# Keyphrase Generation for Scientific Document Retrieval

Florian Boudin1, Ygor Gallina1, Akiko Aizawa2
1LS2N, Université de Nantes, France<EMAIL_ADDRESS>
2National Institute of Informatics, Tokyo, Japan<EMAIL_ADDRESS>

###### Abstract

Sequence-to-sequence models have lead to significant progress in keyphrase
generation, but it remains unknown whether they are reliable enough to be
beneficial for document retrieval. This study provides empirical evidence that
such models can significantly improve retrieval performance, and introduces a
new extrinsic evaluation framework that allows for a better understanding of
the limitations of keyphrase generation models. Using this framework, we point
out and discuss the difficulties encountered with supplementing documents with
–not present in text– keyphrases, and generalizing models across domains. Our
code is available at https://github.com/boudinfl/ir-using-kg.

##  1 Introduction

With the exponential growth of the scientific literature Bornmann and Mutz
(2015), retrieving relevant scientific papers becomes increasingly difficult.
Keywords, also referred to as keyphrases, provide an effective way to
supplement paper indexing and improve retrieval effectiveness in scientific
digital libraries Barker et al. (1972); Zhai (1997); Gutwin et al. (1999); Lu
and Kipp (2014). However, only few documents have assigned keyphrases, and
those who do were, for the most part, self-labeled by their authors, thus
exhibiting annotation inconsistencies Strader (2011); Suzuki et al. (2011).
This has motivated an active line of research on automatic keyphrase
extraction (see Hasan and Ng (2014) for an overview) and, more recently,
keyphrase generation Meng et al. (2017), where the task is to find a set of
words and phrases that represents the main content of a document.

Although models for predicting keyphrases have been extensively evaluated on
their ability to reproduce author’s keywords, it still remains unclear whether
they can be usefully applied in information retrieval. One reason for this
lack of evidence may have been their relatively low performance discouraging
attempts at using them for indexing Liu et al. (2010); Hasan and Ng (2014).
Yet, recently proposed models not only achieve much better performance, but
also display a property that may have a significant impact on retrieval
effectiveness: the capacity to generate keyphrases that do not appear in the
source text. These absent keyphrases do not just highlight the topics that are
most relevant, but provide some form of semantic expansion by adding new
content (e.g. synonyms, semantically related terms) to the index Greulich
(2011). The goal of this paper is two-fold: to gather empirical evidence as to
whether current keyphrase generation models are good enough to improve
scientific document retrieval, and to gain further insights into the
performance of these models from an extrinsic perspective. Our contributions
are listed as follows:

  * •

We report significant improvements for strong retrieval models on a standard
benchmark collection, showing that keyphrases produced by state-of-the-art
models are consistently helpful for document retrieval, even, to our surprise,
when author keywords are provided.

  * •

We introduce a new extrinsic evaluation framework for keyphrase generation
that allows for a deeper understanding of the limitations of current models.
Using it, we discuss the difficulties associated with domain generalization
and absent keyphrase prediction.

##  2 Methodology

This section presents our methodology for assessing the usefulness (§2.3) of
keyphrase generation (§2.2) in scientific document retrieval (§2.1).

###  2.1 Scientific Document Retrieval

Here, we focus on the task of searching through a collection of scientific
papers for relevant documents. All of our experiments are conducted on the
NTCIR-2 test collection Kando (2001) which is, to our knowledge, the only
available benchmark dataset for that task. It contains 322,058
documents111Scientific abstracts and summaries of research results. (title and
abstract pairs) and 49 search topics (queries) with relevance judgments. Most
of the documents (98.6%) include author keywords (4.8 per doc. on avg.), which
we later use to investigate the performance of keyphrase generation models.

Documents cover a broad range of domains from pure science to social sciences
and humanities, although half of the documents are about engineering and
computer science. Queries are also categorized into one or more research
fields (e.g. science, chemistry, engineering), the original intent being to
help retrieval models in narrowing down the search space. We follow common
practice and use short222<description> field of topic description. queries
with binary relevance judgments (i.e. without “partially relevant” documents).

We consider two standard ad-hoc retrieval models to rank documents against
queries: BM25 and query likelihood (QL), both implemented in the Anserini IR
toolkit Yang et al. (2017). These models use unsupervised techniques based on
corpus statistics for term weighting, and will therefore be straightforwardly
affected when keyphrases are added to a document. We further apply a pseudo-
relevance feedback method, known as RM3 Abdul-Jaleel et al. (2004), on top of
the models to achieve strong, near state-of-the-art retrieval results Lin
(2019); Yang et al. (2019). For all models, we use Anserini’s default
parameters.

To verify the effectiveness of the adopted retrieval models, we compared their
performance with that of the best participating systems in NTCIR-2. Retrieval
performance is measured using mean average precision (MAP) and precision at 10
retrieved documents (P@10). MAP measures the overall ranking quality and P@10
reflects the number of relevant documents on the first page of search results.
Documents are indexed with author keywords, same as for participating systems.
Results are presented in Table 1. We see that the considered retrieval models
achieve strong performance, even outperforming the best participating system
by a substantial margin. Note that the two best-performing systems use pseudo-
relevance feedback, and that the second-ranked system is based on BM25.

Model | MAP | P@10
---|---|---
BM25+RM3 | 35.17 | 38.57
QL+RM3 | 33.00 | 34.90
1st Fujita (2001) | 31.93 | 37.35
BM25 | 31.38 | 36.33
2nd Murata et al. (2001) | 31.31 | 36.12
QL | 30.63 | 34.08
3rd Chen et al. (2001) | 26.24 | 33.88
Table 1: Retrieval effectiveness of the considered models and the best
participating systems on NTCIR-2.

###  2.2 Keyphrase Generation

Keyphrase generation is the task of producing a set of words and phrases that
best summarise a document Evans and Zhai (1996). In contrast with most
previous work that formulates this task as an extraction problem (a.k.a.
keyphrase extraction), which can be seen as ranking phrases extracted from a
document, recent neural models for keyphrase generation are based on sequence-
to-sequence learning Sutskever et al. (2014); Bahdanau et al. (2014), thus
potentially allowing them to generate any phrase, also beyond those that
appear verbatim in the text. In this study, we consider the following two
neural keyphrase generation models:

seq2seq+copy

Meng et al. (2017) is a sequence-to-sequence model with attention, augmented
with a copying mechanism Gu et al. (2016) to predict phrases that rarely
occur. The model is trained with document-keyphrase pairs and uses beam search
decoding for inference.

seq2seq+corr

Chen et al. (2018) extends the aforementioned model with correlation
constraints. It employs a coverage mechanism Tu et al. (2016) that diversifies
attention distributions to increase topic coverage, and a review mechanism to
avoid generating duplicates.

We implemented the models in PyTorch Paszke et al. (2017) using AllenNLP
Gardner et al. (2018). Models are trained on the KP20k dataset Meng et al.
(2017), which contains 567,830 scientific abstracts with gold-standard,
author-assigned keywords (5.3 per doc. on avg.). We use the parameters
suggested by the authors for each model.

To validate the effectiveness of our implementations, we conducted an
intrinsic evaluation by counting the number of exact matches between predicted
and gold keyphrases. We adopt the standard metric and compute the f-measure at
top 5, as it corresponds to the average number of keyphrases in KP20k and
NTCIR-2, that is, 5.3 and 4.8, respectively. We also examine cross-domain
generalization using the KPTimes news dataset Gallina et al. (2019), and
include a state-of-the-art unsupervised keyphrase extraction model (Boudin,
2018, henceforth mp-rank) for comparison purposes. This latter baseline also
provides an additional relevance signal based on graph-based ranking whose
usefulness in retrieval will be tested in subsequent experiments. Results are
reported in Table 2. Overall, our results are consistent with those reported
in Meng et al. (2017); Chen et al. (2018), demonstrating the superiority of
well-trained neural models over unsupervised ones, and stressing their lack of
robustness across domains. Rather surprisingly, seq2seq+corr is outperformed
by seq2seq+copy which indicates that relevant, yet possibly redundant,
keyphrases are filtered out by the added mechanisms for promoting diversity in
the output.

Model |  KP20k | NTCIR-2 | KPTimes
---|---|---|---
s2s+copy | 27.75 | 23.90 | 16.47
s2s+corr | 23.78 | 22.27 | 11.73
mp-rank | 14.67 | 18.10 | 14.59
Table 2: f-measure at top-$5$ predicted keyphrases. Stemming is applied to
reduce the number of mismatches.

###  2.3 Extrinsic Evaluation Framework

Our goal is to find out whether the keyphrase generation models described
above are reliable enough to be beneficial for document retrieval. To do so,
we contrast the performance of the retrieval models with and without
automatically predicted keyphrases. Two initial indexing configurations are
also examined: title and abstract only ($T$+$A$), and title, abstract and
author keywords ($T$+$A$+$K$). The idea here is to investigate whether
generated keyphrases simply act as a proxy for author keywords, or instead
supplement them.

Unless mentioned otherwise, the top-5 predicted keyphrases are used to expand
documents, which is in accordance with the average number of author keywords
in NTCIR-2. We evaluate retrieval performance in terms of MAP and omit P@10
for brevity. We use the Student’s paired t-test to assess statistical
significance of our retrieval results at $p<0.05$ Smucker et al. (2007).

##  3 Results

Results for retrieval models using keyphrase generation are reported in Table
3. We note that indexing keyphrases generated by seq2seq+copy, which performs
best in our intrinsic evaluation, significantly improves retrieval
effectiveness for all models. More interestingly, gains in effectiveness are
also significant when both keyphrases and author keywords are indexed,
indicating they complement each other well. This important finding suggests
that predicted keyphrases are consistently helpful for document retrieval, and
should be used even when author keywords are provided. Another important
observation is that while both keyphrase generation models perform reasonably
well in our intrinsic evaluation on NTCIR-2 (cf. Table 2, column 3), their
impact on retrieval effectiveness are quite different, as only s2s+copy
reaches consistent significance. This finding advocates for the importance of
using document retrieval as an extrinsic evaluation task for keyphrase
generation.

Index | BM25 | +RM3 | QL | +RM3
---|---|---|---|---
$T$+$A$ | 29.16 | 31.93 | 28.98 | 31.47
\+ s2s+copy |  †30.54† |  †34.30† |  †30.58† |  †33.26†
\+ s2s+corr |  †30.30† | 33.24 | 29.76 | 31.38
\+ mp-rank | 29.24 | 32.27 | 29.57 | 32.29
$T$+$A$+$K$ | 31.38 | 35.17 | 30.63 | 33.00
\+ s2s+copy | 31.55 |  ‡36.53‡ |  ‡31.70‡ |  ‡35.15‡
\+ s2s+corr | 31.37 | 35.84 | 31.14 | 33.65
\+ mp-rank | 31.38 | 35.18 | 31.23 | 33.47

Table 3: MAP scores for retrieval models using various indexing
configurations. $\dagger$ and $\ddagger$ indicate significance over $T$+$A$
and $T$+$A$+$K$, respectively.

Overall, BM25+RM3 achieves the best retrieval effectiveness, confirming
previous findings on ad-hoc retrieval in limited data scenarios Lin (2019).
For clarity and conciseness, we focus on this model in the rest of this paper.
Encouraging diversity in keyphrases seems not to be appropriate for retrieval,
as seq2seq+corr consistently gives lower results than seq2seq+copy. It is also
interesting to see that the effectiveness gains of query expansion (RM3) and
document expansion are additive, suggesting that they provide different but
complementary relevance signals. Moreover, our results show that query
expansion is more effective, which is in line with past work Billerbeck and
Zobel (2005).

One hyper-parameter that we have deliberately left untouched so far is the
number $N$ of predicted keyphrases that directly controls the precision-recall
trade-off of keyphrase generation models. To understand how this parameter
affects retrieval effectiveness, we repeated our experiments by varying $N$
within the range $[0,9]$, and plotted the results in Figure 1. Without author
keywords, we observe that all models achieve gains, but only seq2seq+copy does
yield significant improvements. With author keywords, seq2seq+copy is again
the only model that achieves significance, while the others show mixed
results, sometimes even degrading scores. One likely explanation for this is
that these models produce keyphrases that cause documents to drift away from
their original meaning. We note that results are close to optimal for $N=5$,
supporting our initial setting for this parameter.

$32$$33$$34$$T$+$A$ (a)

$N$=0 123456789$35$$36$$37$$T$+$A$+$K$ s2s+copy s2s+corr mp${-}$rank (b)

Figure 1: MAP scores for BM25+RM3 w.r.t. the number $N$ of predicted
keyphrases. $\square$ denotes significance.

From our experiments, it appears that unsupervised keyphrase extraction is not
effective enough to significantly improve retrieval effectiveness. The fact
that keyphrase generation does so, suggests that the ability to predict absent
keyphrases may be what enables better performance. Yet counter-intuitively, we
found that most of the gains in retrieval effectiveness are due to the high
extractive accuracy of keyphrase generation models. Results in Table 4 show
that expanding documents with only absent keyphrases is at best useless and at
worst harmful, while using only present keyphrases brings significant
improvements. We draw two conclusions from this. First, absent keyphrases may
not be useful in practice unless they are tied to some form of domain
terminology to prevent semantic drift. Second, as generation does not yield
improvements, keyphrase extraction models may be worth further investigation.
In particular, supervised models could theoretically provide similar results
while being easier to train.

Model |  $T$+$A$ (cf. 31.93) |  $T$+$A$+$K$ (cf. 35.17)
---|---|---
pres. | abs. | pres. | abs.
s2s+copy |  †34.17† |  32.14 |  †36.30† |  34.97
s2s+corr | 32.97 | 31.96 | 36.09 | 34.77

Table 4: MAP scores for BM25+RM3 using the top-5 present or absent keyphrases.
$\dagger$ indicates significance over indexing without predicted keyphrases.

Neural models for keyphrase generation exhibit a limited generalization
ability, which means that their performance degrades on documents that differ
from the ones encountered during training (cf. Table 2, columns 3 and 4). To
quantify how much this affects retrieval effectiveness, we divided the queries
into two disjoint sets: in-domain for those that belong to research fields
present in KP20k, and out-domain for the others. Results are presented in
Table 5. The first thing we notice is the overall lower performance of out-
domain queries, which may be explained by the unbalanced distribution of
domains in the NTCIR-2 collection. Most importantly, out-domain queries on
full indexing (i.e. $T$+$A$+$K$) is the only configuration in which no
significant gains in retrieval effectiveness are achieved. This last
experiment shows that expanding documents using existing keyphrase generation
models may be ineffective in the absence of in-domain training data, and
stresses the need of domain adaptation for keyphrase generation.

Model |  $T$+$A$ |  $T$+$A$+$K$
---|---|---
I (32.70) |  O (30.99) |  I (36.18) |  O (33.93)
s2s+copy |  †35.40† |  †32.96† |  †38.13† | 34.55
s2s+corr | 33.49 | 32.92 | 37.13 | 34.25
mp-rank | 32.73 | 31.71 | 36.74 | 33.26

Table 5: MAP scores for BM25+RM3 on in-domain (I) and out-domain (O) queries.
$\dagger$ indicates significance over w/o keyphrases whose scores are in
parentheses.

##  4 Conclusion

We presented the first study of the usefulness of keyphrase generation for
scientific document retrieval. Our results show that keyphrases can
significantly improve retrieval effectiveness, and also highlight the
importance of evaluating keyphrase generation models from an extrinsic
perspective. Other retrieval tasks may also benefit from using keyphrase
information and we expect our results to serve as a basis for further
improvements.

## Acknowledgements

We thank the anonymous reviewers for their valuable comments. This work was
supported by the IKEBANA project (grant of Atlanstic 2020) and the French
National Research Agency (ANR) through the DELICES project
(ANR-19-CE38-0005-01).

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  * Zhai (1997) Chengxiang Zhai. 1997.  Fast statistical parsing of noun phrases for document indexing.  In _Fifth Conference on Applied Natural Language Processing_ , pages 312–319, Washington, DC, USA. Association for Computational Linguistics.
  * Zhao and Zhang (2019) Jing Zhao and Yuxiang Zhang. 2019.  Incorporating linguistic constraints into keyphrase generation.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5224–5233, Florence, Italy. Association for Computational Linguistics.

##  Appendix A Supplementary material

###  A.1 Related Work

#### Keyphrase extraction and generation

Identifying keyphrases for a given document is a long standing task in NLP.
Earlier work typically involves two steps: 1) extracting keyphrase candidates;
and 2) ranking those candidates by importance. Models mainly differ in how
they do the latter, commonly used techniques being supervised learning Witten
et al. (1999); Turney (2003); Nguyen and Kan (2007); Jiang et al. (2009) and
graph-based methods Mihalcea and Tarau (2004); Wan and Xiao (2008); Bougouin
et al. (2013); Florescu and Caragea (2017). These models are, however,
inherently limited in the sense that they can only output keyphrases that
appear in the text. To allow the prediction of keyphrases describing implicit
topics or using different wordings, previous work relied on external resources
like controlled vocabularies Witten and Medelyan (2006); Bougouin et al.
(2016), while recent attempts leveraged neural generative models Meng et al.
(2017); Chen et al. (2018); Zhao and Zhang (2019).

#### Biomedical indexing

Also related to our work is the research done on biomedical semantic indexing
using MeSH333https://www.nlm.nih.gov/mesh/, a hierarchically-organized
controlled vocabulary. Automated methods for assigning MeSH terms make use of
all sorts of techniques, such as pattern matching Aronson et al. (2004) or
learning to rank Liu et al. (2015); Peng et al. (2016).

#### Document expansion

Our work is similar in nature to previous research on document expansion Tao
et al. (2006); Efron et al. (2012), and is closely related to recent work on
document expansion using automatically generated queries Nogueira et al.
(2019).

###  A.2 Parameters

Table 6 displays the model parameters we use for seq2seq+copy and
seq2seq+corr.

Parameter | Value
---|---
Network | bi-GRU
Vocabulary size | 50K
Word embedding size | 150
Hidden layer | 2
Hidden layer size | 300
Optimizer | Adam
Initial learning rate | $10^{-4}$
Gradient clipping | 0.1
Dropout | 0.5
Beam depth | 6
Beam size | 200
Table 6: Model Parameters.

Table 7 presents the research fields used for dividing queries into two sets.

Research field | In | Out
---|---|---
Electricity, information and control | ✓ | -
Chemistry | ✓ | -
Architecture, civil engineering | - | ✓
Biology and agriculture | - | ✓
Science | ✓ | -
Engineering | ✓ | -
Medicine and dentistry | - | ✓
Cultural and social science | - | ✓
# of queries | 27 | 22
Table 7: Research fields for in- and out-domain queries.

###  A.3 Example

An example of document along with automatically generated keyphrases is shown
in Table A.3.

title |  Grammatical Inference for Concept Acquisition from Documents.
---|---
abstract |  The purpose of this study is to acquire knowledge from large scale natural language documents. There are two types of knowledge in the documents. One is explicitly represented knowledge which is acquired using natural language processing. The other is implicit constrain. In this paper, how to acquire implicit constraint using grammatical inference from the documents is described. We propose a grammatical inference system which uses inference rules based on logic, and show that the system can learn easy pattern of character lists. We also discuss its application to knowledge acquisition from real documents.
gold |  grammatical inference // knowledge acquisition // logic :progamming // concept learning
s2s+copy |  grammatical inference // knowledge // grammatical // knowledge representation // natural language processing
s2s+corr |  grammatical inference // knowledge acquisition // concept acquisition // inference rules // natural language processing
mp-rank |  grammatical inference // documents // knowledge // large scale // concept acquisition

Table 8: Example document (id: gakkai-e-0001014453 from ntc2-e1g) with author
keywords (gold) and automatically generated keyphrases. We note a typo in the
gold annotation (progamming).


# A Carleson type measure and a family of Möbius invariant function spaces

Guanlong Bao and Fangqin Ye  Guanlong Bao
Shantou University
Shantou, Guangdong 515063, China<EMAIL_ADDRESS>Fangqin Ye
Shantou University
Shantou 515063, Guangdong, China<EMAIL_ADDRESS>
###### Abstract.

For $0<s<1$, let $\\{z_{n}\\}$ be a sequence in the open unit disk such that
$\sum_{n}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson measure. In
this paper, we consider the connections between this $s$-Carleson measure and
the theory of Möbius invariant $F(p,p-2,s)$ spaces by the Volterra type
operator, the reciprocal of a Blaschke product, and second order complex
differential equations having a prescribed zero sequence.

###### Key words and phrases:

$F(p,p-2,s)$ space, Carleson type measure, Blaschke product, Volterra type
operator, complex differential equation

###### 2010 Mathematics Subject Classification:

30H25; 30J10; 34C10; 47G10

The work was supported by NNSF of China (No. 12001352 and No. 12271328) and
Guangdong basic and applied basic research foundation (No. 2022A1515012117).

##  1\. Introduction

Let $\mathbb{D}$ be the open unit disk in the complex plane $\mathbb{C}$. A
function $\varphi$ is said to be a Möbius map if $\varphi$ is a one-to-one
analytic function that maps $\mathbb{D}$ onto itself. Möbius maps form a group
with respect to the composition of mappings. This group is said to be the
Möbius group and denoted by ${\rm Aut}(\mathbb{D})$. For $a\in\mathbb{D}$, the
following special Möbius map defined by

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

exchanges points $0$ and $a$. It is well known that every $\varphi$ in ${\rm
Aut}(\mathbb{D})$ can be represented as $\varphi=e^{i\theta}\sigma_{a}$ for
some real number $\theta$ and some $a$ in $\mathbb{D}$. Clearly, the Möbius
group ${\rm Aut}(\mathbb{D})$ is homeomorphic to
$\mathbb{D}\times\partial\mathbb{D}$.

A classical topic in complex analysis is to investigate the theory of Möbius
invariant function spaces. Let $H(\mathbb{D})$ be the space of functions
analytic in $\mathbb{D}$. A space $X$ contained in $H(\mathbb{D})$ is a Möbius
invariant function space if it is equipped with a semi-norm $\rho$ such that
$f\circ\varphi\in X$ and $\rho(f\circ\varphi)\lesssim\rho(f)$ for all $f\in X$
and all $\varphi\in{\rm Aut}(\mathbb{D})$. In this case, it is known that
there is another semi-norm $\rho^{\prime}$ on $X$ satisfying that $\rho$ and
$\rho^{\prime}$ are equivalent, and
$\rho^{\prime}(f\circ\varphi)=\rho^{\prime}(f)$ for all $f\in X$ and
$\varphi\in{\rm Aut}(\mathbb{D})$. See [2] for the theory on Möbius invariant
function spaces.

In this paper, we consider Möbius invariant function spaces $F(p,p-2,s)$. For
$0<p<\infty$ and $0<s<\infty$, the space $F(p,p-2,s)$ is the set of functions
$f\in H(\mathbb{D})$ for which

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

where $dm(z)=1/\pi dxdy$, $z=x+iy$. It is known that if $p+s\leq 1$, then
$F(p,p-2,s)$ is trivial; that is, $F(p,p-2,s)$ consists only of constant
functions. Since

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

for every $f\in F(p,p-2,s)$ and $\varphi\in\text{Aut}(\mathbb{D})$,
$F(p,p-2,s)$ is Möbius invariant. For $p=2$ and $s=1$, $F(p,p-2,s)$ is equal
to $BMOA$, the well-known space of analytic functions in the Hardy space
$H^{1}$ whose boundary values having bounded mean oscillation on
$\partial\mathbb{D}$ (cf. [4, 19]). For $p=2$, $F(p,p-2,s)$ is the well-
studied space $\mathcal{Q}_{s}$ (cf. [3, 36, 37]). For $s>1$, all $F(p,p-2,s)$
spaces are the same and equal to the Bloch space $\mathcal{B}$. Recall that
$\mathcal{B}$ is the space of functions $f\in H(\mathbb{D})$ such that

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

$F(p,p-2,s)$ spaces are special cases of a general family of analytic function
spaces $F(p,q,s)$ that were introduced by R. Zhao [40] and further
investigated in [35]. See [41] for a recent survey on $F(p,q,s)$ spaces. We
also refer to [5, 7, 34] for some recent investigations on $F(p,p-2,s)$
spaces.

Carleson type measures are key tools for the modern function theory and
operator theory. For an arc $I$ of $\partial\mathbb{D}$ with arclength $|I|$,
the Carleson box $S(I)$ is given by

$S(I)=\left\\{r\zeta\in\mathbb{D}:1-\frac{|I|}{2\pi}<r<1,\ \zeta\in
I\right\\}.$

For $0<s<\infty$, a nonnegative Borel measure $\mu$ on $\mathbb{D}$ is said to
be an $s$-Carleson measure if $\mu(S(I))\lesssim|I|^{s}$ for all
$I\subseteq\partial\mathbb{D}$. When $s=1$, we obtain the classical Carleson
measure that was introduced by L. Carleson [9, 10] to solve the problem of
interpolation by functions in $H^{\infty}$ and the corona problem. Here
$H^{\infty}$ is the space of bounded analytic functions in $\mathbb{D}$ and we
will write $\|f\|_{\infty}=\sup_{z\in\mathbb{D}}|f(z)|$ for $f\in H^{\infty}$.
It is also well known (cf. [36]) that $\mu$ is an $s$-Carleson measure if and
only if

$\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\left(\frac{1-|a|^{2}}{|1-\overline{a}z|^{2}}\right)^{s}d\mu(z)<\infty.$
(1.1)

For $X\subseteq H^{\infty}$ and a sequence $\\{z_{n}\\}$ in $\mathbb{D}$,
$\\{z_{n}\\}$ is called an interpolating sequence for $X$ if for every bounded
sequence $\\{\zeta_{n}\\}$ of complex numbers, there is $f\in X$ with
$f(z_{n})=\zeta_{n}$ for each $n$. By L. Carleson [9], $\\{z_{n}\\}$ is an
interpolating sequence for $H^{\infty}$ if and only if $\\{z_{n}\\}$ is
uniformly separated; that is, there exists a positive constant $\gamma$
satisfying

$\inf_{m}\prod_{n\neq
m}\left|\frac{z_{m}-z_{n}}{1-\overline{z_{m}}z_{n}}\right|\geq\gamma.$ (1.2)

A sequence $\\{z_{n}\\}$ in $\mathbb{D}$ is said to be separated if
$\inf_{n\not=k}\rho(z_{n},z_{k})>0$, where
$\rho(z_{n},z_{k})=|\sigma_{z_{n}}(z_{k})|$ is the pseudo-hyperbolic metric
between $z_{n}$ and $z_{k}$. For $0<r<1$ and $a\in\mathbb{D}$, denote by
$\Delta(a,r)=\\{z\in\mathbb{D}:|\rho(a,z)|<r\\}$ the pseudo-hyperbolic disk of
center $a$ and radius $r$. It is well known (cf. [18]) that $\\{z_{n}\\}$ in
$\mathbb{D}$ is uniformly separated if and only if $\\{z_{n}\\}$ is separated
and $\sum_{n}(1-|z_{n}|)\delta_{z_{n}}$ is a Carleson measure, where
$\delta_{z_{n}}$ is the unit point-mass measure at $z_{n}$. For a sequence
$\\{z_{n}\\}$ in $\mathbb{D}$, it is known (cf. [30]) that
$\sum_{n}(1-|z_{n}|)\delta_{z_{n}}$ is a Carleson measure if and only if
$\\{z_{n}\\}$ is finite unions of interpolation sequences for $H^{\infty}$.
See [14, 28, 29] for the study of $\\{z_{n}\\}$ in $\mathbb{D}$ satisfying
that $\sum_{n}(1-|z_{n}|)\delta_{z_{n}}$ is a Carleson measure.

For $0<s<1$, let $\\{z_{n}\\}$ be a sequence in $\mathbb{D}$ such that
$\sum_{n}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson measure. This
kind of Carleson type measures defined by sequences is very useful in the
study of some theory of $F(p,p-2,s)$ spaces. F. Pérez-González and J. Rättyä
[32] used them to characterize inner functions in some $F(p,p-2,s)$ spaces. G.
Bao and J. Pau [5] constructed some examples based on these Carleson type
measures to show that the multiplier from some $F(p,p-2,s)$ to another is only
the zero function. Ch. Yuan and C. Tong [39], and R. Qian and F. Ye [34] used
these Carleson type measures to describe interpolating sequences for
$H^{\infty}\cap F(p,p-2,s)$ with certain ranges of parameters $p$ and $s$.
These Carleson type measures are also used to investigate solutions of second
order complex differential equations having prescribed zeros (cf. [22, 38]).
Recently, in [7] the authors applied these Carleson type measures to construct
Blaschke products for considering the proper inclusion relation associated
with intersections and unions of some $F(p,p-2,s)$ spaces.

The aim of this paper is to consider further the connections between the
$F(p,p-2,s)$ theory and sequence $\\{z_{n}\\}$ in $\mathbb{D}$ such that
$\sum_{n}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson measure. Our
results involve the Volterra type operator from $H^{\infty}$ to $F(p,p-2,s)$,
the reciprocal of a related Blaschke product, and solutions of second order
complex differential equations having prescribed zeros $\\{z_{n}\\}$.

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
$a\lesssim b\lesssim a$.

##  2\. Volterra type operators from $H^{\infty}$ to $F(p,p-2,s)$

In this section, we give a relation between the Volterra type operator from
$H^{\infty}$ to $F(p,p-2,s)$, and $\\{a_{k}\\}$ in $\mathbb{D}$ satisfying
that $\sum_{k=1}^{\infty}(1-|a_{k}|^{2})^{s}\delta_{a_{k}}$ is an $s$-Carleson
measure. Our proof also yields that the range of the Cesàro operator acting on
$H^{\infty}$ is contained in every non-trivial $F(p,p-2,s)$ space, which
strengthens some previous results from the literature.

The Volterra type operator $J_{g}$ with symbol $g\in H(\mathbb{D})$ defined on
$H(\mathbb{D})$ by

$J_{g}f(z)=\int_{0}^{z}f(w)g^{\prime}(w)dw,\ \ z\in\mathbb{D},\ \ f\in
H(\mathbb{D}).$

For the companion operator

$I_{g}f(z)=\int_{0}^{z}f^{\prime}(w)g(w)dw,\ \ z\in\mathbb{D},\ \ f\in
H(\mathbb{D}),$

and the multiplier operator

$M_{g}f(z)=g(z)f(z),\ \ z\in\mathbb{D},\ \ f\in H(\mathbb{D}),$

it is clear that

$M_{g}f(z)=f(0)g(0)+J_{g}f(z)+I_{g}f(z).$ (2.1)

Ch. Pommerenke [33] first studied $J_{g}$ and proved that $J_{g}$ is a bounded
operator on the Hardy space $H^{2}$ if and only if $g\in BMOA$. Later, the
operator $J_{g}$ was studied systematically in [1]. Operators $J_{g}$ and
$I_{g}$ have attracted a lot of interest. Here we mention the paper [31]
studying these operators related to $F(p,q,s)$ spaces.

An inner function $f$ is an $H^{\infty}$ function whose boundary values
satisfy $|f(e^{i\theta})|=1$ almost everywhere. A sequence $\\{a_{k}\\}$ in
$\mathbb{D}$ is called a Blaschke sequence if

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

This condition ensures the convergence of the Blaschke product:

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

All Blaschke products are inner functions. If the zero sequence of a Blaschke
product $B$ is uniformly separated, then $B$ is said to be an interpolating
Blaschke product. A Blaschke product is called a Carleson-Newman Blaschke
product if it can be expressed as a product of finitely many interpolating
Blaschke products. Recall that a Blaschke product associated with a sequence
$\\{a_{k}\\}$ in $\mathbb{D}$ is a Carleson-Newman Blaschke product if and
only if $\sum_{k}(1-|a_{k}|^{2})\delta_{a_{k}}$ is a 1-Carleson measure.

The following result is Theorem 1.4 in [32] characterizing inner functions in
some $F(p,p-2,s)$ spaces.

######  Theorem A.

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 $\\{z_{k}\\}_{k=1}^{\infty}$ in $\mathbb{D}$ which satisfies that
$\sum_{k}(1-|z_{k}|)^{s}\delta_{z_{k}}$ is an $s$-Carleson measure.

We give the following conclusion first.

######  Theorem 2.1.

Suppose $g\in H(\mathbb{D})$, $0<p<\infty$ and $0<s<\infty$ satisfying
$p+s>1$. Then the following statements hold:

  1. (a)

if $s>1$, or $s=1$ and $p\geq 2$, then $I_{g}$ is a bounded operator from
$H^{\infty}$ to $F(p,p-2,s)$ if and only if $g\in H^{\infty}$;

  2. (b)

if $0<s<1$, or $s=1$ and $0<p<2$, then $I_{g}$ is a bounded operator from
$H^{\infty}$ to $F(p,p-2,s)$ if and only if $g=0$;

  3. (c)

$J_{g}$ is a bounded operator from $H^{\infty}$ to $F(p,p-2,s)$ if and only if
$g\in F(p,p-2,s)$;

  4. (d)

if $s>1$, or $s=1$ and $p\geq 2$, then $M_{g}$ is a bounded operator from
$H^{\infty}$ to $F(p,p-2,s)$ if and only if $g\in H^{\infty}$;

  5. (e)

if $0<s<1$, or $s=1$ and $0<p<2$, then $M_{g}$ is a bounded operator from
$H^{\infty}$ to $F(p,p-2,s)$ if and only if $g=0$.

###### Proof.

(a) Note that $F(p_{1},p_{1}-2,s_{1})\subseteq F(p_{2},p_{2}-2,s_{2})$ for all
possible $0<p_{1}\leq p_{2}$ and $0<s_{1}\leq s_{2}$. It is also known that
$H^{\infty}\subseteq BMOA\subseteq\mathcal{B}$, $BMOA=F(2,0,1)$, and
$F(p,p-2,s)=\mathcal{B}$ for $s>1$. Now consider $s>1$, or $s=1$ and $p\geq
2$. Then $H^{\infty}\subseteq F(p,p-2,s)$ for such $p$ and $s$. Let $g\in
H^{\infty}$. For any $f\in H^{\infty}$, then $f$ also belongs to $F(p,p-2,s)$.
Thus

$\|I_{g}f\|^{p}_{F(p,p-2,s)}\leq\|g\|_{\infty}^{p}\|f\|^{p}_{F(p,p-2,s)}\lesssim\|g\|_{\infty}^{p}\|f\|_{\infty}^{p},$

which yields the boundedness of $I_{g}$ from $H^{\infty}$ to $F(p,p-2,s)$. On
the other hand, suppose $I_{g}$ is bounded from $H^{\infty}$ to $F(p,p-2,s)$.
Note that $\sup_{b\in\mathbb{D}}\|\sigma_{b}\|_{\infty}\leq 1$. Then

$\displaystyle\infty$
$\displaystyle>\sup_{b\in\mathbb{D}}\|I_{g}\sigma_{b}\|^{p}_{F(p,p-2,s)}$
$\displaystyle\geq\sup_{b\in\mathbb{D}}\sup_{a\in\mathbb{D}}\int_{\Delta(b,1/2)}\frac{(1-|b|^{2})^{p}}{|1-\overline{b}z|^{2p}}|g(z)|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dm(z).$

It is well known (cf. [42, p. 69] and [42, Lemma 4.30]) that

$1-|z|\thickapprox 1-|b|\thickapprox|1-\overline{b}z|$

for all $z\in\Delta(b,1/2)$, and
$|1-\overline{z}a|\thickapprox|1-\overline{b}a|$ for all $z\in\Delta(b,1/2)$
and all $a\in\mathbb{D}$. Also the area of $\Delta(b,1/2)$ is comparable with
$(1-|b|)^{2}$. Combining these facts and the subharmonicity of $|g|^{p}$, we
obtain

$1\gtrsim\sup_{a\in\mathbb{D}}|g(b)|^{p}(1-|\sigma_{a}(b)|^{2})^{s}=|g(b)|^{p}$

for all $b\in\mathbb{D}$. Thus $g\in H^{\infty}$.

(b) First, consider $0<s<1$. Suppose $I_{g}$ is a bounded operator from
$H^{\infty}$ to $F(p,p-2,s)$. By the proof of (a), we get $g\in H^{\infty}$.
For every $e^{i\theta}\in\partial\mathbb{D}$, from Lemma 3.3 in [5], there
exists a Blaschke sequence $\\{a_{k}\\}_{k=1}^{\infty}$ in $\mathbb{D}$ such
that $e^{i\theta}$ is the unique accumulation point of
$\\{a_{k}\\}_{k=1}^{\infty}$,

$\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}(1-|\sigma_{a}(a_{k})|^{2})^{\frac{1+s}{2}}<\infty,$
(2.2)

and

$\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}(1-|\sigma_{a}(a_{k})|^{2})^{s}=+\infty.$
(2.3)

Condition (2.2) implies that $\sum_{k=1}^{\infty}(1-|a_{k}|)\delta_{a_{k}}$ is
a Carleson measure. Hence $\\{a_{k}\\}_{k=1}^{\infty}$ is finite unions of
interpolation sequences for $H^{\infty}$. Then there exists a positive integer
$m$ and positive real numbers $\gamma_{i}$, $i=1,2,\cdots,m$, such that

$\\{a_{k}\\}=\bigcup_{1\leq i\leq m}\\{a_{ik}\\},\ \ \text{and}\ \
\inf_{\ell}\prod_{j\not=\ell}\rho(a_{ij},a_{il})\geq\gamma_{i},\ \
i=1,2,\cdots,m.$

Because of (2.3), among these sequences $\\{a_{ik}\\}_{k=1}^{\infty}$,
$i=1,2,\cdots,m$, there exists a sequence $\\{a_{i_{0}k}\\}_{k=1}^{\infty}$
such that

$\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}(1-|\sigma_{a}(a_{i_{0}k})|^{2})^{s}=+\infty.$
(2.4)

Of course, $e^{i\theta}$ is also the unique accumulation point of
$\\{a_{i_{0}k}\\}_{k=1}^{\infty}$. Denote by $B$ the Blaschke product
associated with the sequence $\\{a_{i_{0}k}\\}_{k=1}^{\infty}$. By a well-
known fact of interpolating sequences for $H^{\infty}$ (cf. [21, p. 681]), we
know

$\bigcup_{k=1}^{\infty}\Delta\left(a_{i_{0}k},\frac{\gamma_{i_{0}}}{4}\right)\subseteq\left\\{z\in\mathbb{D}:(1-|z|)|B^{\prime}(z)|\geq\frac{\gamma_{i_{0}}(1-\gamma_{i_{0}})}{8}\right\\}.$

Clearly, pseudo-hyperbolic disks
$\Delta\left(a_{i_{0}k},\frac{\gamma_{i_{0}}}{4}\right)$ are pairwise
disjoint. Consequently,

$\displaystyle\infty>$ $\displaystyle\|I_{g}B\|_{F(p,p-2,s)}^{p}$
$\displaystyle=$
$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|B^{\prime}(z)|^{p}|g(z)|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dm(z)$
$\displaystyle\geq$
$\displaystyle\sup_{a\in\mathbb{D}}\int_{\left\\{z\in\mathbb{D}:(1-|z|)|B^{\prime}(z)|\geq\frac{\gamma_{i_{0}}(1-\gamma_{i_{0}})}{8}\right\\}}|B^{\prime}(z)|^{p}|g(z)|^{p}$
$\displaystyle\times(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dm(z)$
$\displaystyle\gtrsim$
$\displaystyle\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}\int_{\Delta\left(a_{i_{0}k},\frac{\gamma_{i_{0}}}{4}\right)}|g(z)|^{p}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)$
$\displaystyle\gtrsim$
$\displaystyle\sup_{a\in\mathbb{D}}\sum_{k=1}^{\infty}|g(a_{i_{0}k})|^{p}(1-|\sigma_{a}(a_{i_{0}k})|^{2})^{s}.$

Combining this with (2.4), we get
$|g(e^{i\theta})|=\lim_{k\rightarrow\infty}|g(a_{i_{0}k})|=0$. Due to the
arbitrariness of $e^{i\theta}$ and the maximum modulus principle, we get
$g\equiv 0$. Conversely, if $g\equiv 0$, it is clear that $I_{g}$ is bounded
from $H^{\infty}$ to $F(p,p-2,s)$.

Second, consider the case of $s=1$ and $0<p<2$. Also, $g\equiv 0$ implies
clearly the boundedness of $I_{g}$ from $H^{\infty}$ to $F(p,p-2,s)$. Next, we
follow a method from [11, p. 177]. Let $I_{g}$ be bounded from $H^{\infty}$ to
$F(p,p-2,s)$. As proved in part (a), $g\in H^{\infty}$. Suppose $g\not\equiv
0$. Then there exists a set $E$ in $[0,2\pi]$ with positive Lebesgue measure
such that $\lim_{r\rightarrow 1}g(re^{i\theta})\not=0$ for every $\theta\in
E$. For $0<p<2$, it is known from [20, Theorem 1] that there exists $f\in
H^{\infty}$ such that

$\int_{0}^{1}|f^{\prime}(re^{i\theta})|^{p}(1-r)^{p-1}dr=+\infty,$ (2.5)

for almost every $\theta\in[0,2\pi]$; that is, (2.5) holds when $\theta$
belongs to some set $F$ in $[0,2\pi]$ and the Lebesgue measure of $F$ is
$2\pi$. Thus for any $\theta\in E\cap F$, there exists an $r(\theta)$ in
$(0,1)$ satisfying that $\inf_{r(\theta)<r<1}|g(re^{i\theta})|>0$ and hence

$\displaystyle\int_{0}^{1}|f^{\prime}(re^{i\theta})|^{p}|g(re^{i\theta})|^{p}(1-r)^{p-1}dr$
$\displaystyle\geq$
$\displaystyle\inf_{r(\theta)<r<1}|g(re^{i\theta})|\int_{r(\theta)}^{1}|f^{\prime}(re^{i\theta})|^{p}(1-r)^{p-1}dr$
$\displaystyle=$ $\displaystyle+\infty.$

Clearly, the Lebesgue measure of $E\cap F$ is also positive. We get

$\int_{\mathbb{D}}|f^{\prime}(z)|^{p}|g(z)|^{p}(1-|z|)^{p-1}dm(z)=+\infty.$
(2.6)

But the boundedness of $I_{g}$ from $H^{\infty}$ to $F(p,p-2,s)$ gives

$\displaystyle\infty>$
$\displaystyle\|I_{g}f\|_{F(p,p-2,1)}^{p}\geq\int_{\mathbb{D}}|f^{\prime}(z)|^{p}|g(z)|^{p}(1-|z|^{2})^{p-1}dm(z),$

which contradicts (2.6). Hence $g\equiv 0$.

(c) Suppose $J_{g}$ is bounded from $H^{\infty}$ to $F(p,p-2,s)$. Set
$f(z)\equiv 1$. Then $J_{g}f\in F(p,p-2,s)$. Since $g(z)=J_{g}f(z)+g(0)$, we
get $g\in F(p,p-2,s)$. Conversely, let $g\in F(p,p-2,s)$. For $h\in
H^{\infty}$, it is clear that
$\|J_{g}h\|_{F(p,p-2,s)}^{p}\leq\|h\|_{\infty}^{p}\|g\|_{F(p,p-2,s)}^{p}$,
which gives the bondedness of $J_{g}$ from $H^{\infty}$ to $F(p,p-2,s)$.

(d) Let $M_{g}$ be a bounded operator from $H^{\infty}$ to $F(p,p-2,s)$.
Because constant functions are in $H^{\infty}$, we get $g\in F(p,p-2,s)$. By
(c), $J_{g}$ is bounded from $H^{\infty}$ to $F(p,p-2,s)$. Because of (2.1),
$I_{g}$ is also bounded from $H^{\infty}$ to $F(p,p-2,s)$. From (a), we get
$g\in H^{\infty}$. Conversely, let $g\in H^{\infty}$. Since $s>1$, or $s=1$
and $p\geq 2$, $g$ also belongs to $F(p,p-2,s)$. By (a), (c) and (2.1), we get
the boundedness of $M_{g}$ from $H^{\infty}$ to $F(p,p-2,s)$.

The proof of (e) is similar to the proof of (d), so we omit it. ∎

The following result gives a connection between the Volterra type operator
from $H^{\infty}$ to $F(p,p-2,s)$ and the Carleson type measures we
considered.

######  Theorem 2.2.

Suppose $B$ is a Blaschke product with zeros $\\{a_{k}\\}_{k=1}^{\infty}$. Let
$0<s<1$ and $p>\max\\{s,1-s\\}$. Then the following conditions are equivalent:

  * (a)

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

  * (b)

$J_{B}$ is a bounded operator from $H^{\infty}$ to $F(p,p-2,s)$.

###### Proof.

This result is clearly from Theorem A and Theorem 2.1. ∎

If $g(z)=-\log(1-z)$, then $J_{g}$ is the modified Cesàro operator
$\widetilde{\mathcal{C}}$, namely,

$\widetilde{\mathcal{C}}f(z)=\int_{0}^{z}\frac{f(w)}{1-w}dw,\ \
z\in\mathbb{D},\ \ f\in H(\mathbb{D}).$

Note that the Cesàro operator $\mathcal{C}$ is defined by

$\mathcal{C}f(z)=\frac{1}{z}\int_{0}^{z}\frac{f(w)}{1-w}dw,\ \
z\in\mathbb{D},\ \ f\in H(\mathbb{D}).$

N. Danikas and A. Siskakis [12] showed that $\mathcal{C}(H^{\infty})\nsubseteq
H^{\infty}$ but $\mathcal{C}(H^{\infty})\subseteq BMOA$. M. Essén and J. Xiao
[16] obtained that $\mathcal{C}(H^{\infty})\subseteq\mathcal{Q}_{s}$ for
$0<s<1$. Note that $F(p,p-2,s)\subsetneqq\mathcal{Q}_{s}$ when $0<p<2$ and
$\max\\{1-p,0\\}<s\leq 1$. We obtain smaller Möbius invariant function spaces
closing to $\mathcal{C}(H^{\infty})$ as follows.

######  Theorem 2.3.

Let $0<p<\infty$ and $0<s<\infty$ with $p+s>1$. Then
$\mathcal{C}(H^{\infty})\subseteq F(p,p-2,s)$.

###### Proof.

It is known (cf. [31]) that the function $g(z)=-\log(1-z)$ belongs to all
nontrivial $F(p,p-2,s)$ spaces. By Theorem 2.1, $J_{g}$ is a bounded operator
from $H^{\infty}$ to $F(p,p-2,s)$; that is,
$\widetilde{\mathcal{C}}(H^{\infty})\subseteq F(p,p-2,s)$. Hence
$\mathcal{C}(H^{\infty})\subseteq F(p,p-2,s)$. ∎

One can refer to [6, 8, 17] for more results on the range of the Cesàro
operator or Cesàro-like operators acting on $H^{\infty}$.

##  3\. $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ via the reciprocal of a
Blaschke product in $F(p,p-2,s)$

Let $\\{z_{n}\\}_{n=1}^{\infty}$ be a separated Blaschke sequence in
$\mathbb{D}$ and let $B$ be the Blaschke product associated with
$\\{z_{n}\\}_{n=1}^{\infty}$. Suppose $0<p<2$. By Theorem C in [29],
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a Carleson measure if
and only if

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{1}{|B(\varphi(z))|^{p}}dm(z)<\infty.$
(3.1)

Clearly, (3.1) is equivalent to

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\left(\frac{1}{|B(\varphi(z))|}-1\right)^{p}dm(z)<\infty.$
(3.2)

In this section, for $0<s<1$ and a separated Blaschke sequence
$\\{z_{n}\\}_{n=1}^{\infty}$, based on the description of Blaschke products in
$F(p,p-2,s)$, we give a characterization of $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ in terms of the
reciprocal of the Blaschke product with zeros $\\{z_{n}\\}$. Indeed, this
Blaschke product belongs to some $F(p,p-2,s)$ spaces.

By (1.1), for $s>0$, $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is
an $s$-Carleson measure if and only if

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\sum_{n=1}^{\infty}\left(1-|\varphi(z_{n})|^{2}\right)^{s}<\infty.$
(3.3)

Let $\phi\in{\rm Aut}(\mathbb{D})$. From (3.3), if
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, then
$\sum_{n=1}^{\infty}(1-|\phi(z_{n})|^{2})^{s}\delta_{\phi(z_{n})}$ is also an
$s$-Carleson measure. In this sense, $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is Möbius invariant.
Hence the equivalent characterization we will give also shows this invariance.

We begin with the following auxiliary result.

######  Lemma 3.1.

Let $0<p<\infty$, $0\leq q<\infty$, and $0<s<1$ such that $p+s>1$. Suppose $B$
is a Blaschke product associated with $\\{z_{n}\\}_{n=1}^{\infty}$ in
$\mathbb{D}$. If

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{\left(1-|B(\varphi(z))|\right)^{p}}{|B(\varphi(z))|^{q}}(1-|z|^{2})^{s-2}dm(z)<\infty,$

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

###### Proof.

The Schwarz-Pick Lemma gives

$(1-|z|^{2})|B^{\prime}(z)|\leq 1-|B(z)|^{2}$

for all $z\in\mathbb{D}$. Combining this with the change of variables, we
deduce

$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|B^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dm(z)$
$\displaystyle\lesssim$
$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\frac{\left(1-|B(z)|\right)^{p}}{|B(z)|^{q}}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)$
$\displaystyle\lesssim$ $\displaystyle\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{\left(1-|B(\varphi(z))|\right)^{p}}{|B(\varphi(z))|^{q}}(1-|z|^{2})^{s-2}dm(z)$
$\displaystyle<$ $\displaystyle\infty.$

Hence $B\in F(p,p-2,s)$. Note that $0<p<\infty$ and $0<s<1$ with $p+s>1$. It
follows from Theorem 4.3 in [32] that
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure. ∎

The following useful lemma is from [27].

######  Lemma B.

If $B(z)$ is the Blaschke product associated with a sequence $\\{z_{n}\\}$
satisfying condition (1.2) for a given $\gamma$, and if $\epsilon>0$ is given,
there exists a constant $\gamma_{0}$ depending only on $\gamma$ and $\epsilon$
such that $|B(z)|\geq\gamma_{0}$ whenever $\rho(z,z_{n})\geq\epsilon$ for all
$n$.

Lemma C below is also well-known; see [38, p. 100] for a brief proof of it.

######  Lemma C.

Let $B(z)$ be the Blaschke product associated with a sequence $\\{z_{n}\\}$
satisfying condition (1.2) for a given $\gamma$. Then there exists a positive
constant $C$ depending only on $\gamma$ such that

$|B(z)|\geq C\rho(z,z_{k})$

for all $z\in\Delta(z_{k},\delta/4)$ and for every $k$.

The following conclusion is Corollary 2.4 in [32].

######  Lemma D.

Let $S$ be an inner function and let $1\leq p<\infty$, $-2<q<\infty$, and
$0\leq s,p^{*}<\infty$ such that $p>q+s+1>0$. Then, for any analytic function
$f$ in $\mathbb{D}$ and $a\in\mathbb{D}$, the following quantities are
comparable:

  * (a)

$\int_{\mathbb{D}}|f(z)|^{p^{*}}(1-|S(z)|^{2})^{p}(1-|z|^{2})^{q-p}(1-|\sigma_{a}(z)|^{2})^{s}dm(z);$

  * (b)

$\int_{\mathbb{D}}|f(z)|^{p^{*}}|S^{\prime}(z)|^{p}(1-|z|^{2})^{q}(1-|\sigma_{a}(z)|^{2})^{s}dm(z).$

Note that if $0<s<1$ and
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, then $\\{z_{n}\\}_{n=1}^{\infty}$ is finite unions of uniformly
separated sequences. Using Lemma B, Lemma C and Lemma D, we prove the
following result.

######  Lemma 3.2.

Suppose $0<s<1$ and $\\{z_{n}\\}_{n=1}^{\infty}$ is a sequence in $\mathbb{D}$
such that $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an
$s$-Carleson measure. Let $B$ be the Blaschke product associated with
$\\{z_{n}\\}_{n=1}^{\infty}$. Assume $1\leq p<\infty$ and $0\leq q<2/k$, where
$k$ is the number of unions of uniformly separated sequences as which
$\\{z_{n}\\}_{n=1}^{\infty}$ can be written. Then

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{\left(1-|B(\varphi(z))|\right)^{p}}{|B(\varphi(z))|^{q}}(1-|z|^{2})^{s-2}dm(z)<\infty.$

###### Proof.

Since $\\{z_{n}\\}_{n=1}^{\infty}$ is $k$ unions of uniformly separated
sequences, there exist positive real numbers $\gamma_{i}$, $i=1,2,\cdots,k$,
such that

$\\{z_{n}\\}_{n=1}^{\infty}=\bigcup_{1\leq i\leq
k}\\{z_{in}\\}_{n=1}^{\infty},$

and

$\inf_{\ell}\prod_{j\not=\ell}\rho(z_{ij},z_{il})\geq\gamma_{i},\ \
i=1,2,\cdots,k.$

Denote by $B_{i}$ the Blaschke product associated with
$\\{z_{in}\\}_{n=1}^{\infty}$.

By the change of variables, we get

$\displaystyle\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\frac{\left(1-|B(\varphi(z))|\right)^{p}}{|B(\varphi(z))|^{q}}(1-|z|^{2})^{s-2}dm(z)$
$\displaystyle=$
$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\frac{\left(1-|B(z)|\right)^{p}}{|B(z)|^{q}}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z).$
(3.4)

From a version of Hölder’s inequality that handles the product of more than
two functions (cf. [42, Theorem 3.4]), we get

$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\frac{\left(1-|B(z)|\right)^{p}}{|B(z)|^{q}}\frac{(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)$
$\displaystyle\leq$
$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\int_{\mathbb{D}}\frac{1}{|B_{i}(z)|^{qx_{i}}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}},$
(3.5)

where $x_{1}$, $\cdots$, $x_{k}$ are positive numbers such that

$\frac{1}{x_{1}}+\cdots+\frac{1}{x_{k}}=1.$ (3.6)

By Lemma B, for fixed $i=1,2,\cdots,k$, there exists a positive real number
$\alpha_{i}$ depending only on $\gamma_{i}$ such that
$|B_{i}(z)|\geq\alpha_{i}$ whenever $\rho(z,z_{in})\geq\gamma_{i}/4$ for all
$n$. Write

$\Omega=\mathbb{D}\setminus\left(\bigcup_{1\leq i\leq
k}\bigcup_{n=1}^{\infty}\Delta(z_{in},\frac{\gamma_{i}}{4})\right).$

Note that $q\geq 0$. Then

$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\int_{\Omega}\frac{1}{|B_{i}(z)|^{qx_{i}}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}$
$\displaystyle\leq$
$\displaystyle\prod_{i=1}^{k}\frac{1}{\alpha_{i}^{q}}\sup_{a\in\mathbb{D}}\left(\int_{\mathbb{D}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}.$
(3.7)

Bear in mind that $0<s<1$ and $1\leq p<\infty$. It follows from Lemma D and
Theorem A that

$\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\thickapprox\|B\|_{F(p,p-2,s)}^{p}<\infty.$
(3.8)

Thus, (3) and (3.8) yield

$\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\int_{\Omega}\frac{1}{|B_{i}(z)|^{qx_{i}}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}<\infty.$
(3.9)

On the other hand, by Lemma C, we deduce

$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\int_{\mathbb{D}\setminus\Omega}\frac{1}{|B_{i}(z)|^{qx_{i}}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}$
$\displaystyle\lesssim$
$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\sum_{i=1}^{k}\sum_{n=1}^{\infty}\int_{\Delta(z_{in},\frac{\gamma_{i}}{4})}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(\rho(z,z_{in}))^{qx_{i}}(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}$
$\displaystyle\lesssim$
$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\sum_{i=1}^{k}\sum_{n=1}^{\infty}\frac{(1-|\sigma_{a}(z_{in})|^{2})^{s}}{(1-|z_{in}|)^{2}}\int_{\Delta(z_{in},\frac{\gamma_{i}}{4})}(\rho(z,z_{in}))^{-qx_{i}}dm(z)\right)^{1/x_{i}}.$

If every $x_{i}$ satisfies $qx_{i}<2$, then a change of variables yields

$\displaystyle\int_{\Delta(z_{in},\frac{\gamma_{i}}{4})}(\rho(z,z_{in}))^{-qx_{i}}dm(z)\thickapprox(1-|z_{in}|)^{2}\int_{0}^{\frac{\gamma_{i}}{4}}r^{1-qx_{i}}dr\thickapprox(1-|z_{in}|)^{2}.$

Condition (3.6) and $qx_{i}<2$ for every $x_{i}$ yield the assumed condition
$q<2/k$. In fact, taking $x_{1}=\cdots=x_{k}=k$ is enough for this proof.
Combining these with (3.3), we get

$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\int_{\mathbb{D}\setminus\Omega}\frac{1}{|B_{i}(z)|^{qx_{i}}}\frac{\left(1-|B(z)|\right)^{p}(1-|\sigma_{a}(z)|^{2})^{s}}{(1-|z|^{2})^{2}}dm(z)\right)^{1/x_{i}}$
$\displaystyle\lesssim$
$\displaystyle\sup_{a\in\mathbb{D}}\prod_{i=1}^{k}\left(\sum_{n=1}^{\infty}(1-|\sigma_{a}(z_{n})|^{2})^{s}\right)^{1/x_{i}}<\infty.$
(3.10)

Joining (3), (3), (3.9) and (3), we get the desired result. ∎

Now we state the main result of this section as follows.

######  Theorem 3.3.

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

  * (a)

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

  * (b)

$\sup_{\varphi\in{\rm
Aut}(\mathbb{D})}\int_{\mathbb{D}}\left(\frac{1}{|B(\varphi(z))|}-1\right)^{p}(1-|z|^{2})^{s-2}dm(z)<\infty.$

###### Proof.

Note that if $\\{z_{n}\\}_{n=1}^{\infty}$ is a separated Blaschke sequence
such that $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an
$s$-Carleson measure for $0<s<1$, then $\\{z_{n}\\}_{n=1}^{\infty}$ is a
uniformly separated sequence. Set $q=p$ in Lemma 3.1 and Lemma 3.2. The
conclusion follows. ∎

Taking $s=1$ in (b) of Theorem 3.3, we do not get C. Nolder’s condition (3.2).
In the proof of Theorem 3.3, the characterization of Blaschke products in
$F(p,p-2,s)$ plays an important role, where the condition $0<s<1$ was used.

##  4\. $s$-Carleson measure
$\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ and $F(p,p-2,s)$ via
$f^{\prime\prime}+Af=0$

In this section, for $0<s<1$, $p>\max\\{s,1-s\\}$, $q>\max\\{s,1-s\\}$ and a
separated sequence $\\{z_{n}\\}_{n=1}^{\infty}$ in $\mathbb{D}$ satisfying
that $\sum_{n=1}^{\infty}(1-|z_{n}|^{2})^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, we show that there exists a function $A$ analytic in $\mathbb{D}$
such that $|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$ is an $s$-Carleson measure and
the equation $f^{\prime\prime}+Af=0$ admits a nontrivial solution $f\in
F(p,p-2,s)\cap H^{\infty}$ whose zero-sequence is
$\\{z_{n}\\}_{n=1}^{\infty}$. From the perspective of larger ranges of
parameters $p$ and $q$ given here, our result improves some previous
conclusions.

For $A\in H(\mathbb{D})$, it is well known that all solutions of the second
order complex differential equation

$f^{\prime\prime}+Af=0$ (4.1)

belong to $H(\mathbb{D})$. For a sequence $\\{z_{n}\\}$ of distinct points in
$\mathbb{D}$, if there exists $A\in H(\mathbb{D})$ such that (4.1) has a
solution $f$ with zeros precisely at the points $z_{n}$, then $\\{z_{n}\\}$ is
said to be a prescribed zero sequence (cf. [24, 26]). See [25] for a
historical review in this area. In fact, by (4.1), $A=-f^{\prime\prime}/f$,
which gives that any $z_{n}$ must be a simple zero of $f$; otherwise $A$ is
not analytic at $z_{n}$. We refer to [23] for some recent results on (4.1)
associated with Carleson measures.

For any positive integer $n$, recall that the Bloch space $\mathcal{B}$ is
also equal to the set of functions $f\in H(\mathbb{D})$ such that

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

For $0\leq\alpha<\infty$, the growth space $H^{\infty}_{\alpha}$ is the set of
functions $g\in H(\mathbb{D})$ satisfying

$\|g\|_{H^{\infty}_{\alpha}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|g(z)|<\infty.$

The following result is Lemma 1 in [22].

######  Lemma E.

Let $g\in H^{\infty}_{\alpha}$ for $0\leq\alpha<\infty$, and let $0<\gamma<1$.
If $g(z_{0})=0$ for some $z_{0}\in\mathbb{D}$, then there exists a positive
constant $C=C(\alpha,\gamma)$ such that

$|g(z)|\leq\frac{C\|g\|_{H^{\infty}_{\alpha}}\rho(z,z_{0})}{(1-|z_{0}|^{2})^{\alpha}},\
\ z\in\Delta(z_{0},\gamma).$

Proposition 4.1 below is of interest only if $H^{\infty}\nsubseteq X$. Its
proof is based on a theoretical abstraction from a well-known method in the
filed of second order complex differential equations with prescribed zero
sequences (cf. [25, p. 47] or [22, pp. 301-302]).

######  Proposition 4.1.

Let $X$ be a vector space of analytic functions in $\mathbb{D}$ satisfying
conditions (a), (b) and (c) below.

  1. (a)

Suppose $B$ is a Blaschke product associated with a sequence
$\\{a_{k}\\}_{k=1}^{\infty}$ in $\mathbb{D}$ satisfying $\inf_{m}\prod_{n\neq
m}\rho(a_{n},a_{m})\geq\gamma$ for some $\gamma\in(0,1)$ and $I$ is a Blaschke
product associated with a sequence $\\{b_{k}\\}_{k=1}^{\infty}$ in
$\mathbb{D}$ such that $\rho(a_{k},b_{k})\leq s$, $k=1,2,\cdots$, for some
constant $s\in(0,\gamma/2)$. If $B\in X$, then $I$ also belongs to $X$.

  2. (b)

If both $B$ and $I$ are interpolating Blaschke products in $X$ and $c$ is a
complex constant, then the function $\exp(cBI)$ belongs to $X$.

  3. (c)

If both $f$ and $g$ are in $H^{\infty}\cap X$, then $fg\in H^{\infty}\cap X$.

If $J$ is an interpolating Blaschke product associated with a sequence
$\\{z_{k}\\}_{k=1}^{\infty}$ in $\mathbb{D}$ and $J\in X$, then there exists a
function $A$ analytic in $\mathbb{D}$ such that
$\sup_{z\in\mathbb{D}}(1-|z|^{2})^{2}|A(z)|<\infty$ and the equation
$f^{\prime\prime}+Af=0$ admits a nontrivial solution $f\in H^{\infty}\cap X$
whose zero-sequence is $\\{z_{k}\\}_{k=1}^{\infty}$.

###### Proof.

Since $J$ is an interpolating Blaschke product associated with
$\\{z_{k}\\}_{k=1}^{\infty}$, there is a positive constant $\gamma_{1}$ such
that

$\inf_{n}\prod_{m\neq
n}\left|\frac{z_{m}-z_{n}}{1-\overline{z_{m}}z_{n}}\right|=\inf_{n}(1-|z_{n}|^{2})|J^{\prime}(z_{n})|\geq\gamma_{1}.$

Note that any bounded analytic function is a Bloch function. Then

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

Hence,

$\sup_{n}\frac{|J^{\prime\prime}(z_{n})|}{|J^{\prime}(z_{n})|^{2}}<\infty.$

Set

$\zeta_{n}=-\frac{J^{\prime\prime}(z_{n})}{2(J^{\prime}(z_{n}))^{2}},\ \ \ \
n=1,2,\cdots$

Then $\\{\zeta_{n}\\}$ is a bounded sequence. By Earl’s constructive proof of
Carleson’s interpolating theorem for $H^{\infty}$(see [15]), there exists a
complex constant $c$ depending only on $\gamma_{1}$ and a Blaschke product

$J_{1}(z)=\prod_{k=1}^{\infty}\frac{|\eta_{k}|}{\eta_{k}}\frac{\eta_{k}-z}{1-\overline{\eta_{k}}z}$

such that

$c(\sup_{k}|\zeta_{k}|)J_{1}(z_{n})=\zeta_{n},\ \ n=1,2,\cdots.$

Moreover, the zeros $\\{\eta_{n}\\}$ of the Blaschke product $J_{1}(z)$ can be
chosen to satisfy

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

By $J\in X$ and condition (a), we see that $J_{1}\in X$.

If $n\not=k$, then

$\displaystyle\rho(\eta_{n},\eta_{k})$ $\displaystyle\geq$
$\displaystyle\rho(z_{n},\eta_{k})-\rho(\eta_{n},z_{n})$ $\displaystyle\geq$
$\displaystyle\rho(z_{n},z_{k})-\rho(z_{k},\eta_{k})-\rho(\eta_{n},z_{n})$
$\displaystyle\geq$ $\displaystyle\gamma_{1}-\frac{2}{3}\gamma_{1}.$

Thus $\\{\eta_{n}\\}$ is separated. Since $\\{z_{k}\\}_{k=1}^{\infty}$ is
uniformly separated, $\sum_{n}(1-|z_{n}|^{2})\delta_{z_{n}}$ is a 1-Carleson
measure, namely,

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

Note that $\rho(z_{n},\eta_{n})\leq\gamma_{1}/3$ for all $n$. It is known (cf.
[42, p. 69] and [42, Lemma 4.30]) that

$1-|z_{n}|^{2}\thickapprox 1-|\eta_{n}|^{2},\ \
|1-\overline{a}z_{n}|\thickapprox|1-\overline{a}\eta_{n}|,$

for all $n$ and $a\in\mathbb{D}$. Consequently,

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

Then $\\{\eta_{n}\\}_{n=1}^{\infty}$ is also uniformly separated and hence
$J_{1}$ is also an interpolating Blaschke product.

Set $f=Je^{h}$, where $h=c(\sup_{k}|\zeta_{k}|)J_{1}J$. Then $f$ satisfies the
equation $f^{\prime\prime}+Af=0$, where $A\in H(\mathbb{D})$ and

$A=-\frac{f^{\prime\prime}}{f}=-\frac{J^{\prime\prime}+2J^{\prime}h^{\prime}}{J}-(h^{\prime})^{2}-h^{\prime\prime}.$

Condition (b) yields that $e^{h}\in X$. Clearly, $e^{h}\in H^{\infty}$. We get
$f\in H^{\infty}\cap X$ from condition (c). It is also clear that the zero set
of $f$ is $\\{z_{k}\\}_{k=1}^{\infty}$.

Since $J$ and $h$ are in $H^{\infty}$ which is a subset of $\mathcal{B}$,
$(1-|z|^{2})^{2}|h^{\prime}(z)|^{2}$ and
$(1-|z|^{2})^{2}|h^{\prime\prime}(z)|$ are uniformly bounded on $\mathbb{D}$.
Write

$\Omega=\mathbb{D}\setminus\left(\bigcup_{n=1}^{\infty}\Delta(z_{n},\frac{\gamma_{1}}{4})\right).$

By Lemma B,

$\sup_{z\in\Omega}(1-|z|^{2})^{2}\left|\frac{J^{\prime\prime}(z)+2J^{\prime}(z)h^{\prime}(z)}{J(z)}\right|<\infty.$

Note that $J^{\prime\prime}(z_{n})+2J^{\prime}(z_{n})h^{\prime}(z_{n})=0$ for
all $n$. It follows from Lemma C and Lemma E that

$\sup_{z\in\mathbb{D}\setminus\Omega}(1-|z|^{2})^{2}\left|\frac{J^{\prime\prime}(z)+2J^{\prime}(z)h^{\prime}(z)}{J(z)}\right|<\infty.$

Hence $\sup_{z\in\mathbb{D}}(1-|z|^{2})^{2}|A(z)|<\infty$. We finish the
proof. ∎

Remark 1. It is easy to see that (a) and (b) in Proposition 4.1 can be
replaced by (c) and (d) below simultaneously.

  1. (c)

If an interpolating Blaschke product $B$ associated with a sequence
$\\{a_{k}\\}_{k=1}^{\infty}$ belongs to $X$, then the Blaschke product $I$
associated with a sequence $\\{b_{k}\\}_{k=1}^{\infty}$ also belongs to $X$,
where $\rho(a_{k},b_{k})<s$, $k=1,2,\cdots$, for some constant $s\in(0,1)$.

  2. (d)

Suppose $B$ is an interpolating Blaschke product in $X$, $I$ is a Carleson-
Newman Blaschke product in $X$ and $c$ is a complex constant. Then the
function $\exp(cBI)$ belongs to $X$.

Remark 2. For $X\subseteq H(\mathbb{D})$, denote by $M(X)$ the set of
multipliers on $X$; that is,

$M(X)=\\{f\in X:\ \ fg\in X\ \text{for all}\ g\in X\\}.$

From [13, Lemma 11], if $X$ is a Banach space of analytic functions on which
point evaluations are bounded, then $M(X)\subseteq H^{\infty}$. Replacing $X$
in Proposition 4.1 by $M(X)$, we get a corresponding result on $M(X)$
immediately.

Remark 3. For a special space $X$ satisfying assumptions in Proposition 4.1,
it is interesting to consider further the pointwise growth condition or
integrated growth condition of the function $A$ in Proposition 4.1.

In 2019 J. Gröhn [22] gave several interesting results for solutions of (4.1)
having prescribed zeros in $\mathbb{D}$. In particular, he obtained the
following result.

######  Theorem F.

Let $0<s\leq 1$. If $\Lambda\subset\mathbb{D}$ is a separated sequence such
that $\sum_{z_{n}\in\Lambda}(1-|z_{n}|)^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, then there exists a function $A$ analytic in $\mathbb{D}$ such that
$|A(z)|^{2}(1-|z|^{2})^{2+s}dm(z)$ is an $s$-Carleson measure and the equation
$f^{\prime\prime}+Af=0$ admits a nontrivial solution $f\in\mathcal{Q}_{s}\cap
H^{\infty}$ whose zero-sequence is $\Lambda$.

For $0<p_{1}<p_{2}<\infty$ and $0<s\leq 1$ with $p_{1}+s>1$, it is known that
$F(p_{1},p_{1}-2,s)\subsetneqq F(p_{2},p_{2}-2,s)$. For $s>1$, all nontrivial
$F(p,p-2,s)$ spaces are equal to the Bloch space. Suppose $0<p<\infty$ and
$0<s<\infty$ with $2p+s>1$. For $A\in H(\mathbb{D})$,
$|A(z)|^{p}(1-|z|^{2})^{2p-2+s}dm(z)$ is an $s$-Carleson measure if and only
if

$\sup_{w\in\mathbb{D}}\int_{\mathbb{D}}|A(z)|^{p}(1-|z|^{2})^{2p-2}(1-|\sigma_{w}(z)|^{2})^{s}dm(z)<\infty.$

Denote by $N_{p,s}$ the space of analytic functions $A$ satisfying the formula
above. Clearly, if $0<p<\infty$ and $0<s<\infty$ with $2p+s\leq 1$, then
$N_{p,s}$ contains only constant functions. For $0<p<\infty$ and $s>1$,
$N_{p,s}$ is equal to the Bloch type space $\mathcal{B}^{3}$ consisting of
functions $f\in H(\mathbb{D})$ with

$\sup_{z\in\mathbb{D}}(1-|z|^{2})^{3}|f^{\prime}(z)|<\infty.$

From [38, Remark 3], if $0<p_{1}<p_{2}<\infty$ and $0<s\leq 1$ with
$2p_{1}+s>1$, then $N_{p_{1},s}\subsetneqq N_{p_{2},s}$. Thus the following
theorem from [38] is a proper generalization of the case of $0<s<1$ in Theorem
F.

######  Theorem G.

Let $0<s<1<p<\infty$ and $1<q<\infty$. If $\Lambda\subset\mathbb{D}$ is a
separated sequence such that
$\sum_{z_{n}\in\Lambda}(1-|z_{n}|)^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, then there exists a function $A$ analytic in $\mathbb{D}$ such that
$|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$ is an $s$-Carleson measure and the
equation $f^{\prime\prime}+Af=0$ admits a nontrivial solution $f\in
F(p,p-2,s)\cap H^{\infty}$ whose zero-sequence is $\Lambda$.

Theorems 4.2 below is a sequel to Theorem F and Theorem G. Form the
explanations after Theorem F, we see that Theorem 4.2 below strengthens
Theorem G and the case of $0<s<1$ in Theorem F.

######  Theorem 4.2.

Suppose $0<s<1$, $p>\max\\{s,1-s\\}$ and $q>\max\\{s,1-s\\}$. If
$\Lambda\subset\mathbb{D}$ is a separated sequence such that
$\sum_{z_{n}\in\Lambda}(1-|z_{n}|)^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, then there exists a function $A$ in $H(\mathbb{D})$ such that
$|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$ is an $s$-Carleson measure and the
equation $f^{\prime\prime}+Af=0$ admits a nontrivial solution $f\in
F(p,p-2,s)\cap H^{\infty}$ whose zero-sequence is $\Lambda$.

###### Proof.

Note that $0<s<1$ and $p>\max\\{s,1-s\\}$. We first show that (a), (b) and (c)
in Proposition 4.1 hold by taking $X=F(p,p-2,s)$. Using Theorem A and checking
the proof of Proposition 4.1, we get that (a) in Proposition 4.1 holds when
$X=F(p,p-2,s)$. Clearly, if $X=F(p,p-2,s)$, then (c) in Proposition 4.1 also
holds. Suppose $B$ and $I$ are interpolating Blaschke products in $F(p,p-2,s)$
and $c$ is a complex constant. Then

$\displaystyle\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}\left|(e^{cB(z)I(z)})^{\prime}\right|^{p}(1-|z|^{2})^{p-2}(1-|\sigma_{a}(z)|^{2})^{s}dA(z)$
$\displaystyle\lesssim$
$\displaystyle\|I\|^{p}_{F(p,p-2,s)}\sup_{z\in\mathbb{D}}\left|e^{cB(z)I(z)}\right|^{p}+\|B\|^{p}_{F(p,p-2,s)}\sup_{z\in\mathbb{D}}\left|e^{cB(z)I(z)}\right|^{p}<\infty,$

which gives $e^{cBI}\in F(p,p-2,s)$. Thus (b) in Proposition 4.1 also holds
when $X=F(p,p-2,s)$.

For convenience, we write $\Lambda=\\{z_{n}\\}_{n=1}^{\infty}$. Denote by $J$
the Blaschke product associated with the sequence
$\\{z_{n}\\}_{n=1}^{\infty}$. Since $\\{z_{n}\\}$ is separated and
$\sum_{z_{n}\in\Lambda}(1-|z_{n}|)^{s}\delta_{z_{n}}$ is an $s$-Carleson
measure, $J$ is an interpolating Blaschke product and it follows from Theorem
A that $J\in F(p,p-2,s)$. By Proposition 4.1, there exists a function $A$
analytic in $\mathbb{D}$ such that the equation $f^{\prime\prime}+Af=0$ admits
a nontrivial solution $f\in H^{\infty}\cap F(p,p-2,s)$ whose zero-sequence is
$\\{z_{n}\\}_{n=1}^{\infty}$. As shown in the proof of Proposition 4.1,

$A=-\frac{J^{\prime\prime}+2J^{\prime}h^{\prime}}{J}-(h^{\prime})^{2}-h^{\prime\prime},\
\ h=CJ_{1}J,$

where $C$ is a complex constant and $J_{1}$ is a Blaschke product whose zero-
sequence $\\{\eta_{n}\\}_{n=1}^{\infty}$ satisfies

$\rho(z_{n},\eta_{n})\leq\frac{\gamma}{3},\ \ n=1,2,\cdots$

for some $\gamma\in(0,1)$. For $q>\max\\{s,1-s\\}$, repeating the arguments of
the proof of Theorem 1.1 in [38] (i.e. Theorem G stated in this paper), we see
that $|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$ is an $s$-Carleson measure. We
finish the proof. ∎

Let $0<s<1$, $q>\max\\{s,1-s\\}$ and $A\in H(\mathbb{D})$. Suppose
$|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$ is an $s$-Carleson measure. By (1.1) and
the subharmonicity of $|A|^{q}$, we deduce

$\displaystyle\infty$
$\displaystyle>\sup_{w\in\mathbb{D}}\int_{\Delta(w,1/2)}\left(\frac{1-|w|^{2}}{|1-\overline{w}z|^{2}}\right)^{s}|A(z)|^{q}(1-|z|^{2})^{2q-2+s}dm(z)$
$\displaystyle\gtrsim\sup_{w\in\mathbb{D}}|A(w)|^{q}(1-|w|^{2})^{2q},$

which implies the condition of $A$ appeared in Proposition 4.1. In other
words, the condition of $A$ in Theorem 4.2 is stronger than that in
Proposition 4.1.

It is known (cf. [32, 40]) that $H^{\infty}\subseteq F(p,p-2,1)$ when $2\leq
p<\infty$, but for $0<p<2$, $H^{\infty}\nsubseteq F(p,p-2,1)$ and
$F(p,p-2,1)\nsubseteq H^{\infty}$. It is natural to consider the case of $s=1$
in Theorem F via replacing $\mathcal{Q}_{1}\cap H^{\infty}$ (i.e.
$H^{\infty}$) by $F(p,p-2,1)\cap H^{\infty}$ for $0<p<2$. For this purpose,
one should characterize interpolating Blaschke products $B$ in $F(p,p-2,1)$
for $0<p<2$ via the distribution of zeros of $B$. It is still open to find
this characterization (cf. [32, p.755]).

Data Availability.

All data generated or analyzed during this study are included in this article
and in its bibliography.

Conflict of Interest.

The authors declared that they have no conflict of interest.

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