# How does Burrows’ Delta work on medieval Chinese poetic texts?

Boris Orekhov111webpage: https://nevmenandr.github.io/homepage/, alternative
email<EMAIL_ADDRESS>
School of Linguistics
HSE University,
Institute of Russian Literature (Pushkin House)
Russian Academy of Sciences
<EMAIL_ADDRESS>
###### Abstract

Burrows’ Delta was introduced in 2002 and has proven to be an effective tool
for author attribution. Despite the fact that these are different languages,
they mostly belong to the same grammatical type and use the same graphic
principle to convey speech in writing: a phonemic alphabet with word
separation using spaces. The question I want to address in this article is how
well this attribution method works with texts in a language with a different
grammatical structure and a script based on different principles. There are
fewer studies analyzing the effectiveness of the Delta method on Chinese texts
than on texts in European languages. I believe that such a low level of
attention to Delta from sinologists is due to the structure of the scientific
field dedicated to medieval Chinese poetry. Clustering based on intertextual
distances worked flawlessly. Delta produced results where clustering showed
that the samples of one author were most similar to each other, and Delta
never confused different poets. Despite the fact that I used an unconventional
approach and applied the Delta method to a language poorly suited for it, the
method demonstrated its effectiveness. Tang dynasty poets are correctly
identified using Delta, and the empirical pattern observed for authors writing
in European standard languages has been confirmed once again.

_Keywords_ tang poetry $\cdot$ burrows’s delta $\cdot$ author attribution
$\cdot$ chinese language

##  1 Introduction

Burrows’ Delta was introduced in 2002 (Burrows, 2002) and has proven to be an
effective tool for author attribution. The psychological and linguistic
foundations of this method are not entirely clear, but numerous successful
tests in various languages have confirmed that it is a viable method for
calculating intertextual distance, which correlates well with a specific
author’s relationship to a text. If several texts are written by the same
person, the Delta distance between them is smaller than between texts written
by different authors.

Although this rule does not work in 100% of cases (Skorinkin and Orekhov,
2023), it covers most cases that scholars in philology are interested in. The
number of studies in which Burrows’ Delta becomes a tool for solving
attribution problems is growing. There are two reasons for this: first, the
effectiveness of the method itself, repeatedly tested on texts in different
languages. Second, the low entry threshold for researchers, provided by the
well-crafted stylo package for the R language (Eder et al., 2016), which
generally follows a zero-coding strategy.

The effectiveness of the method has been proved on the material of different
languages, both new and ancient: English (Hoover, 2004), Old English (Garcia
and Martin, 2006), German (Jannidis and Lauer, 2014), Spanish (Hernández-
Lorenzo and Byszuk, 2023), Italian (Savoy, 2018; Rybicki, 2018), Polish
(Rybicki and Heydel, 2013), Russian (Skorinkin and Bonch-Osmolovskaya, 2016),
Arabic (AbdulRazzaq and Mustafa, 2014), and others. There is a paper comparing
the quality of attribution in different languages: Latin, Polish, English,
German (Eder, 2011).

Despite the fact that these are different languages, they mostly (with the
exception of Arabic) belong to the same grammatical type and use the same
graphic principle to convey speech in writing: a phonemic alphabet with word
separation using spaces. The question I want to address in this article is how
well this attribution method works with texts in a language with a different
grammatical structure and a script based on different principles.

There are fewer studies analyzing the effectiveness of the Delta method on
Chinese texts than on texts in European languages. It is worth noting (du2,
2016) that those studies focus on modern Chinese, whereas I am interested in
medieval Chinese poetry. This is a language without inflection and a script
system without spaces as word delimiters. With such a grammatical structure,
the distribution of function words is different from what is typical in
European-standard languages (Xiao, 2008; Liu et al., 2017). Furthermore, in
the writing systems familiar to Europeans, spaces separate words from each
other. In Chinese, tokenization presents a complex problem (Huang and Wu,
2018), so it is important to check if Delta can be used by focusing only on
the obvious text units, the characters, without resorting to complex
algorithms that can introduce errors into text processing. Although the use of
letter ngrams for Delta attribution is also found in European languages,
typically sequences of several characters are used. Since in Chinese, the
significance of a single character is higher than that of an individual letter
in European languages, we will attempt to rely on individual characters rather
than sequences.

I believe that such a low level of attention to Delta from sinologists is due
to the structure of the scientific field dedicated to medieval Chinese poetry.
Medieval Chinese literature is very different from medieval European
literature, which had many gaps. The Chinese tradition of this period was
highly regulated (Sturgeon, 2018), as was much of Chinese society, and it was
built on the concept of authority, so texts were well documented. As a result,
the material generally poses far fewer problems in terms of determining
authorship. Authorship determination is the area where Delta traditionally
shows the most significant results, which is why it is not in demand among
sinologists specializing in medieval studies, as the authorship of texts is
well known.

##  2 Method and Data

There is no need to recount the calculation algorithm for computing Delta once
again. This has been done dozens of times in the most well-known and
authoritative publications. The intertextual distance is calculated using the
formula:

$\Delta=\sum\limits_{i=1}^{n}\frac{|z(x_{i})-z(y_{i})|}{n}$ (1)

Thus, we determine the distance between each and every text in the research
sample. Since we do not have a reliable method for tokenizing the text, we are
forced to work with individual characters. The Stylo package has an option
that allows switching from words to letter n-grams. This means we treat each
character as a separate letter, which is an intermediate approach between
alphabetic scripts and the script characteristic of the Chinese language,
where one word can be represented by multiple characters.

###  2.1 Data

As the source of texts, the collection stored in the repository at
(snowtraces, 2020) was used. It contains a digitized version of the ”Complete
Tang Poems” (or Quan Tangshi), which is the largest collection of Tang poetry.
Thus, this study examines the effectiveness of the Delta method for
determining authorship in Tang dynasty Chinese poetry.

I collected all texts of a single author together, resulting in 2537 poets.
This is too many for a visual analysis of the results. Therefore, I further
worked only with the 20 most prolific poets of the Tang era. Here is their
list and the number of characters for each, including characters, spaces, line
breaks, and punctuation marks:

  1. 1.

白居易 229346

  2. 2.

杜甫 129391

  3. 3.

李白 103294

  4. 4.

元稹 80968

  5. 5.

韓愈 60013

  6. 6.

劉禹錫 58224

  7. 7.

貫休 48836

  8. 8.

齊己 46381

  9. 9.

陸龜蒙 45104

  10. 10.

韋應物 40776

  11. 11.

孟郊 40582

  12. 12.

李商隱 39589

  13. 13.

皎然 38349

  14. 14.

劉長卿 35301

  15. 15.

皮日休 34357

  16. 16.

杜牧 33483

  17. 17.

王建 31607

  18. 18.

姚合 30887

  19. 19.

錢起 29874

  20. 20.

許渾 29775

For stylometry, volume is important, so I combined the poems of one poet into
several large samples to compare them with each other using Delta. However,
the results can depend on which specific poems we combine. For example, one
combination of poems might yield one version of stylometric distribution,
while another combination might yield a different version. Therefore, it is
necessary to try different combinations of poems within the sample. I did the
following: I randomly mixed the order of poems for one author, split the
volume in half, and presented each half as separate texts (samples), repeating
this five times to create five test corpora. The code for these manipulations
is provided in this repository (Orekhov, 2024).

Next, I applied Delta to all these corpora using the Stylo package,
configuring the package to work with characters instead of words and selecting
the 100 most frequent characters. Here is the list of the 10 most frequent
characters from the analysis:

  1. 1.

不

  2. 2.

人

  3. 3.

無

  4. 4.

一

  5. 5.

日

  6. 6.

山

  7. 7.

風

  8. 8.

有

  9. 9.

何

  10. 10.

來

##  3 Results

In all 5 versions of the corpus, clustering based on intertextual distances
worked flawlessly. Delta produced results where clustering showed that the
samples of one author were most similar to each other, and Delta never
confused different poets. We present only one figure 1 as an illustration, as
the others repeat the same result.

Figure 1: Cluster analysis of the first shuffled test corpus.

Let’s provide a table 1 with a portion of the distances. The full table can be
viewed in the data publication (Orekhov, 2024).

Table 1: Results: text distances | 元稹_1 | 元稹_2 | 劉禹錫_1 | 劉禹錫_2 | 劉長卿_1
---|---|---|---|---|---
元稹_1 | 0 | 0.4319 | 0.8917 | 0.8251 | 1.2953
元稹_2 | 0.4319 | 0 | 0.9412 | 0.8734 | 1.3814
劉禹錫_1 | 0.8917 | 0.9412 | 0 | 0.4782 | 1.1990
劉禹錫_2 | 0.8251 | 0.8734 | 0.4782 | 0 | 1.2040
劉長卿_1 | 1.2953 | 1.3814 | 1.1990 | 1.2040 | 0
劉長卿_2 | 1.4421 | 1.4649 | 1.3361 | 1.3059 | 0.5586

The heatmap (Fig. 2) of distances also records the smallest distances between
samples of the same poet.

Figure 2: Heatmap of the delta text distances in the first shuffled test
corpus.

##  4 Conclusion

Despite the fact that I used an unconventional approach and applied the Delta
method to a language poorly suited for it, the method demonstrated its
effectiveness. Tang dynasty poets are correctly identified using Delta, and
the empirical pattern observed for authors writing in European standard
languages has been confirmed once again.

It should be noted that one character as a token is less than a word for
medieval Chinese, but even 100 tokens were sufficient for accurate author
clustering.

Further experiments can be conducted with a larger number of authors,
different numbers of the most frequent tokens, and smaller sample sizes to
understand the threshold at which the method ceases to work. However, the
overall result is positive. It can be concluded that Delta works not only with
European standard languages but also with such a regulated tradition, where
the individuality of poetic style is tightly constrained by prescribed rules.
Thus, an author’s individual style can be hidden behind the rules imposed by
tradition for expressing poetic ideas.

The confirmed effectiveness of the method has shown that raw texts can be used
for Delta, as-is, without preprocessing with complex and error-prone tools
like tokenizers.

If philologists need it, they can use Delta to solve attribution problems for
medieval Chinese texts.

## Acknowledgements

I am grateful to Mariana Zorkina for her suggestions regarding the text
corpus, specific bibliography points, and some observations about the research
field, which we discussed in the early versions of this work. At the same
time, all inaccuracies and errors in the text remain my responsibility.

## References

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  * Skorinkin and Orekhov [2023] Daniil Skorinkin and Boris Orekhov.  Hacking stylometry with multiple voices: Imaginary writers can override authorial signal in delta.  _Digital Scholarship in the Humanities_ , 38(3):1247–1266, 2023.
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  * Hoover [2004] David L Hoover.  Testing burrows’s delta.  _Literary and linguistic computing_ , 19(4):453–475, 2004.
  * Garcia and Martin [2006] Antonio Miranda Garcia and Javier Calle Martin.  Function words in authorship attribution studies.  _Literary and Linguistic Computing_ , 22(1):49–66, 2006.
  * Jannidis and Lauer [2014] Fotis Jannidis and Gerhard Lauer.  Burrows’s delta and its use in german literary history.  _Distant readings. Topologies of German culture in the long nineteenth century_ , pages 29–54, 2014.
  * Hernández-Lorenzo and Byszuk [2023] Laura Hernández-Lorenzo and Joanna Byszuk.  Challenging stylometry: The authorship of the baroque play la segunda celestina.  _Digital Scholarship in the Humanities_ , 38(2):544–558, 2023.
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  * AbdulRazzaq and Mustafa [2014] Ammar Adil AbdulRazzaq and Tareef Kamil Mustafa.  Burrows-delta method fitness for arabic text authorship stylometric detection.  _International Journal of Computer Science and Mobile Computing_ , 3(6):69–78, 2014.
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#  $L^{p}$-continuity of wave operators for higher order Schrödinger operators
with threshold eigenvalues in high dimensions

M. Burak Erdoğan, William R. Green, Kevin LaMaster  Department of Mathematics
University of Illinois
Urbana, IL 61801, U.S.A<EMAIL_ADDRESS>Department of Mathematics
Rose-Hulman Institute of Technology
Terre Haute, IN 47803, U.S.A<EMAIL_ADDRESS>Department of Mathematics
University of Illinois
Urbana, IL 61801, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We consider the higher order Schrödinger operator $H=(-\Delta)^{m}+V(x)$ in
$n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in\mathbb{N}$.
We adapt our recent results for $m>1$ to show that when $H$ has a threshold
eigenvalue the wave operators are bounded on $L^{p}(\mathbb{R}^{n})$ for the
natural range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The
approach used works without distinguishing even and odd cases, and matches the
range of boundedness in the classical case when $m=1$. The proof applies in
the classical $m=1$ case as well and simplifies the argument.

The first and third authors were partially supported by the NSF grant
DMS-2154031. The second author is partially supported by Simons Foundation
Grant 511825.

##  1\. Introduction

We continue the study of wave operators for higher order Schrödinger operators
related to equations of the form

$\displaystyle i\psi_{t}=(-\Delta)^{m}\psi+V\psi,\qquad
x\in\mathbb{R}^{n},\quad m\in\mathbb{N}.$

Here $V$ is a real-valued potential with polynomial decay,
$|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some sufficiently large
$\beta>0$, and some smoothness conditions, see [4] or Assumption 1.2 below.
Note that when $m=1$ this is the classical Schrödinger equation. We consider
the case when $(-\Delta)^{m}+V$ has an eigenvalue at zero energy in high
dimensions $n>4m$. It is well known that there are no threshold obstructions
other than eigenfunctions when $n>4m$.

Denote the free operator by $H_{0}=(-\Delta)^{m}$ and the perturbed operator
by $H=(-\Delta)^{m}+V$. We study the $L^{p}$ boundedness of the wave
operators, which are defined by

$W_{\pm}=s\text{\ --}\lim_{t\to\pm\infty}e^{itH}e^{-itH_{0}}.$

For the class of potentials we consider, the wave operators exist and are
asymptotically complete, [17, 19, 1, 11, 18]. Furthermore, the intertwining
identity

$f(H)P_{ac}(H)=W_{\pm}f((-\Delta)^{m})W_{\pm}^{*}$

holds for these potentials where $P_{ac}(H)$ is the projection onto the
absolutely continuous spectral subspace of $H$, and $f$ is any Borel function.
The intertwining identity and $L^{p}$ continuity of the wave operators allows
one to obtain $L^{p}$-based mapping properties of operators of the form
$f(H)P_{ac}(H)$ from those of the much simpler operators $f((-\Delta)^{m})$.

As usual, we begin with the stationary representation of the wave operators

(1) $\displaystyle W_{+}u$ $\displaystyle=u-\frac{1}{2\pi
i}\int_{0}^{\infty}\mathcal{R}_{V}^{+}(\lambda)V[\mathcal{R}_{0}^{+}(\lambda)-\mathcal{R}_{0}^{-}(\lambda)]u\,d\lambda,$

where $\mathcal{R}_{V}(\lambda)=((-\Delta)^{m}+V-\lambda)^{-1}$,
$\mathcal{R}_{0}(\lambda)=((-\Delta)^{m}-\lambda)^{-1}$, and the ‘+’ and ‘-’
denote the usual limiting values as $\lambda$ approaches the positive real
line from above and below, [1, 6]. As in previous works, [4, 5], we consider
$W_{+}$, bounds for $W_{-}$ follow by conjugation since
$W_{-}=\mathcal{C}W_{+}\mathcal{C}$, where $\mathcal{C}u(x)=\overline{u}(x)$.
Since the identity operator is bounded on all $L^{p}$ spaces, it suffices to
control the contribution of the integral involving the resolvent operators.

In dimensions $n>4m$, there are no resonances at the threshold (zero energy),
[6]. This mirrors the case of dimensions $n>4$ in the classical ($m=1$)
Schrödinger operator where threshold resonances cannot exist. In the classical
case, the existence of threshold eigenvalues limits the upper range of
$L^{p}(\mathbb{R}^{n})$ boundedness of the wave operators, generically to
$1\leq p<\frac{n}{2}$, [24, 9, 25]. Here we prove the analogous result for the
higher order Schrödinger operators in Theorem 1.3 below. Our proof doesn’t
distinguish between $m=1$ and $m>1$, and hence applies to the classical case
as well where it simplifies the existing arguments.

Our main result is to control the low energy portion of the evolution when
there is a threshold eigenvalue. Our strategy builds on the approach in [5],
extending the argument to control the singularities in the spectral parameter
caused by the eigenvalue. Using resolvent identity, one has

$\mathcal{R}_{V}^{+}=\sum_{j=0}^{2k-1}(-1)^{j}\mathcal{R}_{0}^{+}(V\mathcal{R}_{0}^{+})^{j}+(\mathcal{R}_{0}^{+}V)^{k}\mathcal{R}_{V}^{+}(V\mathcal{R}_{0}^{+})^{k}.$

The contribution of the $j$th term of the finite sum to (1) is denoted by
$W_{j}$ and the contribution of the remainder by $W_{r,k}$. $W_{j}$ is
unaffected by the existence of threshold obstructions, while $W_{r,k}$ is
affected only when $\lambda$ is in a neighborhood of zero. To make this more
precise, take a smooth cut-off function $\chi\in C^{\infty}_{0}$ for a
sufficiently small neighborhood of zero, and let $\widetilde{\chi}=1-\chi$ be
the complementary cut-off away from a neighborhood of zero. Define

$W_{low,k}u=\frac{1}{2\pi
i}\int_{0}^{\infty}\chi(\lambda)(\mathcal{R}_{0}^{+}(\lambda)V)^{k}\mathcal{R}_{V}^{+}(\lambda)(V\mathcal{R}_{0}^{+}(\lambda))^{k}V[\mathcal{R}_{0}^{+}(\lambda)-\mathcal{R}_{0}^{-}(\lambda)]u\,d\lambda,$
$W_{high,k}u=\frac{1}{2\pi
i}\int_{0}^{\infty}\widetilde{\chi}(\lambda)(\mathcal{R}_{0}^{+}(\lambda)V)^{k}\mathcal{R}_{V}^{+}(\lambda)(V\mathcal{R}_{0}^{+}(\lambda))^{k}V[\mathcal{R}_{0}^{+}(\lambda)-\mathcal{R}_{0}^{-}(\lambda)]u\,d\lambda.$

Throughout the paper, we write $\langle x\rangle$ to denote
$(1+|x|^{2})^{\frac{1}{2}}$, $A\lesssim B$ to say that there exists a constant
$C$ with $A\leq CB$, and write $a-:=a-\epsilon$ and $a+:=a+\epsilon$ for some
$\epsilon>0$. Our main technical result is

######  Theorem 1.1.

Let $n>4m\geq 4$. Assume that $|V(x)|\lesssim\langle x\rangle^{-\beta}$, where
$V$ is a real-valued potential on $\mathbb{R}^{n}$ and $\beta>n+4$ when $n$ is
odd and $\beta>n+3$ when $n$ is even. If $H=(-\Delta)^{m}+V(x)$ has an
eigenvalue at zero, but no positive eigenvalues, then $W_{low,k}$ extends to a
bounded operator on $L^{p}(\mathbb{R}^{n})$ for all $1\leq p<\frac{n}{2m}$
provided that $k$ is sufficiently large.

We need sufficiently large $k$ when $n>4m$ due to local singularities of the
free resolvents that are not square integrable. The main novelty is that the
arguments are fairly streamlined, we avoid long operator-valued expansions of
the perturbed resolvent by adapting the methods in [4, 5] to control the
singularity as $\lambda\to 0$ that occurs when there is a zero energy
eigenvalue. We further note that no additional decay is needed on the
potential compared to the regular case, [4].

To put this result in the context, recall the first $L^{p}$ boundedness result
in the seminal paper of Yajima, [20], for $m=1$ and $1\leq p\leq\infty$ for
small potentials. For large potentials, the main difficulty is in controlling
the contribution of $W_{low,k}$. The behavior of this operator differs in even
and odd dimensions. In [20, 21, 22], Yajima removed smallness or positivity
assumptions on the potential for all dimensions $n\geq 3$. These arguments
were simplified and Yajima further considered the effect of zero energy
eigenvalues and/or resonances in [23] when $n$ is odd and with Finco in [7]
when $n$ is even for $n>4$ to establish boundedness of the wave operators when
$\frac{n}{n-2}<p<\frac{n}{2}$. These results were further extended to show
that the range of $p$ is generically $1\leq p<\frac{n}{2}$ in the presence of
a zero energy eigenvalue, and that the upper range of $p$ may be larger under
certain orthogonality conditions, [24, 9, 25].

We now give more details in the case $m>1$ to state the new corollary of our
result above on the $L^{p}$ boundedness of wave operators. Let
$\mathcal{F}({f})$ denote the Fourier transform of $f$.

######  Assumption 1.2.

Fix $n>4m$ and $m\geq 1$. For some $0<\delta\ll 1$,
$\sigma>\frac{2n-4m}{n-1-\delta}+\delta$, assume that the real-valued
potential $V$ satisfies the condition

$\big{\|}\mathcal{F}(\langle\cdot\rangle^{\sigma}V(\cdot))\big{\|}_{L^{\frac{n-1-\delta}{n-2m-\delta}}}<\infty.$

In [4], by adapting Yajima’s $m=1$ argument in [20], the first two authors
showed that the contribution of the terms of the Born series may be bounded by

$\displaystyle\|W_{j}\|_{L^{p}\to L^{p}}\leq
C^{j}\big{\|}\mathcal{F}(\langle\cdot\rangle^{\sigma}V(\cdot))\big{\|}_{L^{\frac{n-1-\delta}{n-2m-\delta}}}^{j},$

for some constant $C>0$. In addition, it was shown that if
$|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n+5$ when $n$ is odd
and $\beta>n+4$ when $n$ is even and if $k$ is sufficiently large (depending
on $m$ and $n$), then $W_{high,k}$ is a bounded operator on $L^{p}$ for all
$1\leq p\leq\infty$ provided there are no positive eigenvalues. The lack of
positive eigenvalues is a common assumption for higher order operators since
there may be positive eigenvalues even for smooth, compactly supported
potentials, see [6].

Combining these facts with Theorem 1.1, we have the following result.

######  Corollary 1.3.

Fix $m\geq 1$ and let $n>4m$. Assume that $V$ satisfies Assumption 1.2 and in
addition

  1. i)

$|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n+5$ when $n$ is odd
and for some $\beta>n+4$ when $n$ is even,

  2. ii)

$H=(-\Delta)^{m}+V(x)$ has an eigenvalue at zero energy, but no positive
eigenvalues.

Then, the wave operators extend to bounded operators on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p<\frac{n}{2m}$.

By applying the intertwining identity and the known $L^{p}\to L^{p^{\prime}}$
dispersive bounds when $p^{\prime}$ is the Hölder conjugate of $p$ for the
free solution operator $e^{-it(-\Delta)^{m}}$, we obtain the corollary below.

######  Corollary 1.4.

Under the assumptions of Corollary 1.3, for $\frac{n}{n-2m}<p^{\prime}\leq 2$
we obtain the dispersive estimates

$\|e^{-itH}P_{ac}(H)f\|_{p}\lesssim|t|^{-\frac{n}{m}(\frac{1}{p^{\prime}}-\frac{1}{2})}\|f\|_{p^{\prime}}.$

The study of the $L^{p}$ boundedness of the wave operators in the higher order
$m>1$ case has only recently begun. In the case when there are no eigenvalues
or resonances in the ac spectrum, the case $m=2$ and $n=3$ was studied by
Goldberg and the second author, [10]. The case $n>2m$ was studied by the first
two authors in [4, 5]. In [2] the first two authors and Goldberg showed that a
certain amount of smoothness of the potentials is necessary to control the
large energy behavior in the $L^{p}$ boundedness.

In [14], Mizutani, Wan, and Yao considered the case of $m=4$ and $n=1$ showing
that the wave operators are bounded when $1<p<\infty$, but not when
$p=1,\infty$, where weaker estimates involving the Hardy space or BMO were
proven depending on the type of threshold obstruction. More recently in [8],
Galtbayar and Yajima considered the case $m=2$ and $n=4$ showing that the wave
operators are bounded on $1<p<\infty$ if zero is regular with restrictions on
the upper range of $p$ if zero is not regular depending on the type of
resonance at zero. Mizutani, Wan, and Yao [15, 16] studied the endpoint
behavior and the effect of zero energy resonances when $m=2$ and $n=3$. This
recent work on higher order, $m>1$, Schrödinger operators has roots in the
work of Feng, Soffer, Wu and Yao [6] which considered time decay estimates
between weighted $L^{2}$ spaces.

The paper is organized as follows. In Section 2 we recall important resolvent
expansions and prove Proposition 2.1, which shows $L^{p}$ boundedness for a
class of operators of the form needed in Theorem 1.1. In Section 3 we prove
further resolvent expansions tailored to the case of zero energy eigenvalues
to show that Proposition 2.1 applies to the operator $W_{low,k}$ for large
enough $k$, which suffices to prove Theorem 1.1 and consequently Theorem 1.3.
Finally in Section 4 we provide technical lemmas needed to prove Proposition
2.1.

##  2\. Operator Bounds

In this section we reduce proving Theorem 1.1 to showing that a certain family
of operators extend to bounded operators on $L^{p}(\mathbb{R}^{n})$ on the
desired range of $p$. To do this we utilize various resolvent expansions and
adapt the argument from the case when zero is regular to account for the extra
singularity in the spectral parameter that arises when $(-\Delta)^{2m}+V$ has
a zero energy eigenvalue.

It is convenient to use a change of variables $\lambda\mapsto\lambda^{2m}$ to
represent $W_{low,k}$

$\frac{m}{\pi
i}\int_{0}^{\infty}\chi(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{k}\mathcal{R}_{V}^{+}(\lambda^{2m})(V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{k}V[\mathcal{R}_{0}^{+}(\lambda^{2m})-\mathcal{R}_{0}^{-}(\lambda^{2m})]\,d\lambda.$

We begin by using the symmetric resolvent identity on the perturbed resolvent
$\mathcal{R}_{V}^{+}(\lambda^{2m})$. With $v=|V|^{\frac{1}{2}}$, $U(x)=1$ if
$V(x)\geq 0$ and $U(x)=-1$ if $V(x)<0$, we define
$M^{+}(\lambda)=U+v\mathcal{R}_{0}^{+}(\lambda^{2m})v$.

Using the symmetric resolvent identity, one has

(2)
$\mathcal{R}_{V}^{+}(\lambda^{2m})V=\mathcal{R}_{0}^{+}(\lambda^{2m})vM^{+}(\lambda)^{-1}v.$

Therefore, we have

$W_{low,k}=\frac{m}{\pi
i}\int_{0}^{\infty}\chi(\lambda)\lambda^{-1}\mathcal{R}_{0}^{+}(\lambda^{2m})v\Gamma_{k}(\lambda)v[\mathcal{R}_{0}^{+}(\lambda^{2m})-\mathcal{R}_{0}^{-}(\lambda^{2m})]\,d\lambda,$

where $\Gamma_{0}(\lambda):=\lambda^{2m}M^{+}(\lambda)^{-1}$ and for $k\geq 1$

(3) $\Gamma_{k}(\lambda)\\\
:=Uv\mathcal{R}_{0}^{+}(\lambda^{2m})\big{(}V\mathcal{R}_{0}^{+}(\lambda^{2m})\big{)}^{k-1}v[\lambda^{2m}M^{+}(\lambda)^{-1}]v\big{(}\mathcal{R}_{0}^{+}(\lambda^{2m})V\big{)}^{k-1}\mathcal{R}_{0}^{+}(\lambda^{2m})vU.$

When zero energy is an eigenvalue, the resolvent $\mathcal{R}_{V}$ becomes
unbounded as $\lambda\to 0$. Under the change of variables the singularity is
of order $\lambda^{-2m}$. The definition of $\Gamma_{k}(\lambda)$ above
multiplies $M^{-1}(\lambda)$ by $\lambda^{2m}$ to account for this
singularity.

To state our main result, we recall the following terminology from previous
works involving wave operators and dispersive estimates. An operator
$T:L^{2}\to L^{2}$ with integral kernel $T(x,y)$ is absolutely bounded if the
operator with kernel $|T(x,y)|$ is bounded as an operator on
$L^{2}(\mathbb{R}^{n})$. We recall that finite rank operators and Hilbert-
Schmidt operators are absolutely bounded, where $T$ is Hilbert-Schmidt if

$\|T\|_{HS}^{2}=\int_{\mathbb{R}^{2n}}|T(x,y)|^{2}\,dx\,dy<\infty.$

######  Proposition 2.1.

Fix $n>4m\geq 4$ and let $\Gamma$ be a $\lambda$ dependent operator. Assume
that

$\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\Big{[}\sup_{0\leq\ell\leq\lceil\tfrac{n}{2}\rceil+1}\big{|}\lambda^{\ell}\partial_{\lambda}^{\ell}\Gamma(\lambda)(x,y)\big{|}\Big{]}$

satisfies the pointwise estimate

(4) $\widetilde{\Gamma}(x,y)\lesssim\langle x\rangle^{-\frac{n}{2}-}\langle
y\rangle^{-\frac{n}{2}-}.$

If $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n_{\star}$ where
$n_{\star}=n+4$ if $n$ is odd and $n_{\star}=n+3$ if $n$ is even, then the
operator with kernel

(5)
$K(x,y)=\int_{0}^{\infty}\chi(\lambda)\lambda^{-1}\big{[}\mathcal{R}_{0}^{+}(\lambda^{2m})v\Gamma(\lambda)v[\mathcal{R}_{0}^{+}(\lambda^{2m})-\mathcal{R}_{0}^{-}(\lambda^{2m})]\big{]}(x,y)d\lambda$

is bounded on $L^{p}(\mathbb{R}^{n})$ for $1\leq p<\frac{n}{2m}$.

The claim of Theorem 1.1 follows in Section 3 by showing that
$\Gamma_{k}(\lambda)$ defined in (3) satisfies the hypotheses of this
proposition. To prove the proposition we need the following representations of
the free resolvent, these are Lemma 2.3, 2.4, and Remark 2.5 in [5], which
have their roots in [4].

######  Lemma 2.2.

Let $n>2m\geq 2$. Then, we have the representations

$\mathcal{R}_{0}^{+}(\lambda^{2m})(y,u)=\frac{e^{i\lambda|y-u|}}{|y-u|^{n-2m}}F(\lambda|y-u|).$

and

$[\mathcal{R}_{0}^{+}(\lambda^{2m})-\mathcal{R}_{0}^{-}(\lambda^{2m})](y,u)=\lambda^{n-2m}\big{[}e^{i\lambda|y-u|}F_{+}(\lambda|y-u|)+e^{-i\lambda|y-u|}F_{-}(\lambda|y-u|)\big{]},$

With the bounds

(6) $|\partial_{\lambda}^{N}F(\lambda r)|\lesssim\lambda^{-N}\langle\lambda
r\rangle^{\frac{n+1}{2}-2m},\,\,\,\,\,\,\,|\partial_{\lambda}^{N}F_{\pm}(\lambda
r)|\lesssim\lambda^{-N}\langle\lambda r\rangle^{\frac{1-n}{2}}.$

for all $N\geq 0$.

We say an operator $K$ with integral kernel $K(x,y)$ is admissible if

$\sup_{x\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|K(x,y)|\,dy+\sup_{y\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|K(x,y)|\,dx<\infty.$

By the Schur test, it follows that an operator with admissible kernel is
bounded on $L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$. We are now
ready to prove Proposition 2.1.

###### Proof of Proposition 2.1.

Using the representations in Lemma 2.2 with $r_{1}=|x-z_{1}|$ and
$r_{2}:=|z_{2}-y|$ we see that $K(x,y)$ is the difference of

(7) $K_{\pm}(x,y)\\\
=\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})}{r_{1}^{n-2m}}\int_{0}^{\infty}e^{i\lambda(r_{1}\pm
r_{2})}\chi(\lambda)\lambda^{n-2m-1}\Gamma(\lambda)(z_{1},z_{2})F(\lambda
r_{1})F_{\pm}(\lambda r_{2})d\lambda dz_{1}dz_{2}.$

We write

$K(x,y)=:\sum_{j=1}^{4}K_{j}(x,y),$

where the $K_{j}$ are $K$ restricted to different regions. $K_{1}$ is the
portion of $K$ restricted to the set $r_{1},r_{2}\lesssim 1$, $K_{2}$ to the
set $r_{1}\approx r_{2}\gg 1$, $K_{3}$ to the set $r_{2}\gg\langle
r_{1}\rangle$, and $K_{4}$ to the set $r_{1}\gg\langle r_{2}\rangle$. We
define $K_{j,\pm}$ analogously.

Since both $\lambda$ and $r_{j}\lesssim 1$ we have $\lambda r_{j}\lesssim 1$.
Using the bounds of Lemmas 2.2 for we bound the contribution of
$|K_{1,\pm}(x,y)|$ by

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{1},r_{2}\lesssim
1}}{r_{1}^{n-2m}}\widetilde{\Gamma}(z_{1},z_{2})\int^{1}_{0}\lambda^{n-2m-1}\
\\!d\lambda dz_{1}dz_{2}\\\
\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{1},r_{2}\lesssim
1}}{r_{1}^{n-2m}}\widetilde{\Gamma}(z_{1},z_{2})dz_{1}dz_{2},$

since $n-2m-1>-1$. Using the pointwise decay of $v$ and $\widetilde{\Gamma}$
in (4), we obtain

$\int|K_{1,\pm}(x,y)|dy\lesssim\int\langle z_{1}\rangle^{-n-}\langle
z_{2}\rangle^{-n-}r_{1}^{2m-n}dz_{1}dz_{2}\lesssim 1,$

uniformly in $x$. By the symmetry between $x$ and $y$ which implies that
$K_{1}$ is admissible.

For $K_{2}$, we consider the contribution of $K_{2,-}$ since the $K_{2,+}$ is
simpler. We integrate by parts twice in the $\lambda$ integral when
$\lambda|r_{1}-r_{2}|\gtrsim 1$ (using (6) and the definition of
$\widetilde{\Gamma}$) and estimate directly when $\lambda|r_{1}-r_{2}|\ll 1$
to obtain

$|K_{2,-}(x,y)|\lesssim\\\
\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\chi_{r_{1}\approx
r_{2}\gg
1}}{r_{1}^{n-2m}}\int_{0}^{\infty}\chi(\lambda)\lambda^{n-2m-1}\chi(\lambda|r_{1}-r_{2}|)\langle\lambda
r_{1}\rangle^{1-2m}d\lambda dz_{1}dz_{2}\\\
+\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\chi_{r_{1}\approx
r_{2}\gg
1}}{r_{1}^{n-2m}}\int_{0}^{\infty}\frac{\chi(\lambda)\lambda^{n-2m-3}\widetilde{\chi}(\lambda|r_{1}-r_{2}|)\langle\lambda
r_{1}\rangle^{1-2m}}{|r_{1}-r_{2}|^{2}}d\lambda dz_{1}dz_{2}\\\
\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\chi_{r_{1}\approx
r_{2}\gg
1}}{r_{1}^{n-2m}}\int_{0}^{\infty}\frac{\chi(\lambda)\lambda^{n-2m-1}\langle\lambda
r_{1}\rangle^{1-2m}}{\langle\lambda(r_{1}-r_{2})\rangle^{2}}d\lambda
dz_{1}dz_{2}.$

Noting that $1-2m<0$, we can integrating this bound with respect to $x$ after
converting to polar coordinates centered around $z_{1}$ to bound by

$\int|K_{2,-}(x,y)|dx\lesssim\\\
\int_{\mathbb{R}^{2n}}\int_{0}^{1}\int_{r_{1}\approx r_{2}\gg
1}v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})r_{1}^{n-1}\frac{\lambda^{n-2m-1}(\lambda
r_{1})^{1-2m}}{r_{1}^{n-2m}\langle\lambda(r_{1}-r_{2})\rangle^{2}}dr_{1}d\lambda
dz_{1}dz_{2}\\\ \lesssim\int_{\mathbb{R}^{2n}}\int_{0}^{1}\int_{r_{1}\approx
r_{2}\gg
1}v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\frac{\lambda^{n-4m}}{\langle\lambda(r_{1}-r_{2})\rangle^{2}}dr_{1}d\lambda
dz_{1}dz_{2}\\\
\lesssim\int_{\mathbb{R}^{2n}}\int_{0}^{1}\int_{\mathbb{R}}v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\frac{\lambda^{n-4m-1}}{\langle\eta\rangle^{2}}d\eta
d\lambda dz_{1}dz_{2}\lesssim 1,$

uniformly in $y$. In the second line we defined $\eta=\lambda(r_{1}-r_{2})$ in
the $r_{1}$ integral and used $n-4m-1\geq 0$. Since $r_{1}\approx r_{2}$, the
integral in $y$ can be bounded uniformly in $x$ similarly and hence the
contribution of $K_{2}$ is admissible.

We now consider the contribution of

(8)
$K_{4,\pm}(x,y)=\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{1}^{n-2m}}\\\ \int_{0}^{\infty}e^{i\lambda(r_{1}\pm
r_{2})}F(\lambda
r_{1})\chi(\lambda)\Gamma(\lambda)(z_{1},z_{2})\lambda^{n-2m-1}F_{\pm}(\lambda
r_{2})\ d\lambda dz_{1}dz_{2}.$

When $\lambda r_{1}\lesssim 1$, using (6), we bound $|F_{\pm}(\lambda
r_{2})|,|F(\lambda r_{1})|\lesssim 1$ and estimate the $\lambda$ integral by
$r_{1}^{2m-n}\widetilde{\Gamma}(z_{1},z_{2})$, whose contribution to $K_{4}$
is bounded by

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{1}^{n-2m+(n-2m)}}dz_{1}dz_{2}\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{2}^{n-2m+(n-2m)}}dz_{1}dz_{2}.$

Where we used that $r_{1}\gg r_{2}$ allows us to replace $r_{1}$ in the
denominator with $r_{2}$ and apply Lemma 4.1 with $\ell=n-2m$ and $k=n-2m$ to
show that that $K_{4,\pm}$ are admissible kernels.

When $\lambda r_{1}\gtrsim 1$, we integrate by parts
$N=\lceil\frac{n}{2}\rceil+1$ times (using (6)) to obtain the bound

$\displaystyle\frac{1}{|r_{1}\pm
r_{2}|^{N}}\int_{0}^{\infty}\Big{|}\partial_{\lambda}^{N}\big{[}F(\lambda
r_{1})\widetilde{\chi}(\lambda
r_{1})\chi(\lambda)\lambda^{n-2m-1}\Gamma(\lambda)(z_{1},z_{2})F_{\pm}(\lambda
r_{2})\big{]}\Big{|}d\lambda$ $\displaystyle\lesssim r_{1}^{-N}\sum_{0\leq
j_{1}+j_{2}+j_{3}+j_{4}\leq N,\,j_{i}\geq
0}\int_{\frac{1}{r_{1}}}^{1}\lambda^{\frac{n+1}{2}-2m-j_{1}}r_{1}^{\frac{n+1}{2}-2m}\lambda^{n-2m-1-j_{2}}\big{|}\partial_{\lambda}^{j_{3}}\Gamma(\lambda)(z_{1},z_{2})\big{|}\frac{\lambda^{-j_{4}}}{\langle\lambda
r_{2}\rangle^{\frac{n-1}{2}}}d\lambda$ $\displaystyle\lesssim
r_{1}^{\frac{n+1}{2}-2m-N}\widetilde{\Gamma}(z_{1},z_{2})\sum_{0\leq
j_{1}+j_{2}+j_{3}+j_{4}\leq N,\,j_{i}\geq
0}\int_{\frac{1}{r_{1}}}^{1}\lambda^{\frac{3n-1}{2}-4m-j_{1}-j_{2}-j_{3}-j_{4}}\langle\lambda
r_{2}\rangle^{\frac{1-n}{2}}d\lambda$ $\displaystyle\lesssim
r_{1}^{\frac{n+1}{2}-2m-N}\widetilde{\Gamma}(z_{1},z_{2})\int_{\frac{1}{r_{1}}}^{1}\frac{\lambda^{\frac{3n-1}{2}-4m-N}}{\langle\lambda
r_{2}\rangle^{\frac{n-1}{2}}}d\lambda$

Since $\frac{n-1}{2}>0$ the contribution of $\langle\lambda r_{2}\rangle$ can
be ignored. Since $n>4m$, $n-4m\geq 1$ and

$\frac{3n-1}{2}-4m-N\geq
1+\frac{n}{2}-\frac{1}{2}-\left\lceil\frac{n}{2}\right\rceil-1\geq-1.$

This gives us that the contribution from the $\lambda$ integral is bounded by
$\log(r_{1})\lesssim r_{1}^{\frac{1}{4}}$. This gives us the total
contribution to $K_{4}$ of

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})}{r_{1}^{\frac{n}{2}-\frac{3}{4}+N}}.$

Noting that $\lceil\frac{n}{2}\rceil+\frac{n}{2}+1-\frac{3}{4}\geq
n+\frac{1}{4}$, again using that $r_{1}\gg r_{2}$, the contribution of this to
(8) is bounded by

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{2}^{n+\frac{1}{4}}}dz_{1}dz_{2},$

which is admissible by Lemma 4.1.

We now consider $K_{3}$ which restricts the upper portion of the range of $p$
to $1\leq p<\frac{n}{2m}$. Using Lemma 2.2 we write

(9)
$K_{3}(x,y):=\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}}\\\ \int_{0}^{\infty}e^{i\lambda(r_{1}\pm
r_{2})}F(\lambda
r_{1})\chi(\lambda)\lambda^{n-2m-1}\Gamma(\lambda)(z_{1},z_{2})F_{\pm}(\lambda
r_{2})\ d\lambda dz_{1}dz_{2}.$

When $\lambda r_{2}\lesssim 1$, we can apply (6) to obtain the bound

$\displaystyle\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}}$
$\displaystyle\widetilde{\Gamma}(z_{1},z_{2})\int^{r_{2}^{-1}}_{0}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}\lambda^{n-2m-1}\langle\lambda
r_{2}\rangle^{\frac{1-n}{2}}\ d\lambda dz_{1}dz_{2}$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}}\widetilde{\Gamma}(z_{1},z_{2})\int^{r_{2}^{-1}}_{0}\lambda^{n-2m-1}\
d\lambda dz_{1}dz_{2}$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}r_{2}^{n-2m}}\widetilde{\Gamma}(z_{1},z_{2}),$

which is bounded when $1\leq p<\frac{n}{2m}$ by Lemma 4.3 with $k=\ell=0$.

When $\lambda r_{2}\gtrsim 1$, we integrate by parts
$N=\lceil\frac{n}{2}\rceil+1$ times (using (6)) to obtain the bound

$\displaystyle\frac{1}{|r_{1}\pm
r_{2}|^{N}}\int_{0}^{\infty}\Big{|}\partial_{\lambda}^{N}\big{[}F(\lambda
r_{1})\widetilde{\chi}(\lambda
r_{1})\chi(\lambda)\lambda^{n-2m-1}\Gamma(\lambda)(z_{1},z_{2})F_{\pm}(\lambda
r_{2})\big{]}\Big{|}d\lambda$ $\displaystyle\lesssim r_{2}^{-N}\sum_{0\leq
j_{1}+j_{2}+j_{3}+j_{4}\leq N,\,j_{i}\geq
0}\int_{\frac{1}{r_{2}}}^{1}\lambda^{-j_{1}}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}\lambda^{n-2m-1-j_{2}}\big{|}\partial_{\lambda}^{j_{3}}\Gamma(\lambda)(z_{1},z_{2})\big{|}\frac{\lambda^{\frac{1-n}{2}-j_{4}}}{r_{2}^{\frac{n-1}{2}}}d\lambda$
$\displaystyle\lesssim
r_{2}^{\frac{1-n}{2}-N}\widetilde{\Gamma}(z_{1},z_{2})\sum_{0\leq
j_{1}+j_{2}+j_{3}+j_{4}\leq N,\,j_{i}\geq
0}\int_{\frac{1}{r_{2}}}^{1}\lambda^{\frac{n-1}{2}-2m-j_{1}-j_{2}-j_{3}-j_{4}}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}d\lambda$ $\displaystyle\lesssim
r_{2}^{\frac{1-n}{2}-N}\widetilde{\Gamma}(z_{1},z_{2})\int_{\frac{1}{r_{2}}}^{1}\lambda^{\frac{n-1}{2}-2m-N}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}d\lambda.$

We consider the cases when $\lambda r_{1}\gtrsim 1$ and $\lambda r_{1}\lesssim
1$ as follows:

$r_{2}^{\frac{1-n}{2}-N}\widetilde{\Gamma}(z_{1},z_{2})\int_{\frac{1}{r_{2}}}^{1}\lambda^{\frac{n-1}{2}-2m-N}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}d\lambda\\\ \lesssim
r_{2}^{-\frac{n-1}{2}-N}\widetilde{\Gamma}(z_{1},z_{2})\Big{(}\int_{\frac{1}{r_{2}}}^{\min(\frac{1}{r_{1}},1)}\lambda^{\frac{n}{2}-2m-N-\frac{1}{2}}\chi(\lambda
r_{1})d\lambda\\\
+\int_{\min(\frac{1}{r_{1}},1)}^{1}r_{1}^{\frac{n+1}{2}-2m}\lambda^{n-4m-N}\widetilde{\chi}(\lambda
r_{1})d\lambda\Big{)}.$

Since
$\frac{n}{2}-2m-N-\frac{1}{2}=\frac{n}{2}-\lceil\frac{n}{2}\rceil-2m-\frac{3}{2}\leq-\frac{3}{2}$,
the first integral is at most $r_{2}^{2m-\frac{n}{2}+N-\frac{1}{2}}$, so its
contribution to (9) is at most

(10) $\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}r_{2}^{n-2m}}\widetilde{\Gamma}(z_{1},z_{2})dz_{1}dz_{2},$

which is bounded for $1\leq p<\frac{n}{2m}$ by Lemma 4.3.

Similarly, after multiplying the second integral by $(\lambda
r_{1})^{N-1}\gtrsim 1$,

$\int_{\min(\frac{1}{r_{1}},1)}^{1}r_{1}^{\frac{n+1}{2}-2m}\lambda^{n-4m-N}\widetilde{\chi}(\lambda
r_{1})d\lambda\lesssim\int_{\min(\frac{1}{r_{1}},1)}^{1}r_{1}^{\frac{n+1}{2}-2m+N-1}\lambda^{n-4m-1}d\lambda,$

since $n-4m-1\geq 0$, the $\lambda$ integral is bounded and the contribution
is bounded by $r_{1}^{\frac{n+1}{2}-2m+N-1}$. Letting $\\{n/2\\}=\lceil
n/2\rceil-n/2$, the second integral’s contribution to (9) is at most

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}r_{1}^{N-\frac{n}{2}-\frac{1}{2}}}{r_{2}^{N+\frac{n}{2}-\frac{1}{2}}}\widetilde{\Gamma}(z_{1},z_{2})dz_{1}dz_{2}\\\
=\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}r_{1}^{\\{n/2\\}+\frac{1}{2}}}{r_{2}^{n+\\{n/2\\}+\frac{1}{2}}}\widetilde{\Gamma}(z_{1},z_{2})dz_{1}dz_{2},$

which is admissible for all $1\leq p\leq\infty$ by Lemma 4.2. ∎

##  3\. Resolvent and Inverse Expansions

It remains only to prove that the operators $\Gamma_{k}(\lambda)$ defined in
(3) satisfy the bounds needed to apply Proposition 2.1. In this section we
develop different resolvent expansions and develop an expansion of
$M^{+}(\lambda)^{-1}$ for small $\lambda$ when there is a threshold
eigenvalue. Throughout this section we consider the ‘+’ limiting operators and
omit the superscript.

Recall that $n_{\star}=n+4$ if $n$ is odd and $n_{\star}=n+3$ if $n$ is even.
The bounds in Lemma 2.2 imply that the operator $R_{\ell}$ with kernel

(11)
$R_{\ell}(x,y):=v(x)v(y)\sup_{0<\lambda<\lambda_{0}}|\lambda^{\ell}\partial_{\lambda}^{\ell}\mathcal{R}_{0}^{+}(\lambda^{2m})(x,y)|$

satisfies

$R_{\ell}(x,y)\lesssim
v(x)v(y)\big{(}|x-y|^{2m-n}+|x-y|^{\ell-(\frac{n-1}{2})}\big{)},\,\,\,\ell\geq
0.$

This pointwise bound implies that $R_{\ell}$ is bounded on
$L^{2}(\mathbb{R}^{n})$ for $0\leq\ell\leq\lceil\frac{n}{2}\rceil+1$ provided
that $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n_{\star}$, see
[4, 5].

We write the iterated resolvent operators

(12) $\displaystyle
A(\lambda,z_{1},z_{2})=\big{[}\big{(}\mathcal{R}_{0}^{+}(\lambda^{2m})V\big{)}^{k-1}\mathcal{R}_{0}^{+}(\lambda^{2m})\big{]}(z_{1},z_{2}).$

For odd dimensions $n>4m$, if $k$ is sufficiently large depending on $n,m$ and
$|V(x)|\lesssim\langle x\rangle^{-n_{\star}-}$, then

$\displaystyle\sup_{0<\lambda<1}|\lambda^{\ell}\partial_{\lambda}^{\ell}A(\lambda,z_{1},z_{2})|$
$\displaystyle\lesssim\langle z_{1}\rangle^{2}\langle z_{2}\rangle^{2},$

for $0\leq\ell\leq\frac{n+3}{2}=\lceil\frac{n}{2}\rceil+1$. This follows from
the pointwise bounds on $R_{\ell}$ above. The iteration of the resolvents
smooths out the local singularity $|x-\cdot|^{2m-n}$. Each iteration improves
the local singularity by $2m$, so that after $\ell$ iterations the local
singularity is of size $|x-\cdot|^{2m\ell-n}$. Selecting $k$ large enough
ensures that the local singularity is completely integrated away. See the
proofs of Propositions 5.3 and 6.5 in [4] for more details. For even $n>4m$,
since we need fewer derivatives we have

$\sup_{0<\lambda<1}|\lambda^{\ell}\partial_{\lambda}^{\ell}A(\lambda,z_{1},z_{2})|\lesssim\langle
z_{1}\rangle^{\frac{3}{2}}\langle z_{2}\rangle^{\frac{3}{2}},$

for $0\leq\ell\leq\frac{n+2}{2}=\lceil\frac{n}{2}\rceil+1$. More compactly, we
have that

$|A(\lambda,z_{1},z_{2})|\lesssim\langle
z_{1}\rangle^{\\{\frac{n}{2}\\}+\frac{3}{2}}\langle
z_{2}\rangle^{\\{\frac{n}{2}\\}+\frac{3}{2}},\qquad
0\leq\ell\leq\lceil\frac{n}{2}\rceil+1.$

Finally, recalling that

$\Gamma_{k}(\lambda)=UvA(\lambda)v\lambda^{2m}M^{-1}(\lambda)vA(\lambda)vU,$

our goal is to show that
$\sup_{0<\lambda<\lambda_{0}}|\lambda^{\ell+2m}\partial_{\lambda}^{\ell}[M(\lambda)]^{-1}(x,y)|$
is bounded on $L^{2}$ for each $0\leq\ell\leq\lceil\frac{n}{2}\rceil+1$.

We then define the following operators or their integral kernels

$\displaystyle G_{0}(x,y):=$ $\displaystyle
a_{0}|x-y|^{2m-n}=\mathcal{R}_{0}(0)(x,y),$ $\displaystyle T_{0}:=$
$\displaystyle U+vG_{0}v.$

Here $a_{0}\neq 0$ is a real constant depending on $m$ and $n$. Recall that
the invertibility of $T_{0}$ on $L^{2}(\mathbb{R}^{n})$ is equivalent to the
lack of a threshold eigenvalue when $n>4m$, see [6]. We further define $S_{1}$
to be the Riesz projection onto the kernel of $T_{0}$ so that the operator
$T_{0}+S_{1}$ is invertible, and we define

$\displaystyle D_{0}:=$ $\displaystyle(T_{0}+S_{1})^{-1}.$

From Proposition 2.4 in [6], we have

(13)
$\mathcal{R}_{0}(\lambda^{2m})(x,y)=\sum^{N-1}_{j=0}\lambda^{2mj}G_{j}(x,y)+c_{n,m}\lambda^{n-2m}+E_{0}(\lambda)(x,y)$

with $N=\lfloor\frac{n}{2m}\rfloor$. Here

$G_{j}(x,y)=c_{j,n,m}|x-y|^{2m-n+2mj},j=0,...,N-1.$

The exact value of these constants is unimportant for our purposes. We note
that these expansions follow from those of the Schrödinger resolvents
$R_{0}(z)=(-\Delta-z)^{-1}$ and the splitting identity

$\mathcal{R}_{0}(z)(x,y):=((-\Delta)^{m}-z)^{-1}(x,y)=\frac{1}{mz^{1-\frac{1}{m}}}\sum_{\ell=0}^{m-1}\omega_{\ell}R_{0}(\omega_{\ell}z^{\frac{1}{m}})(x,y),$

where $\omega_{\ell}=\exp(i2\pi\ell/m)$ are the $m^{th}$ roots of unity. The
expansions utilize a significant amount of cancellation obtained from the
splitting identity and the sum over the roots of unity. See also the proofs of
Lemma 4.2 and 6.2 in [4].

We note that the error bounds for $E_{0}(\lambda)$ in [6] don’t suffice for
our purposes, instead we develop shorter expansions with more detailed control
of the error term. We make this more precise with the following lemmas.

######  Lemma 3.1.

When $0<\lambda<\lambda_{0}$ and $0\leq\ell\leq\lceil\frac{n}{2}\rceil+1$, we
have

$\mathcal{R}_{0}(\lambda)=G_{0}+\lambda^{2m}G_{1}+E(\lambda)$

where for any $0<\epsilon<1$ we have

$\lambda^{\ell}|\partial_{\lambda}^{\ell}E(\lambda)(x,y)|\lesssim\lambda^{2m+\epsilon}[|x-y|^{\\{\frac{n}{2}\\}+\frac{3}{2}}+|x-y|^{4m-n+\epsilon}],$

where $\\{n/2\\}=\lceil n/2\rceil-n/2$.

###### Proof.

We need to consider cases as the resolvents behave differently in even and odd
dimensions and based on the size of $\lambda|x-y|$. When $n$ is odd and
$\lambda|x-y|\ll 1$, we have from Lemma 2.3, equation (2.2), and Remark 2.2 in
[6] that

(14)
$\mathcal{R}_{0}(\lambda^{2m})(x,y)=\sum^{N-1}_{j=0}c_{j,n,m}\lambda^{2mj}|x-y|^{2m-n+2mj}+\sum_{j=0}^{\infty}c_{j,n,m}\lambda^{2N+j}|x-y|^{2m-n+2N+j}$

with $N=\lfloor\frac{n}{2m}\rfloor$. Truncating after the first two terms of
the series yields the expansion with

$E(\lambda)=\sum^{\infty}_{k=1}c_{k}\lambda^{2m+k}|x-y|^{4m-n+k},\,\,\,\lambda|x-y|\ll
1.$

Note that many $c_{k}$ may be zero, this doesn’t affect our bounds. This
implies that for any $\ell$ we have

$\lambda^{\ell}\big{|}\partial^{\ell}_{\lambda}E(\lambda)(x,y)\chi(\lambda|x-y|)\big{|}\lesssim\lambda^{2m+1}|x-y|^{4m-n+1},$

which suffices for small $\lambda|x-y|$.

When $n$ is even, there are logarithms when $\lambda|x-y|\ll 1$ that we must
account for. Using Lemma 2.1 and Lemma 2.3 in [6], (see also the expansion
after Lemma 6.2 in [4]) we have the following representation

$\displaystyle\mathcal{R}_{0}(\lambda^{2m})=$
$\displaystyle\sum^{\lceil\frac{n}{2m}\rceil-1}_{k=0}a_{k}\lambda^{2mk}|x-y|^{2m(k+1)-n}+\sum^{\infty}_{j=0}b_{j}\lambda^{n-2m+2j}|x-y|^{2j}$
$\displaystyle+\sum^{\infty}_{k=\lceil\frac{n}{2m}\rceil}c_{k}\lambda^{2m(k-1)}|x-y|^{2mk-n}\Bigl{(}\ln(|x-y|)+\ln\lambda\Bigr{)}.$

However, since the first logarithm appears with a power of $\lambda$ at least
as large as $n-2m$ since $\lceil\frac{n}{2m}\rceil\geq 3$, the logarithms
don’t affect the required bound and

$\displaystyle E(\lambda)$
$\displaystyle=\sum^{\lceil\frac{n}{2m}\rceil-1}_{k=2}a_{k}\lambda^{2mk}|x-y|^{2m(k+1)-n}+\sum^{\infty}_{j=0}b_{j}\lambda^{n-2m+2j}|x-y|^{2j}$
$\displaystyle+\sum^{\infty}_{k=\lceil\frac{n}{2m}\rceil}c_{k}\lambda^{2m(k-1)}|x-y|^{2mk-n}\Bigl{(}\ln(|x-y|)+\ln\lambda\Bigr{)}.$

The first two sums with only powers of $\lambda|x-y|$ are controlled as in the
odd $n$ argument. For the logarithms, since $k-1\geq 2$, the largest possible
contribution is of the form

$\lambda^{4m}|x-y|^{6m-n}\ln(\lambda|x-y|)\lesssim\lambda^{4m-1}|x-y|^{6m-n-1},$

using $\log(z)\lesssim z^{-1}$ when $z\lesssim 1$. We may further divide by
powers of $(\lambda|x-y|)$ to match the polynomial bound as needed. Since
$m\geq 1$, we have

$\lambda^{\ell}\big{|}\partial^{\ell}_{\lambda}E(\lambda)(x,y)\chi(\lambda|x-y|)\big{|}\lesssim\lambda^{2m+1}|x-y|^{4m-n+1},$

which suffices for small $\lambda|x-y|$.

The bound on the error term when $\lambda|x-y|\gtrsim 1$ follows from the
bounds in Lemma 2.3 of [5] and the definition of the kernels of $G_{0},G_{1}$,
we have

$\displaystyle\lambda^{\ell}|\partial_{\lambda}^{\ell}E(\lambda)|$
$\displaystyle=\lambda^{\ell}\partial_{\lambda}^{\ell}\big{[}\mathcal{R}_{0}(\lambda^{2m})-G_{0}(x,y)-\lambda^{2m}G_{1}(x,y)\big{]}$
$\displaystyle\lesssim\lambda^{\frac{n+1}{2}-2m+\ell}|x-y|^{\frac{1-n}{2}+\ell}+|x-y|^{2m-n}+\lambda^{2m}|x-y|^{4m-n}.$

We note that
$\frac{1-n}{2}+\ell\leq\frac{1-n}{2}+\lceil\frac{n}{2}\rceil+1=\\{\frac{n}{2}\\}+\frac{3}{2}$,
and $\frac{n+1}{2}-2m+\lceil\frac{n}{2}\rceil+1>2m$. Since
$\lambda|x-y|\gtrsim 1$ and $\lambda\lesssim 1$, we can bound this by

$\lambda^{\ell}|\partial_{\lambda}^{\ell}E(\lambda)\widetilde{\chi}(\lambda|x-y|)|\lesssim\lambda^{2m+\epsilon}[|x-y|^{\\{\frac{n}{2}\\}+\frac{3}{2}}+|x-y|^{4m-n+\epsilon}].$

∎

To invert $M(\lambda)$ in a neighborhood of the threshold, we utilize the
Jensen-Nenciu inversion scheme in [13], which requires some set up. We
introduce some notation to help streamline the upcoming statements and proofs.
For an absolutely bounded operator $T(\lambda)$ on $L^{2}(\mathbb{R}^{n})$ we
write $T(\lambda)=O_{N}(\lambda^{j})$ to mean that

$\|\lambda^{\ell}|\partial_{\lambda}^{\ell}T(\lambda)|\,\|_{L^{2}\to
L^{2}}\lesssim\lambda^{j},\qquad 0\leq\ell\leq N\quad\text{for
}0<\lambda<\lambda_{0}.$

We note that, if $S,T$ satisfy $S(\lambda)=O_{N}(\lambda^{j})$ and
$T(\lambda)=O_{N}(\lambda^{k})$, by the product rule the composition of the
operators $ST$ is absolutely bounded and satisfies

$S(\lambda)T(\lambda)=O_{N}(\lambda^{j+k})\qquad\text{on }(0,\lambda_{0}).$

Similarly,

$\lambda^{j}T(\lambda)=O_{N}(\lambda^{j+k})\qquad\text{on }(0,\lambda_{0}).$

In particular, if $R$ is an absolutely bounded operator on $L^{2}$, then

$T(\lambda)R,\,RT(\lambda)=O_{N}(\lambda^{k})\qquad\text{on }(0,\lambda_{0}).$

Of particular use is the observation that if
$T(\lambda)=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{0})$, then

$\sup_{0<\lambda<\lambda_{0}}\sup_{0\leq\ell\leq\lceil\frac{n}{2}\rceil+1}\lambda^{\ell}|\partial_{\lambda}^{\ell}T(\lambda)|$

is a bounded operator on $L^{2}(\mathbb{R}^{n})$.

Finally, we have

######  Lemma 3.2.

If $\Gamma$ is a $\lambda$-independent, invertible, absolutely bounded
operator on $L^{2}$ with an absolutely bounded inverse, then for $\epsilon>0$,
we have (for sufficiently small $\lambda_{0}$)

$[\Gamma+O_{N}(\lambda^{\epsilon})]^{-1}=\Gamma^{-1}+O_{N}(\lambda^{\epsilon})=O_{N}(\lambda^{0}).$

###### Proof.

This is just a Neumann series expansion for $N=0$. For derivatives
($1\leq\ell\leq N$), note that
$\lambda^{\ell}\partial^{\ell}_{\lambda}\big{[}[\Gamma+O_{N}(\lambda^{\epsilon})]^{-1}-\Gamma^{-1}\big{]}$
is a linear combination of operators of the form

$[\Gamma+O_{N}(\lambda^{\epsilon})]^{-1}\prod_{j=1}^{J}\big{[}\lambda^{\ell_{j}}\partial^{\ell_{j}}_{\lambda}O_{N}(\lambda^{\epsilon})[\Gamma+O_{N}(\lambda^{\epsilon})]^{-1}\big{]}=O_{0}(\lambda^{\epsilon}).$

Here $J\geq 1$, $1\leq\ell_{j}\leq N$ with $\sum\ell_{j}=\ell$. ∎

Our main result is the following proposition:

######  Proposition 3.3.

If $\beta>n_{\star}$ and there is a zero energy eigenvalue, the operator
$M(\lambda)$ is invertible on $L^{2}$ for sufficiently small
$0<\lambda<\lambda_{0}$. Furthermore, we have

$M^{-1}(\lambda)=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{-2m}).$

By the discussion above, these bounds on $M^{-1}(\lambda)$ will suffice to
allow us to establish that the operators $\Gamma_{k}$ satisfy the hypotheses
of Proposition 2.1. To this end, we have the following series of lemmas.

######  Lemma 3.4.

If $\beta>n_{\star}$, then for sufficiently small $\lambda$ the operator
$M(\lambda)+S_{1}$ is invertible on $L^{2}$ with

$(M(\lambda)+S_{1})^{-1}=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{0}).$

Further, for any $0<\epsilon<1$, on $0<\lambda<\lambda_{0}$ we have

$(M+S_{1})^{-1}=D_{0}-\lambda^{2m}D_{0}T_{1}D_{0}+O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m+\epsilon}),$

where $T_{1}=vG_{1}v.$

###### Proof.

For the first claim, recall that by Lemma 3.1, we have

$\displaystyle
M(\lambda)=U+v\mathcal{R}_{0}(\lambda^{2m})v=U+v(G_{0}+\lambda^{2m}G_{1}+E(\lambda))v,$

where for $0\leq\ell\leq\lceil\frac{n}{2}\rceil+1$

$\lambda^{\ell}|\partial_{\lambda}^{\ell}vE(\lambda)v|\lesssim\lambda^{2m+\epsilon}v(x)|x-y|^{\\{\frac{n}{2}\\}+\frac{3}{2}}v(y)+\lambda^{2m+\epsilon}v(x)|x-y|^{4m-n+\epsilon}v(y).$

The first term is Hilbert-Schmidt when $\beta>n_{\star}$ since

$[v(x)|x-y|^{\\{\frac{n}{2}\\}+\frac{3}{2}}v(y)]^{2}\leq\langle
x\rangle^{-n-}\langle y\rangle^{-n-}.$

In dimensions $4m<n\leq 8m$, the second term is Hilbert-Schmidt under weaker
decay assumptions than $\beta>n_{\star}$. On the other hand, if $n>8m$, the
resulting kernel is too singular to be Hilbert-Schmidt, instead we identify
$|x-y|^{4m-n+\epsilon}$ as a scalar multiple of the fractional integral
operator $I_{4m+\epsilon}$. By Lemma 2.3 in [12], $I_{4m+\epsilon}$ is bounded
from $L^{2,s}\to L^{2,-s}$ provided $s>2m+\frac{\epsilon}{2}$. This suffices
to show the $L^{2}$ boundedness of
$v(x)\lambda^{\ell}|\partial_{\lambda}^{\ell}E(\lambda)|v(y)$ in dimensions
$n>8m$ since $v(x)\lesssim\langle x\rangle^{-\frac{\beta}{2}}$ with
$\beta>n_{*}>4m+3$. We conclude that

$M(\lambda)=T_{0}+\lambda^{2m}T_{1}+O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m+\epsilon}).$

We claim that $D_{0}$ is an absolutely bounded operator. To see this, first
note that the argument of Lemma 4.3 in [3] may be adapted since, by the
argument above, $vG_{0}v$ is absolutely bounded. Here one needs to iterate the
resolvent identity sufficiently since $(vG_{0}v)^{k}$ is Hilbert-Schmidt for
sufficiently large $k$ while $vG_{0}v$ itself is not. We leave the details to
the interested reader.

Since $T_{0}+S_{1}$ is invertible on $L^{2}$ with an absolutely bounded
inverse, by Lemma 3.2, we have

$[M(\lambda)+S_{1}]^{-1}=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{0}).$

To obtain the second claim, we utilize the resolvent identity
$A^{-1}=B^{-1}+B^{-1}(B-A)A^{-1}$ with $A=(M(\lambda)+S_{1})$ and
$B=T_{0}+S_{1}$. From this, we see that

$[M(\lambda)+S_{1}]^{-1}=D_{0}-D_{0}[M(\lambda)-T_{0}]D_{0}+(D_{0}[M(\lambda)-T_{0}])^{2}[M(\lambda)+S_{1}]^{-1}.$

Note that
$M(\lambda)-T_{0}=\lambda^{2m}T_{1}+O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m+\epsilon})=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m})$.
Therefore,

$[M(\lambda)+S_{1}]^{-1}=D_{0}-\lambda^{2m}D_{0}T_{1}D_{0}+O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m+\epsilon})+O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{4m}),$

which suffices. ∎

To utilize the Jensen-Nenciu inversion machinery, Corollary 2.2 in [13], we
need to invert the operator

$\frac{1}{\lambda^{2m}}(S_{1}-S_{1}(M(\lambda)+S_{1})^{-1}S_{1})$

on $S_{1}L^{2}$. To do so, we note that $S_{1}D_{0}=D_{0}S_{1}=S_{1}$ so that
the leading $S_{1}$ cancels and consider the operator

(15)
$B(\lambda)=\lambda^{-2m}[S_{1}-S_{1}(M(\lambda)+S_{1})^{-1}S_{1}]=S_{1}T_{1}S_{1}-S_{1}\lambda^{-2m}O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{2m+\epsilon})S_{1}\\\
=S_{1}T_{1}S_{1}+S_{1}O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{\epsilon})S_{1}.$

We now have

######  Lemma 3.5.

If $\beta>n_{\star}$, for sufficiently small $\lambda$ the operator
$B(\lambda)$ is invertible on $S_{1}L^{2}$. Further, we have

$B^{-1}(\lambda)=O_{\lceil\frac{n}{2}\rceil+1}(\lambda^{0})$

as an operator on $S_{1}L^{2}$.

###### Proof.

Using the expansion in (15), we note that $T_{1}$ is an invertible operator on
$S_{1}L^{2}$ (c.f. Definition 2.6 and Remark 2.7 in [6]). Since $S_{1}L^{2}$
is finite dimensional, $T_{1}$ and its inverse are absolutely bounded. The
claim now follows from Lemma 3.2. ∎

We are now ready to prove Proposition 3.3:

###### Proof.

By the Jensen-Nenciu inversion technique, [13], we have

(16) $\displaystyle
M^{-1}(\lambda)=(M(\lambda)+S_{1})^{-1}+\lambda^{-2m}(M(\lambda)+S_{1})^{-1}S_{1}B^{-1}(\lambda)S_{1}(M(\lambda)+S_{1})^{-1},$

provided that $B(\lambda)$ is invertible on $S_{1}L^{2}(\mathbb{R}^{n})$.
Therefore, the claim follows from Lemmas 3.4 and 3.5. ∎

We are now ready to prove the main technical result.

###### Proof of Theorem 1.1.

Proposition 3.3 shows that
$\sup_{0<\lambda<\lambda_{0}}|\lambda^{\ell+2m}\partial_{\lambda}^{\ell}[M(\lambda)]^{-1}(x,y)|$
is $L^{2}$ bounded for all $0\leq\ell\leq\lceil\frac{n}{2}\rceil+1$. By (2),
the definition of $W_{low,k}$, $\Gamma_{k}(\lambda)$ and the discussion
following (12), we see that the operator $\Gamma_{k}(\lambda)$ satisfies the
hypotheses of Proposition 2.1.

∎

##  4\. Technical Lemmas

For completeness, we include necessary lemmas about the
$L^{p}(\mathbb{R}^{n})$ boundedness of certain integral kernels that are
needed in the proof of Proposition 2.1. We use the two Lemmas below about
admissible kernels, which are Lemmas 4.1 and 4.2 in [5] respectively.

######  Lemma 4.1.

Let $K$ be an operator with integral kernel $K(x,y)$ that satisfies the bound

$|K(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})\chi_{\\{|y-z_{2}|\gg\langle
z_{1}-x\rangle\\}}}{|x-z_{1}|^{n-2m-k}|z_{2}-y|^{n+\ell}}\,dz_{1}\,dz_{2}$

for some $0\leq k\leq n-2m$ and $\ell>0$. Then, under the hypotheses of Lemma
2.1, the kernel of $K$ is admissible, and consequently $K$ is a bounded
operator on $L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$.

######  Lemma 4.2.

Let $K$ be an operator with integral kernel $K(x,y)$ that satisfies the bound

$|K(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})\chi_{\\{|y-z_{2}|\gg\langle
z_{1}-x\rangle\\}}|x-z_{1}|^{\ell}}{|z_{2}-y|^{n+\ell}}\,dz_{1}\,dz_{2}$

for some $\ell>0$. Then, under the hypotheses of Lemma 2.1, the kernel of $K$
is admissible, and consequently $K$ is a bounded operator on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$.

We also need the following new bounds for the analysis of $K_{3}$.

######  Lemma 4.3.

Let $K$ be an operator with integral kernel $K(x,y)$ that satisfies the bound

$|K(x,y)|\lesssim\int_{\mathbb{R}^{n}}\frac{v(z_{1})v(z_{2})\widetilde{\Gamma}(z_{1},z_{2})\chi_{\\{|y-z_{2}|\gg\langle
z_{1}-x\rangle\\}}}{|x-z_{1}|^{n-2m-\ell}|y-z_{2}|^{n-2m+k}}\ \\!dz_{1}dz_{2}$

for some $-2m<k<2m$ with $\ell<k+n-4m$. Then, under the hypotheses of
Proposition 2.1, $K$ is a bounded operator on $L^{p}(\mathbb{R}^{n})$ for
$1\leq p<\frac{n}{2m-k}$.

###### Proof.

We show that $\left\lVert K(x,y)\right\rVert_{L^{p^{\prime}}_{y}L^{p}_{x}}$ is
bounded. Using the assumptions on $\widetilde{\Gamma}$ in Proposition 2.1,
provided that $\frac{n}{n-2m+k}<p^{\prime}\leq\infty$ or equivalently $1\leq
p<\frac{n}{2m-k}$, we may take the $L^{p^{\prime}}_{y}$ norm to bound (recall
that $r_{1}=|x-z_{z}|$ and $r_{2}=|z_{2}-y|$)

$\displaystyle\left\lVert
K(x,y)\right\rVert_{L^{p^{\prime}}_{y}L^{p}_{x}}\lesssim$
$\displaystyle\left\lVert\langle z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}r_{1}^{2m+\ell-n}\langle
r_{1}\rangle^{2m-n-k+n/p^{\prime}}\right\rVert_{L^{1}_{z_{1}}L^{1}_{z_{2}}L^{p}_{x}}$
(17) $\displaystyle\lesssim$ $\displaystyle\left\lVert\langle
z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}r_{1}^{2m-n+\ell}\langle
r_{1}\rangle^{2m-k-n/p}\right\rVert_{L^{1}_{z_{1}}L^{1}_{z_{2}}L^{p}_{x}}.$

We then consider cases to control the upper bound in (17); when $r_{1}\lesssim
1$ and $r_{1}\gtrsim 1$. If $r_{1}\gtrsim 1$, we can directly take the
$L^{p}_{x}$ norm since $4m-n-(k-\ell)<0$ to see

$\displaystyle\Big{\lVert}\langle z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}r_{1}^{2m-n+\ell}$ $\displaystyle\langle
r_{1}\rangle^{2m-k-n/p}\chi_{r_{1}\gg
1}\Big{\rVert}_{L^{p}_{x}L^{1}_{z_{1}}L^{1}_{z_{2}}}$ $\displaystyle\lesssim$
$\displaystyle\Big{\lVert}\langle z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}r_{1}^{4m-n-n/p-(k-\ell)}\chi_{r_{1}\gg
1}\Big{\rVert}_{L^{p}_{x}L^{1}_{z_{1}}L^{1}_{z_{2}}}$ $\displaystyle\lesssim$
$\displaystyle\left\lVert\langle z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}\right\rVert_{L^{1}_{z_{1}}L^{1}_{z_{2}}}\lesssim
1,$

since $\beta>n$.

On the other hand, when $r_{1}\lesssim 1$, we leverage the decay of $v(z_{1})$
and since all quantities are non-negative take the $L_{z_{1}}^{1}$ norm first
by converting to polar coordinates centered around $x$.

$\Big{\lVert}\langle z_{1}\rangle^{-\frac{\beta+n}{2}}\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}r_{1}^{2m-n+\ell}\langle
r_{1}\rangle^{2m-n-k+n-n/p}\chi_{r_{1}\lesssim
1}\Big{\rVert}_{L^{1}_{z_{1}}L^{1}_{z_{2}}L^{p}_{x}}\\\
\lesssim\left\lVert\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}\int_{\mathbb{R}^{n}}\langle
x+r\rangle^{-\frac{\beta+n}{2}}r^{2m+\ell-1}\chi_{r\lesssim 1}\
\\!dr\right\rVert_{L^{1}_{z_{2}}L^{p}_{x}}\\\ \lesssim\left\lVert\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}\langle
x\rangle^{-\frac{\beta+n}{2}}\int_{\mathbb{R}^{n}}r^{2m+\ell-1}\chi_{r\lesssim
1}\ \\!dr\right\rVert_{L^{1}_{z_{2}}L^{p}_{x}}\lesssim\left\lVert\langle
z_{2}\rangle^{-\frac{\beta+n}{2}}\langle
x\rangle^{-\frac{\beta+n}{2}}\right\rVert_{L^{1}_{z_{2}}L^{p}_{x}}\lesssim 1,$

since $2m+\ell-1>-1$ and $\beta>n$.

Therefore the kernel of $K$ is a bounded operator on $L^{p}(\mathbb{R}^{n})$
for $1\leq p<\frac{n}{2m-k}$. ∎

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