne \emptyset \} \end{aligned}$$

of I. We then define

$$\begin{aligned} W_{k,j} = Q_{k,j}\cup \{\cup _{(\ell ,i)\in F_{k,j}}Q_{\ell ,i}\}, \end{aligned}$$

$(k,j) \in I$. By (3.6) and (3.7), we have $W_{k,j} \supset \{ru : 1 - 2^{-2k} \le r < 1, u \in B(u_{k,j},3\cdot 2^{-k})\}$.

Lemma A.1

There is a constant $C_{\text {A}.1}$ which depends only on n and $\alpha$ such that

$$\begin{aligned} \Phi (\{A^s(f;W_{k,j})\}_{(k,j)\in I}) \le {C_{\text {A}.1}\over 1 - 2^{-(1+\alpha )s}}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$, every symmetric gauge function $\Phi$, and every $0 < s \le 1$.

As in [20], for each $(k,j) \in I$ we define

$$\begin{aligned} H_{k,j} = \{(t,h) \in I : 0 \le t \le k, 1 \le h \le m(t), B(u_{t,h},2^{-t})\cap B(u_{k,j},2^{-k}) \ne \emptyset \}. \end{aligned}$$

Lemma A.2

Given any $i \in \mathbf{Z}_+$, there is a constant $C_{\text {A}.2}$ which depends only on n, $\alpha$ and i such that the following estimate holds: Let $(k,j) \in I$ and $z \in T_{k,j}$. Then there exist $(\ell ,\nu (\ell )) \in H_{k,j}$ for $\ell = 0,\dots ,k$ such that for every $f \in L^2(\mathbf{B},dv_\alpha )$, we have

$$\begin{aligned} \Vert f\psi _{z,i}\Vert \le C_{\text {A}.2}\sum ^k_{\ell =0}2^{-(n+1+\alpha +2i)(k-\ell )}A(f;W_{\ell ,\nu (\ell )}). \end{aligned}$$

Lemma A.3

Let $0 < s \le 1$ be given, and let $i \in \mathbf{Z}_+$ satisfy the condition $s(n+1+\alpha +2i) > 2n$. Then there exists a constant $0< C_{\text {A}.3} < \infty$ which depends only on n, $\alpha$, s and i such that the following estimate holds: Let $z(k,j) \in T_{k,j}$ for each $(k,j) \in I$. Then for each $f \in L^2(\mathbf{B},dv_\alpha )$ and each symmetric gauge function $\Phi$, we have

$$\begin{aligned} \Phi (\{\Vert f\psi _{z(k,j),i}\Vert ^s\}_{(k,j)\in I}) \le C_{\text {A}.3}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}). \end{aligned}$$

Proof of Proposition 3.7

Let $0 < s \le 1$, and let $i \in \mathbf{Z}_+$ satisfy the condition $s(n+1+\alpha +2i) > 2n$. Given $0< a < \infty$, let K be the natural number provided by Lemma 3.6. According to that lemma, each a-separated set $\Gamma$ is the union of pairwise disjoint subsets $\Gamma _1, \dots , \Gamma _K$ such that $\text {card}(\Gamma _\mu \cap T_{k,j}) \le 1$ for all $\mu \in \{1, \dots , K\}$ and $(k,j) \in I$. Thus for each $\mu \in \{1, \dots , K\}$, it follows from Lemma A.3 that

$$\begin{aligned} \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma _\mu }) \le C_{\text {A}.3}\Phi (\{A^s(f;Q_{k,j})\}_{(k,j)\in I}) \end{aligned}$$

for every $f \in L^2(\mathbf{B},dv_\alpha )$ and every symmetric gauge function $\Phi$. Since $\Gamma _1\cup \cdots \cup \Gamma _K = \Gamma$, we have

$$\begin{aligned} \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma }) \le \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma _1}) + \cdots + \Phi (\{\Vert f\psi _{z,i}\Vert ^s\}_{z\in \Gamma _K}). \end{aligned}$$

Hence Proposition 3.7 holds for the constant $C_{3.7} = KC_{\text {A}.3}$. $\square$