# Chinese Grammatical Correction Using BERT-based Pre-trained Model

Hongfei Wang, Michiki Kurosawa, Satoru Katsumata  Currently at Nomura Research
Institute, Ltd. Currently at Retrieva, Inc. Mamoru Komachi
Tokyo Metropolitan University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In recent years, pre-trained models have been extensively studied, and several
downstream tasks have benefited from their utilization. In this study, we
verify the effectiveness of two methods that incorporate a BERT-based pre-
trained model developed by Cui et al. (2020) into an encoder-decoder model on
Chinese grammatical error correction tasks. We also analyze the error type and
conclude that sentence-level errors are yet to be addressed.

##  1 Introduction

Grammatical error correction (GEC) can be regarded as a sequence-to-sequence
task. GEC systems receive an erroneous sentence written by a language learner
and output the corrected sentence. In previous studies that adopted neural
models for Chinese GEC (Ren et al., 2018; Zhou et al., 2018), the performance
was improved by initializing the models with a distributed word
representation, such as Word2Vec (Mikolov et al., 2013). However, in these
methods, only the embedding layer of a pre-trained model was used to
initialize the models.

In recent years, pre-trained models based on Bidirectional Encoder
Representations from Transformers (BERT) have been studied extensively (Devlin
et al., 2019; Liu et al., 2019), and the performance of many downstream
Natural Language Processing (NLP) tasks has been dramatically improved by
utilizing these pre-trained models. To learn existing knowledge of a language,
a BERT-based pre-trained model is trained on a large-scale corpus using the
encoder of Transformer (Vaswani et al., 2017). Subsequently, for a downstream
task, a neural network model is initialized with the weights learned by a pre-
trained model that has the same structure and is fine-tuned on training data
of the downstream task. Using this two-stage method, the performance is
expected to improve because downstream tasks are informed by the knowledge
learned by the pre-trained model.

Recent works (Kaneko et al., 2020; Kantor et al., 2019) show that BERT helps
improve the performance on the English GEC task. As the Chinese pre-trained
models are developed and released continuously (Cui et al., 2020; Zhang et
al., 2019), the Chinese GEC task may also benefit from using those pre-trained
models.

Figure 1: Two methods for incorporating a pre-trained model into the GEC
model.

In this study, as shown in Figure 1, we develop a Chinese GEC model based on
Transformer with a pre-trained model using two methods: first, by initializing
the encoder with the pre-trained model (BERT-encoder); second, by utilizing
the technique proposed by Zhu et al. (2020), which uses the pre-trained model
for additional features (BERT-fused); on the Natural Language Processing and
Chinese Computing (NLPCC) 2018 Grammatical Error Correction shared task test
dataset (Zhao et al., 2018), our single models obtain $\mathrm{F_{0.5}}$
scores of 29.76 and 29.94 respectively, which is similar to the performance of
ensemble models developed by the top team of the shared task. Moreover, using
a 4-ensemble model, we obtain an $\mathrm{F_{0.5}}$ score of 35.51, which
outperforms the results from the top team by a large margin. We annotate the
error types of the development data; the results show that word-level errors
dominate all error types and that sentence-level errors remain challenging and
require a stronger approach.

##  2 Related Work

Given the success of the shared tasks on English GEC at the Conference on
Natural Language Learning (CoNLL) (Ng et al., 2013, 2014), a Chinese GEC
shared task was performed at the NLPCC 2018. In this task, approximately one
million sentences from the language learning website
Lang-8111https://lang-8.com/ were used as training data and two thousand
sentences from the PKU Chinese Learner Corpus (Zhao et al., 2018) were used as
test data. Here, we briefly describe the three methods with the highest
performance.

First, Fu et al. (2018) combined a 5-gram language model-based spell checker
with subword-level and character-level encoder-decoder models using
Transformer to obtain five types of outputs. Then, they re-ranked these
outputs using the language model. Although they reported a high performance,
several models were required, and the combination method was complex.

Second, Ren et al. (2018) utilized a convolutional neural network (CNN), such
as in Chollampatt and Ng (2018). However, because the structure of the CNN is
different from that of BERT, it cannot be initialized with the weights learned
by the BERT.

Last, Zhao and Wang (2020) proposed a dynamic masking method that replaces the
tokens in the source sentences of the training data with other tokens (e.g.
[PAD] token). They achieved state-of-the-art results on the NLPCC 2018 Grammar
Error Correction shared task without using any extra knowledge. This is a data
augmentation method that can be a supplement for our study.

##  3 Methods

In the proposed method, we construct a correction model using Transformer, and
incorporate a Chinese pre-trained model developed by Cui et al. (2020) in two
ways as described in the following sections.

###  3.1 Chinese Pre-trained Model

We use a BERT-based model as our pre-trained model. BERT is mainly trained
with a task called Masked Language Model. In the Masked Language Model task,
some tokens in a sentence are replaced with masked tokens ([MASK]) and the
model has to predict the replaced tokens.

In this study, we use the Chinese-RoBERTa-wwm-ext model developed by Cui et
al. (2020). The main difference between Chinese-RoBERTa-wwm-ext and the
original BERT is that the latter uses whole word masking (WWM) to train the
model. In WWM, when a Chinese character is masked, other Chinese characters
that belong to the same word should also be masked.

###  3.2 Grammatical Error Correction Model

In this study, we use Transformer as the correction model. Transformer has
shown excellent performance in sequence-to-sequence tasks, such as machine
translation, and has been widely adopted in recent studies on English GEC
(Kiyono et al., 2019; Junczys-Dowmunt et al., 2018).

However, a BERT-based pre-trained model only uses the encoder of Transformer;
therefore, it cannot be directly applied to sequence-to-sequence tasks that
require both an encoder and a decoder, such as GEC. Hence, we incorporate the
encoder-decoder model with the pre-trained model in two ways as described in
the following subsections.

#### BERT-encoder

We initialize the encoder of Transformer with the parameters learned by
Chinese-RoBERTa-wwm-ext; the decoder is initialized randomly. Finally, we
fine-tune the initialized model on Chinese GEC data.

#### BERT-fused

Zhu et al. (2020) proposed a method that uses a pre-trained model as the
additional features. In this method, input sentences are fed into the pre-
trained model and representations from the last layer of the pre-trained model
are acquired first. Then, the representations will interact with the encoder
and decoder by using attention mechanism. Kaneko et al. (2020) verified the
effectiveness of this method on English GEC tasks.

##  4 Experiments

###  4.1 Experimental Settings

#### Data

In this study, we use the data provided by the NLPCC 2018 Grammatical Error
Correction shared task. We first segment all sentences into characters because
the Chinese pre-trained model we used is character-based. In the GEC task,
source and target sentences do not tend to change significantly. Considering
this, we filter the training data by excluding sentence pairs that meet the
following criteria: i) the source sentence is identical to the target
sentence; ii) the edit distance between the source sentence and the target
sentence is greater than 15; iii) the number of characters of the source
sentence or the target sentence exceeds 64. Once the training data were
filtered, we obtained 971,318 sentence pairs.

Because the NLPCC 2018 Grammatical Error Correction shared task did not
provide development data, we opted to randomly extract 5,000 sentences from
the training data as the development data following Ren et al. (2018).

The test data consist of 2,000 sentences extracted from the PKU Chinese
Learner Corpus. According to Zhao et al. (2018), the annotation guidelines
follow the minimum edit distance principle (Nagata and Sakaguchi, 2016), which
selects the edit operation that minimizes the edit distance from the original
sentence.

#### Model

We implement the Transformer model using fairseq
0.8.0.222https://github.com/pytorch/fairseq and load the pre-trained model
using pytorch_transformer 2.2.0.333https://github.com/huggingface/transformers

We then train the following models based on Transformer.

Baseline: a plain Transformer model that is initialized randomly without using
a pre-trained model.

BERT-encoder: the correction model introduced in Section 3.2.

BERT-fused: the correction model introduced in Section 3.2. We use the
implementation provided by Zhu et al. (2020).444https://github.com/bert-
nmt/bert-nmt

Finally, we train a 4-ensemble BERT-encoder model and a 4-ensemble BERT-fused
model.

More details on the training are provided in the appendix A.

#### Evaluation

As the evaluation is performed on word-unit, we strip all delimiters from the
system output sentences and segment the sentences using the pkunlp555
http://59.108.48.12/lcwm/pkunlp/downloads/libgrass-ui.tar.gz provided in the
NLPCC 2018 Grammatical Error Correction shared task.

Based on the setup of the NLPCC 2018 Grammatical Error Correction shared task,
the evaluation is conducted using MaxMatch
(M2).666https://github.com/nusnlp/m2scorer

###  4.2 Evaluation Results

Table 1 summarizes the experimental results of our models. We run the single
models four times, and report the average score. For comparison, we also cite
the result of the state-of-the-art model (Zhao and Wang, 2020) and the results
of the models developed by two teams in the NLPCC 2018 Grammatical Error
Correction shared task.

[Our models] | P | R | $\mathrm{F_{0.5}}$
---|---|---|---
Baseline | 25.14 | 14.34 | 21.85
BERT-encoder | 32.67 | 22.19 | 29.76
BERT-fused | 32.11 | 23.57 | 29.94
BERT-encoder (4-ensemble) | 41.94 | 22.02 | 35.51
BERT-fused (4-ensemble) | 32.20 | 23.16 | 29.87
[SOTA Result] |  |  |
Zhao and Wang (2020) | 44.36 | 22.18 | 36.97
[NLPCC 2018] |  |  |
Fu et al. (2018) | 35.24 | 18.64 | 29.91
Ren et al. (2018) | 41.73 | 13.08 | 29.02
Ren et al. (2018) (4-ensemble) | 47.63 | 12.56 | 30.57
Table 1: Experimental results on the NLPCC 2018 Grammatical Error Correction shared task. src |  持 别 是 北京 ， 没有 “ 自然 ” 的 感觉 。 | 人们 在 一 辈子 经验 很多 事情 。
---|---|---
gold |  特别 是 北京 ， 没有 “ 自然 ” 的 感觉 。 | 人们 在 一 辈子 经历 很多 事情 。
baseline |  持 别 是 北京 ， 没有 “ 自然 ” 的 感觉 。 | 人们 在 一辈子 经历 了 很多 事情 。
BERT-encoder |  特别 是 北京 ， 没有 “ 自然 ” 的 感觉 。 | 人们 一辈子 会 经历 很多 事情 。
Translation |  Especially in Beijing, there is no natural feeling. | People experience many things in their lifetime.
Table 2: Source sentence, gold edit, and output of our models. Error Type |  Number of errors | Examples
---|---|---
B | 9 |  最后 ， 要 关主{关注} 一些 关于 天气 预报 的 新闻 。 (Finally, pay attention to some weather forecast news.)
CC | 35 |  有 一 天 晚上 他 下 了 决定{决心} 向 富丽 堂皇 的 宫殿 里 走 ， 偷偷 的{地} 进入 宫内 。 (One night he decided to walk to the magnificent palace, and sneaked in it secretly.)
CQ | 30 |  在 上海 我 总是 住 NONE{在} 一家 特定 NONE{的} 酒店 。 (I always stay in the same hotel in Shanghai.)
CD | 21 |  我 很 喜欢 念{NONE}读 小说 . (I like to read novels.)
CJ | 35 |  …… 但是 同时 也 对 环境 问题{NONE} 日益 严重 造成 了{造成 了 日益 严重 的} 空气 污染 问题 。 (But on the meanwhile, it also aggravated the problem of air pollution.)
Table 3: Examples of each error type. The underlined tokens are detected errors that should be replaced with the tokens in braces. Type | Detection | Correction
---|---|---
P | R | $\mathrm{F_{0.5}}$ | P | R | $\mathrm{F_{0.5}}$
BERT-encoder
B | 80.0 | 55.6 | 73.5 | 80.0 | 55.6 | 73.5
CC | 62.5 | 31.4 | 52.2 | 43.8 | 20.0 | 35.4
CQ | 65.0 | 43.3 | 59.1 | 45.0 | 30.0 | 40.9
CD | 58.3 | 28.6 | 48.3 | 50.0 | 28.6 | 43.5
CJ | 56.5 | 42.9 | 53.1 |  4.3 |  2.9 |  3.9
BERT-fused
B | 80.0 | 44.4 | 69.0 | 80.0 | 44.4 | 69.0
CC | 61.9 | 42.9 | 56.9 | 38.1 | 22.9 | 33.6
CQ | 69.0 | 63.3 | 67.8 | 44.8 | 46.7 | 45.2
CD | 71.4 | 42.9 | 63.0 | 57.1 | 38.1 | 51.9
CJ | 63.2 | 34.3 | 54.1 | 15.8 |  8.6 | 13.5
Table 4: Detection and correction performance of BERT-encoder and BERT-fused
models on each type of error.

The performances of BERT-encoder and BERT-fused are significantly superior to
that of the baseline model and are comparable to those achieved by the two
teams in the NLPCC 2018 Grammatical Error Correction shared task, indicating
the effectiveness of adopting the pre-trained model.

The BERT-encoder (4-ensemble) model yields an $\mathrm{F_{0.5}}$ score nearly
5 points higher than the highest-performance model in the NLPCC 2018
Grammatical Error Correction shared task. However, there is no improvement for
the BERT-fused (4-ensemble) model compared with the single BERT-fused model.
We find that the performance of the BERT-fused model depends on the warm-up
model. Compared with Kaneko et al. (2020) using a state-of-the-art model to
warm-up their BERT-fused model, we did not use a warm-up model in this work.
The performance noticeably drops when we try to warm-up the BERT-fused model
from a weak baseline model, therefore, the BERT-fused model may perform better
when warmed-up from a stronger model (e.g., the model proposed by Zhao and
Wang (2020)).

For the state-of-the-art result achieved by Zhao and Wang (2020), both the
precision and the recall are comparatively high, and they therefore obtain the
best $\mathrm{F_{0.5}}$ score.

Additionally, the precision of the models that used a pre-trained model is
lower than that of the models proposed by the two teams; conversely, the
recall is significantly higher.

##  5 Discussion

#### Case Analysis

Table 2 shows the sample outputs.

In the first example, the spelling error 持别 is accurately corrected to 特别
(which means especially) by the proposed model, whereas it is not corrected by
the baseline model. Hence, it appears that the proposed model captures context
more efficiently by using the pre-trained model through the WWM strategy.

In the second example, the output of the proposed model is more fluent,
although the correction made by the proposed model is different from the gold
edit. The proposed model not only changed the wrong word 经验 (which usually
means the noun experience) to 经历 (which usually means the verb experience),
but also added a new word 会 (would, could); this addition makes the sentence
more fluent. It appears that the proposed model can implement additional
changes to the source sentence because the pre-trained model is trained with a
large-scale corpus. However, this type of change may affect the precision
because the gold edit in this dataset followed the principle of minimum edit
distance (Zhao et al., 2018).

#### Error Type Analysis

To understand the error distribution of Chinese GEC, we annotate 100 sentences
of development data and obtain 130 errors (one sentence may contain more than
one error). We refer to the annotation of the HSK learner
corpus777http://hsk.blcu.edu.cn/ and adopt five categories of error: B, CC,
CQ, CD, and CJ. B denotes character-level errors, which are mainly spelling
and punctuation errors. CC, CQ, and CD are word-level errors, which are word
selection, missed word, and redundant word errors, respectively. CJ denotes
sentence-level errors which contain several complex errors, such as word order
and lack of subject errors. Several examples are presented in Table 3. Based
on the number of errors, it is evident that word-level errors (CC, CQ, and CD)
are the most frequent.

Table 4 lists the detection and correction results of the BERT-encoder and
BERT-fused models for each error type. The two models perform poorly on
sentence-level errors (CJ), which often involve sentence reconstructions,
demonstrating that this is a difficult task. For character-level errors (B),
the models achieve better performance than for other error types. Compared
with the correction performance, the systems indicate moderate detection
performance, demonstrating that the systems address error positions
appropriately. With respect to the difference in performance of the two
systems on each error type, we can conclude that BERT-encoder performs better
on character-level errors (B), and BERT-fused performs better on other error
types.

##  6 Conclusion

In this study, we incorporated a pre-trained model into an encoder-decoder
model using two methods on Chinese GEC tasks. The experimental results
demonstrate the usefulness of the BERT-based pre-trained model in the Chinese
GEC task. Additionally, our error type analysis showed that sentence-level
errors remain to be addressed.

## Acknowledgments

This work has been partly supported by the programs of the Grant-in-Aid for
Scientific Research from the Japan Society for the Promotion of Science (JSPS
KAKENHI) Grant Numbers 19K12099 and 19KK0286.

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##  Appendix A Appendices

Table 5 shows the training details for each model.

Baseline
---
Architecture | Encoder (12-layer), Decoder (12-layer)
Learning rate | $1\times 10^{-5}$
Batch size | 32
Optimizer | Adam ($\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=1\times 10^{-8}$)
Max epochs | 20
Loss function | cross-entropy
Dropout | 0.1
BERT-encoder
Architecture | Encoder (12-layer), Decoder (12-layer)
Learning rate | $3\times 10^{-5}$
Batch size | 32
Optimizer | Adam ($\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=1\times 10^{-8}$)
Max epochs | 20
Loss function | cross-entropy
Dropout | 0.1
BERT-fused
Architecture | Transformer (big)
Learning rate | $3\times 10^{-5}$
Batch size | 32
Optimizer | Adam ($\beta_{1}=0.9,\beta_{2}=0.98,\epsilon=1\times 10^{-8}$)
Max epochs | 20
Loss function | label smoothed cross-entropy ($\epsilon_{ls}=0.1$)
Dropout | 0.3
Table 5: Training details for each model.


#  $L_{x}^{p}\rightarrow L^{q}_{x,u}$ estimates for dilated averages over
planar curves 00footnotetext:  2020 Mathematics Subject Classification.
Primary 42B10; Secondary 42B15. Key words and phrases. generalised Radon
transforms, $L_{x}^{p}-L^{q}_{x,u}$ estimates, local smoothing estimates,
Fourier integral operators. Junfeng Li was supported by Natural Science
Foundation of China (No. 12071052) and the Fundamental Research Funds for the
Central Universities. Naijia Liu was supported by China Postdoctoral Science
Foundation (No. 2022M723673). Zengjian Lou was supported by Natural Science
Foundation of China (No. 12071272). Haixia Yu was supported by Natural Science
Foundation of China (No. 12201378), Guangdong Basic and Applied Basic Research
Foundation (No. 2023A1515010635) and STU Scientific Research Foundation for
Talents (No. NTF21038).

Junfeng Li, Naijia Liu, Zengjian Lou and Haixia Yu111Corresponding author.

###### Abstract

In this paper, we consider the $L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ estimate for the operator $T$ along a
dilated plane curve $(ut,u\gamma(t))$, where

$Tf(x,u):=\int_{0}^{1}f(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t,$

$x:=(x_{1},x_{2})$ and $\gamma$ is a general plane curve satisfying some
suitable smoothness and curvature conditions. We show that $T$ is
$L_{x}^{p}(\mathbb{R}^{2})$ to $L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
bounded whenever
$(\frac{1}{p},\frac{1}{q})\in\square\cup\\{(0,0)\\}\cup\\{(\frac{2}{3},\frac{1}{3})\\}$
and $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$, where the trapezium
$\square:=\\{(\frac{1}{p},\frac{1}{q}):\
\frac{2}{p}-1\leq\frac{1}{q}\leq\frac{1}{p},\frac{1}{q}>\frac{1}{3p},\frac{1}{q}>\frac{1}{p}-\frac{1}{3}\\}$
and $\omega:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$. This
result is sharp except for some borderline cases. On the other hand, in a
smaller $(\frac{1}{p},\frac{1}{q})$ region, we also obtain the almost sharp
estimate $T:L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$
uniformly for $u\in[1,2]$. These results imply that the operator $T$ has the
so called local smoothing phenomenon, i.e., the $L^{q}$ integral about $u$ on
$[1,2]$ extends the region of $(\frac{1}{p},\frac{1}{q})$ in uniform estimate
$T:L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$.

##  1 Introduction

It is well known that the theory of maximal functions plays a central role in
harmonic analysis and partial differential equations. Which has attracted a
lot of attention in the last decades. Our topic in this paper originated in
the estimate for maximal function. Let us begin with a definition of the
_maximal function_ $M_{t}$ along the straight line $(t,t)$,

$\displaystyle
M_{t}f(x):=\sup_{u\in(0,\infty)}\frac{1}{u}\left|\int_{0}^{u}f(x_{1}-t,x_{2}-t)\,\textrm{d}t\right|$

for appropriate functions $f$ on $\mathbb{R}^{2}$. Here and hereafter,
$x:=(x_{1},x_{2})$. From Stein [36], it follows that $M_{t}$ is bounded on
$L^{p}(\mathbb{R}^{2})$ for all $p\in(1,\infty]$ because $M_{t}$ is
essentially the Hardy-Littlewood maximal function. By a straightforward change
of variables, we can rewrite

$\displaystyle
M_{t}f(x)=\sup_{u\in(0,\infty)}\left|\int_{0}^{1}f(x_{1}-ut,x_{2}-ut)\,\textrm{d}t\right|.$

It is natural to replace the straight line $(t,t)$ in $M_{t}$ by a general
plane curve $(t,\gamma(t))$, such as the parabola $(t,t^{2})$, where $\gamma$
satisfies some suitable smoothness and curvature conditions. We now consider
the _maximal function_ $M_{\gamma}$ along a curve $(t,\gamma(t))$,

$\displaystyle
M_{\gamma}f(x):=\sup_{u\in(0,\infty)}\left|\int_{0}^{1}f(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t\right|.$

For notational convenience, we define the operator $T$ along a dilated plane
curve $(ut,u\gamma(t))$ as

(1.1) $\displaystyle
Tf(x,u):=\int_{0}^{1}f(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t,$

then

$\displaystyle M_{\gamma}f(x)=\sup_{u\in(0,\infty)}\left|Tf(x,u)\right|.$

It is natural to ask whether the $L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x}^{q}(\mathbb{R}^{2})$ estimate for $M_{\gamma}$ is valid or not. However,
by Liu and Yu [24, Remark 1.3], the maximal function $M_{\gamma}$ does not map
$L_{x}^{p}(\mathbb{R}^{2})$ to $L_{x}^{q}(\mathbb{R}^{2})$ (even when
$\sup_{u\in(0,\infty)}$ in $M_{\gamma}$ is replaced by $\sup_{u\in(0,1)}$),
unless $p=q$. Related works can be found in [2, 35, 16, 18] and references
therein. However, if we take supremum over $u\in[1,2]$ in $M_{\gamma}$, i.e.,

$\displaystyle\bar{M}_{\gamma}f(x):=\sup_{u\in[1,2]}\left|Tf(x,u)\right|.$

Then $\bar{M}_{\gamma}$ is actually bounded from $L_{x}^{p}(\mathbb{R}^{2})$
to $L_{x}^{q}(\mathbb{R}^{2})$ for some $q>p$, see [24]. Indeed, this result
can be rewritten as

(1.2)
$\displaystyle\left\|Tf\right\|_{L_{x}^{q}L_{u}^{\infty}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L^{p}(\mathbb{R}^{2})}.$

This phenomenon is called $L^{p}(\mathbb{R}^{2})$-improving. We refer the
reader to [43, 30, 19, 20] for the details of this phenomenon.

Our goal in this paper is to establish the
$L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
estimate for $T$. By $\textrm{H}\ddot{\textrm{o}}\textrm{lder}$’s inequality,
it is easy to see that, when $(\frac{1}{p},\frac{1}{q})$ verifies the
inequality (1.2), we have the $L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ estimate for $T$. However, the
$(\frac{1}{p},\frac{1}{q})$ region for $T:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
becomes large. We now state our main results.

######  Theorem 1.1.

Assume $\gamma\in C^{N}(0,1]$ with $N\in\mathbb{N}$ large enough,
$\lim_{t\rightarrow 0^{+}}\gamma(t)=0$ and $\gamma$ is monotonic on $(0,1]$.
Moreover, $\gamma$ satisfies the following two conditions:

  1. (i)

there exist positive constants $\\{C^{(j)}_{1}\\}_{j=1}^{2}$ such that
$|\frac{t^{j}\gamma^{(j)}(t)}{\gamma(t)}|\geq C^{(j)}_{1}$ for any
$t\in(0,1]$;

  2. (ii)

there exist positive constants $\\{C^{(j)}_{2}\\}_{j=1}^{N}$ such that
$|\frac{t^{j}\gamma^{(j)}(t)}{\gamma(t)}|\leq C^{(j)}_{2}$ for any
$t\in(0,1]$.

Then, there exists a positive constant $C$ such that for all $f\in
L_{x}^{p}(\mathbb{R}^{2})$,

$\displaystyle\left\|Tf\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\leq
C\|f\|_{L_{x}^{p}(\mathbb{R}^{2})},$

if
$(\frac{1}{p},\frac{1}{q})\in\square\cup\\{(0,0)\\}\cup\\{(\frac{2}{3},\frac{1}{3})\\}$
and $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$. Here and hereafter, the
trapezium $\square:=\\{(\frac{1}{p},\frac{1}{q}):\
\frac{2}{p}-1\leq\frac{1}{q}\leq\frac{1}{p},\frac{1}{q}>\frac{1}{3p},\frac{1}{q}>\frac{1}{p}-\frac{1}{3}\\}$
and $\omega:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$.

We show the conditions of $(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 are
necessary, which means that Theorem 1.1 is sharp except for some borderline
cases. Moreover, we can also find that the region of
$(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 remains unchanged when
$\omega\in(0,2)$, but decreases as $\omega$ increases when
$\omega\in[2,\infty)$. Since the curvature of a curve is key in the proofs,
which may provide an underlying cause of this phenomenon. At the same time,
for the infinitely flat curve222A curve $\gamma$ is called a infinitely flat
curve if it satisfies $\gamma^{(k)}(0)=0$ for any $k\in\mathbb{N}_{0}$, such
as $\gamma(t):=e^{-1/t}$ or $e^{-1/t^{2}}$. These curves become very close to
being a “line” at the origin., it is reasonable to conjecture that the
estimate $T:\ L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ holds only for $p=q\in[1,\infty]$.

######  Theorem 1.2.

Let $\gamma$ be defined as in Theorem 1.1. Then, there exists a positive
constant $C$ such that for all $f\in L_{x}^{p}(\mathbb{R}^{2})$, the estimate

$\displaystyle\left\|Tf\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\leq
C\|f\|_{L_{x}^{p}(\mathbb{R}^{2})},$

holds, only if the following conditions are satisfied:

  1. (i)

$(\frac{1}{p},\frac{1}{q})$ satisfy $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})\geq
0$;

  2. (ii)

$(\frac{1}{p},\frac{1}{q})$ satisfy
$\frac{2}{p}-1\leq\frac{1}{q}\leq\frac{1}{p}$;

  3. (iii)

$(\frac{1}{p},\frac{1}{q})$ satisfy $\frac{1}{q}\geq\frac{1}{3p}$;

  4. (iv)

$(\frac{1}{p},\frac{1}{q})$ satisfy $\frac{1}{q}\geq\frac{1}{p}-\frac{1}{3}$.

For these results, we would like to give the following some remarks.

######  Remark 1.3.

Here are some examples of curves satisfying the conditions (i) and (ii) of
Theorem 1.1. We may add a characteristic function
$\chi_{(0,\varepsilon_{0}]}(t)$ to these curves if necessary, where
$\varepsilon_{0}$ is small enough.

  1. (1)

$\gamma_{1}(t):=t^{d}$, where $d\in(0,\infty)$ and $d\neq 1$;

  2. (2)

$\gamma_{2}(t):=t^{d-1}\ln(1+t)$, where $d\in(1,\infty)$;

  3. (3)

$\gamma_{3}(t):=a_{d}t^{d}+a_{d+1}t^{d+1}+\cdots+a_{d+m}t^{d+m}$, where $d\geq
2$, $d\in\mathbb{N}$ and $m\in\mathbb{N}_{0}$ , $a_{d}\neq 0$, i.e.,
$\gamma_{3}$ is a polynomial of degree at least $d$ with no linear term and
constant term;

  4. (4)

$\gamma_{4}(t):=\sum_{i=1}^{d}\beta_{i}t^{\alpha_{i}}$, where
$\alpha_{i}\in(0,\infty)$ for all $i=1,2,\cdots,d$,
$\min_{i\in\\{1,2,\cdots,d\\}}\\{\alpha_{i}\\}_{i=1}^{d}\neq 1$ and
$d\in\mathbb{N}$;

  5. (5)

$\gamma_{5}(t):=1-\sqrt{1-t^{2}}$, or $t\sin t$, or $t-\sin t$, or $1-\cos t$,
or $e^{t}-t-1$;

  6. (6)

$\gamma_{6}(t)$ is a smooth function on $[0,1]$ satisfying
$\gamma(0)=\gamma^{\prime}(0)=\cdots=\gamma^{(d-1)}(0)=0$ and
$\gamma^{(d)}(0)\neq 0$, where $d\geq 2$ and $d\in\mathbb{N}$. Note that
$\gamma_{6}$ is finite type $d$ at the origin (see, Iosevich [17]).
$\gamma_{3}$ and $\gamma_{5}$ provide some special cases of $\gamma_{6}$.

######  Remark 1.4.

If $\gamma(t):=\gamma_{1}(t)=t^{d}$ with $d\in(0,\infty)$ and $d\neq 1$, it
follows that $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$ is equivalent to
$(\frac{1}{p},\frac{1}{q})\in\\{(\frac{1}{p},\frac{1}{q}):\
\frac{1}{q}>\frac{1}{p}-\frac{1}{d+1}\\}$. As a result, we can express the
region of $(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 as in Figure 1. On the
other hand, by a simple calculation, $\gamma_{2}$, $\gamma_{3}$ and
$\gamma_{6}$ share the same region with $\gamma_{1}$.

Figure 1: The region of $(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 for
$\gamma=\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ or $\gamma_{6}$.

In [13], Hickman obtained the almost sharp (up to an endpoint)
$L_{x}^{p}(\mathbb{R}^{n})\rightarrow L_{x,u}^{q}(\mathbb{R}^{n}\times[1,2])$
estimate for the $n$-dimensional version of $T$ along the moment curve
$(t,t^{2},\cdots,t^{n})$ by establishing a weak-type endpoint estimate. Our
main results in Theorems 1.1 and 1.2 are the two-dimensional case, but with a
general plane curve including the parabola $\gamma(t):=t^{2}$. However, it is
worth pointing out that our proofs are completely different. In particular, we
establish a fixed time estimate and a local smoothing estimate for a general
Fourier integral operator, based on the works of Seeger, Sogge and Stein [32],
Stein [36, Chapter IX] and, Gao, Liu, Miao and Xi [9]. We remark that [9] can
be viewed as a generalization of the local smoothing conjecture in $2+1$
dimensions obtained in Guth, Wang and Zhang [12]. Furthermore, as stated
above, the region of $(\frac{1}{p},\frac{1}{q})$ in estimate $T:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
depends on the value $\omega=\limsup_{t\rightarrow
0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$, this is a new situation, to the best of
our knowledge. At the same time, we also show that $T$ has the local smoothing
phenomenon by obtaining the almost sharp estimate
$T:L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$ uniformly
for $u\in[1,2]$ in Theorem 2.1.

We should point out some historical backgrounds about the study of the
operator $T$. When $\gamma(t):=t^{2}$, based on the method of our proofs, we
can not show the $L_{x}^{2}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{6}(\mathbb{R}^{2}\times[1,2])$ boundedness of $T$. However, the
estimate $T:\ L_{x}^{2}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{6}(\mathbb{R}^{2}\times[1,2])$ is already known to hold essentially
by Strichartz [42], Schlag and Sogge [31]. We also remark that this estimate
for dilated averages over circles is equivalent to the famous Stein-Tomas
Fourier restriction theorem for a conic surface. On the other hand, for other
rangers of $(\frac{1}{p},\frac{1}{q})$, the estimates $T:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
about $(t,t^{2})$ obtained in Gressman [10, 11] by using the combinatorial
argument, which can be traced back to Christ [7].

The motivation for our curved variant also comes from the _maximal function_
$\mathcal{M}$ defined by taking averages over curve $(t,\gamma(t))$,

$\displaystyle\mathcal{M}f(x):=\sup_{\epsilon\in(0,\infty)}\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}\left|f(x_{1}-t,x_{2}-\gamma(t))\right|\,\textrm{d}t,$

which is also been extensively studied. For $\gamma(t):=t^{2}$, Nagel, Riviere
and Wainger [27] established the $L_{x}^{p}(\mathbb{R}^{2})$ boundedness of
$\mathcal{M}$ for any $p\in(1,\infty]$. Stein [37] obtained this boundedness
for homogeneous curves. Stein and Wainger [38] studied some smooth curves.
Later it was extended to more general families of curves; see, for example,
[39, 4, 5].

Let us now give an overview of the proofs of Theorems 1.1 and 1.2. By a non-
isotropic dilation, the proof is reduced to an uniform estimate for the
averages operator $\widetilde{T}_{j}$ along a dilated plane curve
$(ut,u\Gamma_{j}(t))$, where
$\Gamma_{j}(t):=\frac{\gamma(2^{j}t)}{\gamma(2^{j})}$ behaves uniformly in the
parameter $j$. Specifically, the second derivative of $\Gamma_{j}(t)$ is
bounded below uniformly in $j$. This non-isotropic dilation seems necessary,
which yields the slightly different notation $\omega=\limsup_{t\rightarrow
0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$. For the estimate of $\widetilde{T}_{j}$,
by interpolation, it suffices to obtain an endpoint estimate at
$D(\frac{2}{3},\frac{1}{3})$ and an estimate at the half open line $(C,M]$ in
Figure 3. For the $D(\frac{2}{3},\frac{1}{3})$ case, the corresponding
estimate can be established by Lemmas 2.3 and 2.4. For the half open line
$(C,M]$ case, by using the theory of oscillatory integrals and stationary
phase estimates, it is enough to obtain a local smoothing estimate for

$\displaystyle
F_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}e^{i\Psi_{j}(x,u,\xi)}a_{j,k}(x,u,\xi)\hat{f}(\xi)\,\textrm{d}\xi,$

which is defined in (3.5). The operator $F_{j,k}$ is related to a class of
Fourier integral operators studied in many papers (see, for instance, [15, 8,
34, 1]). These Fourier integral operators are originated from the study of
pseudo-differential operators or half-wave propagator. Furthermore, $F_{j,k}$
is also essentially a Fourier integral operator with phase function
$x\cdot\xi-u|\xi|$, and the local smoothing estimate for $F_{j,k}$ will be
reduced to a decoupling inequality for cones due to Bourgain and Demeter [3].
Here, for a general $n$ dimension Fourier integral operator in (3.8)
satisfying the cinematic curvature condition, we establish a fixed time
estimate about $P_{k}f$ in Proposition 3.2. When $n=2$, we also obtain a
corresponding local smoothing estimate in Proposition 3.3 based on the work of
Gao, Liu, Miao and Xi [9]. Our local smoothing estimates for $F_{j,k}$ rely on
Propositions 3.2 and 3.3. It is worth noticing that to verify the so called
cinematic curvature condition we need some complicated calculations. In order
to obtain the necessity of the region of $(\frac{1}{p},\frac{1}{q})$, we
construct four examples. In particular, the proofs of (i) and (iv) of Theorem
1.2 are a bit tricky.

This paper is structured as follows:

  1. $\diamond$

In Section 2, we show the almost sharp estimate
$T:L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$ uniformly
for $u\in[1,2]$ in a smaller $(\frac{1}{p},\frac{1}{q})$ region comparing to
that in Theorem 1.1. Furthermore, by the $L^{q}$ integral about $u$ on
$[1,2]$, Theorem 2.1 implies an important estimate at endpoint
$D(\frac{2}{3},\frac{1}{3})$. It also implies that the operator $T$ has the so
called local smoothing phenomenon.

  2. $\diamond$

In Section 3, we give a proof of Theorem 1.1. An estimate at the half open
line $(C,M]$ in Figure 3 plays a key role, whose proof relies heavily on the
local smoothing estimates.

  3. $\diamond$

In Section 4, we consider the necessary conditions for Theorem 1.1, i.e.,
Theorem 1.2. We will see that the estimate
$T:L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ in Theorem 1.1 is almost sharp except
for some endpoints.

Finally, we make some convention on the notations. Throughout this paper, the
letter “$C$” will denote a _positive constant_ , independent of the essential
variables, but whose value may vary from line to line. Moreover, we use
$C_{(u)}$ to denote a _positive constant_ depending on the indicated parameter
$u$. The notation $a\lesssim b$ (or $a\gtrsim b$) will used to indicate the
existence of a finite positive constant $C$ such that $a\leq Cb$ (or $a\geq
Cb$). $a\approx b$ means $a\lesssim b$ and $b\lesssim a$. For any
$x\in\mathbb{R}^{n}$ and $r\in(0,\infty)$, let
$B(x,r):=\\{y\in\mathbb{R}^{n}:\ |x-y|<r\\}$ and $B^{\complement}(x,r)$ be its
_complement_ in $\mathbb{R}^{n}$. $\hat{f}$ and $f^{\vee}$ shall denote the
_Fourier transform_ and the _inverse Fourier transform_ of $f$, respectively.
For $1<q\leq\infty$, we will denote $q^{\prime}$ the _adjoint number_ of $q$,
i.e., ${1}/{q}+{1}/{q^{\prime}}=1$. Let
$\mathbb{Z}_{0}^{-}:=\\{0,\,-1,\,-2,...\\}$, $\mathbb{N}:=\\{1,\,2,...\\}$,
$\mathbb{N}_{0}:=\\{0,\,1,\,2,...\\}$ and $\mathbb{R}^{+}:=(0,\infty)$. For
any set $E$, we use $\chi_{E}$ to denote the _characteristic function_ of $E$.

##  2 The local smoothing phenomenon

Recall that

$\displaystyle Tf(x,u)=\int_{0}^{1}f(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t$

with $u\in[1,2]$, which is an operator of convolution type. When $u=1$,
$Tf(\cdot,1)$ is a classical operator in harmonic analysis. For some related
results, we refer to [21, 28, 29, 40, 41] and the references therein. In this
section, our main goal is to determine the almost optimal range of exponents
$(\frac{1}{p},\frac{1}{q})$ such that the
$L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$ estimate for
$T$ holds uniformly for $u\in[1,2]$. We now state our result.

######  Theorem 2.1.

Assume $\gamma\in C^{2}(0,1]$, $\lim_{t\rightarrow 0^{+}}\gamma(t)=0$ and
$\gamma$ is monotonic on $(0,1]$. Moreover, $\gamma$ satisfies the following
two conditions:

  1. (i)

there exist positive constants $\\{C^{(j)}_{1}\\}_{j=1}^{2}$ such that
$|\frac{t^{j}\gamma^{(j)}(t)}{\gamma(t)}|\geq C^{(j)}_{1}$ for any
$t\in(0,1]$;

  2. (ii)

there exists a positive constant $C^{(1)}_{2}$ such that
$|\frac{t\gamma^{\prime}(t)}{\gamma(t)}|\leq C^{(1)}_{2}$ for any $t\in(0,1]$.

Then, there exists a positive constant $C$ uniformly for $u\in[1,2]$ such that
for all $f\in L^{p}(\mathbb{R}^{2})$,

(2.1) $\displaystyle\left\|Tf(\cdot,u)\right\|_{L_{x}^{q}(\mathbb{R}^{2})}\leq
C\|f\|_{L_{x}^{p}(\mathbb{R}^{2})},$

if

$(\frac{1}{p},\frac{1}{q})\in\ \vartriangle$ and $(\frac{1}{p},\frac{1}{q})$
satisfy $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$.

On the other hand, (2.1) holds only if

$(\frac{1}{p},\frac{1}{q})\in\ \vartriangle$ and $(\frac{1}{p},\frac{1}{q})$
satisfy $1+(1+\omega)(\frac{1}{q}-\frac{1}{p})\geq 0$.

Here and hereafter, the triangle $\vartriangle:=\\{(\frac{1}{p},\frac{1}{q}):\
\frac{1}{2p}\leq\frac{1}{q}\leq\frac{1}{p},\frac{1}{q}\geq\frac{2}{p}-1\\}$.

######  Remark 2.2.

_The local smoothing phenomenon_. Since the estimate in (2.1) is uniform for
$u\in[1,2]$. By taking a $L^{q}$ integral about $u$ over $[1,2]$, it leads to
the $L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ estimate for $T$. Therefore, Theorem
1.1 will hold in the region of $(\frac{1}{p},\frac{1}{q})$ in Theorem 2.1.

On the other hand, from definitions of $\square$ and $\vartriangle$, we see
that the trapezium $\square$ given by the open convex hull of the points
$\\{O,A,D,C\\}$ but with the half open lines $(O,A]$ and $(D,A]$333For the
convenience of statement of Theorem 1.1, we set $\square$ does not contain the
points $O$ and $D$., and the triangle $\vartriangle$ given by the closed
convex hull of the points $\\{O,A,D\\}$, see the following Figure 2. As a
result, we have
$\vartriangle\subsetneqq\square\cup\\{(0,0)\\}\cup\\{(\frac{2}{3},\frac{1}{3})\\}$,
which leads to the fact that the $L^{q}$ integral about $u$ on $[1,2]$ will
extend the region of $(\frac{1}{p},\frac{1}{q})$ in $T:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$.
This phenomenon is the so called local smoothing phenomenon. It is well known
that this local smoothing phenomenon is very important in harmonic analysis,
such as the local smoothing estimates originated from the works of
Mockenhaupt, Seeger and Sogge [25, 26].

If we consider $\gamma=\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ or
$\gamma_{6}$, we can express the region of $(\frac{1}{p},\frac{1}{q})$ in
Theorem 2.1 as the shaded region in Figure 2. Then, there is an additional
triangle region $OEF^{\prime}$, which is produced by the local smoothing
phenomenon.

Figure 2: The region of $(\frac{1}{p},\frac{1}{q})$ in Theorem 2.1 for
$\gamma=\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ or $\gamma_{6}$.

The proof of Theorem 2.1 relies on the following two lemmas.

######  Lemma 2.3.

([24, Lemma 2.3]) Recall that $\omega=\limsup_{t\rightarrow
0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$. Then, for all $(\frac{1}{p},\frac{1}{q})$
satisfying $\frac{1}{q}\leq\frac{1}{p}$ and
$1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$, we have

$\displaystyle\sum_{j\in\mathbb{Z}_{0}^{-}}2^{j}|2^{j}\gamma(2^{j})|^{\frac{1}{q}-\frac{1}{p}}<\infty.$

######  Lemma 2.4.

([21] or [6, Lemma 2.1]) Let $I$ be a compact interval and let $\phi:\
I\rightarrow\mathbb{R}$ be a $C^{2}$ function. Suppose there exists a positive
constant $C$ such that $|\phi^{\prime\prime}(t)|\geq C$ for $t\in I$. Then
$T^{\phi}_{e}$ given by

$\displaystyle
T^{\phi}_{e}f(x):=\int_{I}f(x_{1}-t,x_{2}-\phi(t))\,\textrm{d}t$

satisfies

$\displaystyle\|T^{\phi}_{e}f\|_{L^{3}(\mathbb{R}^{2})}\leq
C^{\prime}\|f\|_{L^{3/2}(\mathbb{R}^{2})}$

with some constants $C^{\prime}$ depending only on $C$.

We now turn to the proof of Theorem 2.1. Without loss of generality, we can
assume that $f\geq 0$. We split $T$ into pieces via a standard partition of
unity. Let $\psi:\ \mathbb{R}^{+}\rightarrow\mathbb{R}$ be a smooth function
supported on $\\{t\in\mathbb{R}:\ \frac{1}{2}\leq t\leq 2\\}$ with
$0\leq\psi(t)\leq 1$ and $\Sigma_{j\in\mathbb{Z}}\psi_{j}(t)=1$ for any $t>0$,
where $\psi_{j}(t):=\psi(2^{-j}t)$. We conclude that $T$ can be bounded by the
sum of $T_{j}$ over $j\in\mathbb{Z}_{0}^{-}$, where

$T_{j}f(x,u):=\int_{0}^{\infty}f(x_{1}-ut,x_{2}-u\gamma(t))\psi_{j}(t)\,\textrm{d}t,$

which can be rewritten as, by a change of variables,

$\int_{0}^{\infty}f(x_{1}-u2^{j}t,x_{2}-u\gamma(2^{j}t))\psi(t)2^{j}\,\textrm{d}t.$

Let us set

(2.2)
$\displaystyle\widetilde{T}_{j}f(x,u):=\int_{0}^{\infty}f(x_{1}-ut,x_{2}-u\Gamma_{j}(t))\psi(t)\,\textrm{d}t$

with $\Gamma_{j}(t):=\frac{\gamma(2^{j}t)}{\gamma(2^{j})}$, and we denote the
non-isotropic dilation $\delta_{j}$ by
$\delta_{j}f(x):=f(2^{j}x_{1},\gamma(2^{j})x_{2})$ for any
$j\in\mathbb{Z}_{0}^{-}$. Note that

$|2^{j}\gamma(2^{j})|^{\frac{1}{q}}\|\delta_{j}f\|_{L^{q}(\mathbb{R}^{2})}=\|f\|_{L^{q}(\mathbb{R}^{2})}~{}~{}~{}\textrm{and}~{}~{}~{}\delta_{j}\left(T_{j}f\right)=2^{j}\widetilde{T}_{j}(\delta_{j}f)$

holds for all $j\in\mathbb{Z}_{0}^{-}$, it suffices to prove that

$\displaystyle\sum_{j\in\mathbb{Z}_{0}^{-}}2^{j}|2^{j}\gamma(2^{j})|^{\frac{1}{q}-\frac{1}{p}}\left\|\widetilde{T}_{j}\right\|_{L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x}^{q}(\mathbb{R}^{2})}\lesssim 1$

uniformly in $u\in[1,2]$. From Lemma 2.3, if $(\frac{1}{p},\frac{1}{q})$ lies
in the region of Theorem 2.1, it follows that the series
$\sum_{j\in\mathbb{Z}_{0}^{-}}2^{j}|2^{j}\gamma(2^{j})|^{\frac{1}{q}-\frac{1}{p}}$
is bounded. Therefore, for all $(\frac{1}{p},\frac{1}{q})\in\ \vartriangle$,
we need only give an estimate $\widetilde{T}_{j}:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x}^{q}(\mathbb{R}^{2})$ uniformly in
$u\in[1,2]$ and $j\in\mathbb{Z}_{0}^{-}$.

By Minkowski’s inequality, for $p\in[1,\infty]$, it is easy to obtain the
$L_{x}^{p}(\mathbb{R}^{2})$ boundedness of $\widetilde{T}_{j}$ uniformly in
$u\in[1,2]$ and $j\in\mathbb{Z}_{0}^{-}$. It is the desired estimate on the
closed line $[O,A]$ in Figure 2. By interpolation, it suffices to prove the
uniform $L_{x}^{3}(\mathbb{R}^{2})\rightarrow L_{x}^{3/2}(\mathbb{R}^{2})$
estimate for $\widetilde{T}_{j}$, i.e., the estimate at point $D$ in Figure 2.
We will use Lemma 2.4 to complete the proof. Indeed, from [22, Remark 1.4] or
[23, Lemma 2.1], we have the following doubling condition,

(2.3) $\displaystyle e^{C^{(1)}_{1}/2}\leq\frac{\gamma(2t)}{\gamma(t)}\leq
e^{C^{(1)}_{2}}$

for any $t\in(0,1]$, which will also be used in obtaining the necessity of
$1+(1+\omega)(\frac{1}{q}-\frac{1}{p})\geq 0$. By a change of variables,
$\widetilde{T}_{j}$ can be pointwise bounded by $T^{\phi}_{e}$ with
$I:=[\frac{1}{2},4]$ and $\phi(t):=u\Gamma_{j}\left(\frac{t}{u}\right)$,
uniformly in $u\in[1,2]$ and $j\in\mathbb{Z}_{0}^{-}$. From (2.3) and the
following $(\textrm{iii})$ of Lemma 3.1, we have the uniform estimate that
$|\phi^{\prime\prime}(t)|\gtrsim 1$ for any $t\in I$. In view of Lemma 2.4, we
obtain

(2.4)
$\displaystyle\|\widetilde{T}_{j}f\|_{L^{3}(\mathbb{R}^{2})}\lesssim\|f\|_{L^{3/2}(\mathbb{R}^{2})}$

holds uniformly in $u\in[1,2]$ and $j\in\mathbb{Z}_{0}^{-}$. As a result, we
finish the proof of the sufficiency parts in Theorem 1.2.

We now show the necessity. From Remark 2.2 and Theorem 1.2, since the estimate
is uniform in $u\in[1,2]$, we just need to show444Note: The necessity of
$\frac{1}{q}\leq\frac{1}{p}$ can be obtained by a theorem of
$\textrm{H}\ddot{\textrm{o}}\textrm{rmander}$ [14]. The necessity of
$\frac{1}{q}\geq\frac{2}{p}-1$ can also be established by the adjoint operator
$T_{u}^{*}$ of $T_{u}f(x):=Tf(x,u)$, which is essentially the operator
$T_{u}$. the necessity of $\frac{1}{2p}\leq\frac{1}{q}$. Indeed, let
$f:=\chi_{A}$, where $A$ is the $\epsilon$ neighborhood of the curve
$(-ut,-u\gamma(t))$ with $t\in(0,1]$, $u\in[1,2]$ and $\epsilon>0$ small
enough. Then

$\displaystyle\int_{0}^{1}\chi_{A}(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t=1$

for any $(x_{1},x_{2})\in B(0,\epsilon)$. It further follows that

$\displaystyle\left\|\int_{0}^{1}\chi_{A}(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t\right\|_{L_{x}^{q}(B(0,\epsilon))}\gtrsim\epsilon^{\frac{2}{q}}~{}~{}\textrm{and}~{}~{}\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}\approx\epsilon^{\frac{1}{p}}.$

Apply the estimate (2.1) to deduce that
$\epsilon^{\frac{2}{q}-\frac{1}{p}}\lesssim 1$. As a consequence, we have that
$\frac{1}{2p}\leq\frac{1}{q}$ since $\epsilon>0$ may tend to $0$. This
concludes the necessity parts in Theorem 2.1.

##  3 Proof of Theorem 1.1

In this section, we devote to the proof of Theorem 1.1. Let us begin by
introducing the following lemma.

######  Lemma 3.1.

([22, Lemma 2.2]) Let $t\in[\frac{1}{2},2]$ and
$\Gamma_{j}(t)=\frac{\gamma(2^{j}t)}{\gamma(2^{j})}$ with
$j\in\mathbb{Z}_{0}^{-}$. The following inequalities hold uniformly in $j$,

  1. (i)

$e^{-C^{(1)}_{2}}\leq\Gamma_{j}(t)\leq e^{C^{(1)}_{2}}$;

  2. (ii)

$\frac{C^{(1)}_{1}}{2e^{C^{(1)}_{2}}}\leq|\Gamma_{j}^{\prime}(t)|\leq
2e^{C^{(1)}_{2}}C^{(1)}_{2}$;

  3. (iii)

$\frac{C^{(2)}_{1}}{4e^{C^{(1)}_{2}}}\leq|\Gamma_{j}^{\prime\prime}(t)|\leq
4e^{C^{(1)}_{2}}C^{(2)}_{2}$;

  4. (iv)

$|\Gamma_{j}^{(k)}(t)|\leq 2^{k}e^{C^{(1)}_{2}}C^{(k)}_{2}$ for all $2\leq
k\leq N$ and $k\in\mathbb{N}$;

  5. (v)

$|((\Gamma_{j}^{\prime})^{-1})^{(k)}(t)|\lesssim 1$ for all $0\leq k<N$ and
$k\in\mathbb{N}$, where $(\Gamma_{j}^{\prime})^{-1}$ is the inverse function
of $\Gamma_{j}^{\prime}$.

We now turn to the proof of Theorem 1.1. Similarly as in the proof of Theorem
2.1, using the fact that

$\displaystyle\delta_{j}\left(\left\|T_{j}f\right\|_{L_{u}^{q}([1,2])}\right)=2^{j}\left\|\widetilde{T}_{j}(\delta_{j}f)\right\|_{L_{u}^{q}([1,2])}$

holds for all $j\in\mathbb{Z}_{0}^{-}$, it is enough to show

$\displaystyle\sum_{j\in\mathbb{Z}_{0}^{-}}2^{j}|2^{j}\gamma(2^{j})|^{\frac{1}{q}-\frac{1}{p}}\left\|\widetilde{T}_{j}\right\|_{L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim 1.$

From Lemma 2.3, we should only obtain $\widetilde{T}_{j}:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
uniformly in $j\in\mathbb{Z}_{0}^{-}$ for all
$(\frac{1}{p},\frac{1}{q})\in\square\cup\\{(0,0)\\}\cup\\{(\frac{2}{3},\frac{1}{3})\\}$.
By interpolation, it suffices to establish the corresponding uniform estimate
with $(\frac{1}{p},\frac{1}{q})$ lies in Figure 3. For the closed line
$[O,A]$, by Minkowski’s inequality, we may obtain the
$L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$
estimate for $\widetilde{T}_{j}$ uniformly in $j\in\mathbb{Z}_{0}^{-}$ for all
$p=q\in[1,\infty]$. For the point $D(\frac{2}{3},\frac{1}{3})$,
$L_{u}^{q}([1,2])$ integration in (2.4) will leads to the uniform estimate
$\widetilde{T}_{j}:\ L_{x}^{3/2}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{3}(\mathbb{R}^{2}\times[1,2])$ as desired. Therefore, it suffices to
consider the case of the half open line $(C,M]$ in Figure 3.

Figure 3: The framework to obtain the uniform estimate $\widetilde{T}_{j}:\
L_{x}^{p}(\mathbb{R}^{2})\rightarrow L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$.

After taking a Fourier transform in (2.2), let $\xi:=(\xi_{1},\xi_{2})$, we
see that

$\widetilde{T}_{j}f(x,u)=\int_{\mathbb{R}^{2}}\hat{f}(\xi)e^{ix\cdot\xi}H_{j}(u,\xi)\,\textrm{d}\xi~{}~{}~{}~{}~{}\textrm{with}~{}~{}~{}~{}~{}H_{j}(u,\xi):=\int_{\mathbb{R}}e^{-iu\xi_{1}t-iu\xi_{2}\Gamma_{j}(t)}\psi(t)\,\textrm{d}t.$

Let $\Psi^{0}:=\sum_{k\leq 0}\psi_{k}$, we decompose

(3.1)
$\displaystyle\widetilde{T}_{j}f(x,u)=\widetilde{T}^{0}_{j}f(x,u)+\sum_{k\geq
1}\widetilde{T}_{j,k}f(x,u)$

and try to estimate the each decomposed pieces, where

$\displaystyle\left\\{\begin{aligned}
\widetilde{T}^{0}_{j}f(x,u):=\int_{\mathbb{R}^{2}}\hat{f}(\xi)e^{ix\cdot\xi}H_{j}(u,\xi)\Psi^{0}(u|\xi|)\,\textrm{d}\xi;\\\
\widetilde{T}_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}\hat{f}(\xi)e^{ix\cdot\xi}H_{j}(u,\xi)\psi_{k}(u|\xi|)\,\textrm{d}\xi.\end{aligned}\right.$

Consider $\widetilde{T}^{0}_{j}$. Noting that $\Psi^{0}$ is a compactly
supported smooth function, by Lemma 3.1, we have
$|\partial_{u}^{\alpha}\partial^{\beta}_{\xi}(H(u,\xi)\Psi^{0}(u|\xi|))|\lesssim
1/(1+|\xi|)^{3}$ for all $\alpha\in\mathbb{N}^{2}_{0}$ with $|\alpha|\leq 3$,
where the implicit constant is independent of $u\in[1,2]$ and
$j\in\mathbb{Z}_{0}^{-}$, which further leads to
$(H(u,\cdot)\Psi^{0}(u|\cdot|))^{\vee}\in L^{r}(\mathbb{R}^{2})$ for all
$1\leq r\leq\infty$. Thus, by Young’s inequality, we assert that555Here and
hereafter, our estimates are all uniformly in $j\in\mathbb{Z}_{0}^{-}$, we
will not repeat it again.

$\displaystyle\left\|\widetilde{T}^{0}_{j}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$

for all $q\geq p\geq 1$.

Consider $\widetilde{T}_{j,k}$. The phase function in $H_{j}(u,\xi)$ is
$\varphi_{j}(u,\xi,t):=-u\xi_{1}t-u\xi_{2}\Gamma_{j}(t)$. Differentiate in $t$
to obtain

$\displaystyle\partial_{t}\varphi_{j}(u,\xi,t)=-u\xi_{1}-u\xi_{2}\Gamma^{\prime}_{j}(t),~{}~{}\partial^{2}_{t}\varphi_{j}(u,\xi,t)=-u\xi_{2}\Gamma^{\prime\prime}_{j}(t),~{}~{}\partial^{2}_{t}\varphi_{j}(u,\xi,t)=-u\xi_{2}\Gamma^{\prime\prime\prime}_{j}(t).$

By $(\textrm{ii})$ of Lemma 3.1, if $|\xi_{1}|\geq
10e^{C^{(1)}_{2}}C^{(1)}_{2}|\xi_{2}|$ or $|\xi_{2}|\geq
10e^{C^{(1)}_{2}}/C^{(1)}_{1}|\xi_{1}|$, we have
$|\partial_{t}\varphi_{j}(u,\xi,t)|\gtrsim|\xi|$. Let $\rho\in
C^{\infty}_{c}(\mathbb{R}^{+})$ be a function such that $\rho=1$ on
$[C^{(1)}_{1}/(10e^{C^{(1)}_{2}}),10e^{C^{(1)}_{2}}C^{(1)}_{2}]$. Observe that
$|\xi|\approx 2^{k}$, integration by parts shows that
$|\partial_{u}\partial^{\alpha}_{\xi}[(1-\rho(\frac{|\xi_{1}|}{|\xi_{2}|}))H(u,\xi)\psi_{k}(u|\xi|)]|\lesssim
1/(1+|\xi|)^{3}$ for all $|\alpha|\leq 3$, where the implicit constant is
independent of $u\in[1,2]$. We now define

$\displaystyle\left\\{\begin{aligned}
&\widetilde{T}^{1}_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}\hat{f}(\xi)e^{ix\cdot\xi}\rho\left(\frac{|\xi_{1}|}{|\xi_{2}|}\right)H_{j}(u,\xi)\psi_{k}(u|\xi|)\,\textrm{d}\xi;\\\
&\widetilde{T}^{2}_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}\hat{f}(\xi)e^{ix\cdot\xi}\left(1-\rho\left(\frac{|\xi_{1}|}{|\xi_{2}|}\right)\right)H_{j}(u,\xi)\psi_{k}(u|\xi|)\,\textrm{d}\xi.\end{aligned}\right.$

Then we conclude immediately the desired estimate that

(3.2)
$\displaystyle\left\|\widetilde{T}^{2}_{j,k}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim
2^{-k}\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$

for all $q\geq p\geq 1$. This, combined with (3.1), implies that we should
only to bound the operator $\widetilde{T}^{1}_{j,k}$ with a decay in $k$.

Consider $\widetilde{T}^{1}_{j,k}$. If $\xi_{1}\xi_{2}\geq 0$, it is easy to
see that $|\partial_{t}\varphi_{j}(u,\xi,t)|\gtrsim|\xi|$. By the argument as
above, we also obtain the desired estimate (3.2) for
$\widetilde{T}^{1}_{j,k}$. Therefore, from now on, we restrict our attention
to the most difficult case of $\xi_{1}\xi_{2}<0$. Let

$\partial_{t}\varphi_{j}(u,\xi,t_{0})=-u\xi_{1}-u\xi_{2}\Gamma^{\prime}_{j}(t_{0})=0.$

Note that $t_{0}$ is the critical point and
$\Gamma^{\prime}_{j}(t_{0})=-\frac{\xi_{1}}{\xi_{2}}$. Since our estimates
about $t_{0}$ is uniform in $j$, we omit the parameter $j$ in the notation
$t_{0}$. Since $\Gamma^{\prime}_{j}$ is strictly monotonic, we can write
$t_{0}=(\Gamma^{\prime}_{j})^{-1}(-\frac{\xi_{1}}{\xi_{2}})$. We can further
assume that $t_{0}\in(1/2,2)$. Otherwise, it is easy to see that
$|\varphi^{\prime}_{t}(u,\xi,t)|\gtrsim|\xi|$ holds, which leads to the
desired estimate as above.

We consider a one-dimensional oscillatory integral involving a phase function
with a non-degenerate critical point. Based on the spirit of stationary phase
estimates, we rewrite

$\displaystyle
H_{j}(u,\xi)=\int_{\mathbb{R}}e^{i\varphi_{j}(u,\xi,t+t_{0})}\psi(t+t_{0})\,\textrm{d}t.$

By Taylor’s theorem and the fact that $t_{0}$ is the critical point, we
conclude that

$\displaystyle\varphi_{j}(u,\xi,t+t_{0})=\varphi_{j}(u,\xi,t_{0})-u\xi_{2}t^{2}\eta_{j}(t,t_{0}),$

where
$\eta_{j}(t,t_{0}):=\frac{\Gamma^{\prime\prime}_{j}(t_{0})}{2}+\frac{t}{2}\int_{0}^{1}(1-\theta)^{2}\Gamma^{\prime\prime\prime}_{j}(\theta
t+t_{0})\,\textrm{d}\theta$, which can be viewed as a compactly supported
smooth function. Therefore,

(3.3)
$\displaystyle\widetilde{T}^{1}_{j,k}f(x,u)=\int_{\mathbb{R}^{2}}e^{ix\cdot\xi}e^{i\varphi_{j}(u,\xi,t_{0})}m_{j,k}(u,\xi)\hat{f}(\xi)\,\textrm{d}\xi,$

where

$m_{j,k}(u,\xi):=\rho\left(\frac{|\xi_{1}|}{|\xi_{2}|}\right)\psi_{k}(u|\xi|)\int_{\mathbb{R}}e^{-iu\xi_{2}t^{2}\eta_{j}(t,t_{0})}\psi(t+t_{0})\,\textrm{d}t.$

Localization.

$\widetilde{T}^{1}_{j,k}$ can be localized. Indeed, we write

$\displaystyle\widetilde{T}^{1}_{j,k}f(x,u)=\int_{\mathbb{R}^{2}}K_{j,k}(x,u,y)f(y)\,\textrm{d}y$

with the kernel

$\displaystyle
K_{j,k}(x,u,y):=\int_{\mathbb{R}^{2}}e^{i(x-y)\cdot\xi}e^{i\varphi_{j}(u,\xi,t_{0})}m_{j,k}(u,\xi)\,\textrm{d}\xi.$

Notice that $\Gamma^{\prime}_{j}(t_{0})=-\frac{\xi_{1}}{\xi_{2}}$,
$t_{0}\approx 1$ and $\Gamma_{j}(t_{0})\approx 1$, it follows that there
exists a positive constant $\varpi$ large enough such that
$|\nabla_{\xi}[(x-y)\cdot\xi+\varphi_{j}(u,\xi,t_{0})]|\gtrsim|x-y|$ if
$|x-y|\geq\varpi$. Therefore, via an integration by parts, we can bound the
kernel $K_{j,k}$ by $2^{(-\frac{1}{2}-N)k}/|x-y|^{N}$ uniformly in $u\in[1,2]$
for some $N\in\mathbb{N}$ large enough and $|x-y|\geq\varpi$. Furthermore, we
may obtain the desired estimate that

$\displaystyle\left\|\widetilde{T}^{1,a}_{j,k}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim
2^{(-\frac{1}{2}-N)k}\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$

for all $q\geq p\geq 1$, where

$\displaystyle\widetilde{T}^{1,a}_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}\chi_{B^{\complement}(x,\varpi)}(y)K_{j,k}(x,u,y)f(y)\,\textrm{d}y.$

As a result, we should only to consider the following localized operator,

(3.4)
$\displaystyle\widetilde{T}^{1,b}_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}\chi_{B(x,\varpi)}(y)K_{j,k}(x,u,y)f(y)\,\textrm{d}y.$

Reduce to a local smoothing estimate.

We start with some definitions. Let $\Omega:\
\mathbb{R}^{2}\times\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative smooth
function with the properties that $\Omega(x,u)=1$ on
$\\{(x,u)\in\mathbb{R}^{2}\times\mathbb{R}:\ |x|\leq\varpi,u\in[1,2]\\}$ and
vanishing outside $\\{(x,u)\in\mathbb{R}^{2}\times\mathbb{R}:\ |x|\leq
2\varpi,u\in(1/2,5/2)\\}$,
$\Psi_{j}(x,u,\xi):=x\cdot\xi+\varphi_{j}(u,\xi,t_{0})$ and
$a_{j,k}(x,u,\xi):=\Omega(x,u)m_{j,k}(u,\xi)$. Define

(3.5) $\displaystyle
F_{j,k}f(x,u):=\int_{\mathbb{R}^{2}}e^{i\Psi_{j}(x,u,\xi)}a_{j,k}(x,u,\xi)\hat{f}(\xi)\,\textrm{d}\xi.$

In order to estimate $\widetilde{T}^{1,b}_{j,k}$ in (3.4), it suffices to
obtain the following estimate: there exists a positive constant $\varepsilon$
such that

(3.6)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{3})}\lesssim
2^{-\varepsilon k}\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$

for all $(\frac{1}{p},\frac{1}{q})$ in the half open line $(C,M]$ in Figure 3.

Indeed, we can decompose
$\mathbb{R}^{2}=\bigcup_{i\in\mathbb{Z}^{2}}B(x_{i},\varpi)$ such that
$|x_{i}-x_{i^{\prime}}|\approx|i-i^{\prime}|\varpi$ for any $i\neq
i^{\prime}$. Next, we change variables to bound
$\|\widetilde{T}^{1,b}_{j,k}f\|^{q}_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}$
by

$\displaystyle\sum_{i\in\mathbb{Z}^{2}}\left\|\int_{\mathbb{R}^{2}}\chi_{B(x,\varpi)}(y)K_{j,k}(x,u,y)f(y-x_{i})\,\textrm{d}y\right\|^{q}_{L_{x,u}^{q}(B(0,\varpi)\times[1,2])},$

which is further bounded by

(3.7) $\displaystyle\sum_{i\in\mathbb{Z}^{2}}\left\|(E_{j,k}-F_{j,k})(f(\cdot-
x_{i}))\right\|^{q}_{L_{x,u}^{q}(\mathbb{R}^{3})}+\sum_{i\in\mathbb{Z}^{2}}\left\|F_{j,k}(f(\cdot-
x_{i}))\right\|^{q}_{L_{x,u}^{q}(\mathbb{R}^{3})},$

where

$\displaystyle
E_{j,k}f(x,u):=\Omega(x,u)\int_{\mathbb{R}^{2}}\chi_{B(x,\varpi)}(y)K_{j,k}(x,u,y)f(y)\,\textrm{d}y.$

The operator $E_{j,k}-F_{j,k}$ can be handled by a similar argument to
$\widetilde{T}^{1,a}_{j,k}$ since the kernel of $E_{j,k}-F_{j,k}$ is supported
on $\\{(x,y)\in\mathbb{R}^{2}:\ |x-y|\geq\varpi\\}$. Therefore, the first part
of (3.7) can be controlled by

$\displaystyle\sum_{i\in\mathbb{Z}^{2}}2^{(-\frac{1}{2}-N)kq}\|f(\cdot-
x_{i})\chi_{B(0,2\varpi)}(\cdot)\|^{q}_{L_{x}^{p}(\mathbb{R}^{2})}\lesssim
2^{(-\frac{1}{2}-N)kq}\|f\|^{q}_{L_{x}^{p}(\mathbb{R}^{2})}$

for some $N\in\mathbb{N}$ large enough, as desired. For the second part of
(3.7), by (3.6), we can also obtain the following desired estimate,

$\displaystyle\sum_{i\in\mathbb{Z}^{2}}2^{-\varepsilon kq}\|f(\cdot-
x_{i})\chi_{B(0,2\varpi)}(\cdot)\|^{q}_{L_{x}^{p}(\mathbb{R}^{2})}\lesssim
2^{-\varepsilon kq}\|f\|^{q}_{L_{x}^{p}(\mathbb{R}^{2})}.$

Putting things together, by (3.7), we can establish
$\widetilde{T}^{1,b}_{j,k}:\ L_{x}^{p}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])$ with bound
$2^{(-\frac{1}{2}-N)k}+2^{-\varepsilon k}$, this is the desired estimate.
Thus, we can reduce our considerations to proving a local smoothing estimate
in (3.6).

A key local smoothing estimate.

Consider the _Fourier integral operators_

(3.8)
$\displaystyle\mathcal{F}f(z):=\int_{\mathbb{R}^{n}}e^{i\phi(z,\xi)}a(z,\xi)\hat{f}(\xi)\,\textrm{d}\xi,$

where $a(z,\xi)$ is a symbol of order $\sigma$. Here and hereafter, $z$ is
used to denote a vector in $\mathbb{R}^{n}\times\mathbb{R}$ comprised of the
space-time variables $(x,u)$. Suppose that $\textrm{supp}~{}a(\cdot,\xi)$ is
contained in a fixed compact set and $\phi(z,\cdot)$ is a homogeneous function
of degree $1$. We say that $\mathcal{F}$ satisfies the _cinematic curvature
condition_ if the following conditions are satisfied:

  1. $\bullet$

Non-degeneracy condition. For all $(z,\xi)\in\textrm{supp}~{}a$

$\displaystyle\textrm{rank}~{}\partial^{2}_{z\xi}\phi(z,\xi)=n.$

  2. $\bullet$

Cone condition. Consider the Gauss map $G:\
\textrm{supp}~{}a\rightarrow\mathbb{S}^{n}$ by
$G(z,\xi):=\frac{G_{0}(z,\xi)}{|G_{0}(z,\xi)|}$, where

$\displaystyle
G_{0}(z,\xi):=\bigwedge_{j=1}^{n}\partial_{\xi_{j}}\partial_{z}\phi(z,\xi).$

The curvature condition

$\displaystyle\textrm{rank}~{}\partial^{2}_{\xi\xi}\langle\partial_{z}\phi(z,\xi),G(z,\xi_{0})\rangle|_{\xi=\xi_{0}}=n-1$

holds for all $(z,\xi_{0})\in\textrm{supp}~{}a$.

Geometrically, the curvature condition of course means that, for fixed
$z_{0}$, the cone

$\Gamma_{z_{0}}:=\left\\{\partial_{z}\phi(z_{0},\eta):\
\eta\in\mathbb{R}^{n}\backslash\\{0\\}~{}\textrm{in}~{}\textrm{a}~{}\textrm{conic}~{}\textrm{neighbourhood}~{}\textrm{of}~{}\eta_{0}\right\\}$

is a smooth conic manifold of dimension $n$ with $n-1$ non-vanishing principal
curvatures at every point. We can see that the phase featured in the half-wave
propagator, i.e., $\phi(z,\xi)=x\cdot\xi-u|\xi|$, satisfies the cinematic
curvature condition.

Now, we show some local smoothing estimates for $\mathcal{F}$ in (3.8), which
will be used to prove (3.6). To state the following propositions we introduce
some definitions. Let $\bar{\psi}:\ \mathbb{R}^{+}\rightarrow\mathbb{R}$ be a
smooth function supported on $\\{t\in\mathbb{R}:\ \frac{1}{8}\leq t\leq 8\\}$
with $0\leq\psi(t)\leq 1$ and $\bar{\psi}(t)=1$ on $\\{t\in\mathbb{R}:\
\frac{1}{4}\leq t\leq 4\\}$. We then use $P_{k}$ to denote the operator on
$\mathbb{R}^{n}$ with multiplier $\bar{\psi}_{k}(|\xi|)$, where
$\bar{\psi}_{k}(|\xi|):=\bar{\psi}(2^{-k}|\xi|)$ and $k\in\mathbb{N}$.

######  Proposition 3.2.

Let $1\leq p\leq q\leq\infty$ and $s\in\mathbb{R}$. Let $\mathcal{F}$ be the
Fourier integral operators in (3.8) with symbol of order $\sigma$ and satisfy
the cinematic curvature condition, together with the special assumption that
$\textrm{rank}~{}\partial^{2}_{\xi\xi}\phi(z,\xi)=n-1$ when $\xi\neq 0$. Then,
if $q=p\in[1,\infty)$, we require
$s\geq(n-1)|\frac{1}{p}-\frac{1}{2}|+\sigma$; if $q=\infty$, we require
$s\geq\frac{n-1}{2}+\frac{1}{p}+\sigma$, there exists a positive constant
$C_{(u)}$ such that

(3.9)
$\displaystyle\left\|\mathcal{F}P_{k}f\right\|_{L_{x}^{q}(\mathbb{R}^{n})}\leq
C_{(u)}\|f\|_{W_{x}^{s,p}(\mathbb{R}^{n})}.$

Note that the case $1\leq p<q<\infty$ can be proved by interpolation, we omit
the details.

###### Proof of Proposition 3.2..

For the case $q=p\in[1,\infty)$, which follows from the fixed time estimate in
Seeger, Sogge and Stein [32]. Here, for the case $p=q=1$, we used the estimate
$\|P_{k}f\|_{H_{x}^{1}(\mathbb{R}^{n})}\lesssim\|f\|_{L_{x}^{1}(\mathbb{R}^{n})}$.
For the case $q=\infty$, by interpolation, it suffices to show

(3.10)
$\displaystyle\left\|\mathcal{F}P_{k}f\right\|_{L_{x}^{\infty}(\mathbb{R}^{n})}\leq
C_{(u)}\left\|f\right\|_{W_{x}^{\frac{n+1}{2}+\sigma,1}(\mathbb{R}^{n})}$

and

(3.11)
$\displaystyle\left\|\mathcal{F}P_{k}f\right\|_{L_{x}^{\infty}(\mathbb{R}^{n})}\leq
C_{(u)}\left\|f\right\|_{W_{x}^{\frac{n-1}{2}+\sigma,\infty}(\mathbb{R}^{n})}.$

(3.10) follows from Stein [36, Chapter IX, 6.14] and the estimate
$\|P_{k}f\|_{H_{x}^{1}(\mathbb{R}^{n})}\lesssim\|f\|_{L_{x}^{1}(\mathbb{R}^{n})}$.
We now prove (3.11). First of all, Seeger, Sogge and Stein [32, Theorem 2.2]
obtained the $H_{x}^{1}(\mathbb{R}^{n})\rightarrow L_{x}^{1}(\mathbb{R}^{n})$
estimate for the Fourier integral operators with symbol of order
$-\frac{n-1}{2}$. From Stein [36, Chapter IX, Section 4], this result also
holds for its adjoint operator, which further leads to the corresponding
$L_{x}^{\infty}(\mathbb{R}^{n})\rightarrow BMO_{x}(\mathbb{R}^{n})$ estimate
for these Fourier integral operators with symbol of order $-\frac{n-1}{2}$.
Consequently

(3.12)
$\displaystyle\left\|\mathcal{F}P_{k}f\right\|_{BMO_{x}(\mathbb{R}^{n})}\leq
C_{(u)}\left\|f\right\|_{W_{x}^{\frac{n-1}{2}+\sigma,\infty}(\mathbb{R}^{n})}.$

Notice that $\|\mathcal{F}P_{k}f\|_{L_{x}^{\infty}(\mathbb{R}^{n})}$ can be
bounded by

$\displaystyle\left\|\tilde{P}_{k}\mathcal{F}P_{k}f\right\|_{L_{x}^{\infty}(\mathbb{R}^{n})}+\left\|(I-\tilde{P}_{k})\mathcal{F}P_{k}f\right\|_{L_{x}^{\infty}(\mathbb{R}^{n})}\leq
C_{(u)}\left(\left\|\mathcal{F}P_{k}f\right\|_{BMO_{x}(\mathbb{R}^{n})}+\left\|f\right\|_{W_{x}^{\frac{n-1}{2}+\sigma,\infty}(\mathbb{R}^{n})}\right),$

where $\tilde{P}_{k}f:=(\tilde{\psi}_{k}(|\cdot|)\hat{f}(\cdot))^{\vee}$,
$\tilde{\psi}$ is a compactly supported $C^{\infty}$ function on
$\mathbb{R}\backslash\\{0\\}$ and $\tilde{\psi}=1$ on $[C^{-1},C]$ for a
positive constant $C$ sufficiently large. This, combined with (3.12), leads to
(3.11). Therefore, the proof is complete. ∎

We give the local smoothing estimates for $\mathcal{F}$ about $P_{k}f$ in the
following Proposition 3.3. Since Theorem 1.1 is almost sharp and relies
heavily on this proposition, we may see that the result in Proposition 3.3 is
also almost sharp.

######  Proposition 3.3.

Let $n=2$, $1\leq p\leq q\leq\infty$ and $s\in\mathbb{R}$. Let $\mathcal{F}$
be the Fourier integral operators in (3.8) with symbol of order $\sigma$ and
satisfy the cinematic curvature condition, together with the special
assumption that $\textrm{rank}~{}\partial^{2}_{\xi\xi}\phi(z,\xi)=1$ when
$\xi\neq 0$. If $q=p\in[1,\infty)$, we require:
$s\geq\frac{1}{p}-\frac{1}{2}+\sigma$ for $p\in[1,2]$; $s>\sigma$ for
$p\in(2,4)$; $s>\frac{1}{2}-\frac{2}{p}+\sigma$ for $p\in[4,\infty)$. If
$q=\infty$, we require $s\geq\frac{1}{2}+\frac{1}{p}+\sigma$. Then, there
exists a positive constant $C$ such that

(3.13)
$\displaystyle\left\|\mathcal{F}P_{k}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\leq
C\|f\|_{W_{x}^{s,p}(\mathbb{R}^{2})}.$

Note that the case $1\leq p<q<\infty$ can be obtained by interpolation, we
omit the details.

###### Proof of Proposition 3.3..

For the case $q=p\in[1,2]$, since $C_{(u)}\lesssim 1$ for any $u\in[1,2]$ from
Seeger, Sogge and Stein [32], it follows that (3.13) can be obtained by
Proposition 3.2. For the case $q=p\in[4,\infty)$, (3.13) follows from Gao,
Liu, Miao and Xi [9, Theorem 1.4], which is a generalization of the local
smoothing conjecture in $2+1$ dimensions obtained in Guth, Wang and Zhang [12,
Theorem 1.2]. For the case $q=p\in(2,4)$, by interpolation, we may establish
(3.13). It is easy to see that (3.13) also follows from Proposition 3.2 for
the case $q=\infty$, since $C_{(u)}\lesssim 1$ for any $u\in[1,2]$. This
finishes the proof. ∎

In order to obtain (3.6) by the key Proposition 3.3. We need to verify that
the Fourier integral operators $F_{j,k}$ in (3.5) satisfy all of the
conditions stated about $\mathcal{F}$ and the special assumption in
Proposition 3.3:

  1. $\circ$

We can directly check that
$\textrm{supp}~{}a_{j,k}\subset\\{x\in\mathbb{R}^{2}:\ |x|\leq
2\varpi\\}\times[1/2,5/2]\times\\{\xi\in\mathbb{R}^{2}:\
|\xi_{1}|\approx|\xi_{2}|\approx 2^{k}\\}$ and the phase function
$\Psi_{j}(x,u,\cdot)$ is a homogeneous function of degree $1$.

  2. $\circ$

$a_{j,k}(x,u,\xi)$ is a symbol of order $-\frac{1}{2}$. Indeed, by a similar
argument in Stein [36, P. 335, Proposition 3] or Sogge [33, P. 44, Lemma
1.1.2], one may verify that

(3.14) $\displaystyle\left|\int_{\mathbb{R}}e^{-i\lambda
t^{2}\eta_{j}(t,t_{0})}t^{2l}\varrho(t,t_{0})\,\textrm{d}t\right|\lesssim|\lambda|^{-\frac{1}{2}-l},$

where $|\lambda|\geq 1$, $l\in\mathbb{N}_{0}$ and $\varrho$ is a compactly
supported smooth function. Here, we used the uniform properties in Lemma 3.1,
and leave the details to the reader. Let $I_{j,l}$ be the integral in (3.14)
and

$\displaystyle\zeta_{j}(\lambda,t_{0}):=\int_{\mathbb{R}}e^{-i\lambda
t^{2}\eta_{j}(t,t_{0})}\psi(t+t_{0})\,\textrm{d}t.$

A calculation gives that $\partial_{t_{0}}^{b}\zeta_{j}$ can be written as a
finite sum of $\lambda^{l}I_{j,l}$ for all $b<N-2$. Then, (3.14) leads to
$|\partial_{t_{0}}^{b}\zeta_{j}|\lesssim|\lambda|^{-\frac{1}{2}}$ for all
$b<N-2$. Furthermore, by the Leibniz rule, we can write
$\partial_{\lambda}^{a}\partial_{t_{0}}^{b}\zeta_{j}$ as a finite sum of
$\lambda^{-a}\lambda^{l}I_{j,l}$ for all $a,b\in\mathbb{N}$ and $b<N-2$.
Therefore, for $|\lambda|\geq 1$, by (3.14), we deduce that

(3.15)
$\displaystyle\left|\partial_{\lambda}^{a}\partial_{t_{0}}^{b}\zeta_{j}(\lambda,t_{0})\right|\lesssim|\lambda|^{-\frac{1}{2}-a}$

for all $a,b\in\mathbb{N}$ and $b<N-2$. On the other hand,
$t_{0}=(\Gamma^{\prime}_{j})^{-1}(-\frac{\xi_{1}}{\xi_{2}})$, $|\xi|\approx
2^{k}\geq 1$ and $\frac{|\xi_{1}|}{|\xi_{2}|}\approx 1$, together with
$(\textrm{v})$ of Lemma 3.1 yield
$|\partial_{\xi}^{\alpha}t_{0}|\lesssim(1+|\xi|)^{-|\alpha|}$ for all
$\alpha\in\mathbb{N}_{0}^{2}$ with $|\alpha|<N$. This, combined with (3.15),
shows that

$\displaystyle\left|\partial_{u}^{\beta}\partial_{\xi}^{\alpha}m_{j,k}(u,\xi)\right|\lesssim(1+|\xi|)^{-\frac{1}{2}-|\alpha|}$

for all $(\alpha,\beta)\in\mathbb{N}^{2}_{0}\times\mathbb{N}_{0}$ with
$|\alpha|+|\beta|<N$. Consequently,

$\displaystyle\left|\partial_{x}^{\gamma}\partial_{u}^{\beta}\partial_{\xi}^{\alpha}a_{j,k}(x,u,\xi)\right|\lesssim(1+|\xi|)^{-\frac{1}{2}-|\alpha|}$

for all
$(\alpha,\beta,\gamma)\in\mathbb{N}^{2}_{0}\times\mathbb{N}_{0}\times\mathbb{N}^{2}_{0}$
with $|\alpha|+|\beta|+|\gamma|<N$, where the implicit constant is independent
of $j$ and $k$. Therefore, $a_{j,k}(x,u,\xi)$ is a symbol of order
$-\frac{1}{2}$.

  3. $\circ$

$F_{j,k}$ in (3.5) satisfies the non-degeneracy condition with $n=2$. Recall
that
$\Psi_{j}(z,\xi)=x_{1}\xi_{1}+x_{2}\xi_{2}-u\xi_{1}t_{0}-u\xi_{2}\Gamma_{j}(t_{0})$
and $t_{0}=(\Gamma^{\prime}_{j})^{-1}(-\frac{\xi_{1}}{\xi_{2}})$. Then

$\displaystyle\partial^{2}_{z\xi}\Psi_{j}(z,\xi)=\left(\begin{array}[]{ccc}1{}&~{}0{}&~{}-t_{0}\\\
0{}&~{}1{}&~{}-\Gamma_{j}(t_{0})\end{array}\right).$

It is clear that the rank of the mixed Hessian of $\Psi_{j}$ is $2$.

  4. $\circ$

For the cone condition with $n=2$, one observes that
$\partial_{z}\Psi_{j}(z,\xi)=(\xi_{1},\xi_{2},-\xi_{1}t_{0}-\xi_{2}\Gamma_{j}(t_{0}))$.
Let $G:=(G_{1},G_{2},G_{3})\in\mathbb{S}^{2}$ be the direction (unique up to
sign) satisfying $\nabla_{\xi}\langle\partial_{z}\Psi_{j}(z,\xi),G\rangle=0$,
combining $\Gamma^{\prime}_{j}(t_{0})=-\frac{\xi_{1}}{\xi_{2}}$, we then have
$G_{1}-G_{3}t_{0}=0$ and $G_{2}-G_{3}\Gamma_{j}(t_{0})=0$, which further leads
to
$G=G(z,\xi)=(t_{0},\Gamma_{j}(t_{0}),1)/[t_{0}^{2}+\Gamma_{j}(t_{0})^{2}+1]^{1/2}$.
On the other hand, by calculation, we obtain

$\displaystyle\left\\{\begin{aligned}
&\partial^{2}_{\xi_{1}\xi_{1}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=-G_{3}(z,\xi_{0})\partial_{\xi_{1}}t_{0},\\\
&\partial^{2}_{\xi_{1}\xi_{2}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=-G_{3}(z,\xi_{0})\partial_{\xi_{2}}t_{0},\\\
&\partial^{2}_{\xi_{2}\xi_{1}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=-G_{3}(z,\xi_{0})\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{1}}t_{0},\\\
&\partial^{2}_{\xi_{2}\xi_{2}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=-G_{3}(z,\xi_{0})\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{2}}t_{0}.\end{aligned}\right.$

As a consequence, it follows that

$\displaystyle\partial^{2}_{\xi\xi}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=\left(\begin{array}[]{ccc}-G_{3}(z,\xi_{0})\partial_{\xi_{1}}t_{0}{}&~{}-G_{3}(z,\xi_{0})\partial_{\xi_{2}}t_{0}\\\
-G_{3}(z,\xi_{0})\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{1}}t_{0}{}&~{}-G_{3}(z,\xi_{0})\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{2}}t_{0}\end{array}\right).$

Observe that
$\partial_{\xi_{1}}t_{0}=-\frac{1}{\xi_{2}}\frac{1}{\Gamma^{\prime\prime}_{j}(t_{0})}$
and
$\partial_{\xi_{2}}t_{0}=\frac{\xi_{1}}{\xi_{2}^{2}}\frac{1}{\Gamma^{\prime\prime}_{j}(t_{0})}$,
one obtains

$\displaystyle\partial^{2}_{\xi_{1}\xi_{2}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=\partial^{2}_{\xi_{2}\xi_{1}}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}},$

which leads to
$\textrm{rank}~{}\partial^{2}_{\xi\xi}\langle\partial_{z}\Psi_{j}(z,\xi),G(z,\xi_{0})\rangle~{}|_{~{}\xi=\xi_{0}}=1$.
Thus the cone condition is satisfied.

  5. $\circ$

The special assumption that
$\textrm{rank}~{}\partial^{2}_{\xi\xi}\Psi_{j}(z,\xi)=1$ when $\xi\neq 0$ is
also satisfied. This part of the proof is similar to the proof of the cone
condition with $n=2$. Notice that
$\Gamma^{\prime}_{j}(t_{0})=-\frac{\xi_{1}}{\xi_{2}}$, a similar calculation
gives

$\displaystyle\partial^{2}_{\xi\xi}\Psi_{j}(z,\xi)=\left(\begin{array}[]{ccc}-u\partial_{\xi_{1}}t_{0}{}&~{}-u\partial_{\xi_{2}}t_{0}\\\
-u\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{1}}t_{0}{}&~{}-u\Gamma^{\prime}_{j}(t_{0})\partial_{\xi_{2}}t_{0}\end{array}\right).$

Then, by the facts that
$\partial_{\xi_{1}}t_{0}=-\frac{1}{\xi_{2}}\frac{1}{\Gamma^{\prime\prime}_{j}(t_{0})}$
and
$\partial_{\xi_{2}}t_{0}=\frac{\xi_{1}}{\xi_{2}^{2}}\frac{1}{\Gamma^{\prime\prime}_{j}(t_{0})}$,
we obtain that $\textrm{rank}~{}\partial^{2}_{\xi\xi}\Psi_{j}(z,\xi)=1$ when
$\xi\neq 0$.

Therefore, $F_{j,k}$ satisfies the conditions in Proposition 3.3 with
$\sigma=-\frac{1}{2}$. By $F_{j,k}f=F_{j,k}P_{k}f$ for any $k\in\mathbb{N}$
and $j\in\mathbb{Z}_{0}^{-}$, which follows from the definition in (3.5), then
Proposition 3.3 implies

(3.16)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{4}(\mathbb{R}^{3})}\lesssim
2^{\left(-\frac{1}{2}+\vartheta_{1}\right)k}\|f\|_{L_{x}^{4}(\mathbb{R}^{2})}$

for any $\vartheta_{1}\in(0,\infty)$, and

(3.17)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{\infty}(\mathbb{R}^{3})}\lesssim
2^{k}\|f\|_{L_{x}^{1}(\mathbb{R}^{2})}.$

Using interpolation between (3.16) and (3.17), we obtain that

(3.18)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{6}(\mathbb{R}^{3})}\lesssim
2^{\vartheta_{2}k}\|f\|_{L_{x}^{2}(\mathbb{R}^{2})}$

for any $\vartheta_{2}\in(0,\infty)$. Furthermore, by interpolation between
(3.16) and (3.18) with
$\vartheta_{1}:=\vartheta_{2}:=\frac{1}{2}-\frac{1}{p}>0$, we obtain

(3.19)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{q}(\mathbb{R}^{3})}\lesssim
2^{-\left(\frac{1}{2}-\frac{1}{p}\right)k}\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$

for all $(\frac{1}{p},\frac{1}{q})$ lies in the line segment
$(C(\frac{1}{2},\frac{1}{6}),M(\frac{1}{4},\frac{1}{4})]$ in Figure 3. This is
the desired estimate (3.6) with $\varepsilon:=\frac{1}{2}-\frac{1}{p}>0$.
Therefore, we finish the proof of Theorem 1.1.

######  Remark 3.4.

We can also obtain some other endpoint estimates for $F_{j,k}$ on the
trapezium $\square$. Indeed, by Proposition 3.3, we have

(3.20)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{1}(\mathbb{R}^{3})}\lesssim\|f\|_{L_{x}^{1}(\mathbb{R}^{2})},$
(3.21)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{2}(\mathbb{R}^{3})}\lesssim
2^{-\frac{1}{2}k}\|f\|_{L_{x}^{2}(\mathbb{R}^{2})},$

and

(3.22)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{\infty}(\mathbb{R}^{3})}\lesssim\|f\|_{L_{x}^{\infty}(\mathbb{R}^{2})}.$

Using interpolation between (3.17) and (3.21), we obtain that

(3.23)
$\displaystyle\left\|F_{j,k}f\right\|_{L_{x,u}^{3}(\mathbb{R}^{3})}\lesssim\|f\|_{L_{x}^{3/2}(\mathbb{R}^{2})}.$

Therefore, the point $D(\frac{2}{3},\frac{1}{3})$ is also an endpoint of
$\widetilde{T}_{j}$, but not $L_{x}^{3/2}(\mathbb{R}^{2})$ to
$L_{x,u}^{3}(\mathbb{R}^{2}\times[1,2])$ estimate. As in the point
$C(\frac{1}{2},\frac{1}{6})$, by interpolation between $D$ with $M$, we can
obtain the corresponding estimates on $(D,M]$. This, combined with the
estimates on $(C,M]$ and $[O,A]$, leads to the desired estimates (3.6) except
for the case of $[D,A)$. In this paper, we use other methods and obtain the
estimate $\widetilde{T}_{j}:\ L_{x}^{3/2}(\mathbb{R}^{2})\rightarrow
L_{x,u}^{3}(\mathbb{R}^{2}\times[1,2])$ by Lemma 2.4, which implies that our
desired estimates can be established on $[D,A)$.

##  4 Proof of Theorem 1.2

In this section, we show the necessity of the region of
$(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1, i.e., Theorem 1.2. We split the
proof into four parts.

Proof of (i) of Theorem 1.2.

Since $\omega=\inf_{\tau\in(0,1)}\sup_{t\in(0,\tau)}\frac{\ln|\gamma(t)|}{\ln
t}$, then, for any $\epsilon>0$, there exists $\tau\in(0,1)$ such that
$\sup_{t\in(0,\tau)}\frac{\ln|\gamma(t)|}{\ln t}\in[\omega,\omega+\epsilon)$.
Moreover, there exists $t\in(0,\tau)$ such that $\frac{\ln|\gamma(t)|}{\ln
t}\in(\omega-\epsilon,\omega+\epsilon)$, which further implies that

$\sup_{s\in(0,t)}\frac{\ln|\gamma(s)|}{\ln
s}\leq\sup_{s\in(0,\tau)}\frac{\ln|\gamma(s)|}{\ln
s}<\omega+\epsilon~{}~{}\textrm{and}~{}~{}\sup_{s\in(0,t)}\frac{\ln|\gamma(s)|}{\ln
s}\geq\omega.$

Hence, there exists $t^{\prime}\in(0,t)$ such that
$\frac{\ln|\gamma(t^{\prime})|}{\ln
t^{\prime}}\in(\omega-\epsilon,\omega+\epsilon)$. Repeating the same
procedure, there exists a sequence $\\{t_{i}\\}_{i=1}^{\infty}\subset(0,1)$
such that $\frac{\ln|\gamma(t_{i})|}{\ln
t_{i}}\in(\omega-\epsilon,\omega+\epsilon)$, where
$\\{t_{i}\\}_{i=1}^{\infty}$ is strictly decreasing and
$\lim_{i\rightarrow\infty}t_{i}=0$ . Then we clearly have

(4.1)
$\displaystyle\left|\gamma(t_{i})\right|={t_{i}}^{\log_{t_{i}}{|\gamma(t_{i})|}}={t_{i}}^{\frac{\ln|\gamma(t_{i})|}{\ln
t_{i}}}\in\left({t_{i}}^{\omega+\epsilon},{t_{i}}^{\omega-\epsilon}\right).$

On the other hand, we claim that

(4.2) $\displaystyle
t_{i}\left(t_{i}\left|\gamma(t_{i})\right|\right)^{\frac{1}{q}-\frac{1}{p}}\lesssim
1,$

where the implicit constant is independent of $i$. Indeed, let
$S_{t_{i}}:=[-t_{i},t_{i}]\times[-|\gamma(t_{i})|,|\gamma(t_{i})|]$ and

$\Lambda_{\chi_{S_{t_{i}}}}(x,u):=\left|\left\\{t\in[0,1]:\
(x_{1}-ut,x_{2}-u\gamma(t))\in S_{t_{i}}\right\\}\right|.$

Then, for any $x\in S_{t_{i}}/2$, by the doubling condition (2.3), there
exists $\delta\in(0,1)$ small enough such that, for any $t\in[0,\delta
t_{i}]$, we have $|x_{1}-ut|\leq t_{i}/2+2\delta t_{i}\leq t_{i}$ and
$|x_{2}-u\gamma(t)|\leq|\gamma(t_{i})|/2+2|\gamma(\delta
t_{i})|\leq|\gamma(t_{i})|$, which further leads to the pointwise estimate
$\Lambda_{\chi_{S_{t_{i}}}}(x,u)\geq\delta t_{i}\chi_{S_{t_{i}}/2}(x)$.
Noticing that $T(\chi_{S_{t_{i}}})(x,u)\geq\Lambda_{\chi_{S_{t_{i}}}}(x,u)$
and the assumption that
$\|Tf\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$,
we conclude that

$\displaystyle|S_{t_{i}}|\lesssim\left|\left\\{(x,u)\in\mathbb{R}^{2}\times[1,2]:\
T(\chi_{S_{t_{i}}})(x,u)\geq\delta
t_{i}\right\\}\right|\lesssim\left(t_{i}^{-1}\left\|\chi_{S_{t_{i}}}\right\|_{L_{x}^{p}(\mathbb{R}^{2})}\right)^{q}.$

Combining this with the fact that $|S_{t_{i}}|\approx t_{i}|\gamma(t_{i})|$
and
$\|\chi_{S_{t_{i}}}\|_{L_{x}^{p}(\mathbb{R}^{2})}\approx(t_{i}|\gamma(t_{i})|)^{\frac{1}{p}}$,
we obtain (4.2).

A theorem of $\textrm{H}\ddot{\textrm{o}}\textrm{rmander}$ [14] implies that
$\frac{1}{q}\leq\frac{1}{p}$. Combining (4.1) and (4.2), we obtain

$\displaystyle 1\gtrsim
t_{i}\left(t_{i}\left|\gamma(t_{i})\right|\right)^{\frac{1}{q}-\frac{1}{p}}\gtrsim
t_{i}\left(t_{i}\left({t_{i}}^{\omega-\epsilon}\right)\right)^{\frac{1}{q}-\frac{1}{p}}={t_{i}}^{1+(1+\omega-\epsilon)\left(\frac{1}{q}-\frac{1}{p}\right)}.$

Notice that the sequence $\\{t_{i}\\}_{i=1}^{\infty}\subset(0,1)$ is strictly
decreasing and $\lim_{i\rightarrow\infty}t_{i}=0$, so that

(4.3) $\displaystyle
1+(1+\omega-\epsilon)\left(\frac{1}{q}-\frac{1}{p}\right)\geq 0$

for all $\epsilon>0$. Let $\epsilon\rightarrow 0$ in (4.3), it follows that
$1+(1+\omega)(\frac{1}{q}-\frac{1}{p})\geq 0$, and so we have (i) of Theorem
1.2.

Proof of (ii) of Theorem 1.2.

As above, the condition $\frac{1}{q}\leq\frac{1}{p}$ follows from a theorem of
$\textrm{H}\ddot{\textrm{o}}\textrm{rmander}$ [14]. Hence, we should only to
show that $\frac{2}{p}-1\leq\frac{1}{q}$. For any $\epsilon>0$, let $D$ be the
$\epsilon$ neighborhood of the curve $(ut,u\gamma(t))$ with $t\in(0,1]$ and
$u\in[1,2]$. Then, for any $x\in D$, there exists $t_{0}\in[0,1]$ such that
$|(x_{1}-ut_{0},x_{2}-u\gamma(t_{0}))|\leq\epsilon$. Therefore, for any
$t\in(t_{0}-\epsilon,t_{0}+\epsilon)$, there exists $\delta\in(0,\infty)$ such
that $|(x_{1}-ut,x_{2}-u\gamma(t))|\leq\delta\epsilon$. Here, we used the
triangle inequality and the estimate that
$|\gamma(t)-\gamma(t_{0})|\lesssim|t-t_{0}|\lesssim\epsilon$. Furthermore, we
have $T(\chi_{B(0,\delta\epsilon)})(x,u)\gtrsim\epsilon\chi_{D}(x)$. As an
immediate consequence of
$\|Tf\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$,
we obtain

$\displaystyle|D|\lesssim\left|\left\\{(x,u)\in\mathbb{R}^{2}\times[1,2]:\
T(\chi_{B(0,\delta\epsilon)})(x,u)\gtrsim\epsilon\right\\}\right|\lesssim\left(\epsilon^{-1}\left\|\chi_{B(0,\delta\epsilon)}\right\|_{L_{x}^{p}(\mathbb{R}^{2})}\right)^{q}.$

It is easy to check that $|D|\approx\epsilon$ and
$\|\chi_{B(0,\delta\epsilon)}\|_{L_{x}^{p}(\mathbb{R}^{2})}\approx\epsilon^{\frac{2}{p}}$.
This readily implies that $\epsilon^{\frac{1}{q}-\frac{2}{p}+1}\lesssim 1$
holds for all $\epsilon>0$. Thus, via letting $\epsilon\rightarrow 0$ we
obtain that $\frac{2}{p}-1\leq\frac{1}{q}$ and, hence, we have (ii) of Theorem
1.2.

Proof of (iii) of Theorem 1.2.

A simple calculation gives that the adjoint operator $T^{*}$ of $T$ can be
written as

$\displaystyle
T^{*}g(x):=\int_{1}^{2}\int_{0}^{1}g(x_{1}+ut,x_{2}+u\gamma(t),u)\,\textrm{d}t\,\textrm{d}u,$

and the boundedness
$\|Tf\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$
is equivalent to

(4.4)
$\displaystyle\left\|T^{*}g\right\|_{L_{x}^{p^{\prime}}(\mathbb{R}^{2})}\lesssim\|g\|_{L_{x,u}^{q^{\prime}}(\mathbb{R}^{2}\times[1,2])}.$

For any $\epsilon>0$, let $L$ be the $\epsilon$ neighborhood of the curve
$(t,\gamma(t))$ with $t\in(0,1]$. Then, for any $x\in L$, there exists
$t_{0}\in[0,1]$ such that $|(x_{1}-t_{0},x_{2}-\gamma(t_{0}))|\leq\epsilon$.
Therefore, for any $-x\in L$, $t\in(t_{0}-\epsilon,t_{0}+\epsilon)$ and
$u\in[1,1+\epsilon]$, there exists $\delta\in(0,\infty)$ such that
$|(x_{1}+ut,x_{2}+u\gamma(t))|\leq\delta\epsilon$. Here, we used the triangle
inequality and the estimates that
$|\gamma(t)-\gamma(t_{0})|\lesssim|t-t_{0}|\lesssim\epsilon$ and
$|\gamma(t_{0})|\lesssim 1$. As a result, by setting
$W:=B(0,\delta\epsilon)\times[1,1+\epsilon]$, for $-x\in L$, we have

$\displaystyle
T^{*}\chi_{W}(x)\geq\int_{1}^{1+\epsilon}\int_{t_{0}-\epsilon}^{t_{0}+\epsilon}\chi_{W}(x_{1}+ut,x_{2}+u\gamma(t),u)\,\textrm{d}t\,\textrm{d}u\gtrsim\epsilon^{2},$

i.e., we have $T^{*}\chi_{W}(x)\gtrsim\epsilon^{2}\chi_{L}(-x)$. According to
(4.4), we assert that

$\displaystyle|L|\lesssim\left|\left\\{x\in\mathbb{R}^{2}:\
T^{*}\chi_{W}(x)\gtrsim\epsilon^{2}\right\\}\right|\lesssim\left(\epsilon^{-2}\left\|\chi_{W}\right\|_{L_{x,u}^{q^{\prime}}(\mathbb{R}^{2}\times[1,2])}\right)^{p^{\prime}}.$

This, combined with the fact that $|L|\approx\epsilon$ and
$\|\chi_{W}\|_{L_{x,u}^{q^{\prime}}(\mathbb{R}^{2}\times[1,2])}\approx\epsilon^{\frac{3}{q^{\prime}}}$,
implies that $\epsilon^{\frac{3}{q}-\frac{1}{p}}\lesssim 1$ holds for all
$\epsilon>0$. Then, we get $\frac{1}{q}\geq\frac{1}{3p}$ by letting
$\epsilon\rightarrow 0$, which is (iii) of Theorem 1.2.

Proof of (iv) of Theorem 1.2.

It is easy to see that (i) implies (iv) of Theorem 1.2, when
$\omega\in[2,\infty)$. Furthermore, from (ii) and (iii) of Theorem 1.2, the
region of $(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 remains unchanged, when
$\omega\in(0,\frac{3}{2})$. On the other hand, the points
$D(\frac{2}{3},\frac{1}{3})$ and $C(\frac{1}{2},\frac{1}{6})$ are the
endpoints in the proof. Therefore, we may conjecture that the region of
$(\frac{1}{p},\frac{1}{q})$ in Theorem 1.1 also remains unchanged for all
$\omega\in(0,2)$, which further implies that we can expect (iv) of Theorem
1.2. Therefore, the $\omega=2$ is a critical point, and we may need a Taylor’s
expansion of order $2$ as in (4.5).

We now give the proof of (iv) of Theorem 1.2. From the condition (i) of
Theorem 1.1, there exists $\tau_{0}\in(0,1]$ such that
$\gamma^{\prime}(\tau_{0})\neq\gamma(\tau_{0})$. One may assume without loss
of generality that $\tau_{0}=1$, $\gamma(t)$ is increasing on $(0,1]$ and
$\gamma(1)=1$. Then $\gamma^{\prime}(1)\neq 1$. Let
$e_{1}:=(-1,-\gamma^{\prime}(1))/\sqrt{1+\gamma^{\prime}(1)^{2}}$ and
$e_{2}:=(-\gamma^{\prime}(1),1)/\sqrt{1+\gamma^{\prime}(1)^{2}}$ be two
orthogonal unit vectors. For any $\epsilon>0$, set

$Q:=\left\\{x\in\mathbb{R}^{2}:\ |x\cdot e_{1}|\leq
6\epsilon~{}~{}\textrm{and}~{}~{}|x\cdot
e_{2}|\leq[3+2|\gamma^{\prime\prime}(1)|]\epsilon^{2}\right\\}.$

We choose a subset $\\{u_{i}\\}\subset[1,2]$ such that
$|u_{i+1}-u_{i}|=2\epsilon^{2}$ for all $i$. For each $u_{i}$, denote

$\Omega_{u_{i}}:=\left\\{x\in\mathbb{R}^{2}:\
|\left(x-(u_{i},u_{i})\right)\cdot
e_{1}|\leq\epsilon~{}\textrm{and}~{}|\left(x-(u_{i},u_{i})\right)\cdot
e_{2}|\leq\epsilon^{2}\right\\}.$

For any $x\in\Omega_{u_{i}}$ and
$t\in[1-\epsilon/\sqrt{1+\gamma^{\prime}(1)^{2}},1]$, Taylor’s expansion
yields

(4.5)
$\displaystyle\gamma(t)=1+\gamma^{\prime}(1)(t-1)+\frac{\gamma^{\prime\prime}(1)}{2}(t-1)^{2}+o\left((t-1)^{2}\right),$

where $o((t-1)^{2})$ means $o((t-1)^{2})/(t-1)^{2}\rightarrow 0$ as
$t\rightarrow 1^{-}$. Furthermore, by (4.5), a simple calculation gives
$|((u_{i}t,u_{i}\gamma(t))-(u_{i},u_{i}))\cdot e_{1}|\leq 4\epsilon$ and
$|((u_{i}t,u_{i}\gamma(t))-(u_{i},u_{i}))\cdot e_{2}|\leq
2|\gamma^{\prime\prime}(1)|\epsilon^{2}$ for $\epsilon>0$ small enough. By the
triangle inequality, we may conclude that $|(x-(u_{i}t,u_{i}\gamma(t)))\cdot
e_{1}|\leq 5\epsilon$ and $|(x-(u_{i}t,u_{i}\gamma(t)))\cdot
e_{2}|\leq[1+2|\gamma^{\prime\prime}(1)|]\epsilon^{2}$. Therefore, for any
$u\in(u_{i}-\epsilon^{2},u_{i}+\epsilon^{2})$, the triangle inequality also
gives $|(x-(ut,u\gamma(t)))\cdot e_{1}|\leq 6\epsilon$ and
$|(x-(ut,u\gamma(t)))\cdot
e_{2}|\leq[3+2|\gamma^{\prime\prime}(1)|]\epsilon^{2}$, where $\epsilon>0$
small enough.

The above argument implies that $(x_{1}-ut,x_{2}-u\gamma(t))\in Q$ if
$x\in\Omega_{u_{i}}$, $u\in(u_{i}-\epsilon^{2},u_{i}+\epsilon^{2})$ and
$t\in[1-\epsilon/\sqrt{1+\gamma^{\prime}(1)^{2}},1]$. Consequently, for
$\epsilon>0$ small enough, one may obtain

$\displaystyle
T\chi_{Q}(x,u)\geq\int_{1-\frac{\epsilon}{\sqrt{1+\gamma^{\prime}(1)^{2}}}}^{1}\chi_{Q}(x_{1}-ut,x_{2}-u\gamma(t))\,\textrm{d}t=\frac{\epsilon}{\sqrt{1+\gamma^{\prime}(1)^{2}}}$

for any $x\in\Omega_{u_{i}}$ and
$u\in(u_{i}-\epsilon^{2},u_{i}+\epsilon^{2})$. By $\gamma^{\prime}(1)\neq 1$,
we have $\Omega_{u_{i}}\cap\Omega_{u_{i^{\prime}}}=\emptyset$ for any $i\neq
i^{\prime}$. Notice that the number of the elements in
$\\{u_{i}\\}\subset[1,2]$ is equivalent to $\epsilon^{-2}$. Then clearly

$\displaystyle\left\|T\chi_{Q}\right\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\geq\left(\sum_{i}\int_{u_{i}-\epsilon^{2}}^{u_{i}+\epsilon^{2}}\int_{\Omega_{u_{i}}}\left|T\chi_{Q}\right|^{q}\,\textrm{d}x\,\textrm{d}u\right)^{\frac{1}{q}}\gtrsim\epsilon^{1+\frac{3}{q}}.$

This, combined with
$\|\chi_{Q}\|_{L^{p}(\mathbb{R}^{2})}\approx\epsilon^{\frac{3}{p}}$ and
$\|Tf\|_{L_{x,u}^{q}(\mathbb{R}^{2}\times[1,2])}\lesssim\|f\|_{L_{x}^{p}(\mathbb{R}^{2})}$,
leads to that $\epsilon^{1+\frac{3}{q}-\frac{3}{p}}\lesssim 1$ holds for all
$\epsilon>0$ small enough. Let $\epsilon\rightarrow 0$, it follows that
$\frac{1}{q}\geq\frac{1}{p}-\frac{1}{3}$. Thus (iv) of Theorem 1.2 is proved.

Therefore, putting things together we finish the proof of Theorem 1.2.

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Junfeng Li

School of Mathematical Sciences, Dalian University of Technology, Dalian,
116024, People’s Republic of China

E-mail<EMAIL_ADDRESS>

Naijia Liu

School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s
Republic of China

E-mail<EMAIL_ADDRESS>

Zengjian Lou and Haixia Yu (Corresponding author)

Department of Mathematics, Shantou University, Shantou, 515821, People’s
Republic of China

E-mails<EMAIL_ADDRESS>(Z. Lou)

E-mails<EMAIL_ADDRESS>(H. Yu)