# Interpolating sequences for some subsets of analytic Besov type spaces

Ruishen Qian and Fangqin Ye  Ruishen Qian
Lingnan Normal University
Zhanjiang 524048, Guangdong, China<EMAIL_ADDRESS>Fangqin Ye
Shantou University
Shantou 515063, Guangdong, China<EMAIL_ADDRESS>
###### Abstract.

Let $B_{p}(s)$ be an analytic Besov type space. Let $M(B_{p}(s))$ be the class
of multipliers of $B_{p}(s)$ and let $F(p,p-2,s)$ be the Möbius invariant
subspace generated by $B_{p}(s)$. In this paper, when $0<s<1$ and
$\max\\{s,1-s\\}<p\leq 1$, we give a completed description of interpolating
sequences for $M(B_{p}(s))$ and $F(p,p-2,s)\cap H^{\infty}$. We also consider
certain condition appeared in this description by an $L^{p}$ characterization
and the closure of $F(p,p-2,s)$ in the Bloch space.

###### Key words and phrases:

Interpolating sequence, multiplier, Besov type space, $F(p,p-2,s)$ space.

###### 2010 Mathematics Subject Classification:

30E05, 30H05, 30H25

R. Qian was supported by NNSF of China (No. 11801250 and No. 11871257),
Overseas Scholarship Program for Elite Young and Middle-aged Teachers of
Lingnan Normal University, the Key Subject Program of Lingnan Normal
University (No. 1171518004 and No. LZ1905), and the Department of Education of
Guangdong Province (No. 2018KTSCX133). F. Ye was supported by NNSF of China
(No. 12001352) and Guangdong Basic and Applied Basic Research Foundation (No.
2019A1515110178).

##  1\. Introduction

A classical topic in complex analysis is to characterize interpolating
sequences for some spaces of bounded analytic functions. Let $H^{\infty}$ be
the space of bounded analytic functions on the open unit disk $\mathbb{D}$ of
the complex plane $\mathbb{C}$. For $X\subseteq H^{\infty}$, a sequence
$\\{z_{n}\\}$ in $\mathbb{D}$ is said to be an interpolating sequence for $X$
if for each bounded sequence $\\{\zeta_{n}\\}$ of complex values, there exists
a function $f\in X$ such that $f(z_{n})=\zeta_{n}$ for every $n$. A notable
result of L. Carleson [11] stated that $\\{z_{n}\\}$ is an interpolating
sequence for $H^{\infty}$ if and only if $\\{z_{n}\\}$ is uniformly separated,
namely there exists a $\delta>0$ with

$\inf_{m}\prod_{n\neq
m}\left|\frac{z_{m}-z_{n}}{1-\overline{z_{m}}z_{n}}\right|\geq\delta.$

We refer to [12, 15] for more results related to this topic.

This paper is devoted to consider interpolating sequences for certain spaces
of bounded analytic functions related to analytic Besov type spaces. Next we
recall some definitions and notations.

For an arc $I$ of the unit circle $\mathbb{T}$, the Carleson box $S(I)$ is

$S(I)=\\{r\zeta\in\mathbb{D}:1-|I|<r<1,\ \zeta\in I\\},$

where $|I|$ is the normalized arclength of $I$. For $0<s<\infty$, a
nonnegative Borel measure $\mu$ on $\mathbb{D}$ is said to be an $s$-Carleson
measure if

$\sup_{I\subseteq\mathbb{T}}\frac{\mu(S(I))}{|I|^{s}}<\infty.$

For $s=1$, we get the classical Carleson measure. Also, if $\mu$ is a
$1$-Carleson measure, we write that $\mu$ is a Carleson measure.

Recall that for $a$ and $z$ in $\mathbb{D}$,
$\rho(a,z)=|a-z|/|1-\overline{a}z|$ is the pseudo-hyperbolic metric between
$a$ and $z$. A sequence $\\{z_{n}\\}_{n=1}^{\infty}$ in $\mathbb{D}$ is called
separated if

$\inf_{n\not=k}\rho(z_{n},z_{k})>0.$

It is well known (cf. [18]) that $\\{z_{n}\\}_{n=1}^{\infty}$ in $\mathbb{D}$
is an interpolating sequence for $H^{\infty}$ if and only if
$\\{z_{n}\\}_{n=1}^{\infty}$ is separated and
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson measure.

Denote by $H(\mathbb{D})$ the space of analytic functions in $\mathbb{D}$. For
$0<p<\infty$ and $s>1-p$, let $B_{p}(s)$ be the Besov type space consisting of
functions $f\in H(\mathbb{D})$ with

$\|f\|_{B_{p}(s)}=|f(0)|+\left(\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}dA(z)\right)^{1/p}<\infty,$

where $dA$ is the normalized area measure in $\mathbb{D}$. It is easy to check
that when $s\leq 1-p$, the integral above is finite only if $f$ is constant.
If $s>1$, then $B_{p}(s)$ is the Bergman space $A^{p}_{s-2}$ (see [37]). Besov
type spaces $B_{p}(s)$ have been studied extensively. N. Arcozzi, D. Blasi and
J. Pau [2] characterized universal interpolating sequences for $B_{p}(s)$
spaces. D. Blasi and J. Pau [10] studied Hankel-type operators associated with
$B_{p}(s)$ spaces. See [13, 30] for more results of $B_{p}(s)$ spaces.

Denote by $M(B_{p}(s))$ the class of multipliers of the space $B_{p}(s)$.
Namely

$M(B_{p}(s))=\\{f\in H(\mathbb{D}):fg\in B_{p}(s)\ \ \text{for all}\ \ g\in
B_{p}(s)\\}.$

A nonnegative Borel measure $\mu$ on $\mathbb{D}$ is called a Carleson measure
for $B_{p}(s)$ if there is a positive constant $C$ such that

$\int_{\mathbb{D}}|f(z)|^{p}d\mu(z)\leq C\|f\|_{B_{p}(s)}^{p}$

for all $f\in B_{p}(s)$.

Next we recall the Möbius invariant function space generated by $B_{p}(s)$.
For $a\in\mathbb{D}$, let

$\sigma_{a}(z)=\frac{a-z}{1-\overline{a}z},\qquad z\in\mathbb{D}.$

Namely $\sigma_{a}$ is a Möbius transformation of $\mathbb{D}$ interchanging
the points $0$ and $a$. Recall that

$\text{Aut}(\mathbb{D})=\\{e^{i\theta}\sigma_{a}:\ \ a\in\mathbb{D}\
\text{and}\ \theta\ \ \text{is real}\\}$

is the group of Möbius maps of $\mathbb{D}$. For $0<p<\infty$ and
$0<s<\infty$, the space $F(p,p-2,s)$ consists of those functions $f\in
H(\mathbb{D})$ such that

$\|f\|_{F(p,p-2,s)}^{p}=\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)<\infty.$

Namely, for $f\in F(p,p-2,s)$ if and only if
$|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}$ is an $s$-Carleson measure. It is
also known that $F(p,p-2,s)$ contains only constant functions if $p+s\leq 1$.
Hence, to study $F(p,p-2,s)$ spaces, we let $p+s>1$. $F(p,p-2,s)$ spaces and a
general family of analytic function spaces were introduced by R. Zhao [35] and
further investigated in [29]. If $p=2$ and $s=1$, then $F(p,p-2,s)$ is $BMOA$,
the space of analytic functions whose boundary values having bounded mean
oscillation on the unit circle. If $s>1$, all nontrivial $F(p,p-2,s)$ spaces
are the same and equal to the Bloch space $\mathcal{B}$ consisting of those
functions $f\in H(\mathbb{D})$ with

$\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

$F(p,p-2,s)$ spaces are Möbius invariant in the sense that

$\|f\circ\phi-f(\phi(0))\|_{F(p,p-2,s)}=\|f\|_{F(p,p-2,s)}$

for every $f\in F(p,p-2,s)$ and $\phi\in\text{Aut}(\mathbb{D})$. Indeed,
$F(p,p-2,s)$ is equal to the Möbius invariant subspace generated by
$B_{p}(s)$; that is, $F(p,p-2,s)$ is the space of those functions $f\in
H(\mathbb{D})$ with

$\sup_{\phi\in\text{Aut}(\mathbb{D})}\left\|f\circ\phi-f(\phi(0))\right\|_{B^{p}_{s}}<\infty.$

See [3] for the general theory of Möbius invariant function spaces. If $p=2$,
then $F(p,p-2,s)$ is the well-known Möbius invariant space $\mathcal{Q}_{s}$
(see [5, 31, 32]).

As usual, let $\delta_{a}$ be the unit point-mass measure at $a\in\mathbb{D}$.
N. Arcozzi, D. Blasi and J. Pau [2] gave a series of interesting results on
$B_{p}(s)$ and $M(B_{p}(s))$. In particular, they obtained the following
theorem.

######  Theorem A.

Suppose $1<p<\infty$, $0<s<1$ and $\\{z_{n}\\}_{n=1}^{\infty}$ is a sequence
in $\mathbb{D}$. Then the following conditions are equivalent:

  1. (a)

$\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for $M(B_{p}(s))$;

  2. (b)

$\\{z_{n}\\}_{n=1}^{\infty}$ is a separated sequence and
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is a Carleson measure
for $B_{p}(s)$.

Ch. Yuan and C. Tong [34] characterized interpolating sequences for
$F(p,p-2,s)\cap H^{\infty}$ which generalizes the case of $\mathcal{Q}_{s}\cap
H^{\infty}$ in [24]. We recall Ch. Yuan and C. Tong’s result as follows.

######  Theorem B.

Suppose $1<p<\infty$, $0<s<1$ and $\\{z_{n}\\}_{n=1}^{\infty}$ is a sequence
in $\mathbb{D}$. Then the following conditions are equivalent:

  1. (a)

$\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for $F(p,p-2,s)\cap
H^{\infty}$;

  2. (b)

$\\{z_{n}\\}_{n=1}^{\infty}$ is a separated sequence and
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure.

In light of Theorems A and B, it is quite natural to characterize
interpolating sequences for $M(B_{p}(s))$ and $F(p,p-2,s)\cap H^{\infty}$ when
$0<s<1$ and $1-s<p\leq 1$. In this paper, we get the corresponding result for
$0<s<1$ and $\max\\{s,1-s\\}<p\leq 1$. In fact, we observe that
$F(p,p-2,s)\cap H^{\infty}=M(B_{p}(s))$ when $0<p\leq 1$ and $1-p<s\leq 1$.
For $0<s<1$ and certain $\\{z_{n}\\}_{n=1}^{\infty}$ in $\mathbb{D}$,
$s$-Carleson measures $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$
play an important role in our interpolating theorem. This inspires us to study
them further. In Section 3, we consider these $s$-Carleson measures via an
$L^{p}$ description. In Section 4, we apply these $s$-Carleson measures to
investigate proper inclusion relation related to the closure of $F(p,p-2,s)$
in the Bloch space.

Throughout this paper, we write $a\lesssim b$ if there exists a positive
constant $C$ such that $a\leq Cb$. The symbol $a\thickapprox b$ means that
$a\lesssim b\lesssim a$.

##  2\. Interpolating sequence for $M(B_{p}(s))$ and $F(p,p-2,s)\cap
H^{\infty}$ with small exponents

Suppose $0<s<1$ and $1-s<p\leq 1$. The function $\log(1-z)\in
F(p,p-2,s)\backslash H^{\infty}$. From [27, Proposition A], there exists a
function in $H^{\infty}\backslash F(p,p-2,s)$. Hence $F(p,p-2,s)\nsubseteq
H^{\infty}$ and $H^{\infty}\nsubseteq F(p,p-2,s)$.

The main result in this section is the following theorem.

######  Theorem 2.1.

Let $0<s<1$, $\max\\{s,1-s\\}<p\leq 1$ and $t>0$. Suppose
$\\{z_{n}\\}_{n=1}^{\infty}$ is a sequence in $\mathbb{D}$. Then the following
conditions are equivalent:

  1. (a)

$\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for $M(B_{p}(s))$;

  2. (b)

$\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for $F(p,p-2,s)\cap
H^{\infty}$;

  3. (c)

$\\{z_{n}\\}_{n=1}^{\infty}$ is a separated sequence and
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure;

  4. (d)

$\\{z_{n}\\}_{n=1}^{\infty}$ is a separated sequence and

$\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{t}(1-|z_{n}|^{2})^{s}}{|1-\overline{a}z_{n}|^{s+t}}<\infty.$

A function $f\in H(\mathbb{D})$ is called an inner function if it is an
$H^{\infty}$-function with radial limits of modulus one almost everywhere on
$\mathbb{T}$. A sequence $\\{a_{k}\\}_{k=1}^{\infty}$ in $\mathbb{D}$ is said
to be a Blaschke sequence if

$\sum_{k=1}^{\infty}(1-|a_{k}|)<\infty.$

This condition ensures the convergence of the corresponding Blaschke product
$B$ which is

$B(z)=\prod_{k=1}^{\infty}\frac{|a_{k}|}{a_{k}}\frac{a_{k}-z}{1-\overline{a_{k}}z}.$

Every Blaschke product is an inner function. A Blaschke product is said to be
an interpolating Blaschke product if its zero sequence is an interpolating
sequence for $H^{\infty}$.

F. Pérez-González and J. Rättyä [27] described inner functions in $F(p,p-2,s)$
spaces as follows.

######  Theorem C.

Let $0<s<1$ and $p>\max\\{s,1-s\\}$. Then an inner function belongs to
$F(p,p-2,s)$ if and only if it is a Blaschke product associated with a
sequence $\\{z_{k}\\}_{k=1}^{\infty}\subseteq\mathbb{D}$ which satisfies that
$\sum_{k}(1-|z_{k}|)^{s}\delta_{z_{k}}$ is an $s$-Carleson measure; that is

$\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}\left(1-|\sigma_{a}(z_{k})|^{2}\right)^{s}<\infty.$

We also recall the following well-known description of Carleson type measures
(cf. [9] ).

######  Lemma D.

Suppose $s>0$, $t>0$ and $\mu$ is a nonnegative Borel measure on $\mathbb{D}$.
Then $\mu$ is an $s$-Carleson measure if and only if

$\sup_{z\in\mathbb{D}}\int_{\mathbb{D}}\frac{(1-|z|^{2})^{t}}{|1-\overline{z}w|^{s+t}}d\mu(w)<\infty.$

The following (a) and (b) of Theorem E are from Theorem 3.3 and Theorem 4.2 in
[30], respectively.

######  Theorem E.

The following statements are true:

  1. (a)

let $0<p<\infty$, $0\leq s<\infty$ and $p+s>1$. Then $g\in M(B_{p}(s))$ if and
only if $g\in H^{\infty}$ and $|g^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}dA(z)$ is
a Carleson measure for $B_{p}(s)$;

  2. (b)

let $0<p\leq 1$ and $1-p<s\leq 1$. Then a nonnegative Borel measure $\mu$ on
$\mathbb{D}$ is a Carleson measure for $B_{p}(s)$ if and only if

$\int_{\mathbb{D}}\frac{d\mu(w)}{|1-\overline{w}z|^{2}}=O((1-|z|^{2})^{s-2}).$

When $0<p<\infty$ and $s>1$, recall that $F(p,p-2,s)$ is the Bloch space
$\mathcal{B}$, and $B_{p}(s)$ is the Bergman space $A^{p}_{s-2}$ (cf. [37]).
Hence, for $0<p<\infty$ and $s>1$, $M(B_{p}(s))=F(p,p-2,s)\cap
H^{\infty}=H^{\infty}$.

From [4, Theorem 5.2], if $0<s\leq 1$, then
$M(B_{2}(s))\subseteq\mathcal{Q}_{s}$. By [2, Lemma 5.1], if $1<p\leq 2$ and
$0<s<1$, then $M(B_{p}(s))\subseteq\mathcal{Q}_{s}\cap H^{\infty}$.

Suppose $1<p\leq 2$ and $0<s<1$. Proposition 2.2 below gives
$M(B_{p}(s))\subseteq F(p,p-2,s)\cap H^{\infty}$. It is known that (cf. [33,
Remark 2]) $F(p,p-2,s)\cap H^{\infty}\subseteq\mathcal{Q}_{s}\cap H^{\infty}$.
Thus we also get that $M(B_{p}(s))\subseteq\mathcal{Q}_{s}\cap H^{\infty}$
when $1<p\leq 2$ and $0<s<1$.

######  Proposition 2.2.

The following statements hold:

  1. (a)

if $0<p\leq 1$ and $1-p<s\leq 1$, then

$M(B_{p}(s))=F(p,p-2,s)\cap H^{\infty};$

  2. (b)

if $p>1$ and $0<s\leq 1$, then

$M(B_{p}(s))\subseteq F(p,p-2,s)\cap H^{\infty}.$

###### Proof.

(a) From Theorem E, $g\in M(B_{p}(s))$ if and only if $g\in H^{\infty}$ and

$\int_{\mathbb{D}}\frac{|g^{\prime}(w)|^{p}(1-|w|^{2})^{p-2+s}dA(w)}{|1-\overline{w}z|^{2}}=O((1-|z|^{2})^{s-2}).$
(1)

Taking $t=2-s$ and $d\mu(w)=|g^{\prime}(w)|^{p}(1-|w|^{2})^{p-2+s}dA(w)$ in
Lemma D, we see that (1) holds if and only if
$|g^{\prime}(w)|^{p}(1-|w|^{2})^{p-2+s}dA(w)$ is an $s$-Carleson measure. In
other words, (1) holds if and only if $g\in F(p,p-2,s)$. Hence
$M(B_{p}(s))=F(p,p-2,s)\cap H^{\infty}$.

(b) Let $g\in M(B_{p}(s))$. For $a\in\mathbb{D}$, set

$f_{a}(z)=\frac{(1-|a|^{2})^{s/p}}{(1-\overline{a}z)^{\frac{2s}{p}}},\ \
z\in\mathbb{D}.$

Note that $p>1$ and $0<s\leq 1$. We see that (cf. [10, p. 407])

$\sup_{a\in\mathbb{D}}\|f_{a}\|_{B_{p}(s)}<\infty.$

Bear in mind that $|g^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}dA(z)$ is a Carleson
measure for $B_{p}(s)$. Then

$\displaystyle\int_{\mathbb{D}}|f_{a}(z)|^{p}|g^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}dA(z)\lesssim\|f_{a}\|_{B_{p}(s)}^{p}$

for all $a\in\mathbb{D}$. This gives

$\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\frac{(1-|a|^{2})^{s}}{|1-\overline{a}z|^{2s}}|g^{\prime}(z)|^{p}(1-|z|^{2})^{p-2+s}dA(z)<\infty.$

Thus $g\in F(p,p-2,s)$. Hence $M(B_{p}(s))\subseteq F(p,p-2,s)\cap
H^{\infty}$. ∎

Proof of Theorem 2.1. The equivalence of (a) and (b) is from Proposition 2.2.
By Lemma D, (c) and (d) are also equivalent.

(b)$\Rightarrow$(c). Note that $0<s<1$ and $\max\\{s,1-s\\}<p\leq 1$. By a
simple observation (cf. [7, p. 207]), we see that $F(p,p-2,s)\subseteq
F(2,0,s)$. Since $\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for
$F(p,p-2,s)\cap H^{\infty}$, $\\{z_{n}\\}_{n=1}^{\infty}$ is also an
interpolating sequence for $F(2,0,s)\cap H^{\infty}$. By Theorem B, we get
that (c) holds.

(c)$\Rightarrow$(b). Suppose $\\{z_{n}\\}_{n=1}^{\infty}$ is a separated
sequence and $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an
$s$-Carleson measure. Then

$\sum_{z_{n}\in S(I)}(1-|z_{n}|^{2})\leq|I|^{1-s}\sum_{z_{n}\in
S(I)}(1-|z_{n}|^{2})^{s}\lesssim|I|$

for all $I\subseteq\mathbb{T}$. Hence
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson measure. Thus
$\\{z_{n}\\}_{n=1}^{\infty}$ is an interpolating sequence for $H^{\infty}$.
Then

$\inf_{k}\prod_{j\neq k}\rho(z_{j},z_{k})=:\delta>0.$

Let $\\{a_{n}\\}_{n=1}^{\infty}$ be any bounded sequence of complex numbers.
From J. Earl’s constructive proof of L. Carleson’s interpolating theorem for
$H^{\infty}$ (see [15]), there exists a complex constant $C$ and a Blaschke
product $B$ with zeros $\\{\zeta_{n}\\}_{n=1}^{\infty}$ such that

$f(z)=:C\sup_{n}\\{a_{n}\\}B(z)$

satisfies

$f(z_{n})=a_{n},\ \ n=1,2,\cdots,$

and

$\rho(z_{n},\zeta_{n})\leq\frac{\delta}{3},\ n=1,2,\cdots.$ (2)

It suffices to prove that $B\in F(p,p-2,s)$. Because of (2), we know from [37,
p. 69] and [37, Lemma 4.30] that

$1-|z_{n}|\thickapprox 1-|\zeta_{n}|\thickapprox|1-\overline{z_{n}}\zeta_{n}|$
(3)

for all positive integers $n$, and

$|1-\overline{z_{n}}w|\thickapprox|1-\overline{\zeta_{n}}w|$ (4)

for all positive integers $n$ and all $w\in\mathbb{D}$. Also, the equivalence
of (c) and (d) yields

$\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{s}(1-|z_{n}|^{2})^{s}}{|1-\overline{a}z_{n}|^{2s}}<\infty.$
(5)

Joining (3), (4) and (5), we get

$\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{s}(1-|\zeta_{n}|^{2})^{s}}{|1-\overline{a}\zeta_{n}|^{2s}}<\infty.$

Combining this with Theorem C, we obtain $B\in F(p,p-2,s)$. This finishes the
proof. $\square$

For $0<s\leq 1/2$, the assumption $\max\\{s,1-s\\}<p\leq 1$ in Theorem 2.1 is
sharp because $\max\\{s,1-s\\}=1-s$ for such case, and the condition
$1-s<p\leq 1$ is necessary because of the non-triviality of $B^{p}_{s}$ and
$F(p,p-2,s)$. When $1/2<s<1$ and $1-s<p\leq s$, we do not have a good answer
for the characterization of interpolating sequence for $M(B_{p}(s))$ and
$F(p,p-2,s)\cap H^{\infty}$.

##  3\. An $L^{p}$ description related to $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ with $0<s<1$

Suppose $\\{z_{n}\\}_{n=1}^{\infty}$ is a sequence in $\mathbb{D}$. For
$0<s<1$, the $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ plays a key role in our
Theorem 2.1, which characterizes interpolating sequence for $M(B_{p}(s))$ and
$F(p,p-2,s)\cap H^{\infty}$ for certain range of parameters $p$ and $s$. In
this section, we consider further this kind of $s$-Carleson measures. In fact,
these $s$-Carleson measures defined by sequences $\\{z_{n}\\}_{n=1}^{\infty}$
were used to characterize inner functions in $\mathcal{Q}_{s}$ and
$F(p,p-2,s)$ spaces (see[16, 27]), and also used to study solutions of second
order complex differential equation having certain pre-given zeros (cf. [21,
33]). We refer to [24, 28] for more results associated with these $s$-Carleson
measures.

Now we recall the classical case of $s=1$. Namely we let
$\\{z_{n}\\}_{n=1}^{\infty}$ be a sequence in $\mathbb{D}$ such that
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson measure. From
[22], this is equivalent to say that $\\{z_{n}\\}_{n=1}^{\infty}$ is a union
of finitely many uniformly separated sequences in $\mathbb{D}$. By an $L^{p}$
description, C. Nolder [25, Theorem D] characterized the union of finitely
many uniformly separated sequences. We refer to P. Duren and A. Schuster [14]
and A. Nicolau [23] for more interesting results on these sequences.

Give an $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$, where $0<s\leq 1$ and
$\\{z_{n}\\}_{n=1}^{\infty}$ is certain sequence in $\mathbb{D}$. Then it is
easy to see that $\\{z_{n}\\}_{n=1}^{\infty}$ is also a Blaschke sequence. For
$s=1$, C. Nolder [25, Theorem D] obtained an $L^{p}$ characterization related
to these $s$-Carleson measures as follows.

######  Theorem F.

Suppose $0<p<2$ and $\\{z_{n}\\}_{n=1}^{\infty}$ is a Blaschke sequence in
$\mathbb{D}$. Let $B$ be the Blaschke product whose zero sequence is
$\\{z_{n}\\}_{n=1}^{\infty}$. Then the following conditions are equivalent:

  1. (a)

$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson measure;

  2. (b)

$\sup_{\phi\in\text{Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{1}{|B(\phi(z))|^{p}}dA(z)<\infty.$

In this section, motivated by C. Nolder’s result above, we obtain an $L^{p}$
description associated with these $s$-Carleson measures with $0<s<1$. As shown
by Theorem 3.1 below, the case of $0<s<1$ seems to be more delicate.

Next we recall some well-known estimates (cf. [37, Lemma 3.10]).

######  Lemma G.

Suppose $z\in\mathbb{D}$, $c$ is real and $t>-1$. Then

$\int_{\mathbb{D}}\frac{(1-|w|^{2})^{t}}{|1-\bar{z}w|^{2+t+c}}dA(w)\thickapprox\begin{cases}1&\enspace\text{if}\
\ \ c<0,\\\ \log\frac{2}{1-|z|^{2}}&\enspace\text{if}\ \ \ c=0,\\\
\frac{1}{(1-|z|^{2})^{c}}&\enspace\text{if}\ \ \ c>0,\end{cases}$

as $|z|\rightarrow 1^{-}$.

Now we give our result in this section.

######  Theorem 3.1.

Suppose $0<s<1$, $\max\\{s,1-s\\}<p\leq 1$ and $\\{z_{n}\\}_{n=1}^{\infty}$ is
a Blaschke sequence in $\mathbb{D}$. Let $B$ be the Blaschke product whose
zero sequence is $\\{z_{n}\\}_{n=1}^{\infty}$. Then the following conditions
are equivalent:

  1. (a)

$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure;

  2. (b)

$\sup_{\phi\in\text{Aut}(\mathbb{D})}\int_{\mathbb{D}}\left|\frac{B^{\prime}(\phi(z))}{B(\phi(z))}\right|^{p}(1-|\phi(z)|^{2})^{p}(1-|z|^{2})^{s-2}dA(z)<\infty.$

###### Proof.

By a change of variables, (b) is equivalent to

$\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\left|\frac{B^{\prime}(z)}{B(z)}\right|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)<\infty.$
(6)

For the sake of comparison with Theorem F, we use (b) in the expressions of
this theorem.

First, let (b) hold. Then (6) is true. Combining this with the fact that
$|B(z)|\leq 1$ for all $z\in\mathbb{D}$, we see that

$\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|B^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)<\infty,$

which yields $B\in F(p,p-2,s)$. Note that $0<s<1$ and $\max\\{s,1-s\\}<p\leq
1$. By Theorem C, $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an
$s$-Carleson measure.

On the other hand, let $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$
be an $s$-Carleson measure. By Lemma D, we get

$\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{s}(1-|z_{n}|^{2})^{s}}{|1-\overline{a}z_{n}|^{2s}}<\infty.$
(7)

Note that

$\frac{B^{\prime}(z)}{B(z)}=\sum_{n=1}^{\infty}\frac{|z_{n}|^{2}-1}{(z_{n}-z)(1-\overline{z_{n}}z)}$

for $z\in\mathbb{D}\setminus\\{z_{n}\\}_{n=1}^{\infty}$. Since
$0<\max\\{s,1-s\\}<p\leq 1$, we have

$\left|\frac{B^{\prime}(z)}{B(z)}\right|^{p}\leq\sum_{n=1}^{\infty}\frac{(1-|z_{n}|^{2})^{p}}{|z_{n}-z|^{p}|1-\overline{z_{n}}z|^{p}}.$

Hence, for each $a\in\mathbb{D}$, we deduce that

$\displaystyle\int_{\mathbb{D}}\left|\frac{B^{\prime}(z)}{B(z)}\right|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)$
(8) $\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{p}\int_{\mathbb{D}}\frac{(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}}{|z_{n}-z|^{p}|1-\overline{z_{n}}z|^{p}}dA(z)$
$\displaystyle=$
$\displaystyle(1-|a|^{2})^{s}\sum_{n=1}^{\infty}\int_{\mathbb{D}}\frac{(1-|z_{n}|^{2})^{p}(1-|z|^{2})^{p-2+s}}{|1-\overline{a}z|^{2s}|1-\overline{z_{n}}z|^{2p}|\sigma_{z_{n}}(z)|^{p}}dA(z).$

Making a change of variables $w=\sigma_{z_{n}}(z)$ in the integral in (8), we
get

$\displaystyle\int_{\mathbb{D}}\frac{(1-|z|^{2})^{p-2+s}}{|1-\overline{a}z|^{2s}|1-\overline{z_{n}}z|^{2p}|\sigma_{z_{n}}(z)|^{p}}dA(z)$
(9) $\displaystyle=$
$\displaystyle\int_{\mathbb{D}}\frac{(1-|\sigma_{z_{n}}(w)|^{2})^{p+s}}{|w|^{p}(1-|w|^{2})^{2}|1-\overline{a}\sigma_{z_{n}}(w)|^{2s}|1-\overline{z_{n}}\sigma_{z_{n}}(w)|^{2p}}dA(w)$
$\displaystyle=$
$\displaystyle(1-|z_{n}|^{2})^{s-p}\int_{\mathbb{D}}\frac{(1-|w|^{2})^{p+s-2}}{|w|^{p}|1-\overline{z_{n}}w|^{2s}|1-\overline{a}\sigma_{z_{n}}(w)|^{2s}}dA(w).$

For $0<p\leq 1$, $dA(w)/|w|^{p}=r^{1-p}drd\theta$, where $w=re^{i\theta}$.
Thus the integral in (9) is convergent. In fact, from [37, Lemma 4.26], we
know

$\displaystyle\int_{\mathbb{D}}\frac{(1-|w|^{2})^{p+s-2}}{|w|^{p}|1-\overline{z_{n}}w|^{2s}|1-\overline{a}\sigma_{z_{n}}(w)|^{2s}}dA(w)$
(10) $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}\frac{(1-|w|^{2})^{p+s-2}}{|1-\overline{z_{n}}w|^{2s}|1-\overline{a}\sigma_{z_{n}}(w)|^{2s}}dA(w)$

for all $z_{n}$ and $a\in\mathbb{D}$. We compute that

$\displaystyle 1-\overline{a}\sigma_{z_{n}}(w)$ $\displaystyle=$
$\displaystyle\frac{1-\overline{a}z_{n}-w(\overline{z_{n}}-\overline{a})}{1-\overline{z_{n}}w}$
$\displaystyle=$
$\displaystyle(1-\overline{a}z_{n})\frac{1-w\overline{\sigma_{z_{n}}(a)}}{1-\overline{z_{n}}w}.$

Combining this with Lemma G, we obtain

$\displaystyle\int_{\mathbb{D}}\frac{(1-|w|^{2})^{p+s-2}}{|1-\overline{z_{n}}w|^{2s}|1-\overline{a}\sigma_{z_{n}}(w)|^{2s}}dA(w)$
(11) $\displaystyle=$
$\displaystyle\frac{1}{|1-\overline{a}z_{n}|^{2s}}\int_{\mathbb{D}}\frac{(1-|w|^{2})^{p+s-2}}{|1-w\overline{\sigma_{z_{n}}(a)}|^{2s}}dA(w)$
$\displaystyle\thickapprox$
$\displaystyle\frac{1}{|1-\overline{a}z_{n}|^{2s}},$

where we use $0<\max\\{s,1-s\\}<p$.

Consequently, formulas (7), (8), (9), (10) and (11) yield

$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\left|\frac{B^{\prime}(z)}{B(z)}\right|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)$
$\displaystyle\lesssim$
$\displaystyle\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{s}(1-|z_{n}|^{2})^{s}}{|1-\overline{a}z_{n}|^{2s}}<\infty.$

The proof is complete. ∎

##  4\. An application of $s$-Carleson measures
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ to
$C_{\mathcal{B}}(F(p,p-2,s))$

In 1974, J. Anderson, J. Clunie and Ch. Pommerenke [1] posed the following
problem: what is the closure in the Bloch norm of $H^{\infty}$? This problem
remains open. For a space $X$ of functions analytic in $\mathbb{D}$, let
$C_{\mathcal{B}}(X)$ be the closure of $X$ in the Bloch norm. An important
result in this field is P. Jones’ characterization of $C_{B}(BMOA)$ (see
[19]). Later, R. Zhao [36] described $C_{\mathcal{B}}(F(p,p-2,s))$ when
$0<s\leq 1$ and $1\leq p<\infty$. R. Zhao’s characterization of
$C_{\mathcal{B}}(F(p,p-2,s))$ is independent of the parameter $p$. In this
section, we consider the proper inclusion relation related to
$C_{\mathcal{B}}(F(p,p-2,s))$ via interpolating Blaschke products associated
with $s$-Carleson measures
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$.

For $f\in H(\mathbb{D})$ and $\epsilon>0$, let
$\Omega_{\epsilon}(f)=\\{z\in\mathbb{D}:|f^{\prime}(z)|(1-|z|^{2})\geq\epsilon\\}$.
We recall R. Zhao’s characterization of $C_{\mathcal{B}}(F(p,p-2,s))$ as
follows.

######  Theorem H.

Let $0<s\leq 1$, $1\leq p<\infty$ and $0\leq t<\infty$. Suppose
$f\in\mathcal{B}$. Then $f\in C_{\mathcal{B}}(F(p,p-2,s))$ if and only if for
every $\epsilon>0$,

$\sup_{a\in\mathbb{D}}\int_{\Omega_{\epsilon}(f)}|f^{\prime}(z)|^{t}(1-|z|^{2})^{t-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)<\infty.$

Based on some arguments of $s$-Carleson measures
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$, we obtain the
following proper inclusion relation, which is the main result in this section.

######  Theorem 4.1.

Let $0<s\leq 1$ and $1\leq p<\infty$. Then

$C_{\mathcal{B}}(F(p,p-2,s))\subsetneqq\bigcap_{s<t<\infty}C_{\mathcal{B}}(F(p,p-2,t)).$

Before the proof of Theorem 4.1, we need some preliminary results. The
following useful inequality is well known (cf. [26, Lemma 2.5]).

######  Lemma I.

Suppose $s>-1$, $r>0$, $t>0$, and $t<s+2<r$. Then there exists a positive
constant $C$ such that

$\int_{\mathbb{D}}\frac{(1-|w|^{2})^{s}}{|1-\overline{w}z|^{r}|1-\overline{w}\zeta|^{t}}dA(w)\leq
C\frac{(1-|z|^{2})^{2+s-r}}{|1-\overline{\zeta}z|^{t}}$

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

We also need the following result, which characterizes interpolating Blaschke
products in $C_{\mathcal{B}}(F(p,p-2,s))$.

######  Lemma 4.2.

Suppose $0<s<1$, $1\leq p<\infty$, and $I$ is an interpolating Blaschke
product with zero set $\\{z_{n}\\}_{n=1}^{\infty}$. Then $I\in
C_{\mathcal{B}}(F(p,p-2,s))$ if and only if
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure.

###### Proof.

First, let $I\in C_{\mathcal{B}}(F(p,p-2,s))$. We adapt an argument from [17,
p. 1220] (see also [6]). Since $I$ is an interpolating Blaschke product, there
exists $\gamma>0$ such that

$\inf_{n}(1-|z_{n}|^{2})|I^{\prime}(z_{n})|\geq\gamma.$

Take $\epsilon\in(0,\frac{\gamma}{4})$. From [20, p. 681]), there exists a
positive constant $\rho$ such that

$(1-|z|^{2})|I^{\prime}(z)|\geq\epsilon$

on every $D_{h}(z_{n},\rho)$, where $D_{h}(z_{n},\rho)$ is the hyperbolic disk
with center $z_{n}$ and radius $\rho$. Namely,

$D_{h}(z_{n},\rho)=\\{z\in\mathbb{D}:\frac{1}{2}\log\frac{1+|\sigma_{z_{n}}(z)|}{1-|\sigma_{z_{n}}(z)|}<\rho\\}.$

Then

$\bigcup_{n=1}^{\infty}D_{h}(z_{n},\rho)\subseteq\\{z\in\mathbb{D}:(1-|z|^{2})|I^{\prime}(z)|\geq\epsilon\\}.$

Since $\\{a_{n}\\}$ is uniformly separated, we can select $\rho$ such that
these hyperbolic disks are pairwise disjoints. Combining these with Theorem H,
we see that

$\displaystyle\infty$ $\displaystyle>$
$\displaystyle\sup_{w\in\mathbb{D}}\int_{\\{z\in\mathbb{D}:\
(1-|z|^{2})|I^{\prime}(z)|\geq\epsilon\\}}\frac{(1-|\sigma_{w}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dA(z)$
$\displaystyle\geq$
$\displaystyle\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}\int_{D_{h}(z_{n},\rho)}\frac{(1-|\sigma_{w}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dA(z).$

By [37, Proposition 4.5] and [37, Lemma 4.30], we know that

$1-|z|\thickapprox 1-|z_{n}|\thickapprox|1-\overline{z_{n}}z|,\ \
|1-\overline{w}z|\thickapprox|1-\overline{w}z_{n}|,$

for all $z\in D_{h}(z_{n},\rho)$ and all $w\in\mathbb{D}$. Therefore,

$\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}(1-|\sigma_{w}(z_{n})|^{2})^{s}<\infty,$

which gives that $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an
$s$-Carleson measure.

On the other hand, let $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$
be an $s$-Carleson measure. Bear in mind that

$|I^{\prime}(z)|\leqslant\sum_{n=1}^{\infty}\frac{1-|z_{n}|^{2}}{|1-\overline{z_{n}}z|^{2}},\
\ z\in\mathbb{D}.$

Then for every $\epsilon>0$, one gets that

$\displaystyle\sup_{a\in\mathbb{D}}\int_{\Omega_{\epsilon}(I)}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dA(z)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\epsilon}\sup_{a\in\mathbb{D}}\int_{\Omega_{\epsilon}(I)}(1-|z|^{2})|I^{\prime}(z)|\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dA(z)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\epsilon}\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\int_{\Omega_{\epsilon}(I)}\frac{1-|z_{n}|^{2}}{|1-\overline{z_{n}}z|^{2}}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{1-|z|^{2}}dA(z)$
$\displaystyle\leq$
$\displaystyle\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}\frac{(1-|a|^{2})^{s}(1-|z_{n}|^{2})}{\epsilon}\int_{\mathbb{D}}\frac{(1-|z|^{2})^{s-1}}{|1-\overline{z_{n}}z|^{2}|1-\overline{a}z|^{2s}}dA(z).$

Note that $0<s<1$. By Lemma I, we see that

$\int_{\mathbb{D}}\frac{(1-|z|^{2})^{s-1}}{|1-\overline{z_{n}}z|^{2}|1-\overline{a}z|^{2s}}dA(z)\lesssim\frac{(1-|z_{n}|^{2})^{s-1}}{|1-\overline{z_{n}}a|^{2s}}.$

Thus,

$\sup_{a\in\mathbb{D}}\int_{\Omega_{\epsilon}(I)}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dA(z)\lesssim\sup_{a\in\mathbb{D}}\sum_{n=1}^{\infty}(1-|\sigma_{a}(z_{n})|^{2})^{s}<\infty.$

By Theorem H, one gets $I\in C_{\mathcal{B}}(F(p,p-2,s))$. The proof is
complete. ∎

Proof of Theorem 4.1. Note that for $1\leq p<\infty$ and $0<s_{1}\leq
s_{2}<\infty$, $F(p,p-2,s_{1})\subseteq F(p,p-2,s_{2})$. Hence
$C_{\mathcal{B}}(F(p,p-2,s_{1}))\subseteq C_{\mathcal{B}}(F(p,p-2,s_{2}))$.
Thus for $0<s\leq 1$ and $1\leq p<\infty$, we have

$C_{\mathcal{B}}(F(p,p-2,s))\subseteq\bigcap_{s<t<\infty}C_{\mathcal{B}}(F(p,p-2,t)).$

If $s=1$, then $C_{\mathcal{B}}(F(p,p-2,s))=C_{\mathcal{B}}(BMOA)$ and

$\bigcap_{s<t<\infty}C_{\mathcal{B}}(F(p,p-2,t))=\mathcal{B}.$

It is known from [19] that $C_{\mathcal{B}}(BMOA)\subsetneqq\mathcal{B}$. Thus

$C_{\mathcal{B}}(F(p,p-2,1))\subsetneqq\bigcap_{1<t<\infty}C_{\mathcal{B}}(F(p,p-2,t)).$

Let $0<s<1$. From Lemma 2.1 in [8], there exists a Blaschke sequence
$\\{z_{n}\\}_{n=1}^{\infty}$ such that
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is not an $s$-Carleson
measure and

$\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}\left(1-|\sigma_{w}(z_{n})|^{2}\right)^{s}\left(\frac{1}{\ln\frac{2}{1-|\sigma_{w}(z_{n})|^{2}}}\right)^{2}<\infty.$
(12)

Denote by $I$ the Blaschke product whose zero sequence is
$\\{z_{n}\\}_{n=1}^{\infty}$. Formula (12) implies that

$\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}\left(1-|\sigma_{w}(z_{n})|^{2}\right)<\infty;$

that is, $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson
measure. Hence $I$ is an interpolating Blaschke product. By Lemma 4.2,

$I\not\in C_{\mathcal{B}}(F(p,p-2,s)).$ (13)

If $0<s<t<1$, it is clear that

$\displaystyle\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}\left(1-|\sigma_{w}(z_{n})|^{2}\right)^{t}$
$\displaystyle\lesssim$
$\displaystyle\sup_{w\in\mathbb{D}}\sum_{n=1}^{\infty}\left(1-|\sigma_{w}(z_{n})|^{2}\right)^{s}\left(\frac{1}{\ln\frac{2}{1-|\sigma_{w}(z_{n})|^{2}}}\right)^{2}<\infty,$

which means that $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{t}\delta_{z_{n}}$ is a
$t$-Carleson measure. From Lemma 4.2 again, $I\in C_{\mathcal{B}}(F(p,p-2,t))$
for any $t\in(s,1)$. Hence

$I\in\bigcap_{s<t<1}C_{\mathcal{B}}(F(p,p-2,t)).$ (14)

Clearly,

$\bigcap_{s<t<1}C_{\mathcal{B}}(F(p,p-2,t))=\bigcap_{s<t<\infty}C_{\mathcal{B}}(F(p,p-2,t)).$
(15)

From (13), (14) and (15), we get

$C_{\mathcal{B}}(F(p,p-2,s))\subsetneqq\bigcap_{s<t<\infty}C_{\mathcal{B}}(F(p,p-2,t)).$

The proof of Theorem 4.1 is finished. $\square$

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