#  $L^{p}$-boundedness of wave operators for bi-Schrödinger operators on the
line

Haruya Mizutani  Department of Mathematics, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan<EMAIL_ADDRESS>,
Zijun Wan  Department of Mathematics, Central China Normal University, Wuhan,
430079, P.R. China<EMAIL_ADDRESS>and  Xiaohua Yao  Department of
Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central
China Normal University, Wuhan, 430079, P.R. China<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to establishing several types of $L^{p}$-boundedness of
wave operators $W_{\pm}=W_{\pm}(H,\Delta^{2})$ associated with the bi-
Schrödinger operators $H=\Delta^{2}+V(x)$ on the line $\mathbb{R}$. Given
suitable decay potentials $V$, we firstly prove that the wave and dual wave
operators are bounded on $L^{p}(\mathbb{R})$ for all $1<p<\infty$:

$\|W_{\pm}f\|_{L^{p}(\mathbb{R})}+\|W_{\pm}^{*}f\|_{L^{p}(\mathbb{R})}\lesssim\|f\|_{L^{p}(\mathbb{R})},$

which are further extended to the $L^{p}$-boundedness on the weighted spaces
$L^{p}(\mathbb{R},w)$ with general even $A_{p}$-weights $w$ and to the
boundedness on the Sobolev spaces $W^{s,p}(\mathbb{R})$. For the limiting
case, we prove that $W_{\pm}$ are bounded from $L^{1}({\mathbb{R}})$ to
$L^{1,\infty}({\mathbb{R}})$ as well as bounded from the Hardy space
${\mathcal{H}}^{1}({\mathbb{R}})$ to $L^{1}({\mathbb{R}})$. These results
especially hold whatever the zero energy is a regular point or a resonance of
$H$. We also obtain that $W_{\pm}$ are bounded from $L^{\infty}({\mathbb{R}})$
to ${\mathop{\mathrm{BMO}}}({\mathbb{R}})$ if zero is a regular point or a
first kind resonance of $H$. Next, we show that $W_{\pm}$ are neither bounded
on $L^{1}(\mathbb{R})$ nor on $L^{\infty}(\mathbb{R})$ even if zero is a
regular point of $H$. Moreover, if zero is a second kind resonance of $H$,
then $W_{\pm}$ are shown to be even not bounded from
$L^{\infty}({\mathbb{R}})$ to ${\mathop{\mathrm{BMO}}}({\mathbb{R}})$ in
general. In particular, we remark that our results give a complete picture of
the validity of $L^{p}$-boundedness of the wave operators for all $1\leq
p\leq\infty$ in the regular case. Finally, as applications, we deduce the
$L^{p}$-$L^{q}$ decay estimates for the propagator
$e^{-itH}P_{\mathrm{ac}}(H)$ with pairs $(1/p,1/q)$ belonging to a certain
region of $\mathbb{R}^{2}$, as well as establish the Hörmander-type
$L^{p}$-boundedness theorem for the spectral multiplier $f(H)$.

###### Key words and phrases:

$L^{p}$-boundedness, Wave operator, bi-Schrödinger operators, Zero resonances

##  1\. Introduction and main results

###  1.1. Introduction

Let $\Delta^{2}=\frac{d^{4}}{dx^{4}}$ be the bi-Laplacian and
$H=\Delta^{2}+V(x)$ be the (fourth-order) bi-Schrödinger operator on
$\mathbb{R}$, where $V(x)$ is a real-valued potential satisfying

$|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$

with some $\mu>0$ specified later and ${\langle}x{\rangle}=\sqrt{1+x^{2}}$. By
the Kato–Rellich theorem, $\Delta^{2}$ and $H$ are realized as self-adjoint
operators on $L^{2}(\mathbb{R})$ with domain
$D(\Delta^{2})=D(H)=H^{4}({\mathbb{R}})$, generating the associated unitary
groups $e^{-it\Delta^{2}}$ and $e^{-itH}$ on $L^{2}({\mathbb{R}})$,
respectively, where $H^{4}({\mathbb{R}})$ is the $L^{2}$-Sobolev space of
order $4$.

For $\mu>1$, it is well-known (see e.g. [1, 46, 52]) that the wave operators

$\displaystyle
W_{\pm}=W_{\pm}(H,\Delta^{2}):=\operatorname*{s-lim}_{t\to\pm\infty}e^{itH}e^{-it\Delta^{2}}$
( 1.1)

exist as partial isometries from $L^{2}(\mathbb{R})$ to
$\mathcal{H}_{\mathop{\mathrm{ac}}\nolimits}(H)$ and are asymptotically
complete, i.e.
$\mathop{\mathrm{Ran}}(W_{\pm})=\mathcal{H}_{\mathop{\mathrm{ac}}\nolimits}(H)$,
where $\mathcal{H}_{\mathop{\mathrm{ac}}\nolimits}(H)$ is the absolutely
continuous spectral subspace of $H$. Moreover, the absolutely continuous
spectrum $\sigma_{\mathop{\mathrm{ac}}\nolimits}(H)$ coincides with
$[0,\infty)$ and the singular continuous spectrum $\sigma_{\mathrm{sc}}(H)$ is
absent. In particular, the inverse (dual) wave operators

$W_{\pm}(\Delta^{2},H):=\operatorname*{s-lim}_{t\to\pm\infty}e^{it\Delta^{2}}e^{-itH}P_{\mathrm{ac}}(H)$

also exist and satisfy $W_{\pm}(\Delta^{2},H)=W_{\pm}(H,\Delta^{2})^{*}$,
where $P_{\mathop{\mathrm{ac}}\nolimits}(H)$ is the projection onto
$\mathcal{H}_{\mathop{\mathrm{ac}}\nolimits}(H)$. The point spectrum
$\sigma_{\mathrm{p}}(H)$ consists of finitely many negative eigenvalues and
possible embedded eigenvalues in $[0,\infty)$. Throughout the paper, we always
assume that $H$ has no embedded eigenvalue in $(0,\infty)$ (see Subsection 1.3
below for some sufficient conditions to ensure the absence of embedded
eigenvalues of $H$).

$W_{\pm}$ and $W_{\pm}^{*}$ are clearly bounded on $L^{2}({\mathbb{R}})$. Then
the main purpose of this paper is the following $L^{p}$-bounds of $W_{\pm}$
and $W_{\pm}^{*}$ for $p\neq 2$:

$\displaystyle\|W_{\pm}\phi\|_{L^{p}(\mathbb{R})}\lesssim\|\phi\|_{L^{p}(\mathbb{R})},\quad\|W_{\pm}^{*}\phi\|_{L^{p}(\mathbb{R})}\lesssim\|\phi\|_{L^{p}(\mathbb{R})}.$
( 1.2)

To explain the importance of these bounds, we recall that $W_{\pm}$ satisfy
the following identities

$\displaystyle W_{\pm}W_{\pm}^{*}=P_{\mathop{\mathrm{ac}}\nolimits}(H),\ \
W_{\pm}^{*}W_{\pm}=I,$

and the intertwining property $f(H)W_{\pm}=W_{\pm}f(\Delta^{2})$, where $f$ is
any Borel measurable function on $\mathbb{R}$. These formulas especially imply

$\displaystyle
f(H)P_{\mathop{\mathrm{ac}}\nolimits}(H)=W_{\pm}f(\Delta^{2})W_{\pm}^{*},$ (
1.3)

which we also call the intertwining property. By virtue of 1.3, the
$L^{p}$-boundedness of $W_{\pm},W_{\pm}^{*}$ can immediately be used to reduce
the $L^{p}$-$L^{q}$ estimates for the perturbed operator $f(H)$ to the same
estimates for the free operator $f(\Delta^{2})$ as follows:

$\displaystyle\|f(H)P_{\mathop{\mathrm{ac}}\nolimits}(H)\|_{L^{p}\to
L^{q}}\leq\|W_{\pm}\|_{L^{q}\to L^{q}}\ \|f(\Delta^{2})\|_{L^{p}\to L^{q}}\
\|W_{\pm}^{*}\|_{L^{p}\to L^{p}}.$ ( 1.4)

For many cases, under suitable conditions on $f$, it is accessible to
establish the $L^{p}$-$L^{q}$ bounds of $f(\Delta^{2})$ by Fourier multiplier
methods. Thus, in order to obtain the inequality 1.4, it is a key problem to
prove the $L^{p}$-bounds 1.2 of $W_{\pm}$ and $W_{\pm}^{*}$. Note that this
observation applies to not only the $L^{p}$-$L^{q}$ bounds, but also general
$X$-$Y$ bounds, namely one has

$\displaystyle\|f(H)P_{\mathop{\mathrm{ac}}\nolimits}(H)\|_{X\to
Y}\leq\|W_{\pm}\|_{Y\to Y}\ \|f(\Delta^{2})\|_{X\to Y}\ \|W_{\pm}^{*}\|_{X\to
X}.$ ( 1.5)

Because of such a useful feature, the $L^{p}$-boundedness of the wave
operators has been extensively studied for the Schrödinger operator
$-\Delta+V(x)$ and widely recognized as a fundamental tool for studying
various nonlinear dispersive equations, such as the nonlinear Schrödinger and
Klein–Gordon equations with potentials (see e.g. [11, 18, 53, 54, 56]).
Therefore, it is natural and seems to be very important to try extending the
$L^{p}$-boundedness of the wave operators to more general Hamiltonians,
especially to the higher-order elliptic operator $P(D)+V(x)$ which has
attracted increasing attention in the mathematical and mathematical physics
literatures. Since the fourth-order Schrödinger operator $\Delta^{2}+V(x)$ can
be considered as one of primal models of such higher-order operators, it thus
is natural to ask whether the $L^{p}$-boundedness 1.2 for $W_{\pm}$ and
$W_{\pm}^{*}$ holds or not.

Up to now, although some progresses have been made recently by Goldberg–Green
[33] and Erdoğan–Green [21], considerably less is known for the higher-order
case compared with the case of the Schrödinger operator. In particular, to the
author’s best knowledge, there seems to be no results on the
$L^{p}$-boundedness of the wave operators for the higher-order elliptic
operators on ${\mathbb{R}}^{n}$ (including $H=\Delta^{2}+V(x)$) if there is a
zero resonance or in low dimensions $n=1,2$, in contrast with the case of the
Schrödinger operator $-\Delta+V(x)$. We refer to Subsection 1.5 below for more
backgrounds and existing literature.

In light of those observations, the main purpose of the paper is to show that
the wave operators $W_{\pm}$ and $W_{\pm}^{*}$ for $H=\Delta^{2}+V(x)$ on
${\mathbb{R}}$ are bounded on $L^{p}({\mathbb{R}})$ for all $1<p<\infty$,
whatever zero is a regular point or a resonance of $H$ (see Definition 1.1
below). Moreover, we also establish several related interesting results in
both positive and negative directions, complementing to or improving upon this
result, which specifically include:

  * •

Several weak-boundedness in the limiting cases $p=1,\infty$;

  * •

Weighted $L^{p}$-boundedness for any even Muckenhoupt weights $w\in A_{p}$ and
$1<p<\infty$ without assuming any additional condition on $V$;

  * •

$W^{s,p}$-boundedness, where $W^{s,p}$ is the $L^{p}$-Sobolev space of order
$s$;

  * •

Counterexamples of the $L^{1}$\- and $L^{\infty}$-boundedness.

These results particularly give a complete classification for the validity of
$L^{p}$-boundedness of $W_{\pm},W_{\pm}^{*}$ if $H$ is has no non-negative
eigenvalue nor zero resonance. Furthermore, we apply our main theorem to show
the $L^{p}$-$L^{q}$ decay estimates for the propagator
$e^{-itH}P_{\mathop{\mathrm{ac}}\nolimits}(H)$ and the Hörmander-type theorem
of the $L^{p}$-boundedness for the spectral multiplier $f(H)$.

###  1.2. Main results

To state our results, we need to recall the notion of the zero resonances for
the operator $H=\Delta^{2}+V(x)$ on ${\mathbb{R}}$ due to Soffer–Wu–Yao [57].
For $s\in{\mathbb{R}}$, we set $L^{2}_{s}({\mathbb{R}})=\\{f\in
L^{2}_{\mathrm{loc}}(\mathbb{R})\ |\ {\langle}x{\rangle}^{s}f\in
L^{2}(\mathbb{R})\\}$, which is decreasing in $s$. Then we define

$W_{\sigma}(\mathbb{R})=\bigcap_{s>\sigma}L^{2}_{-s}(\mathbb{R}),$

which is increasing in $\sigma$ and satisfies
$L^{2}_{-\sigma}(\mathbb{R})\subset W_{\sigma}(\mathbb{R})$. Note that
$(1+|x|)^{\alpha}\in W_{\sigma}({\mathbb{R}})$ if $\sigma\geq\alpha+1/2$. In
particular, $f\in W_{1/2}({\mathbb{R}})$ and ${\langle}x{\rangle}f\in
W_{3/2}({\mathbb{R}})$ for any $f\in L^{\infty}({\mathbb{R}})$.

######  Definition 1.1.

Let $H=\Delta^{2}+V(x)$ and $|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ for
some $\mu>0$. We say that

  * •

zero is a first kind resonance of $H$ if there exists some nonzero $\phi\in
W_{3/2}({\mathbb{R}})$ but no any non-zero $\phi\in W_{1/2}({\mathbb{R}})$
such that $H\phi=0$ in the distributional sense;

  * •

zero is a second kind resonance of $H$ if there exists some nonzero $\phi\in
W_{1/2}({\mathbb{R}})$ but no any non-zero $\phi\in L^{2}({\mathbb{R}})$ such
that $H\phi=0$ in the distributional sense;

  * •

zero is an eigenvalue of $H$ if there exists some nonzero $\phi\in
L^{2}({\mathbb{R}})$ such that $H\phi=0$ in the distributional sense;

  * •

zero is a regular point of $H$ if $H$ has neither zero eigenvalue nor zero
resonances.

The case when zero is a regular point of $H$ is also called the generic case
and the case when zero is a resonance or an eigenvalue of $H$ is called the
exceptional case in the literature.

######  Remark 1.2.

It was observed by Goldberg [30] (see also [57, Remark 1.2]) that if
$|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ with some $\mu$ satisfying a weaker
condition than 1.6, then $H$ has no zero eigenvalue. Hence in the following
theorems of this paper, we do not need to consider the zero eigenvalue case
(see also Subsection 1.3 below for more related comments).

Let $\mathbb{B}(X,Y)$ be the space of bounded operators from $X$ to $Y$,
namely $A\in\mathbb{B}(X,Y)$ if

$\|Af\|_{Y}\lesssim\|f\|_{X},\quad f\in X.$

We also set $\mathbb{B}(X)=\mathbb{B}(X,X)$. We now state the main result of
this paper as follows.

######  Theorem 1.3.

Let $H=\Delta^{2}+V(x)$ and $V$ satisfy
$|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ for some $\mu>0$ depending on the
following types:

$\displaystyle\mu>\begin{cases}15&\text{if zero is a regular point of $H$},\\\
21&\text{if zero is a first kind resonance of $H$},\\\ 29&\text{if zero is a
second kind resonance of $H$}.\end{cases}$ ( 1.6)

Assume also $H$ has no embedded eigenvalue in $(0,\infty)$. Let
$W_{\pm},W_{\pm}^{*}$ be the wave and inverse (or dual) wave operators defined
by 1.1. Then the following statements hold:

  * (1)

$W_{\pm},W_{\pm}^{*}\in\mathbb{B}(L^{p}({\mathbb{R}}))$ for all $1<p<\infty$.
Moreover, if $V$ is compactly supported, then
$W_{\pm},W_{\pm}^{*}\in\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$.

  * (2)

$W_{\pm}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$ and
$W_{\pm}^{*}\in\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$.
Moreover, if in addition zero is either a regular point or a first kind
resonance of $H$, then
$W_{\pm}\in\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$
and
$W_{\pm}^{*}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$.

  * (3)

Let $1<p<\infty$, $w_{p}\in A_{p}$ and set $\tau f(x)=f(-x)$. Then

$\displaystyle{\|W_{\pm}f\|}_{L^{p}(w_{p})}+{\|W_{\pm}^{*}f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim{\|f\|}_{L^{p}(w_{p})}+{\|\tau f\|}_{L^{p}(w_{p})}.$

In particular, $W_{\pm},W_{\pm}^{*}\in\mathbb{B}(L^{p}(w_{p}))$ if $w_{p}$ is
even. Moreover, if zero is a regular point of $H$ and the operator
$Q_{1}A_{1}^{0}Q_{1}$ appeared in Lemma 2.2 below is finite rank, then
$W_{\pm},W_{\pm}^{*}\in\mathbb{B}(L^{1}(w_{1}),L^{1,\infty}(w_{1}))$ for any
even $w_{1}\in A_{1}$.

Here $A_{p}$ is the Muckenhoupt class (see Appendix A below for more details
and some examples), $L^{p}(w)$, $L^{1,\infty}(w)$,
$\mathcal{H}^{1}({\mathbb{R}})$ and ${\mathop{\mathrm{BMO}}}({\mathbb{R}})$
are the weighted $L^{p}$, weighted weak $L^{1}$, Hardy and Bounded Mean
Oscillation spaces on ${\mathbb{R}}$, respectively (see Subsection 1.8 below).

######  Remark 1.4.

We here make a few remarks (see Subsection 1.3 for more remarks).

  * (1)

In Theorem 1.3, the presence of zero resonances has no effect on the $p$-range
of the $L^{p}$-boundedness of wave operators $W_{\pm},W_{\pm}^{*}$, and only
require that the potentials $V$ satisfy stronger decay conditions than the
regular case.

  * (2)

We in fact prove the following bounds with an explicit dependence on the
weights:

$\displaystyle{\|W_{\pm}f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}}({\|f\|}_{L^{p}(w_{p})}+{\|\tau
f\|}_{L^{p}(w_{p})}),\quad 1<p<\infty,$ ( 1.7)
$\displaystyle{\|W_{\pm}f\|}_{L^{1,\infty}(w_{1})}$
$\displaystyle\lesssim[w_{1}]_{A_{1}}(1+\log[w]_{A_{1}})({\|f\|}_{L^{1}(w_{1})}+{\|\tau
f\|}_{L^{1}(w_{1})}),$ ( 1.8)

where $[w]_{A_{p}}$ is the $A_{p}$-characteristic constant of $w$ (see
Appendix A) and the implicit constants are independent of $w_{p},w_{1}$.
Moreover, the same bounds also hold for $W_{\pm}^{*}$. The estimates of type
1.7 (without ${\|\tau f\|}_{L^{p}(w_{p})}$) are known as the
$A_{p}$-estimates in the theory of Calderón–Zygmund operators and known to be
sharp (see [39]). We also refer to [48] for the estimates of type 1.8 for
Calderón–Zygmund operators.

  * (3)

For the Schrödinger operator $-\Delta+V(x)$ on ${\mathbb{R}}^{3}$, Beceanu [3]
proved a weighted $L^{p}$-boundedness of the wave operators with a specific
weight ${\langle}x{\rangle}^{a}$ for $|a|<1$ under a suitable assumption on
$V$ depending on $a$. Compared with his result, the interesting point of
Theorem 1.3 (2) is that we can take general even (i.e. radial) weight
$w_{p}\in A_{p}$. Moreover, our assumption on $V$ is independent of the choice
of weights.

In Theorem 1.3, we have obtained the desired $L^{p}$ (or even weighted
$L^{p}$) boundedness of $W_{\pm}$ for non-endpoint cases $1<p<\infty$ and some
weak-boundedness for the limiting cases $p=1,\infty$. Then it is natural to
ask whether $W_{\pm}$ are bounded on $L^{1}({\mathbb{R}})$ and
$L^{\infty}({\mathbb{R}})$ or not. The next theorem answers this question
negatively in the regular case, which shows that Theorem 1.3 is sharp (in
general) in terms of the $p$-range of the $L^{p}$-boundedness.

######  Theorem 1.5.

Suppose that $|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ with $\mu>15$,
$V\not\equiv 0$ and that $H$ has no embedded eigenvalue in $(0,\infty)$. Then
we have the following statements:

  * (1)

Suppose that zero is a regular point of $H$. Then
$W_{\pm},W_{\pm}^{*}\notin\mathbb{B}(L^{1}({\mathbb{R}}))\cup\mathbb{B}(L^{\infty}({\mathbb{R}}))$.

  * (2)

Suppose that zero is a second kind resonance of $H$ and $V$ is compactly
supported. If $D_{*}\neq 0$, then
$W_{\pm}\notin\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$
and
$W_{\pm}^{*}\notin\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$,
where the constant $D_{*}$ is defined in Proposition 6.4.

######  Remark 1.6.

One can also obtain some results on the unboundedness in $L^{1}$ and
$L^{\infty}$ for the resonant cases. We refer to Remark 6.3 in Section 6 for
more details.

Finally, we also obtain the $W^{s,p}$-boundedness of $W_{\pm}$, where
$W^{s,p}=W^{s,p}({\mathbb{R}})$ is the $L^{p}$-Sobolev space of order $s$. For
$N\in{\mathbb{N}}$, we set

$\displaystyle B^{N}({\mathbb{R}})=\\{V\in C^{N}({\mathbb{R}})\ |\ V^{(k)}\in
L^{\infty}({\mathbb{R}})\ \text{for all $k=0,1,...,N$}\\}.$ ( 1.9)

######  Theorem 1.7.

Let $1<p<\infty$ and $H=\Delta^{2}+V(x)$ satisfy the same assumption in
Theorem 1.3. Then $W_{\pm},W_{\pm}^{*}\in\mathbb{B}(W^{s,p}({\mathbb{R}}))$
for all $0\leq s\leq 4$. Moreover, if in addition $V\in B^{4N}({\mathbb{R}})$
with some $N\in{\mathbb{N}}$, then
$W_{\pm},W_{\pm}^{*}\in\mathbb{B}(W^{s,p}({\mathbb{R}}))$ for all $0\leq s\leq
4(N+1)$.

Here we summarize the above results in the following Table 1, from which it is
clear that, for the case when zero is a regular point of $H$, our results give
a complete classification of the validity of the $L^{p}$-boundedness for all
$1\leq p\leq\infty$ and weak-boundedness in the framework of $L^{1,\infty}$,
${\mathcal{H}}^{1}$ and ${\mathop{\mathrm{BMO}}}$ for the limiting cases
$p=1,\infty$.

Boundedness | $W_{\pm}(H,\Delta^{2})$ | $W_{\pm}(H,\Delta^{2})^{*}$
---|---|---
$L^{p}({\mathbb{R}})$, $L^{p}(w_{p})$, $W^{s,p}({\mathbb{R}})$ ($1<p<\infty$) | True | True
$L^{1}({\mathbb{R}})\to L^{1,\infty}({\mathbb{R}})$ | True | True
$L^{1}({\mathbb{R}})$, $L^{\infty}({\mathbb{R}})$ | False (R) | False (R)
${\mathcal{H}}^{1}({\mathbb{R}})\to L^{1}({\mathbb{R}})$ | True | True (R, 1st)
$L^{\infty}({\mathbb{R}})\to{\mathop{\mathrm{BMO}}}({\mathbb{R}})$ | True (R, 1st) | True

(R=regular case, 1st=first kind case)

Table 1. Boundedness of $W_{\pm}(H,\Delta^{2})$ and
$W_{\pm}(H,\Delta^{2})^{*}$

###  1.3. Further remarks on eigenvalues and potentials

Here we make further comments on the above theorems, especially the spectral
assumptions and the decay condition on $V$.

####  1.3.1. Zero resonance and zero eigenvalue

We first give two simple examples of $V$ such that $H$ has a zero resonance.
On one hand, zero is a second kind resonance for the free case
$H=\Delta^{2},V\equiv 0$ since any constant function $\phi_{0}\in
W_{1/2}({\mathbb{R}})$ satisfies $\Delta^{2}\phi_{0}=0$. On the other hand, it
is also easy to construct $V\not\equiv 0$ such that $H$ has a zero resonance.
Indeed, let $\phi_{1}\in C^{\infty}(\mathbb{R})$ be a positive function such
that $\phi_{1}(x)=c|x|+d$ for $|x|>1$ with some constants $c,d\geq 0$
satisfying $(c,d)\neq(0,0)$. Then $H\phi_{1}=0$ if taking

$V(x)=-(\Delta^{2}\phi_{1})/\phi_{1},\ \ x\in\mathbb{R}.$

Note that $V\in C_{0}^{\infty}(\mathbb{R})$ and $\phi_{1}\in
W_{3/2}({\mathbb{R}})\setminus W_{1/2}({\mathbb{R}})$ if $c>0$ and
$\phi_{1}\in W_{1/2}({\mathbb{R}})$ if $c=0$. These examples indicate that
zero resonances may occur even for compactly supported potentials.

We next discuss on the zero eigenvalue of $H$. It is again easy to construct
an example of $H$ having zero eigenvalue if $V$ decays sufficiently slowly. In
fact, let $\phi=(1+|x|^{2})^{-s/2}$ and $V(x)=-(\Delta^{2}\phi)/\phi$. Then
$\phi\in H^{4}(\mathbb{R})$ for any $s>1$ and $(\Delta^{2}+V)\phi=0$, which
means $|V(x)|\lesssim{\langle}x{\rangle}^{-4}$ and zero is an eigenvalue of
$H$. However, as already mentioned in Remark 1.2, if
$|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ with some $\mu$ satisfying 1.6,
then zero cannot be an eigenvalue of $H$ in dimension one. We believe such a
decay condition on $V$ may not be sharp, expecting that the decay rate $\mu>4$
is optimal to ensure the absence of zero eigenvalue for $\Delta^{2}+V$ on
${\mathbb{R}}$.

Based on these remarks, and in view of the the fast decay conditions of
potential $V$ in our theorems, we remark that zero eigenvalue can be actually
excluded, while zero resonances must be taken into account. However, we again
emphasize that the presence of zero resonances has no effect on the validity
of $L^{p}$-boundedness of $W_{\pm},W_{\pm}^{*}$ at least for $1<p<\infty$.

####  1.3.2. Embedded positive eigenvalue

In contrast with the zero energy case, the absence of positive eigenvalues of
$H$ are more subtle than that of zero resonance or zero eigenvalue.

It is well-known as Kato’s theorem [44] that if $V$ is bounded and
$V=o(|x|^{-1})$ as $|x|\rightarrow\infty$ then the Schrödinger operator
$-\Delta+V$ has no positive eigenvalues (also see [29, 40, 45] for more
related results and references). By contraries, such a criterion cannot hold
for the fourth-order Schrödinger operator $H=\Delta^{2}+V$, so the assumption
on the absence of positive eigenvalues seems to be indispensable. Indeed, it
is easy to construct a Schwartz function $V(x)$ so that $H$ on ${\mathbb{R}}$
has an eigenvalue $1$.111In fact,
$V(x)=20/\cosh^{2}(x)-24/\cosh^{4}(x)\in{\mathcal{S}}({\mathbb{R}})$ satisfies
$\frac{d^{4}\psi_{0}}{dx^{4}}+V(x)\psi_{0}=\psi_{0}$ where
$\psi_{0}=1/\cosh(x)=2/(e^{x}+e^{-x})\in L^{2}(\mathbb{R})$. Moreover, in any
dimensions $n\geq 1$, one can also easily construct $V\in
C_{0}^{\infty}({\mathbb{R}}^{n})$ so that $H$ has positive eigenvalues (see
Feng et al.[25, Section 7.1]). These results clearly indicate that the absence
of positive eigenvalues for the fourth-order Schrödinger operator would be
more subtle and unstable than the second order cases under the potential
perturbation $V$.

We however stress that if $V\in C^{1}({\mathbb{R}})\cap
L^{\infty}({\mathbb{R}})$ is repulsive, i.e., $xV^{\prime}(x)\leq 0$, then $H$
has no eigenvalues (see [25, Theorem 1.11]). Note that such a criterion also
works for the general higher-order elliptic operator $P(D)+V$ in any
dimensions $n\geq 1$. Besides, we also notice that for a general selfadjoint
operator $\mathcal{H}$ on $L^{2}(\mathbb{R}^{n})$, even if $\mathcal{H}$ has a
simple embedded eigenvalue, Costin–Soffer in [10] have proved that
$\mathcal{H}+{\varepsilon}W$ can kick off the eigenvalue located in a small
interval under certain small perturbation of the potential ${\varepsilon}W$.

####  1.3.3. Decay condition on the potential

The rather fast decay condition 1.6 on the potential $V$ in our theorems is
due to the use of low energy expansions of the resolvent
$(H-\lambda^{4}-i0)^{-1}$ obtained by Soffer–Wu–Yao [57] (see Lemma 2.2
below). In fact, in the regular case for instance, the proof of Theorem 1.3
woks well if $|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ for $\mu>9$ under the
assumption that the expansion (i) holds. Although it is an interesting
problem to improve the assumption of Lemma 2.2, we do not pursue it for the
sake of simplicity.

Note that in the case of the Schrödinger operator $-\Delta+V(x)$, the Jost
functions are known to be very useful tools for studying asymptotic behaviors
of the resolvent (see [14]) and have been widely used in the proof of
$L^{p}$-boundedness of wave operators (see [2, 13, 60]). However, it is not
clear whether the same method can be also applied to the fourth-order case.
Indeed, in view of the explicit formula of the free resolvents
$(\Delta^{2}-\lambda^{4}\mp i0)^{-1}$ (see 2.1 below), we must construct
four Jost functions $f_{\pm}(\lambda,x),g_{\pm}(\lambda,x)$ such that

$f_{\pm}(\lambda,x)\sim e^{\pm i\lambda x},\quad g_{\pm}(\lambda,x)\sim
e^{\mp\lambda x},\quad x\to\pm\infty,$

Hence the situation is very different from the second-order case since
$g_{\pm}(\lambda,x)$ can grow exponentially fast if $\lambda<0$, while the
Jost functions are uniformly bounded in the second-order case. Note that one
needs several global estimates of Jost functions or their Fourier transforms
with respect to $\lambda,x\in{\mathbb{R}}$ in the proof of $L^{p}$-bounds for
the wave operators (see e.g. [60, Section 2]). For readers interested in the
construction of $f_{\pm},g_{\pm}$, we refer to [37] where the potential $V$
has been assumed to be compactly supported.

###  1.4. Two types of applications

By virtue of 1.4, or more generally 1.5, our main estimates may have a lot
of potential applications. We however do not pursue to list them as many as
possible, but focus on a few primal applications which will be important for
further applications to nonlinear equations. More precisely, we prove the
following two types of results (see Section 8 for the precise statements):

  * •

$L^{p}$-$L^{q}$ decay estimates for the propagator
$e^{-itH}P_{\mathop{\mathrm{ac}}\nolimits}(H)$:

${\|e^{-itH}P_{\mathop{\mathrm{ac}}\nolimits}(H)\phi\|}_{L^{q}({\mathbb{R}})}\lesssim|t|^{-\frac{1}{4}(\frac{1}{p}-\frac{1}{q})}{\|\phi\|}_{L^{p}({\mathbb{R}})},\quad
t\neq 0,$

for $(1/p,1/q)$ belonging to a region of $\mathbb{R}^{2}$ (see Figure 1 in
Section 8).

  * •

$L^{p}$-boundedness of the spectral multiplier $f(H)$:

${\|f(H)\phi\|}_{L^{p}({\mathbb{R}})}\lesssim{\|\phi\|}_{L^{p}({\mathbb{R}})},\quad
1<p<\infty,$

where $f\in L^{\infty}({\mathbb{R}})$ satisfies the standard Hörmander
condition (see 8.6).

These $L^{p}$-$L^{q}$ decay estimates for
$e^{-itH}P_{\mathop{\mathrm{ac}}\nolimits}(H)$ generalize the
$L^{1}$-$L^{\infty}$ decay estimate obtained recently by Soffer–Wu–Yao [57].
On the other hand, the new interesting point of this spectral multiplier
theorem for $f(H)$ is that our operator $H=\Delta^{2}+V(x)$ may have negative
eigenvalues as well as zero resonances, so $e^{-tH}$ possibly has no sharp
(generalized) Gaussian kernel bounds. Hence a standard criterion based on
Gaussian kernel bounds (see e.g. Sikora–Yan–Yao [55]) cannot be applied in the
present case.

Furthermore, we notice that in the case of the Schrödinger operator
$-\Delta+V(x)$, the $L^{p}$-boundedness of wave operators, as well as the
$L^{p}$-$L^{p^{\prime}}$ decay estimates for $e^{it(\Delta-V)}$ and the
spectral multiplier theorem for $f(-\Delta+V)$ are very important tools for
studying associated dispersive equations such as the nonlinear Schrödinger
equations with potentials (see e.g. [11, 18, 53, 54, 56] and reference
therein). Hence, we believe that Theorems 1.3 and 1.7, as well as these two
results on $e^{-itH}$ and $f(H)$, will be fundamental tools for studying
several nonlinear dispersive equations associated with $H$, especially for the
following fourth-order nonlinear Schrödinger equation with a potential:

$i\partial_{t}u-\partial_{x}^{4}u-V(x)u=\mu|u|^{p-1}u,\quad
t,x\in{\mathbb{R}}.$

###  1.5. More related backgrounds

In this subsection, we record some known results on the $L^{p}$-boundedness of
the wave operators, comparing them with our theorems. We also discuss some
related results, as well as some backgrounds on the higher-order elliptic
operators.

For the Schrödinger operator $-\Delta+V(x)$ on ${\mathbb{R}}^{n}$ in any
dimensions $n\geq 1$, there exists a great number of works are devoted to
establish the $L^{p}$-boundedness of the wave operators in last almost thirty
years. For instance, Yajima in the seminar work [62] proved the
$L^{p}$-boundedness of wave operators for $n\geq 3$ in the regular case.
Subsequently, the case $n=1$ were studied by Weder [60] and Artbazar–Yajima
[2] independently and the case $n=2$ by Yajima [63]. Since then later, many
further progresses and applications related to the $L^{p}$-boundedness of wave
operators have been made for all the regular, zero resonance and zero
eigenvalue cases under various conditions on the potential $V$ (see [3, 4, 5,
8, 9, 12, 13, 15, 16, 20, 28, 31, 32, 41, 42, 43, 56, 61, 64, 65, 66, 67, 68]
and references therein). Certainly, these works would play indispensable roles
in the studies of higher-order elliptic operators.

The weighted boundedness considerably less is known compared with the
unweighted one. As already mentioned in Remark 1.4 (3), Beceanu [3] obtained
some weighted $L^{p}$-boundedness with polynomial weights
${\langle}x{\rangle}^{a}$. Note that Beceanu–Schlag [4, 5] proved (again for
the Schrödinger operator) $W_{\pm}\in\mathbb{B}(X)$ if $X$ is any Banach space
of measurable functions on ${\mathbb{R}}^{3}$ such that the norm
${\|\cdot\|}_{X}$ is invariant under reflections and translations and that
${\|\chi_{H}f\|}_{X}\leq A{\|f\|}_{X}$ for any half space
$H\subset{\mathbb{R}}^{3}$ with some uniform constant $A$. This result clearly
implies the $L^{p}$-boundedness (even the $W^{s,p}$-boundedness) of $W_{\pm}$,
but not the weighted $L^{p}$-boundedness since weighted $L^{p}$-norms are not
invariant under translations.

Now we shall explain known results for the Schrödinger operator on
${\mathbb{R}}$ more precisely. Weder [60] proved
$W_{\pm}\in\mathbb{B}(L^{p}({\mathbb{R}}))$ for $1<p<\infty$ if
${\langle}x{\rangle}^{\gamma}V\in L^{1}({\mathbb{R}})$ with some $\gamma>3/2$
in the regular case and $\gamma>5/2$ in the zero resonant case.
Artbazar–Yajima [2] also proved independently a similar result under a
slightly stronger decay condition on $V$. Later, the assumption on $V$ has
been weakened to ${\langle}x{\rangle}V\in L^{1}({\mathbb{R}})$ in the regular
case and ${\langle}x{\rangle}^{2}V\in L^{1}({\mathbb{R}})$ in the zero
resonant case by D’Ancona–Fanelli [13], and finally to
${\langle}x{\rangle}V\in L^{1}({\mathbb{R}})$ in the zero resonant case by
Weder [61]. It was also shown by [60] that
$W_{\pm}\in\mathbb{B}(W^{k,p}({\mathbb{R}}))\cap\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))\cap\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$
for general cases and that
$W_{\pm}\in\mathbb{B}(L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}))$
if zero is a resonance and the scattering matrix at $\lambda=0$ is the
identity matrix. It was also mentioned in [60] that $W_{\pm}$ are neither
bounded on $L^{1}({\mathbb{R}})$ nor on $L^{\infty}({\mathbb{R}})$ in general.
The case with a delta potential $V=a\delta$ was studied by
Duchêne–Marzuola–Weinstein [16] and Weder [61]. Weder [61] also studied the
case with matrix Schrödinger operators on the line or the half line. Note
that, in all these papers [2, 13, 16, 60, 61], the proofs heavily rely on the
Jost functions and their properties studied by Deift–Trubowitz [14].

Comparatively speaking, it would be desirable to generalize the
$L^{p}$-boundedness of the wave operators of Schrödinger operators to the
fourth-order Schrödinger operator $H=\Delta^{2}+V(x)$. In fact, until recently
there exist several interesting works to study the resolvent $(H-z)^{-1}$ and
the decay estimates for $e^{-itH}$ (see e.g. [22, 25]). Based on these recent
works, Goldberg–Green [33] first proved the $L^{p}$-boundedness of the wave
operators for the fourth-order Schrödinger operator $\Delta^{2}+V(x)$ on
${\mathbb{R}}^{3}$ under the assumption that zero is a regular point.
Furthermore, this result was extended by Erdoğan–Green [21] to the higher-
order Schrödinger operators $(-\Delta)^{m}+V$ on ${\mathbb{R}}^{n}$ for
$n>2m\geq 4$ again in the regular case.

Compared with those existing literature, the interest of our results are
threefold:

  * (1)

At least in the regular case, we give a complete classification (with detailed
proofs) of the validity of the $L^{p}$-boundedness for all $1\leq p\leq\infty$
and weak-boundedness in the framework of $L^{1,\infty}$, ${\mathcal{H}}^{1}$
and ${\mathop{\mathrm{BMO}}}$ for the limiting cases $p=1,\infty$;

  * (2)

There are no previous results on the $L^{p}$-boundedness for the higher-order
operators (including the fourth-order Schrödinger operator) if zero is not a
regular point;

  * (3)

The weighted $L^{p}$-boundedness with general even $A_{p}$-weight $w_{p}$, as
well as the explicit bounds 1.7 and 1.8, seems to be totally new (see also
Remark 1.4 (3)).

Finally we also mention that there is a huge literature on the study of
higher-order elliptic operators $P(D)+V(x)$ in many topics in mathematics and
mathematical physics. In addition to the aforementioned works [33, 21], we
refer the readers to e.g. [1, 38, 46] for the spectral and scattering theory,
[17, 55] for Harmonic analysis and [19, 22, 25, 26, 27, 36, 49, 50, 57] for
various dispersive properties such as time decay, local energy decay,
Strichartz estimates for $e^{-itH}$, and the asymptotic expansion and uniform
resolvent estimates for $(H-z)^{-1}$.

###  1.6. The outline of the proof

Here we briefly explain the ideas of the proof of the above theorems. For
simplicity, we consider the case when zero is a regular point of $H$ only.

The starting point is the following stationary formula:

$W_{-}=\mathop{\mathrm{Id}}\nolimits-\frac{2}{\pi
i}\int_{0}^{\infty}\lambda^{3}R_{V}^{+}(\lambda^{4})V\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda,$

where $R_{0}^{\pm}(\lambda^{4})=(\Delta^{2}-\lambda^{4}\mp i0)^{-1}$ and
$R_{V}^{\pm}(\lambda^{4})=(H-\lambda^{4}\mp i0)^{-1}$ are the boundary values
of the free and perturbed resolvents. The integral kernels of
$R_{0}^{\pm}(\lambda^{4})$ are explicitly given by

$\displaystyle
R_{0}^{\pm}(\lambda^{4},x,y)=\frac{F_{\pm}(\lambda|x-y|)}{4\lambda^{3}}=\frac{F_{\pm}(\lambda|x|)}{4\lambda^{3}}-\frac{y}{4\lambda^{2}}\int_{0}^{1}\mathop{\mathrm{sgn}}\nolimits(x-\theta
y)F_{\pm}^{\prime}(\lambda|x-\theta y|)d\theta,$ ( 1.10)

where $F_{\pm}(s):=\pm ie^{\pm is}-e^{-s}$ and we have used the Taylor
expansion near $y=0$ in the second line. In particular,
$R_{0}^{\pm}(\lambda^{4})=O(\lambda^{-3})$ at the level of the order of
$\lambda$.

Decompose $W_{-}-\mathop{\mathrm{Id}}\nolimits$ into the low energy
$\\{0\leq\lambda\ll 1\\}$ and the high energy $\\{\lambda\gtrsim 1\\}$ parts.
The high energy part is easier to treat than the low energy part since the
free resolvent does not have singularity for $\lambda\geq 1$, so we here
consider the following low energy part only:

$\displaystyle
W_{-}^{L}:=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)R_{V}^{+}(\lambda^{4})V\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda,$
( 1.11)

where $\chi\in C_{0}^{\infty}({\mathbb{R}})$ such that $\chi\equiv 1$ near
$\lambda=0$. Setting $v(x)=\sqrt{|V(x)|}$,
$U(x)=\mathop{\mathrm{sgn}}\nolimits V(x)$ and
$M(\lambda)=U+vR_{0}^{+}(\lambda^{4})v$, one has the standard symmetric second
resolvent equation:

$R_{V}^{+}(\lambda^{4})V=R_{0}^{+}(\lambda^{4})vM^{-1}(\lambda)v.$

Then one of key tools in our argument is the asymptotic expansion of
$M^{-1}(\lambda)$ as $\lambda\to+0$ obtained recently by [57] which, in the
regular case, is of the form

$\displaystyle M^{-1}(\lambda)$ $\displaystyle=Q_{2}A_{0}^{0}Q_{2}+\lambda
Q_{1}A_{1}^{0}Q_{1}+\lambda^{2}\left(Q_{1}A_{21}^{0}Q_{1}+Q_{2}A_{22}^{0}+A_{23}^{0}Q_{2}\right)$
$\displaystyle\quad+\lambda^{3}\left(Q_{1}A_{31}^{0}+A_{32}^{0}Q_{1}\right)+\lambda^{3}\widetilde{P}+\Gamma^{0}_{4}(\lambda),$
( 1.12)

where $A_{k}^{0},A_{kj}^{0},\widetilde{P},Q_{\alpha}\in\mathbb{B}(L^{2})$,
${\|\Gamma^{0}_{4}(\lambda)\|}_{L^{2}\to L^{2}}=O(\lambda^{4})$ and
$Q_{\alpha}$ satisfies

$\displaystyle Q_{\alpha}(x^{k}v)\equiv
0,\quad{\langle}x^{k}v,Q_{\alpha}f{\rangle}=0,$ ( 1.13)

for any $f\in L^{2}$ and any integer $0\leq k\leq\alpha-1$. The interest of
these properties 1.13 is that, combined with the Taylor expansion formula
1.10, one has

$\displaystyle
Q_{\alpha}vR_{0}^{\pm}(\lambda^{4})=O(\lambda^{-3+\alpha}),\quad
R_{0}^{\pm}(\lambda^{4})vQ_{\alpha}=O(\lambda^{-3+\alpha}),$ ( 1.14)

which are less singular in $\lambda\in(0,1]$ compared with the free resolvent
$R_{0}^{\pm}(\lambda^{4})=O(\lambda^{-3})$. $\bullet$ On the
$L^{p}({\mathbb{R}})$-boundedness. Substituting 1.12 into 1.11 one can
find that $W_{-}^{L}$ is a sum of nine integral operators with integral
kernels of the form

$\displaystyle\int_{0}^{\infty}\lambda^{\ell-\alpha-\beta}\chi(\lambda)\left(R_{0}^{+}(\lambda^{4})vQ_{\alpha}BQ_{\beta}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\right)(x,y)d\lambda,$
( 1.15)

where $B\in\mathbb{B}(L^{2})$ varies from line to line, $\ell=6$ or $7$ and we
set $Q_{0}=\mathop{\mathrm{Id}}\nolimits$. Note that the integrand is of order
$\lambda^{\ell-6}$ by 1.14. Then such nine integral operators are classified
into the following two classes with respect to the order of $\lambda$ of the
integrands of their integral kernels:

  * (I)

$O(\lambda)$:
$Q_{\alpha}BQ_{\beta}=Q_{2}A_{0}^{0}Q_{2},Q_{1}A_{21}^{0}Q_{1},Q_{2}A_{22}^{0},A_{23}^{0}Q_{2},Q_{1}A_{31}^{0},A_{32}^{0}Q_{1}$
and $\lambda^{-4}\Gamma_{4}^{0}(\lambda)$;

  * (II)

$O(1)$: $Q_{\alpha}BQ_{\beta}=Q_{1}A_{1}^{0}Q_{1}$ and $\widetilde{P}$.

The operators in the class (I) can be shown to be bounded on
$L^{p}({\mathbb{R}})$ for any $1\leq p\leq\infty$. We shall explain this for
$Q_{\alpha}BQ_{\beta}=Q_{1}A_{21}^{0}Q_{1}$ as a model case. In such a case,
by using 1.10, 1.14 and the identity

$F_{+}^{\prime}(\lambda|x|)[F_{+}^{\prime}-F_{-}^{\prime}](\lambda|y|)=e^{i\lambda(|x|+|y|)}-e^{i\lambda(|x|-|y|)}+e^{-\lambda(|x|+i|y|)}-e^{-\lambda(|x|-i|y|)},$

we can rewrite 1.15 as a linear combination of following four functions:

$\displaystyle K^{\pm}_{a_{1}}(x,y)$
$\displaystyle=\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\lambda\chi(\lambda)a_{1}^{\pm}(\lambda,x,y)d\lambda,$
$\displaystyle K^{\pm}_{a_{2}}(x,y)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\lambda\chi(\lambda)a_{2}^{\pm}(\lambda,x,y)d\lambda,$

where $a^{\pm}_{j}$ satisfy

$|\partial_{\lambda}^{\ell}a_{1}^{\pm}(\lambda,x,y)|+e^{-\lambda|x|}|\partial_{\lambda}^{\ell}a_{2}^{\pm}(\lambda,x,y)|\lesssim{\|{\langle}x{\rangle}^{4+2\ell}V\|}_{L^{1}},\quad
x,y\in{\mathbb{R}},\ \lambda\geq 0,\ \ell=0,1,2.$

Then we apply integration by parts twice to $K_{a_{j}}^{\pm}$, obtaining

$|K^{\pm}_{a_{j}}(x,y)|\lesssim{\langle}|x|\pm|y|{\rangle}^{-2},\quad
x,y\in{\mathbb{R}},$

where note that $K^{\pm}_{a_{j}}\in L^{\infty}({\mathbb{R}}^{2})$ since
$\chi\in C_{0}^{\infty}({\mathbb{R}})$ and the term
$O({\langle}|x|\pm|y|{\rangle}^{-1})$ does not appear thanks to the fact
$\lambda\chi(\lambda)|_{\lambda=0}=0$. Now the
$L^{p}({\mathbb{R}})$-boundedness for any $1\leq p\leq\infty$ follows from
standard Schur’s lemma since ${\langle}|x|\pm|y|{\rangle}^{-2}\in
L^{\infty}_{y}L^{1}_{x}\cap L^{\infty}_{x}L^{1}_{y}$.

In the above argument, the crucial point is that we have an additional
$\lambda$ in the integrands. For the operators in the class (II) which do not
have such a factor $\lambda$, we need more precise estimates for the integral
kernels to employ the theory of Calderón–Zygmund operators. As a model case,
we shall consider the integral 1.15 with
$Q_{\alpha}BQ_{\beta}=\widetilde{P}$ (in which case $\ell=6$). In such a case,
a similar argument as above yields that 1.15 can be rewritten in the form

$-\sum_{\pm}\int_{0}^{\infty}\left(e^{i\lambda(|x|\pm|y|)}\chi(\lambda)c_{1}^{\pm}(\lambda,x,y)+ie^{-\lambda(|x|\pm
i|y|)}\chi(\lambda)c_{2}^{\pm}(\lambda,x,y)\right)d\lambda$

with some $c_{j}^{\pm}$ satisfying the same estimates as for $a_{j}^{\pm}$.
Applying integration by parts twice, we find that this integral is a sum of
the leading term $\frac{-1+i}{8}g_{1}^{+}(x,y)$, where

$g_{1}^{+}(x,y)=\frac{\psi_{+}}{|x|+|y|}+\frac{\psi_{-}}{|x|-|y|}+\frac{\psi_{-}}{|x|+i|y|}+\frac{\psi_{-}}{|x|-i|y|},$

and the error term $O({\langle}|x|-|y|{\rangle}^{-2})$ which can be dealt as
above, where $\psi_{\pm}=\psi(||x|\pm|y||^{2})$ are smooth cut-off functions
supported in $\\{(x,y)\ |\ ||x|\pm|y||\geq 1\\}$. Although the integral
operator $T_{g_{1}^{+}}$ with the kernel $g_{1}^{+}$ itself is not a
Calderón–Zygmund operator, using the identity

$g_{1}^{+}(x,y)=\left(\chi_{+}(x)+\chi_{-}(x)\right)g_{1}^{+}(x,y)\left(\chi_{+}(y)+\chi_{-}(y)\right)$

with $\chi_{\pm}$ being the indicator function of ${\mathbb{R}}_{\pm}$, one
can write

$\displaystyle
T_{g_{1}^{+}}=\left((\chi_{+}-\chi_{-})T_{\widetilde{k}_{1}}+\chi_{+}T_{\widetilde{k}_{2}^{+}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{2}^{+}}\chi_{-}+\chi_{+}T_{\widetilde{k}_{2}^{-}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{2}^{-}}\chi_{-}\right)(1+\tau),$
( 1.16)

where $\tau:f(x)\mapsto f(-x)$,
$\widetilde{k}_{1}(x,y)=\psi(|x-y|^{2})(x-y)^{-1}$ and
$\widetilde{k}_{2}(x,y)=\psi(|x-y|^{2})(x\pm iy)^{-1}$ so that
$T_{\widetilde{k}_{1}}$ and $T_{\widetilde{k}_{2}^{\pm}}$ can be shown to be
Calderón–Zygmund operators. The abstract theorem for Calderón–Zygmund
operators then shows
$T_{g_{1}^{+}}\in\mathbb{B}(L^{p}({\mathbb{R}}))\cap\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$.
$\bullet$ On the weighted $L^{p}$-boundedness. We shall consider
$T_{g^{+}_{1}}$ as a model case. The theory of Calderón–Zygmund operators
shows
$T_{\widetilde{k}_{1}},T_{\widetilde{k}_{2}^{\pm}}\in\mathbb{B}(L^{p}(w_{p}))$
for any $w_{p}\in A_{p}$. Moreover, recent deep results by [39] for
$1<p<\infty$ imply

$\displaystyle{\|T_{\widetilde{k}_{1}}\|}_{L^{p}(w_{p})\to
L^{p}(w_{p})}+{\|T_{\widetilde{k}_{2}^{\pm}}\|}_{L^{p}(w_{p})\to
L^{p}(w_{p})}\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}},\quad 1<p<\infty.$

If $w_{p}$ and $w_{1}$ are even, then these bounds on
$T_{\widetilde{k}_{1}},T_{\widetilde{k}_{2}^{\pm}}$ and 1.16 yield desired
weighted boundedness of $T_{g^{+}_{1}}$ with explicit operator norm bounds in
terms of $[w_{p}]_{A_{p}}$. $\bullet$ On the ${\mathcal{H}}^{1}$-$L^{1}$ and
$L^{\infty}$-${\mathop{\mathrm{BMO}}}$ boundedness. Let us consider again the
operator $T_{g^{+}_{1}}$. Since ${\mathcal{H}}^{1}$ is not invariant under the
map $f\mapsto\chi_{\pm}f$ (recall that any $f\in{\mathcal{H}}^{1}$ satisfies
$\int fdx=0$), the formula 1.16 is not enough to prove
$T_{g^{+}_{1}}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$,
although
$T_{\widetilde{k}_{1}},T_{\widetilde{k}_{2}^{\pm}}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$
by the abstract theory for Calderón–Zygmund operators. Instead, we prove
$T_{g^{+}_{1}},T_{g^{+}_{1}}^{*}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$
directly by following the classical proof of the ${\mathcal{H}}^{1}$-$L^{1}$
boundedness for Calderón–Zygmund operators based on the atomic decomposition
of ${\mathcal{H}}^{1}$. By the duality,
$({\mathcal{H}}^{1})^{*}={\mathop{\mathrm{BMO}}}$, one also has
$T_{g^{+}_{1}}\in\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$.
$\bullet$ Counterexamples of $L^{1}$ and $L^{\infty}$ boundedness. As seen
above, all the operators in the class (I) are bounded on $L^{p}({\mathbb{R}})$
for all $1\leq p\leq\infty$. Let $T_{K^{0}_{1}}$ (resp. $T_{K^{0}_{33}}$) be
the integral operator in the class (II) associated with $Q_{1}A_{1}^{0}Q_{1}$
(resp. $\widetilde{P}$). Both of them in fact can be shown to be not bounded
on $L^{1}({\mathbb{R}})$ nor $L^{\infty}({\mathbb{R}})$. Although this is not
sufficient to disprove such boundedness properties of $W_{-}$, if we take a
test function $f_{R}=\chi_{[-R,R]}$ then one can prove that
$\sup_{R>0}{\|T_{K^{0}_{1}}f_{R}\|}_{L^{\infty}}<\infty$ and
$T_{K^{0}_{1}}f_{1}\in L^{1}({\mathbb{R}})$, but
$|(T_{K^{0}_{33}}f_{R})(R+2)|\to\infty$ as $R\to\infty$ and
$T_{K^{0}_{33}}f_{1}\notin L^{1}({\mathbb{R}})$. This shows
$W_{-}\notin\mathbb{B}(L^{1}({\mathbb{R}}))\cup\mathbb{B}(L^{\infty}({\mathbb{R}}))$.

$\bullet$ On the $W^{s,p}$-boundedness. Once the
$L^{p}({\mathbb{R}})$-boundedness of $W_{\pm}$ is obtained, its
$W^{s,p}$-boundedness easily follows from the intertwining identity
$(H+E)^{s/4}W_{\pm}=W_{\pm}(\Delta^{2}+E)^{s/4}$ and the inequality
${\|(\Delta^{2}+E)^{s/4}f\|}_{L^{p}({\mathbb{R}})}\lesssim{\|(H+E)^{s/4}f\|}_{L^{p}}$
for sufficiently large $E>0$, which can be shown by a standard method (see
e.g. [7]) based on the generalized Gaussian bound 7.2 for the semi-group
$e^{-t(H+E)}$ proved by [17].

###  1.7. Organizations of the paper

The rest of the paper is devoted to the proof of Theorems 1.3, 1.5 and 1.7 and
their applications, and is organized as follows.

In Section 2, we prepare several preliminary materials, which include the
stationary formula of wave operators (Subsection 2.1), the asymptotic
expansion at low energy of the resolvent and several useful formulas for the
free resolvent (Subsection 2.2).

In Section 3 we prepare a few criterions to obtain several boundedness
properties of integral operators appeared in the stationary formula of the
wave operator $W_{-}$.

The proof of Theorem 1.3 for the low energy part of $W_{-}$ is given by
Section 4, while the proof for high energy part is given by Section 5.

Section 6 is devoted to the proof of Theorem 1.5. Theorem 1.7 is proved in
Section 7. Section 8 is concerned with the applications explained in
Subsection 1.4.

Finally, we give a short review of Calderón–Zygmund operators in Appendix A.

###  1.8. Notations

Throughout the paper we use the following notations:

  * •

$A\lesssim B$ (resp. $A\gtrsim B$) means $A\leq CB$ (resp. $A\geq CB$) with
some constant $C>0$.

  * •

$L^{p}=L^{p}({\mathbb{R}}),L^{1,\infty}=L^{1,\infty}({\mathbb{R}})$ denote the
Lebesgue and weak $L^{1}$ spaces, respectively.

  * •

${\langle}f,g{\rangle}=\int f\overline{g}$ denotes the inner product in
$L^{2}$.

  * •

For $w\in L^{1}_{\mathop{\mathrm{loc}}\nolimits}({\mathbb{R}})$ positive
almost everywhere and $1\leq p<\infty$, $L^{p}(w)=L^{p}({\mathbb{R}},wdx)$
denotes the weighted $L^{p}$-space with the norm

${\|f\|}_{L^{p}(w)}=\left(\int|f(x)|^{p}w(x)dx\right)^{1/p}.$

$L^{1,\infty}(w)$ denotes the weighted weak $L^{1}$ space with the quasi-norm

${\|f\|}_{L^{1,\infty}(w)}=\sup_{\lambda>0}\lambda w(\\{x\ |\
|f(x)|>\lambda\\}).$

  * •

$\mathop{\mathrm{BMO}}=\mathop{\mathrm{BMO}}({\mathbb{R}})$ is the Bounded
Mean Oscillation space: $f\in\mathop{\mathrm{BMO}}$ if $f\in
L^{1}_{\mathop{\mathrm{loc}}\nolimits}({\mathbb{R}})$ and

${\|f\|}_{{\mathop{\mathrm{BMO}}}}:=\sup_{I}\frac{1}{|I|}\int_{I}|f-f_{I}|dx<\infty,$

where the supremum takes over all bounded intervals and
$f_{I}=\frac{1}{|I|}\int_{I}fdx$. Note that
$L^{\infty}\subset{\mathop{\mathrm{BMO}}}$. For instance,
$\log(a|x|+b)\in{\mathop{\mathrm{BMO}}}\setminus L^{\infty}$ for any positive
$a,b$.

  * •

${\mathcal{H}}^{1}={\mathcal{H}}^{1}({\mathbb{R}})$ is the Hardy space:
$f\in{\mathcal{H}}^{1}$ if $f$ is a tempered distribution and

${\|f\|}_{{\mathcal{H}}^{1}}=\int_{\mathbb{R}}\sup_{t>0}|(f*\varphi_{t})(x)|dx<\infty$

with some Schwartz function $\varphi$ satisfying
$\int_{\mathbb{R}}\varphi(x)dx=1$ and $\varphi_{t}(x)=t^{-1}\varphi(x/t)$. It
is known that $({\mathcal{H}}^{1})^{*}={\mathop{\mathrm{BMO}}}$ and
$|{\langle}f,g{\rangle}|\lesssim{\|f\|}_{{\mathcal{H}}^{1}}{\|g\|}_{{\mathop{\mathrm{BMO}}}}$
(see [24]).

  * •

$T_{K}$ denotes the integral operator with the kernel $K(x,y)$, namely

$T_{K}f(x)=\int_{\mathbb{R}}K(x,y)f(y)dy.$

  * •

Let $\\{\varphi_{N}\\}_{N\in{\mathbb{Z}}}$ be a homogeneous dyadic partition
of unity on $(0,\infty)$: $\varphi_{0}\in C_{0}^{\infty}({\mathbb{R}}_{+})$,
$0\leq\varphi\leq 1$,
$\mathop{\mathrm{supp}}\nolimits\varphi\subset[\frac{1}{4},1]$,
$\varphi_{N}(\lambda)=\varphi_{0}(2^{-N}\lambda)$,
$\mathop{\mathrm{supp}}\nolimits\varphi_{N}\subset[2^{N-2},2^{N}]$ and

$\sum_{N\in{\mathbb{Z}}}\varphi_{N}(\lambda)=1,\quad\lambda>0.$

##  2\. Preliminaries

###  2.1. Stationary representation of wave operators

First of all, we observe that it suffices to deal with $W_{-}$ only since
1.1 implies $W_{+}f=\overline{W_{-}\overline{f}}$.

The starting point is the well-known stationary formula 2.2 of $W_{-}$. To
state the formula, we need to introduce some notations. Let

$\displaystyle R_{0}(z)=(\Delta^{2}-z)^{-1},\quad R_{V}(z)=(H-z)^{-1},\quad
z\in{\mathbb{C}}\setminus[0,\infty),$

be the resolvents of $\Delta^{2}$ and $H=\Delta^{2}+V(x)$, respectively. We
denote by $R^{\pm}_{0}(\lambda),R^{\pm}_{V}(\lambda)$ their boundary values
(limiting resolvents) on $(0,\infty)$, namely

$R^{\pm}_{0}(\lambda)=\lim_{{\varepsilon}\searrow 0}R_{0}(\lambda\pm
i{\varepsilon}),\quad R_{V}^{\pm}(\lambda)=\lim_{{\varepsilon}\searrow
0}R_{V}(\lambda\pm i{\varepsilon}),\quad\lambda>0.$

The existence of $R^{\pm}_{0}(\lambda)$ as bounded operators from $L^{2}_{s}$
to $L^{2}_{-s}$ with $s>1/2$ follows from the limiting absorption principle
for the resolvent $(-\partial_{x}^{2}-z)^{-1}$ of the free Schrödinger
operator $-\partial_{x}^{2}$ (see e.g. [1]) and the formula

$R_{0}(z)=\frac{1}{2\sqrt{z}}\left[(-\partial_{x}^{2}-\sqrt{z})^{-1}-(-\partial_{x}^{2}+\sqrt{z})^{-1}\right],\quad
z\in{\mathbb{C}}\setminus[0,\infty),$

which is obtained by the identity
$\partial_{x}^{4}-z=(-\partial_{x}^{2}-\sqrt{z})(-\partial_{x}^{2}+\sqrt{z})$
and the first resolvent equation. This formula also gives the explicit formula
of the kernel of $R_{0}^{\pm}(\lambda^{4})$:

$\displaystyle R_{0}^{\pm}(\lambda^{4},x,y)=\frac{1}{4\lambda^{3}}\Big{(}\pm
ie^{\pm
i\lambda|x-y|}-e^{-\lambda|x-y|}\Big{)}=\frac{F_{\pm}(\lambda|x-y|)}{4\lambda^{3}},$
( 2.1)

where $F_{\pm}(s)=\pm ie^{\pm is}-e^{-s}$. The existence of
$R^{\pm}_{V}(\lambda)$ for $\lambda>0$ under our assumption of Theorem 1.3 has
been also already shown (see [1, 46]).

Then $W_{-}$ has the following stationary representation (see e.g. [46, 52]):

$\displaystyle W_{-}=\mathop{\mathrm{Id}}\nolimits-\frac{2}{\pi
i}\int_{0}^{\infty}\lambda^{3}R_{V}^{+}(\lambda^{4})V\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda.$
( 2.2)

We decompose the second term of $W_{-}$ into the low and high energy parts as
follows: taking $\lambda_{0}>0$ small enough, we let $\chi_{0}\in
C_{0}^{\infty}(\mathbb{R})$ be such that $\chi_{0}\equiv 1$ on
$(-\lambda_{0},\lambda_{0})$ and
$\mathop{\mathrm{supp}}\nolimits\chi_{0}\subset[-2\lambda_{0},2\lambda_{0}]$
and set $\chi(\lambda)=\chi_{0}(\lambda^{2})$. We then define

$\displaystyle W^{L}_{-}$
$\displaystyle=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)R_{V}^{+}(\lambda^{4})V\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda,$
( 2.3) $\displaystyle W^{H}_{-}$
$\displaystyle=\int_{0}^{\infty}\lambda^{3}\Big{(}1-\chi(\lambda)\Big{)}R_{V}^{+}(\lambda^{4})V\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda$
( 2.4)

such that

$\displaystyle W_{-}=\mathop{\mathrm{Id}}\nolimits-\frac{2}{\pi
i}\left(W_{-}^{L}+W_{-}^{H}\right).$ ( 2.5)

We will deal with $W^{L}_{-},W^{H}_{-}$ in Sections 4 and 5, separately.

###  2.2. Resolvent expansion

This subsection is mainly devoted to the study of asymptotic behaviors of the
resolvent $R_{V}^{+}(\lambda^{4})$ at low energy $\lambda\to+0$. We also
prepare some elementary (but useful) lemmas need in the proof of our main
theorems.

We begin with the well known symmetric second resolvent formula for
$R_{V}^{+}(\lambda^{4})$. Let $v(x)=|V(x)|^{1/2}$ and
$U(x)=\mathop{\mathrm{sgn}}\nolimits V(x)$, namely $U(x)=1$ if $V(x)\geq 0$
and $U(x)=-1$ if $V(x)<0$. Let $M(\lambda)=U+vR_{0}^{+}(\lambda^{4})v$ and
$M^{-1}(\lambda):=[M(\lambda)]^{-1}$ as long as it exists.

######  Lemma 2.1.

For $\lambda>0$, $M(\lambda)$ is invertible on $L^{2}$. Moreover,
$R_{V}^{+}(\lambda^{4})V$ has the form

$\displaystyle
R_{V}^{+}(\lambda^{4})V=R_{0}^{+}(\lambda^{4})vM^{-1}(\lambda)v.$ ( 2.6)

###### Proof.

Thanks to the absence of embedded eigenvalues and the Birman-Schwinger
principle, $M(\lambda)$ is invertible. Using the decompositions $V=vUv$ and
$1=U^{2}$, we compute

$\displaystyle R_{V}^{+}(\lambda^{4})v$
$\displaystyle=R_{0}^{+}(\lambda^{4})v-R_{V}^{+}(\lambda^{4})vUvR_{0}^{+}(\lambda^{4})v=R_{0}^{+}(\lambda^{4})v\Big{(}1+UvR_{0}^{+}(\lambda^{4})v\Big{)}^{-1}$
$\displaystyle=R_{0}^{+}(\lambda^{4})v\Big{(}U+vR_{0}^{+}(\lambda^{4})v\Big{)}^{-1}U^{-1}.$

Multiplying $Uv$ from the right, we obtain the desired formula for
$R_{V}^{+}(\lambda^{4})V$. ∎

By virtue of the formula 2.6, $W_{-}^{L}$ defined by 2.3 is rewritten in
the form

$\displaystyle
W^{L}_{-}=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)R_{0}^{+}(\lambda^{4})vM^{-1}(\lambda)v\left(R_{0}^{+}(\lambda^{4})-R_{0}^{-}(\lambda^{4})\right)d\lambda.$
( 2.7)

We now recall the asymptotic expansion of $M^{-1}(\lambda)$ proved by [57],
which plays a crucial role in the paper. To this end, we introduce some
notations. We say that an integral operator
$T_{K}\in\mathbb{B}(L^{2}({\mathbb{R}}))$ with kernel $K$ is absolutely
bounded if $T_{|K|}\in\mathbb{B}(L^{2}({\mathbb{R}}))$. Let

$P:=\frac{{\langle}\cdot,v{\rangle}v}{{\|V\|}_{L^{1}}},\quad\widetilde{P}=-\frac{2(1+i)}{{\|V\|}_{L^{1}}}P=-\frac{2(1+i)}{{\|V\|}_{L^{1}}^{2}}{\langle}\cdot,v{\rangle}v,\quad
Q_{1}:=\mathop{\mathrm{Id}}\nolimits-P.$

Note that $P$ is the orthogonal projection onto the span of $v$ in
$L^{2}({\mathbb{R}})$, i.e. $PL^{2}=\mathrm{\mathop{span}}\\{v\\}$. Let
$G_{0}:=(\Delta^{2})^{-1}$ and $T_{0}:=U+vG_{0}v$ and define the subspaces
$Q_{2}L^{2},Q_{2}^{0}L^{2},Q_{3}L^{2}$ of $L^{2}$ by

$\displaystyle f\in Q_{2}L^{2}\ $ $\displaystyle\Longleftrightarrow\
f\in\mathrm{\mathop{span}}\\{v,xv\\}^{\perp};$ $\displaystyle f\in
Q_{2}^{0}L^{2}\ $ $\displaystyle\Longleftrightarrow\
f\in\mathrm{\mathop{span}}\\{v,xv\\}^{\perp}\ \text{and}\
T_{0}f\in\mathrm{\mathop{span}}\\{v,xv\\};$ $\displaystyle f\in Q_{3}L^{2}\ $
$\displaystyle\Longleftrightarrow\
f\in\mathrm{\mathop{span}}\\{v,xv,x^{2}v\\}^{\perp}\ \text{and}\
T_{0}f\in\mathrm{\mathop{span}}\\{v\\}.$

Note that $Q_{3}L^{2}\subset Q_{2}^{0}L^{2}\subset Q_{2}L^{2}$. Let
$Q_{\alpha}$ and $Q_{2}^{0}$ be the orthogonal projection onto
$Q_{\alpha}L^{2}$ and $Q_{2}^{0}L^{2}$, respectively. Since $v$ is real-
valued, by definition, $Q_{1},Q_{2},Q_{2}^{0},Q_{3}$ satisfy

$\displaystyle
Q_{\alpha}(x^{k}v)=0,\quad{\langle}x^{k}v,Q_{\alpha}f{\rangle}=0,\quad
Q_{2}^{0}(x^{k}v)=0,\quad{\langle}x^{k}v,Q_{2}^{0}f{\rangle}=0.$ ( 2.8)

for $k=0$ for $Q_{1}$, $k=0,1$ for $Q_{2},Q_{2}^{0}$ and $k=0,1,2$ for $Q_{3}$
. Recall that $|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$.

######  Lemma 2.2 ([57, Theorem 1.8 and Remark 1.9]).

There exists $\lambda_{0}>0$ such that $M^{-1}(\lambda)$ satisfies the
following asymptotic expansions on $L^{2}({\mathbb{R}})$ for
$0<\lambda\leq\lambda_{0}$:

  * (i)

If zero is a regular point of $H$ and $\mu>15$, then

$\displaystyle M^{-1}(\lambda)$ $\displaystyle=Q_{2}A_{0}^{0}Q_{2}+\lambda
Q_{1}A_{1}^{0}Q_{1}+\lambda^{2}\left(Q_{1}A_{21}^{0}Q_{1}+Q_{2}A_{22}^{0}+A_{23}^{0}Q_{2}\right)$
$\displaystyle\quad+\lambda^{3}\left(Q_{1}A_{31}^{0}+A_{32}^{0}Q_{1}\right)+\lambda^{3}\widetilde{P}+\Gamma_{4}^{0}(\lambda).$
( 2.9)

  * (ii)

If zero is a first kind resonance of $H$ and $\mu>21$, then

$\displaystyle M^{-1}(\lambda)$
$\displaystyle=\lambda^{-1}Q_{2}^{0}A_{-1}^{1}Q_{2}^{0}+Q_{2}A_{01}^{1}Q_{1}+Q_{1}A_{02}^{1}Q_{2}+\lambda\left(Q_{1}A_{11}^{1}Q_{1}+Q_{2}A_{12}^{1}+A_{13}^{1}Q_{2}\right)$
$\displaystyle\quad+\lambda^{2}\left(Q_{1}A_{21}^{1}+A_{22}^{1}Q_{1}\right)+\lambda^{3}\left(Q_{1}A_{31}^{1}+A_{32}^{1}Q_{1}\right)+\lambda^{3}\widetilde{P}+\Gamma_{4}^{1}(\lambda).$
( 2.10)

  * (iii)

If zero is a second kind resonance of $H$ and $\mu>29$, then

$\displaystyle M^{-1}(\lambda)$
$\displaystyle=\lambda^{-3}Q_{3}A_{-3}^{2}Q_{3}+\lambda^{-2}\left(Q_{3}A_{-21}^{2}Q_{2}+Q_{2}A_{-22}^{2}Q_{3}\right)$
$\displaystyle\quad+\lambda^{-1}\left(Q_{2}A_{-11}^{2}Q_{2}+Q_{3}A_{-12}^{2}Q_{1}+Q_{1}A_{-13}^{2}Q_{3}\right)$
$\displaystyle\quad+Q_{2}A^{2}_{01}Q_{1}+Q_{1}A^{2}_{02}Q_{2}+Q_{3}A^{2}_{03}+A^{2}_{04}Q_{3}+\lambda\left(Q_{1}A_{11}^{2}Q_{1}+Q_{2}A^{2}_{12}+A^{2}_{13}Q_{2}\right)$
$\displaystyle\quad+\lambda^{2}\left(Q_{1}A^{2}_{21}+A^{2}_{22}Q_{1}\right)+\lambda^{3}\left(Q_{1}A_{31}^{2}+A_{32}^{2}Q_{1}\right)+\lambda^{3}\widetilde{P}+\Gamma_{4}^{2}(\lambda).$
( 2.11)

Here $A^{j}_{k}$ and $A^{j}_{k\ell}$ are $\lambda$-independent bounded
operators on $L^{2}$ and $\Gamma_{4}^{j}(\lambda)$ are $\lambda$-dependent
bounded operators on $L^{2}$ such that all the operators appeared in the right
hand sides of (i), (ii) and (iii) are absolutely bounded. Moreover,
$\Gamma_{4}^{j}(\lambda)$ satisfy, for $\ell=0,1,2$,

$\displaystyle{\|\partial_{\lambda}^{\ell}\Gamma^{j}_{4}(\lambda)\|}_{L^{2}\to
L^{2}}\leq C_{\ell}\lambda^{4-\ell},\quad\lambda>0.$ ( 2.12)

######  Remark 2.3.

  * (1)

We have used different notations $Q_{1},Q_{2},Q_{2}^{0},Q_{3}$ in Lemma 2.2
from ones in [57], which is convenient for our purpose. The relation between
our notations and original ones are as follows:
$(Q_{1},Q_{2},Q_{2}^{0},Q_{3})$ correspond to $(Q,S_{0},S_{1},S_{2})$ in [57,
Theorem 1.9].

  * (2)

In [57, Remark 1.9], it was only stated that 2.12 holds for $\ell=0,1$ under
a slightly weaker condition on $V$ than 1.6. However, it can be seen from
the proof of [57, Theorem 1.9] that 2.12 in fact holds for $\ell=0,1,2,3,4$
under the condition 1.6.

We also prepare three elementary (but useful) lemmas.

######  Lemma 2.4 ([57, Lemma 2.5]).

Let $\lambda>0$ and $x,y\in{\mathbb{R}}$.

  * (i)

If $F\in C^{1}({\mathbb{R}}_{+})$ then

$F(\lambda|x-y|)=F(\lambda|x|)-\lambda
y\int_{0}^{1}\mathop{\mathrm{sgn}}\nolimits(x-\theta
y)F^{\prime}(\lambda|x-\theta y|)d\theta.$

  * (ii)

If $F\in C^{2}({\mathbb{R}}_{+})$ and $F^{\prime}(0)=0$, then

$F(\lambda|x-y|)=F(\lambda|x|)-\lambda
y\mathop{\mathrm{sgn}}\nolimits(x)F^{\prime}(\lambda|x|)+\lambda^{2}y^{2}\int_{0}^{1}(1-\theta)F^{\prime\prime}(\lambda|x-\theta
y|)d\theta.$

  * (iii)

If $F\in C^{3}({\mathbb{R}}_{+})$ and $F^{\prime}(0)=F^{\prime\prime}(0)=0$,
then

$\displaystyle F(\lambda|x-y|)$ $\displaystyle=F(\lambda|x|)-\lambda
y\mathop{\mathrm{sgn}}\nolimits(x)F^{\prime}(\lambda|x|)+\frac{\lambda^{2}y^{2}}{2}F^{\prime\prime}(\lambda|x|)$
$\displaystyle\quad-\frac{\lambda^{3}y^{3}}{2}\int_{0}^{1}(1-\theta)^{2}\mathop{\mathrm{sgn}}\nolimits(x-\theta
y)F^{\prime\prime\prime}(\lambda|x-\theta y|)d\theta.$

We will mainly use this lemma for $F_{\pm}(s)=\pm ie^{\pm is}-e^{-s}$.
Combined with 2.1 and 2.8, Lemma 2.4 implies the following formulas, which
will be one of key tools in the paper.

######  Lemma 2.5.

Let $Q_{1},Q_{2},Q_{2}^{0},Q_{3}$ be as above, $\alpha=0,1,2,3$ and
$\lambda>0$. Then:

$\displaystyle[Q_{\alpha}vR_{0}^{\pm}(\lambda^{4})f](x)$
$\displaystyle=\frac{(-1)^{\alpha}\lambda^{-3+\alpha}}{4\cdot(\alpha-1)!}Q_{\alpha}\left(x^{\alpha}v\int\int_{0}^{1}(1-\theta)^{\alpha-1}\left(\mathop{\mathrm{sgn}}\nolimits(y-\theta
x)\right)^{\alpha}F_{\pm}^{(\alpha)}(\lambda|y-\theta x|)f(y)d\theta
dy\right),$ $\displaystyle[R_{0}^{\pm}(\lambda^{4})vQ_{\alpha}f](x)$
$\displaystyle=\frac{(-1)^{\alpha}\lambda^{-3+\alpha}}{4\cdot(\alpha-1)!}\int\int_{0}^{1}(1-\theta)^{\alpha-1}\\!\left(\mathop{\mathrm{sgn}}\nolimits(x-\theta
y)\right)^{\alpha}F_{\pm}^{(\alpha)}(\lambda|x-\theta
y|)y^{\alpha}v(y)(Q_{\alpha}f)(y)d\theta dy,$

where for simplicity we have used the convention that
$(\mathop{\mathrm{sgn}}\nolimits x)^{2}\equiv 1$ for all $x\in{\mathbb{R}}$ .
Moreover, these estimates for $\alpha=2$ also hold with $Q_{2}$ replaced by
$Q_{2}^{0}$.

More precisely speaking, the above formula for
$Q_{\alpha}vR_{0}^{\pm}(\lambda^{4})f$ means

$Q_{\alpha}vR_{0}^{\pm}(\lambda^{4})f=\frac{(-1)^{\alpha}\lambda^{-3+\alpha}}{4\cdot(\alpha-1)!}Q_{\alpha}\widetilde{f}_{\pm,\alpha}$

with

$\widetilde{f}_{\pm,\alpha}(\lambda,x)=x^{\alpha}v\int\int_{0}^{1}(1-\theta)^{\alpha-1}\left(\mathop{\mathrm{sgn}}\nolimits(y-\theta
x)\right)^{\alpha}F_{\pm}^{(\alpha)}(\lambda|y-\theta x|)f(y)d\theta dy.$

Note that the subscript $\alpha$ of $Q_{\alpha}$ coincides with the order of
differentiation for $F_{\pm}$. This is the main reason why we use the
notations $Q_{1},Q_{2},Q_{2}^{0},Q_{3}$ instead of the original ones.

######  Remark 2.6.

At the level of the order with respect to $\lambda$, this lemma shows

$Q_{\alpha}vR^{\pm}_{0}(\lambda^{4}),R^{\pm}_{0}(\lambda^{4})vQ_{\alpha}=O(\lambda^{-3+\alpha}),\quad\lambda\to+0.$

This gain of positive powers of $\lambda$, compared with that of the free
resolvent $R^{\pm}_{0}(\lambda^{4})=O(\lambda^{-3})$, is useful to cancel out
singularities near $\lambda=0$ appeared in the expansion for $M^{-1}(\lambda)$
(see Lemma 2.2). This cancelation properties will be crucial in our argument.

###### Proof of Lemma 2.5.

Since $F_{\pm}^{\prime}(0)=0$, we can apply Lemma 2.4 to $F_{\pm}$ obtaining

$\displaystyle R_{0}^{\pm}(\lambda^{4},x,y)$
$\displaystyle=\frac{F_{\pm}(\lambda|x|)}{4\lambda^{3}}-\frac{y}{4\lambda^{2}}\int_{0}^{1}\mathop{\mathrm{sgn}}\nolimits(x-\theta
y)F_{\pm}^{\prime}(\lambda|x-\theta y|)d\theta$
$\displaystyle=\frac{F_{\pm}(\lambda|x|)}{4\lambda^{3}}-\frac{y\mathop{\mathrm{sgn}}\nolimits(x)F_{\pm}^{\prime}(\lambda|x|)}{4\lambda^{2}}+\frac{y^{2}}{4\lambda}\int_{0}^{1}(1-\theta)F_{\pm}^{\prime\prime}(\lambda|x-\theta
y|)d\theta.$

The cases $\alpha=1,2$ follow from this formula and 2.8. Indeed, we have

$\displaystyle Q_{2}vR_{0}^{\pm}(\lambda^{4})f$
$\displaystyle=\frac{1}{4\lambda^{3}}Q_{2}(v)\int
F_{\pm}(\lambda|y|)f(y)dy-\frac{1}{4\lambda^{2}}Q_{2}(xv)\int\mathop{\mathrm{sgn}}\nolimits(y)F_{\pm}^{\prime}(\lambda|y|)f(y)dy$
$\displaystyle\quad+\frac{1}{4\lambda}Q_{2}\left(x^{2}v\int\int_{0}^{1}(1-\theta)F_{\pm}^{\prime\prime}(\lambda|y-\theta
x|)f(y)d\theta dy\right)$
$\displaystyle=\frac{1}{4\lambda}Q_{2}\left(x^{2}v\int\int_{0}^{1}(1-\theta)F_{\pm}^{\prime\prime}(\lambda|y-\theta
x|)f(y)d\theta dy\right).$

The proofs for the other cases with $Q_{1},Q_{2}^{0}$ are similar. For the
case $\alpha=3$, we write

$4\lambda^{3}R^{\pm}(\lambda^{4},x,y)=F_{\pm}(\lambda|x-y|)=\widetilde{F}_{\pm}(\lambda|x-y|)-\frac{1\pm
i}{2}\lambda^{2}|x-y|^{2},$

where $\widetilde{F}_{\pm}(s)=F_{\pm}(s)+\frac{1\pm i}{2}s^{2}$. Then we can
write

$\displaystyle
Q_{3}vR_{0}^{\pm}(\lambda^{4})f=Q_{3}\left\\{v\int\left(\widetilde{F}_{\pm}(\lambda|x-y|)-\frac{1\pm
i}{2}\lambda^{2}|x-y|^{2}\right)f(y)dy\right\\}.$

For the first term of the right hand side, since
$\widetilde{F}_{\pm}^{\prime}(0)=\widetilde{F}_{\pm}^{\prime\prime}(0)=0$ and
$F_{\pm}^{\prime\prime\prime}\equiv\widetilde{F}_{\pm}^{\prime\prime\prime}$,
we can apply Lemma 2.4 (iii) and 2.8 to compute

$\displaystyle Q_{3}\left(v\int\widetilde{F}_{\pm}(\lambda|x-y|)f(y)dy\right)$
$\displaystyle=-\frac{1}{8}Q_{3}\left(x^{3}v\int\int_{0}^{1}(1-\theta)^{2}\mathop{\mathrm{sgn}}\nolimits(y-\theta
x)F_{\pm}^{\prime\prime\prime}(\lambda|y-\theta x|)f(y)d\theta dy\right),$

while the second part related with $|x-y|^{2}$ vanishes identically by virtue
of 2.8. The proof for $R_{0}^{+}(\lambda^{4})vQ_{3}f$ is analogous. ∎

We will also use often the following simple formula:

######  Lemma 2.7.

Let $F_{\pm}(s)=\pm ie^{\pm is}-e^{-s}$ and
$\alpha,\beta\in{\mathbb{N}}\cup\\{0\\}$. Then

$\displaystyle
f_{\alpha\beta}(\lambda,x,y):=F_{+}^{(\alpha)}(\lambda|x|)[F_{+}^{(\beta)}-F_{-}^{(\beta)}](\lambda|y|)$
$\displaystyle=-i^{\alpha+\beta}\Big{(}e^{i\lambda(|x|+|y|)}+(-1)^{\beta}e^{i\lambda(|x|-|y|)}\Big{)}+(-1)^{\alpha+1}i^{\beta+1}\Big{(}(-1)^{\beta}e^{-\lambda(|x|+i|y|)}+e^{-\lambda(|x|-i|y|)}\Big{)}.$

###### Proof.

A direct calculation yields

$\displaystyle
F_{+}^{(\alpha)}(s)[F_{+}^{(\beta)}-F_{-}^{(\beta)}](t)=\Big{(}i^{\alpha+1}e^{is}+(-1)^{\alpha+1}e^{-s}\Big{)}\Big{(}i^{\beta+1}e^{it}-(-i)^{\beta+1}e^{-it}\Big{)}$
$\displaystyle=-i^{\alpha+\beta}\Big{(}e^{i(s+t)}+(-1)^{\beta}e^{i(s-t)}\Big{)}+(-1)^{\alpha+1}i^{\beta+1}\Big{(}(-1)^{\beta}e^{-(s+it)}+e^{-(s-it)}\Big{)}$

The lemma then follows by letting $s=\lambda|x|$ and $t=\lambda|y|$. ∎

##  3\. Boundedness of some integrals related with wave operators

Recall that $T_{K}$ denotes the integral operator with the kernel $K(x,y)$:

$T_{K}f(x)=\int K(x,y)f(y)dy.$

This section is devoted to preparing some basic criterions to obtain several
boundedness of $T_{K}$ related with the wave operator $W_{-}$.

###  3.1. Classical Schur kernels

We first recall the classical Schur lemma:

######  Lemma 3.1.

$T_{K}\in\mathbb{B}(L^{p}({\mathbb{R}}))$ for all $1\leq p\leq\infty$ if $K$
satisfies

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

We often use this lemma for the kernel satisfying
$|K(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-\rho}$ with some $\rho>1$. In
fact, one can also obtain several weighted boundedness for such operators:

######  Lemma 3.2.

Let $K$ satisfy $|K(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-\rho}$ on
${\mathbb{R}}^{2}$ with some $\rho>1$ and $\tau f(x)=f(-x)$. Let $1<p<\infty$,
$w_{p}\in A_{p}$ and $w_{1}\in A_{1}$. Then $T_{K}$ satisfies the following
bounds:

$\displaystyle{\|T_{K}f\|}_{L^{p}(w_{p})}+{\|T_{K}^{*}f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}}({\|f\|}_{L^{p}(w_{p})}+{\|\tau
f\|}_{L^{p}(w_{p})}),$ ( 3.1)
$\displaystyle{\|T_{K}f\|}_{L^{1,\infty}(w_{1})}+{\|T_{K}^{*}f\|}_{L^{1,\infty}(w_{1})}$
$\displaystyle\lesssim[w_{1}]_{A_{1}}(1+\log[w]_{A_{1}})({\|f\|}_{L^{1}(w_{1})}+{\|\tau
f\|}_{L^{1}(w_{1})}).$ ( 3.2)

###### Proof.

Let $\chi_{\pm}=\chi_{{\mathbb{R}}_{\pm}}$ be the characteristic function of
${\mathbb{R}}_{\pm}$. We decompose $K$ as

$\displaystyle K(x,y)$
$\displaystyle=\left(\chi_{+}(x)+\chi_{-}(x)\right)K(x,y)\left(\chi_{+}(y)+\chi_{-}(y)\right)$
$\displaystyle=\sum_{\pm}\left(\chi_{\pm}(x)K(x,y)\chi_{\pm}(y)+\chi_{\pm}(x)K(x,y)\chi_{\mp}(y)\right)$
$\displaystyle=:\sum_{\pm}\left(K_{\pm,\pm}(x,y)+K_{\pm,\mp}(x,y)\right),$

By assumption, $K_{\pm,\pm}$ and $K_{\pm,\mp}$ satisfy

$|K_{\pm,\pm}(x,y)|\lesssim{\langle}x-y{\rangle}^{-\rho},\quad|K_{\pm,\mp}(x,y)|\lesssim{\langle}x+y{\rangle}^{-\rho}.$

Hence if we set $\widetilde{K}_{\pm,\mp}(x,y)=K_{\pm,\mp}(x,-y)$ and $\tau
f(x)=f(-x)$, then

$\displaystyle|K_{\pm,\pm}(x,y)|+|\widetilde{K}_{\pm,\mp}(x,y)|\lesssim{\langle}x-y{\rangle}^{-\rho}$
( 3.3)

and $T_{K}=T_{K_{+,+}}+T_{K_{-,-}}+(T_{K_{+,-}}+T_{K_{-,+}})\tau$. It follows
from 3.3 and Lemma 3.1 that the integral operator $T$ with the kernel
${\langle}x-y{\rangle}^{-\rho}$ is a Calderón–Zygmund operator (see Appendix
A). Theorem A.1 in Appendix A thus implies, for $1<p<\infty$,

$\displaystyle{\|T_{K}f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim{\|Tf\|}_{L^{p}(w_{p})}+{\|T\tau f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}}({\|f\|}_{L^{p}(w_{p})}+{\|\tau
f\|}_{L^{p}(w_{p})}).$

Similarly, we obtain for $p=1$,

$\displaystyle{\|T_{K}f\|}_{L^{1,\infty}(w_{1})}$
$\displaystyle\lesssim[w_{1}]_{A_{1}}(1+\log[w_{1}]_{A_{1}})({\|f\|}_{L^{1}(w_{1})}+{\|\tau
f\|}_{L^{1}(w_{1})}).$

By virtue of 3.3, the same argument also implies the desired bounds for
$T_{K}^{*}$. ∎

###  3.2. Non-classical kernels related with wave operators

The low energy part $W^{L}_{-}$ of the wave operator $W_{-}$ has several terms
with kernels satisfying $|K(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-1}$ only.
To deal with such terms, we further prepare two lemmas based on the theory of
Caldeŕon–Zygmund operators (see Appendix A for Caldeŕon–Zygmund operators).
The following lemma is concerned with the boundedness on weighted
$L^{p}$-spaces:

######  Lemma 3.3.

Let $1<p<\infty$ and $\psi\in C^{\infty}({\mathbb{R}};{\mathbb{R}})$ be such
that $\psi(s)=0$ for $0\leq s\leq 1$ and $\psi(s)=1$ for $s\geq 2$. Let
$K(x,y)$ be a linear combination of the following four functions

$\displaystyle k_{1}^{\pm}(x,y)=\frac{\psi(||x|\pm|y||^{2})}{|x|\pm|y|},\quad
k_{2}^{\pm}(x,y)=\frac{\psi(||x|-|y||^{2})}{|x|\pm i|y|}.$

Then $T_{K}$ and $T_{K}^{*}$ satisfy the same bounds as 3.1 and 3.2.

###### Proof.

With some constants $a,b,c,d\in{\mathbb{C}}$, we can write

$K=a\frac{\psi(\big{|}|x|+|y|\big{|}^{2})}{|x|+|y|}+b\frac{\psi(\big{|}|x|-|y|\big{|}^{2})}{|x|-|y|}+c\frac{\psi(||x|-|y||^{2})}{|x|+i|y|}+d\frac{\psi(||x|-|y||^{2})}{|x|-i|y|}.$

We set $\chi_{\pm}=\chi_{{\mathbb{R}}_{\pm}}$, $\tau f(x)=f(-x)$ and

$\widetilde{k}_{1}(x,y)=\frac{\psi(|x-y|^{2})}{x-y},\quad\widetilde{k}_{2}^{\pm}(x,y)=\frac{\psi(|x-y|^{2})}{x\pm
iy}.$

Then $T_{K}$ is written in the form

$\displaystyle T_{K}$
$\displaystyle=\left\\{a\left(\chi_{+}T_{\widetilde{k}_{1}}\chi_{-}-\chi_{-}T_{\widetilde{k}_{1}}\chi_{+}\right)+b\left(\chi_{+}T_{\widetilde{k}_{1}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{1}}\chi_{-}\right)\right\\}(1+\tau)$
$\displaystyle\quad+\left\\{c\left(\chi_{+}T_{\widetilde{k}_{2}^{+}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{2}^{+}}\chi_{-}\right)+d\left(\chi_{+}T_{\widetilde{k}_{2}^{-}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{2}^{-}}\chi_{-}\right)\right\\}(1+\tau).$
( 3.4)

Indeed, since $k_{1}^{\pm}(x,y)=\widetilde{k}_{1}(|x|,\mp|y|)$ and
$\widetilde{k}_{1}(-x,-y)=-\widetilde{k}_{1}(x,y)$, we have

$\displaystyle k_{1}^{\pm}(x,y)$
$\displaystyle=\left(\chi_{+}(x)+\chi_{-}(x)\right)k_{1}^{\pm}(x,y)\left(\chi_{+}(y)+\chi_{-}(y)\right)$
$\displaystyle=\chi_{+}(x)\widetilde{k}_{1}(x,\mp
y)\chi_{+}(y)+\chi_{+}(x)\widetilde{k}_{1}(x,\pm y)\chi_{-}(y)$
$\displaystyle\quad-\chi_{-}(x)\widetilde{k}_{1}(x,\pm
y)\chi_{+}(y)-\chi_{-}(x)\widetilde{k}_{1}(x,\mp y)\chi_{-}(y).$

By the change of variable $y\mapsto-y$ in the first and fourth terms for the
“$+$” case and the second and third terms for the “$-"$ case, respectively, we
obtain

$\displaystyle
T_{k_{1}^{\pm}}f(x)=[(\chi_{+}T_{\widetilde{k}_{1}}\chi_{\mp}-\chi_{-}T_{\widetilde{k}_{1}}\chi_{\pm})(1+\tau)]f(x).$

A similar calculation also implies

$\displaystyle k_{2}^{\pm}(x,y)$
$\displaystyle=\chi_{+}(x)\widetilde{k}_{2}^{\pm}(x,y)\chi_{+}(y)+\chi_{+}(x)\widetilde{k}_{2}^{\pm}(x,-y)\chi_{-}(y)$
$\displaystyle\quad-\chi_{-}(x)\widetilde{k}_{2}^{\pm}(x,-y)\chi_{+}(y)-\chi_{-}(x)\widetilde{k}_{2}^{\pm}(x,y)\chi_{-}(y).$

Hence, by the change of variable $y\mapsto-y$ in the second and third terms,
we have

$\displaystyle
T_{k_{2}^{\pm}}f(x)=[(\chi_{+}T_{\widetilde{k}_{2}^{\pm}}\chi_{+}-\chi_{-}T_{\widetilde{k}_{2}^{\pm}}\chi_{-})(1+\tau)]f(x).$

These two formulas imply 3.2. Since both the multiplication operator by
$\chi_{\pm}(x)$ belongs to
$\mathbb{B}(L^{p}(w_{p}))\cap\mathbb{B}(L^{1,\infty}(w_{1}))$ with operator
norms $1$ for all $1\leq p<\infty$, we obtain

$\displaystyle{\|T_{K}f\|}_{\mathcal{Y}}\lesssim\left({\|T_{\widetilde{k}_{1}}\|}_{\mathcal{X}\to\mathcal{Y}}+{\|T_{\widetilde{k}_{2}^{+}}\|}_{\mathcal{X}\to\mathcal{Y}}+{\|T_{\widetilde{k}_{2}^{-}}\|}_{\mathcal{X}\to\mathcal{Y}}\right)\left({\|f\|}_{\mathcal{X}}+{\|\tau
f\|}_{\mathcal{X}}\right)$

if
$(\mathcal{X},\mathcal{Y})\in\\{(L^{p}(w_{p}),L^{p}(w_{p})),(L^{1}(w_{1}),L^{1,\infty}(w_{1}))\\}$.
Moreover, since

$\displaystyle\overline{\widetilde{k}_{1}(y,x)}=-\widetilde{k}_{1}(x,y),\quad\overline{\widetilde{k}_{2}^{\pm}(y,x)}=\pm
i\widetilde{k}_{2}^{\pm}(x,y)$

we have $(T_{\widetilde{k}_{1}})^{*}=-T_{\widetilde{k}_{1}}$ and
$(T_{\widetilde{k}_{2}^{\pm}})^{*}=\pm iT_{\widetilde{k}_{2}^{\pm}}$ and

${\|T_{K}^{*}f\|}_{\mathcal{Y}}\lesssim\left({\|T_{\widetilde{k}_{1}}\|}_{\mathcal{X}\to\mathcal{Y}}+{\|T_{\widetilde{k}_{2}^{+}}\|}_{\mathcal{X}\to\mathcal{Y}}+{\|T_{\widetilde{k}_{2}^{-}}\|}_{\mathcal{X}\to\mathcal{Y}}\right)\left({\|f\|}_{\mathcal{X}}+{\|\tau
f\|}_{\mathcal{X}}\right).$

By virtue of Theorem A.1, it thus is enough to show that
$T_{\widetilde{k}_{1}},T_{\widetilde{k}_{2}^{\pm}}$ are Calderón–Zygmund
operators, namely $\widetilde{k}_{1},\widetilde{k}_{2}^{\pm}$ are standard
kernels and
$T_{\widetilde{k}_{1}},T_{\widetilde{k}_{2}^{\pm}}\in\mathbb{B}(L^{2}({\mathbb{R}}))$
(see Appendix A).

Since $\widetilde{k}_{1},\widetilde{k}_{2}^{\pm}$ are supported away from a
neighborhood of the diagonal line, they are smooth on ${\mathbb{R}}^{2}$.
Moreover, since $1\leq|x-y|^{2}\leq 2$ on
$\mathop{\mathrm{supp}}\nolimits\psi^{\prime}(|x-y|^{2})$, we have

$\displaystyle|\partial_{x}^{\alpha}\partial_{y}^{\beta}\widetilde{k}_{1}(x,y)|+|\partial_{x}^{\alpha}\partial_{y}^{\beta}\widetilde{k}_{2}^{\pm}(x,y)|$
$\displaystyle\lesssim{\langle}x-y{\rangle}^{-1-\alpha-\beta},\quad\alpha,\beta=0,1,2,....$

Hence $\widetilde{k}_{1}$ and $\widetilde{k}_{2}^{\pm}$ are standard kernels.
To show $T_{\widetilde{k}_{1}}\in\mathbb{B}(L^{2}({\mathbb{R}}))$, we observe
that, modulo a rapidly decaying error term, $T_{\widetilde{k}_{1}}$ is
essentially a truncated Hilbert transform $\mathcal{H}^{(2)}$ defined by

$\mathcal{H}^{({\varepsilon})}f(x)=\int_{|x-y|>{\varepsilon}}\frac{f(y)}{x-y}dy,\quad{\varepsilon}>0.$

Indeed, since $\psi(s)=1$ for $s\geq 2$, we have, for any $N\geq 0$,

$\displaystyle\widetilde{k}_{1}(x,y)=(\chi_{\\{|x-y|>2\\}}+\chi_{\\{|x-y|\leq
2\\}})\widetilde{k}_{1}(x,y)=\frac{\chi_{\\{|x-y|>2\\}}}{x-y}+O({\langle}x-y{\rangle}^{-N}).$

Since $\mathcal{H}^{(2)}\in\mathbb{B}(L^{2}({\mathbb{R}}))$ (see [34, Theorems
5.1.12]) and the error term is also bounded on $L^{2}({\mathbb{R}})$ by Lemma
3.1, so is $T_{\widetilde{k}_{1}}$. For the operators
$T_{\widetilde{k}_{2}^{\pm}}$, we compute

$\widetilde{k}_{2}^{\pm}(x,y)=\frac{x}{x^{2}+y^{2}}\psi(|x-y|^{2})\mp
i\frac{y}{x^{2}+y^{2}}\psi(|x-y|^{2})=:\widetilde{k}_{21}(x,y)\mp
i\widetilde{k}_{22}(x,y).$

Then,
$T_{\widetilde{k}_{21}}\in\mathbb{B}(L^{\infty}({\mathbb{R}}))\cap\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$
since

$\sup_{x\in{\mathbb{R}}}\int|\widetilde{k}_{21}(x,y)|dy\lesssim\sup_{x\in{\mathbb{R}}}\int\frac{|x|}{x^{2}+y^{2}+1}dy\lesssim
1,\quad|\widetilde{k}_{21}(x,y)|\lesssim{\langle}x{\rangle}^{-1}\in
L^{1,\infty}({\mathbb{R}}).$

The Marcinkiewicz interpolation theorem then yields
$T_{\widetilde{k}_{21}}\in\mathbb{B}(L^{2}({\mathbb{R}}))$. Since
$T_{\widetilde{k}_{22}}=(T_{\widetilde{k}_{21}})^{*}$,
$T_{\widetilde{k}_{22}}\in\mathbb{B}(L^{2}({\mathbb{R}}))$ by duality. Hence
$T_{\widetilde{k}_{2}^{\pm}}\in\mathbb{B}(L^{2})$. Summarizing these
arguments, we conclude that $T_{\widetilde{k}_{1}}$ and
$T_{\widetilde{k}_{2}^{\pm}}$ are Calderón–Zygmund operators. This completes
the proof. ∎

######  Remark 3.4.

Although the proof is reduced to the theory of Calderón–Zygmund operators, the
operator $T_{K}$ in Lemma 3.3 itself is not a Calderón–Zygmund operator in
general. Indeed, for instance, $\psi(||x|-|y||^{2})(|x|-|y|)^{-1}$ is not a
standard kernel.

The following lemma will be used to prove the ${\mathcal{H}}^{1}$-$L^{1}$ and
$L^{\infty}$-${\mathop{\mathrm{BMO}}}$ boundedness.

######  Lemma 3.5.

Let $k_{1}^{\pm},k_{2}^{\pm}$ be as in Lemma 3.3 and $a,b\in{\mathbb{C}}$.
Define $g_{a,b}^{\pm}=g_{a,b}^{\pm}(x,y)$ by

$g_{a,b}^{\pm}=a(k_{1}^{+}\pm k_{1}^{-})+b(k_{2}^{+}\pm k_{2}^{-})$

and consider the following eight integral kernels

$\displaystyle g_{1,a,b}^{\pm}(x,y)$ $\displaystyle=g_{a,b}^{\pm}(x,y),$
$\displaystyle g_{2,a,-a}^{+}(x,y)$
$\displaystyle=g_{a,-a}^{+}(x,y)\mathop{\mathrm{sgn}}\nolimits y,$
$\displaystyle g_{2,a,b}^{-}(x,y)$
$\displaystyle=g_{a,b}^{-}(x,y)\mathop{\mathrm{sgn}}\nolimits y,$
$\displaystyle g_{3,a,b}^{+}(x,y)$
$\displaystyle=g_{a,b}^{+}(x,y)\mathop{\mathrm{sgn}}\nolimits x,$
$\displaystyle g_{3,a,-ia}^{-}(x,y)$
$\displaystyle=g_{a,-ia}^{-}(x,y)\mathop{\mathrm{sgn}}\nolimits x,$
$\displaystyle g_{4,a,-a}^{+}(x,y)$
$\displaystyle=g_{a,-a}^{+}(x,y)\mathop{\mathrm{sgn}}\nolimits
x\mathop{\mathrm{sgn}}\nolimits y,$ $\displaystyle g_{4,a,b}^{-}(x,y)$
$\displaystyle=g_{a,b}^{-}(x,y)\mathop{\mathrm{sgn}}\nolimits
x\mathop{\mathrm{sgn}}\nolimits y.$

Then
$T_{g_{1}^{\pm}},T_{g_{2}}^{\pm},T_{g_{3}^{\pm}}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$
and
$T_{g_{4}^{\pm}}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$.

######  Remark 3.6.

For simplicity, we often omit the subscript $a,b$ and write simply
$g_{1}^{\pm}(x,y)=g_{1,a,b}^{\pm}(x,y)$ and so on if there is no confusion.
Note that, in contrast with $g_{1}^{\pm}$, $g_{2}^{-}$, $g_{3}^{+}$ and
$g_{4}^{-}$, there are restrictions on the choice of $b$ for the kernels
$g_{2,a,-a}^{+}$, $g_{3,a,-ia}^{-}$ and $g_{4,a,-a}^{+}$.

###### Proof.

We first observe that Lemma 3.3 applies to $T_{g_{j}^{\pm}}$ for all $1\leq
j\leq 4$ since the multiplication by $\mathop{\mathrm{sgn}}\nolimits x$ is
bounded on $L^{p}(w_{p})$ for all $1\leq p<\infty$ and on
$L^{1,\infty}(w_{1})$. We also observe some duality among
$T_{g_{j,a,b}^{\pm}}$. Namely, since a direct calculation yields

$\displaystyle\overline{g_{a,b}^{\pm}(y,x)}$
$\displaystyle=g_{\overline{a},i\overline{b}}^{\mp}(x,y),$

we have

$(T_{g_{1,a,b}^{\pm}})^{*}=T_{g_{1,\overline{a},i\overline{b}}^{\mp}},\quad(T_{g_{2,a,b}^{\pm}})^{*}=T_{g_{3,\overline{a},i\overline{b}}^{\mp}},\quad(T_{g_{3,a,b}^{\pm}})^{*}=T_{g_{2,\overline{a},i\overline{b}}^{\mp}},\quad(T_{g_{4,a,b}^{\pm}})^{*}=T_{g_{4,\overline{a},i\overline{b}}^{\mp}}.$

Since
${\mathop{\mathrm{BMO}}}({\mathbb{R}})={\mathcal{H}}^{1}({\mathbb{R}})^{*}$
(see [24]), it is thus enough to show
$T_{g_{j}^{\pm}}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$
for $1\leq j\leq 4$ with the above restrictions on $b$ for
$g_{2}^{+},g_{3}^{-}$ and $g_{4}^{+}$. Moreover, since the multiplication by
$\mathop{\mathrm{sgn}}\nolimits x$ is bounded on $L^{1}({\mathbb{R}})$, it is
enough to consider $T_{g_{1}^{\pm}}$ and $T_{g_{2}^{\pm}}$ only.

The proof of
$T_{g_{1}^{\pm}},T_{g_{2}^{\pm}}\in\mathbb{B}(\mathcal{H}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))$
follows a classical argument in the proof of the $\mathcal{H}^{1}$-$L^{1}$
boundedness of Calderón–Zygmund operators. We let $f\in\mathcal{H}^{1}$ and
apply the atomic decomposition (see [35, Section 2.3.5]) to obtain

$f=\sum_{j=1}^{\infty}\lambda_{j}a_{j}(x),\quad\sum_{j=1}^{\infty}|\lambda_{j}|\lesssim{\|f\|}_{\mathcal{H}^{1}},$

where $\lambda_{j}\in{\mathbb{C}}$ and $a_{j}$ are $L^{\infty}$-atoms for
$\mathcal{H}^{1}$ satisfying, with some $x_{j}\in{\mathbb{R}}$ and $r_{j}\geq
2$,

$\mathop{\mathrm{supp}}\nolimits
a_{j}\subset(x_{j}-r_{j},x_{j}+r_{j}),\quad{\|a_{j}\|}_{L^{\infty}}\lesssim
r_{j}^{-1},\quad\int a_{j}(x)dx=0.$

Hence, for a given integral operator $T$, once we obtain

$\displaystyle{\|Ta_{j}\|}_{L^{1}({\mathbb{R}})}\lesssim 1$ ( 3.5)

uniformly in $j$, $T$ is bounded from ${\mathcal{H}}^{1}$ to $L^{1}$ since

${\|Tf\|}_{L^{1}({\mathbb{R}})}\leq\sum_{j=1}^{\infty}|\lambda_{j}|{\|Ta_{j}\|}_{L^{1}({\mathbb{R}})}\lesssim{\|f\|}_{{\mathcal{H}}^{1}({\mathbb{R}})}.$

It is thus enough to prove 3.5 for $T=T_{g_{1}^{\pm}},T_{g_{2}^{\pm}}$. Let
$I=(x_{0}-r,x_{0}+r)$ with some fixed $x_{0}\in{\mathbb{R}},r\geq 2$ and take
an $L^{\infty}$-atom $a$ satisfying

$\mathop{\mathrm{supp}}\nolimits a\subset I,\quad{\|a\|}_{L^{\infty}}\lesssim
r^{-1},\quad\int_{I}a(x)dx=0.$

We also let $I_{*}=(x_{0}-3r,x_{0}+3r)\cup(-x_{0}-3r,-x_{0}+3r)$ and decompose

${\|Ta\|}_{L^{1}({\mathbb{R}})}={\|Ta\|}_{L^{1}(I_{*})}+{\|Ta\|}_{L^{1}(I_{*}^{c})}.$

For the first term, Lemma 3.3 and Hölder’s inequality imply

$\displaystyle{\|Ta\|}_{L^{1}(I_{*})}\leq|I_{*}|^{1/2}{\|Ta\|}_{L^{2}(I_{*})}\lesssim
r^{1/2}{\|a\|}_{L^{2}(I_{*})}\lesssim
r^{1/2}{\|a\|}_{L^{\infty}}r^{1/2}\lesssim 1$ ( 3.6)

uniformly in $x_{0}$ and $r$ for all $T=T_{g_{1}^{\pm}},T_{g_{2}^{\pm}}$ and
$a,b\in{\mathbb{C}}$.

To deal with the second term, we first deal with $g_{1}^{\pm}$ and
$g_{2}^{-}$. Let $x\in I_{*}^{c}$ and $y\in\mathop{\mathrm{supp}}\nolimits a$.
Then, since $\mathop{\mathrm{supp}}\nolimits a\subset I$, we have

$\displaystyle||x|\pm|y||\geq\min_{\pm}|x\pm y|\geq r\geq 2$

and hence $\psi(\big{|}|x|\pm|y|\big{|}^{2})=1$ since $\psi\equiv 1$ on
$[2,\infty)$. It also follows that

$\displaystyle|x\pm y|\geq|x\pm x_{0}|-|x_{0}-y|\geq|x\pm x_{0}|-r\geq|x\pm
x_{0}|/2.$

With these remarks at hand, we obtain

$\displaystyle|g_{1}^{\pm}(x,y)-g_{1}^{\pm}(x,x_{0})|$
$\displaystyle\lesssim\sum_{\pm}\left(\left|\frac{1}{|x|\pm|y|}-\frac{1}{|x|\pm|x_{0}|}\right|+\left|\frac{1}{|x|\pm
i|y|}-\frac{1}{|x|\pm i|x_{0}|}\right|\right)$
$\displaystyle\lesssim\sum_{\pm}\left(\left|\frac{|x_{0}|-|y|}{(|x|\pm|y|)(|x|\pm|x_{0}|)}\right|+\left|\frac{|x_{0}|-|y|}{(|x|\pm
i|y|)(|x|\pm i|x_{0}|)}\right|\right)$
$\displaystyle\lesssim\frac{|x_{0}-y|}{\displaystyle\min_{\pm}|x\pm
x_{0}|^{2}}.$

Using the relations

$\displaystyle\frac{\mathop{\mathrm{sgn}}\nolimits
y}{|x|+|y|}-\frac{\mathop{\mathrm{sgn}}\nolimits y}{|x|-|y|}$
$\displaystyle=\frac{-2y}{x^{2}-y^{2}}=\frac{1}{x+y}-\frac{1}{x-y},$
$\displaystyle\frac{\mathop{\mathrm{sgn}}\nolimits
y}{|x|+i|y|}-\frac{\mathop{\mathrm{sgn}}\nolimits y}{|x|-i|y|}$
$\displaystyle=\frac{-2iy}{x^{2}+y^{2}}=\frac{1}{x+iy}-\frac{1}{x-iy},$

we also have, for $x\in I_{*}^{c}$ and $y\in I$,

$\displaystyle|g_{2}^{-}(x,y)-g_{2}^{-}(x,x_{0})|$
$\displaystyle\lesssim\sum_{\pm}\left(\left|\frac{1}{x\pm y}-\frac{1}{x\pm
x_{0}}\right|+\left|\frac{1}{x\pm iy}-\frac{1}{x\pm ix_{0}}\right|\right)$
$\displaystyle\lesssim\frac{|x_{0}-y|}{|x+x_{0}|^{2}}+\frac{|x_{0}-y|}{|x-x_{0}|^{2}}\lesssim\frac{|x_{0}-y|}{\displaystyle\min_{\pm}|x\pm
x_{0}|^{2}}.$

Hence, for the three cases $K=g_{1}^{\pm},g_{2}^{-}$, $T_{K}$ satisfies

$\displaystyle{\|T_{K}a\|}_{L^{1}(I_{*}^{c})}$
$\displaystyle=\int_{I_{*}^{c}}\left|\int_{I}\Big{(}K(x,y)-K(x,x_{0})\Big{)}a(y)dy\right|dx$
$\displaystyle\lesssim{\|a\|}_{L^{\infty}}\int_{I}|x_{0}-y|dy\int_{I_{*}^{c}}\frac{1}{\displaystyle\min_{\pm}|x\pm
x_{0}|^{2}}dx$ $\displaystyle\lesssim
r\int_{I_{*}^{c}}\frac{1}{\displaystyle\min_{\pm}|x\pm x_{0}|^{2}}dx.$

Setting $U_{x_{0}}=\\{x\ |\ |x+x_{0}|\leq|x-x_{0}|\\}$, we further obtain

$\displaystyle\int_{I_{*}^{c}}\frac{1}{\displaystyle\min_{\pm}|x\pm
x_{0}|^{2}}dx$ $\displaystyle=\int_{I_{*}^{c}\cap
U_{x_{0}}}\frac{1}{|x+x_{0}|^{2}}dx+\int_{I_{*}^{c}\cap
U_{x_{0}}^{c}}\frac{1}{|x-x_{0}|^{2}}dx$ $\displaystyle\leq\int_{|x+x_{0}|\geq
3r}\frac{1}{|x+x_{0}|^{2}}dx+\int_{|x-x_{0}|\geq
3r}\frac{1}{|x-x_{0}|^{2}}dx\lesssim r^{-1}.$

It thus follows that ${\|T_{K}a\|}_{L^{1}(I_{*}^{c})}\lesssim 1$ uniformly in
$x_{0},r$. This bound, together with 3.6, implies 3.5 for
$T=T_{g_{1}^{\pm}},T_{g_{2}^{-}}$.

It remains to show 3.5 for $T=T_{g_{2}^{+}}$. To this end, as above it is
enough to check

$\displaystyle|g_{2}^{+}(x,y)-g_{2}^{+}(x,x_{0})|\lesssim\frac{|x_{0}-y|}{\displaystyle\min_{\pm}|x\pm
x_{0}|^{2}}$ ( 3.7)

for $x\in I_{*}^{c}$ and $y\in I$. Let us compute

$\displaystyle|g_{2}^{+}(x,y)-g_{2}^{+}(x,x_{0})|$
$\displaystyle=|g_{a,-a}^{+}(x,y)\mathop{\mathrm{sgn}}\nolimits
y-g_{a,-a}^{+}(x,x_{0})\mathop{\mathrm{sgn}}\nolimits x_{0}|$
$\displaystyle\leq|g_{a,-a}^{+}(x,y)-g_{a,-a}^{+}(x,x_{0})|+|g_{a,-a}^{+}(x,x_{0})(\mathop{\mathrm{sgn}}\nolimits
x_{0}-\mathop{\mathrm{sgn}}\nolimits y)|,$

where the first term is dominated by $C(\min_{\pm}|x\pm x_{0}|)^{-2}|x_{0}-y|$
as above. For the second term, we further calculate

$\displaystyle g_{a,-a}^{+}(x,x_{0})(\mathop{\mathrm{sgn}}\nolimits
x_{0}-\mathop{\mathrm{sgn}}\nolimits y)$
$\displaystyle=a(\mathop{\mathrm{sgn}}\nolimits
x_{0}-\mathop{\mathrm{sgn}}\nolimits
y)\left\\{\frac{1}{|x|+|x_{0}|}+\frac{1}{|x|-|x_{0}|}-\frac{1}{|x|+i|x_{0}|}-\frac{1}{|x|-i|x_{0}|}\right\\}$
$\displaystyle=(i-1)a(\mathop{\mathrm{sgn}}\nolimits
x_{0}-\mathop{\mathrm{sgn}}\nolimits
y)\left\\{\frac{|x_{0}|}{(|x|+|x_{0}|)(|x|+i|x_{0}|)}-\frac{|x_{0}|}{(|x|-|x_{0}|)(|x|-i|x_{0}|)}\right\\}$
$\displaystyle=(i-1)a(x_{0}-|x_{0}|\mathop{\mathrm{sgn}}\nolimits
y)\left\\{\frac{1}{(|x|+|x_{0}|)(|x|+i|x_{0}|)}-\frac{1}{(|x|-|x_{0}|)(|x|-i|x_{0}|)}\right\\},$

where we have

$|x_{0}-|x_{0}|\mathop{\mathrm{sgn}}\nolimits
y|=|x_{0}-y+(|y|-|x_{0}|)\mathop{\mathrm{sgn}}\nolimits y|\leq 2|x_{0}-y|$

and, for $x\in I_{*}^{c}$,

$\min\\{||x|\pm|x_{0}||,||x|\pm i|x_{0}||\\}\geq\min_{\pm}|x\pm x_{0}|.$

Hence we have 3.7, so 3.5 for $T=T_{g_{2}^{+}}$. This completes the proof.
∎

##  4\. Low energy estimate

In this section we consider the low energy part of Theorem 1.3. Namely, we
prove

######  Theorem 4.1.

Under the assumption in Theorem 1.3, the low energy part $W_{-}^{L}$ defined
by 2.3 satisfies the same statement as that in Theorem 1.3.

###  4.1. Regular case

We first consider the regular case. Throughout this subsection, we thus always
assume that $|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ with $\mu>15$ and zero
is a regular point of $H$.

Substituting the expansion (i) into 2.6, we obtain

$\displaystyle R^{+}(\lambda^{4})V$
$\displaystyle=R_{0}^{+}(\lambda^{4})v\Big{\\{}Q_{2}A_{0}^{0}Q_{2}+\lambda
Q_{1}A_{1}^{0}Q_{1}+\lambda^{2}\left(Q_{1}A_{21}^{0}Q_{1}+Q_{2}A_{22}^{0}+A_{23}^{0}Q_{2}\right)$
$\displaystyle\quad+\lambda^{3}\left(Q_{1}A_{31}^{0}+A_{32}^{0}Q_{1}\right)+\lambda^{3}\widetilde{P}+\Gamma_{4}^{0}(\lambda)\Big{\\}}v.$

Then $W^{L}_{-}$ can be written as follows:

$\displaystyle
W^{L}_{-}=T_{K^{0}_{0}}+T_{K^{0}_{1}}+T_{K^{0}_{21}}+T_{K^{0}_{22}}+T_{K^{0}_{23}}+T_{K^{0}_{31}}+T_{K^{0}_{32}}+T_{K^{0}_{33}}+T_{K^{0}_{4}}$
( 4.1)

with the integral kernels

$\displaystyle K^{0}_{0}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{2}A_{0}^{0}Q_{2}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
$\displaystyle K^{0}_{1}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{4}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{1}A_{1}^{0}Q_{1}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
$\displaystyle K^{0}_{kj}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{3+k}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vB^{0}_{kj}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
$\displaystyle K^{0}_{4}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})v\Gamma_{4}^{0}(\lambda)v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$

where $j=1,2,3$, $k=2,3$ and $B^{0}_{kj}$ are given by

  * •

$B^{0}_{21}=Q_{1}A_{21}^{0}Q_{1}$, $B^{0}_{22}=Q_{2}A_{22}^{0}$ and
$B^{0}_{23}=A_{23}^{0}Q_{2}$;

  * •

$B^{0}_{31}=Q_{1}A_{31}^{0}$, $B^{0}_{32}=A_{32}^{0}Q_{1}$ and
$B^{0}_{33}=\widetilde{P}$.

By virtue of this formula for $W^{L}_{-}$, Theorem 4.1 for the regular case
follows from the corresponding boundedness of these nine integral operators.
Note that, since $|v(x)|\lesssim{\langle}x{\rangle}^{-\mu/2}$ with $\mu>15$ by
the assumption on $V$, we have

${\|{\langle}x{\rangle}^{k}vBv{\langle}x{\rangle}^{k}f\|}_{L^{1}}\leq{\|{\langle}x{\rangle}^{k}v\|}_{L^{2}}^{2}{\|B\|}_{L^{2}\to
L^{2}}{\|f\|}_{L^{\infty}}\lesssim{\|{\langle}x{\rangle}^{2k}V\|}_{L^{1}}{\|f\|}_{L^{\infty}}$

for all
$B=Q_{2}A_{0}^{0}Q_{2},Q_{1}A_{1}^{0}Q_{1},B^{0}_{kj},\Gamma^{0}_{4}(\lambda)$
and $k<(\mu-1)/2$. Hence, in all cases,
${\langle}x{\rangle}^{k}vBv{\langle}x{\rangle}^{k}$ is an absolutely bounded
integral operator for any $k\leq 7$ at least, satisfying

$\displaystyle\int_{{\mathbb{R}}^{2}}{\langle}x{\rangle}^{k}|(vBv)(x,y)|{\langle}y{\rangle}^{k}dxdy\lesssim{\|{\langle}x{\rangle}^{2k}V\|}_{L^{1}}<\infty,$
( 4.2)

where, denoting the integral kernel of $B$ by $B(x,y)$, we use the notation

$(vBv)(x,y)=v(x)B(x,y)v(y).$

By virtue of Remark 2.6 (2), these nine operators
$T_{K^{0}_{j}},T_{K^{0}_{kj}}$ are classified into the following two cases
with respect to the order of $\lambda$ of the integrands of their kernels :

  * (I)

$O(\lambda)$:
$T_{K^{0}_{0}},T_{K^{0}_{21}},T_{K^{0}_{22}},T_{K^{0}_{23}},T_{K^{0}_{31}},T_{K^{0}_{32}}$
and $T_{K^{0}_{4}}$.

  * (II)

$O(1)$: $T_{K^{0}_{1}}$ and $T_{K^{0}_{33}}$.

The class (I) is further decomposed into $T_{K^{0}_{4}}$ and otherwise.

######  Remark 4.2.

Note that the two projections $Q_{2},Q_{2}^{0}$ will play completely the same
role in the following arguments. Hence, in what follows, we do not distinguish
them and use the same notation $Q_{2}$ to denote these operators
$Q_{2},Q_{2}^{0}$.

We start dealing with the operators in the class (I), except for the last one
$T_{K^{0}_{4}}$, namely the operators
$T_{K^{0}_{0}},T_{K^{0}_{21}},T_{K^{0}_{22}},T_{K^{0}_{23}},T_{K^{0}_{31}}$
and $T_{K^{0}_{32}}$.

######  Proposition 4.3.

Let
$K\in\\{K^{0}_{0},K^{0}_{21},K^{0}_{22},K^{0}_{23},K^{0}_{31},K^{0}_{32}\\}$.
Then $T_{K}\in\mathbb{B}(L^{p})$ for all $1\leq p\leq\infty$. Moreover, if
$1<p<\infty$, $w_{p}\in A_{p}$ and $w_{1}\in A_{1}$, then 3.1 and 3.2 also
hold.

###### Proof.

All the kernels $K^{0}_{0},K^{0}_{21},K^{0}_{22},K^{0}_{23},K^{0}_{31}$ and
$K^{0}_{32}$ can be written in the form

$\displaystyle\int_{0}^{\infty}\lambda^{7-\alpha-\beta}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{\alpha}BQ_{\beta}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda$
( 4.3)

with some $B\in\mathbb{B}(L^{2})$ so that $Q_{\alpha}BQ_{\beta}$ is absolutely
bounded, $Q_{0}:=1$ and $(\alpha,\beta)$ is give by

$\displaystyle(\alpha,\beta)$
$\displaystyle=\begin{cases}(2,2)&\text{for}\quad K=K^{0}_{0},\\\
(1,1)&\text{for}\quad K=K^{0}_{21},\\\ (2,0)&\text{for}\quad
K=K^{0}_{22},\end{cases}\ (\alpha,\beta)=\begin{cases}(0,2)&\text{for}\quad
K=K^{0}_{23},\\\ (1,0)&\text{for}\quad K=K^{0}_{31},\\\ (0,1)&\text{for}\quad
K=K^{0}_{32}.\end{cases}$

Let $G^{0}_{\alpha\beta}(x,y)$ be the function given by 4.3. Then we shall
show $T_{G^{0}_{\alpha\beta}}$ satisfies the desired assertion for any
$\alpha,\beta\geq 0$. To this end, by Lemmas 3.1 and 3.2, it is enough to show
that

$\displaystyle|G^{0}_{\alpha\beta}(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-2},\quad
x,y\in{\mathbb{R}}.$ ( 4.4)

We consider three cases (i) $\alpha,\beta\neq 0$, (ii) $\beta=0$ and (iii)
$\alpha=0$ separately.

Case (i). We first suppose $\alpha,\beta\neq 0$ and rewrite
$G^{0}_{\alpha\beta}$ as follows. Let

$\widetilde{f}_{\pm,\beta}(\lambda,x)=x^{\beta}v\int\int_{0}^{1}(1-\theta)^{\beta-1}\left(\mathop{\mathrm{sgn}}\nolimits(y-\theta
x)\right)^{\beta}F_{\pm}^{(\beta)}(\lambda|y-\theta x|)f(y)d\theta dy.$

Then Lemma 2.5 and Remark 2.6 (1) imply that

$\displaystyle\lambda^{6-\alpha-\beta}[R_{0}^{+}(\lambda^{4})vQ_{\alpha}BQ_{\beta}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})f](x)$
$\displaystyle=C_{\beta}\lambda^{3-\alpha}(R_{0}^{+}(\lambda^{4})vQ_{\alpha}BQ_{\beta}[\widetilde{f}_{+,\beta}-\widetilde{f}_{-,\beta}])(x)$
$\displaystyle=C_{\alpha}C_{\beta}\int\int_{0}^{1}(1-\theta_{1})^{\alpha-1}(\mathop{\mathrm{sgn}}\nolimits(X_{1}))^{\alpha}F_{+}^{(\alpha)}(\lambda|X_{1}|)u_{1}^{\alpha}v(u_{1})Q_{\alpha}BQ_{\beta}[\widetilde{f}_{+}-\widetilde{f}_{-}](u_{1})d\theta
du_{1}$
$\displaystyle=\int\left(\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{\alpha\beta}(X_{1},Y_{2},\Theta)F_{+}^{(\alpha)}(\lambda|X_{1}|)[F_{+}^{(\beta)}-F_{-}^{(\beta)}](\lambda|Y_{2}|)d\Theta\right)f(y)dy$
$\displaystyle=\int\left(\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{\alpha\beta}(X_{1},Y_{2},\Theta)f_{\alpha\beta}(\lambda,X_{1},Y_{2})d\Theta\right)f(y)dy,$
( 4.5)

where we set $C_{\alpha}=(-1)^{\alpha}/(4\cdot(\alpha-1)!)$, $f_{\alpha\beta}$
is defined in Lemma 2.7 and

$\Theta=(u_{1},u_{2},\theta_{1},\theta_{2}),\quad
X_{1}=x-\theta_{1}u_{1},\quad Y_{2}=y-\theta_{2}u_{2},$

and $M_{\alpha\beta}(x,y,\Theta)$ is defined by

$\displaystyle
M_{\alpha\beta}(x,y,\Theta)=\frac{(-1)^{\alpha+\beta}(1-\theta_{1})^{\alpha-1}(1-\theta_{2})^{\beta-1}(\mathop{\mathrm{sgn}}\nolimits
x)^{\alpha}(\mathop{\mathrm{sgn}}\nolimits
y)^{\beta}u_{1}^{\alpha}u_{2}^{\beta}(vQ_{\alpha}BQ_{\beta}v)(u_{1},u_{2})}{16(\alpha-1)!(\beta-1)!}.$
( 4.6)

Substituting this formula into 4.3, we obtain

$G^{0}_{\alpha\beta}(x,y)=\int_{0}^{\infty}\lambda\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{\alpha\beta}(X_{1},Y_{2},\Theta)f_{\alpha\beta}(\lambda,X_{1},Y_{2})d\Theta\right)d\lambda.$

It follows from Lemma 2.7 that $f_{\alpha\beta}(\lambda,X_{1},Y_{2})$ is given
by a linear combination of $e^{i\lambda(|X_{1}|\pm|Y_{2}|)}$ and
$e^{-\lambda(|X_{1}|\pm i|Y_{2}|)}$ for any $\alpha,\beta$ (not only for the
case (i)). Let

$\displaystyle\Phi^{\pm}_{1}(x,y,\Theta)$ $\displaystyle=i(|X_{1}|-|x|)\pm
i(|Y_{2}|-|y|),$ ( 4.7) $\displaystyle\Phi^{\pm}_{2}(x,y,\Theta)$
$\displaystyle=-|X_{1}|+|x|\mp i(|Y_{2}|-|y|).$

Then
$e^{i\lambda(|X_{1}|\pm|Y_{2}|)}=e^{i\lambda(|x|\pm|y|)}e^{\lambda\Phi^{\pm}_{1}(x,y,\Theta)}$
and $e^{-\lambda(|X_{1}|\pm i|Y_{2}|)}=e^{-\lambda(|x|\pm
i|y|)}e^{\lambda\Phi^{\pm}_{2}(x,y,\Theta)}$. Define

$\displaystyle a_{j}^{\pm}(\lambda,x,y)$
$\displaystyle=\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}e^{\lambda\Phi^{\pm}_{j}(x,y,\Theta)}M_{\alpha\beta}(X_{1},Y_{2},\Theta)d\Theta,$
( 4.8) $\displaystyle K^{\pm}_{a_{1}}(x,y)$
$\displaystyle=\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\lambda\chi(\lambda)a_{1}^{\pm}(\lambda,x,y)d\lambda,$
$\displaystyle K^{\pm}_{a_{2}}(x,y)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\lambda\chi(\lambda)a_{2}^{\pm}(\lambda,x,y)d\lambda.$

Then $G^{0}_{\alpha\beta}$ can be written as a linear combination of
$K^{\pm}_{a_{1}}$ and $K^{\pm}_{a_{2}}$.

Here we summarize several properties of $M_{\alpha\beta}$, $\Phi^{\pm}_{j}$
and $a^{\pm}_{j}$ needed in the proof:

  * •

By 4.2,
${\langle}u_{1}{\rangle}^{\ell}M_{\alpha\beta}(x,y,\Theta){\langle}u_{2}{\rangle}^{\ell}\in
L^{1}({\mathbb{R}}^{2}\times[0,1]^{2};L^{\infty}({\mathbb{R}}^{2}_{x,y}))$ for
$\ell=0,1,2$ and

$\displaystyle\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}\sup_{x,y\in{\mathbb{R}}^{2}}{\langle}u_{1}{\rangle}^{\ell}|M_{\alpha\beta}(x,y,\Theta)|{\langle}u_{2}{\rangle}^{\ell}d\Theta\lesssim{\|{\langle}x{\rangle}^{2(\alpha+\beta+\ell)}V\|}_{L^{1}}.$
( 4.9)

  * •

By the triangle inequality, for all $x,y\in{\mathbb{R}}$, $\lambda\geq 0$ and
$\Theta\in{\mathbb{R}}^{2}\times[0,1]^{2}$,

$\displaystyle|e^{\lambda\Phi^{\pm}_{1}(x,y,\Theta)}|\leq
1,\quad|e^{\lambda\Phi^{\pm}_{2}(x,y,\Theta)}|\leq
e^{\lambda|x|},\quad|\Phi^{\pm}_{j}(x,y,\Theta)|\leq|u_{1}|+|u_{2}|.$ ( 4.10)

  * •

By 4.9 and 4.10, $a_{j}^{\pm}$ are smooth in $\lambda$, satisfying

$\displaystyle|\partial_{\lambda}^{\ell}a_{1}^{\pm}(\lambda,x,y)|+e^{-\lambda|x|}|\partial_{\lambda}^{\ell}a_{2}^{\pm}(\lambda,x,y)|$
$\displaystyle\lesssim{\|{\langle}x{\rangle}^{4+2\ell}V\|}_{L^{1}}$ ( 4.11)

uniformly in $x,y\in{\mathbb{R}}$ and $\lambda\geq 0$, at least for $\ell\leq
2$.

Since $\chi\in C_{0}^{\infty}({\mathbb{R}})$, $K^{\pm}_{a_{1}}$ and
$K^{\pm}_{a_{2}}$ are bounded on ${\mathbb{R}}^{2}$. In particular,

$|K^{\pm}_{a_{1}}(x,y)|+|K^{\pm}_{a_{2}}(x,y)|\lesssim
1\lesssim{\langle}|x|-|y|{\rangle}^{-2}$

if $||x|-|y||\leq 1$. Next, when $||x|-|y||\geq 1$, we apply integration by
parts twice to compute

$\displaystyle K^{\pm}_{a_{1}}(x,y)$
$\displaystyle=-\frac{1}{i(|x|\pm|y|)}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\left(\chi
a_{1}^{\pm}+\lambda\partial_{\lambda}(\chi
a_{1}^{\pm})\right)(\lambda,x,y)d\lambda$
$\displaystyle=-\frac{a_{1}^{\pm}(0,x,y)}{(|x|\pm|y|)^{2}}-\frac{1}{(|x|\pm|y|)^{2}}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\left(2\partial_{\lambda}(\chi
a_{1}^{\pm})+\lambda\partial_{\lambda}^{2}(\chi a_{1}^{\pm})\right)d\lambda$
$\displaystyle=O({\langle}|x|-|y|{\rangle}^{-2}).$

Similarly, it follows from 4.11 and the integration by parts that

$|K^{\pm}_{a_{2}}(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-2}$

Therefore, we have 4.4 for the case $\alpha,\beta\neq 0$.

Case (ii). Let $\beta=0$, $\alpha\neq 0$. As in the case (i), it follows from
2.1 and Lemma 2.5 that

$\displaystyle G^{0}_{\alpha
0}(x,y)=\int_{0}^{\infty}\lambda\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}\times[0,1]}M_{\alpha
0}(X_{1},\Theta_{1})f_{\alpha
0}(\lambda,X_{1},y-u_{2})d\Theta_{1}\right)d\lambda$

where $\Theta_{1}=(u_{1},u_{2},\theta_{1})$, $X_{1}=x-\theta_{1}u_{1}$ and

$\displaystyle M_{\alpha
0}(x,\Theta_{1})=\frac{(-1)^{\alpha}}{16(\alpha-1)!}(1-\theta_{1})^{\alpha-1}(\mathop{\mathrm{sgn}}\nolimits
x)^{\alpha}u_{1}^{\alpha}(vQ_{\alpha}BQ_{0}v)(u_{1},u_{2}).$ ( 4.12)

Define $\widetilde{a}_{j}^{\pm}(\lambda,x)$ by

$\widetilde{a}_{j}^{\pm}(\lambda,x)=\int_{{\mathbb{R}}^{2}\times[0,1]}e^{\lambda\Phi^{\pm}_{j}(x,y,\Theta_{1})}M_{\alpha
0}(X_{1},\Theta_{1})d\Theta_{1}.$

Then $M_{\alpha 0}$ and $\widetilde{a}_{j}^{\pm}$ satisfy the same estimates
as 4.9 and 4.11 for $M_{\alpha\beta}$ and $a_{j}^{\pm}$, respectively.
Moreover, $G^{0}_{\alpha 0}$ is given by a linear combination of the following
four functions

$\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\lambda\chi(\lambda)\widetilde{a}_{1}^{\pm}(\lambda,x)d\lambda,\quad\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\lambda\chi(\lambda)\widetilde{a}_{2}^{\pm}(\lambda,x)d\lambda.$

Hence, it can be shown by the same argument as in the case (i) that
$G^{0}_{\alpha 0}$ also satisfies 4.4.

Case (iii). Let $\alpha=0$, $\beta\neq 0$. Again, it follows from 2.1 and
Lemma 2.5 that

$G^{0}_{0\beta}(x,y)=\int_{0}^{\infty}\lambda\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}\times[0,1]}M_{0\beta}(Y_{2},\Theta_{2})f_{0\beta}(\lambda,x-u_{1},Y_{2})d\Theta_{2}\right)d\lambda,$

where $\Theta_{2}=(u_{1},u_{2},\theta_{2})$, $Y_{2}=y-\theta_{2}u_{2}$ and

$\displaystyle
M_{0\beta}(y,\Theta_{2})=\frac{(-1)^{\beta}}{16(\beta-1)!}(1-\theta_{2})^{\beta-1}(\mathop{\mathrm{sgn}}\nolimits
y)^{\beta}u_{2}^{\beta}(vQ_{0}BQ_{\beta}v)(u_{1},u_{2}).$ ( 4.13)

Then the same argument as above implies 4.4. This completes the proof. ∎

Next we consider the remaining term $T_{K^{0}_{4}}$ in the class (I).

######  Proposition 4.4.

$T_{K^{0}_{4}}$ satisfies the same statement as that in Proposition 4.3.

###### Proof.

We show $K^{0}_{4}=O({\langle}|x|-|y|{\rangle}^{-3/2})$ which, together with
Lemmas 3.1 and 3.2, implies the assertion. The proof is more involved than in
the previous case since $\Gamma^{0}_{4}$ depends on $\lambda$.

A similar computation as before based on Lemma 2.5 implies

$\displaystyle K^{0}_{4}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})v\Gamma_{4}^{0}(\lambda)v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda$
$\displaystyle=\int_{0}^{\infty}\lambda\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}}\widetilde{\Gamma}(\lambda,u_{1},u_{2})f_{00}(\lambda,x-u_{1},y-u_{2})du_{1}du_{2}\right)d\lambda,$

where we set
$\widetilde{\Gamma}(\lambda,u_{1},u_{2})=\frac{1}{16\lambda^{4}}(v\Gamma_{4}^{0}(\lambda)v)(u_{1},u_{2})$
for short and recall (see Lemma 2.7) that

$\displaystyle
f_{00}(\lambda,x-u_{1},y-u_{2})=-\sum_{\pm}\left(e^{i\lambda(|x-u_{1}|\pm|y-u_{2}|)}+ie^{-\lambda(|x-u_{1}|\pm
i|y-u_{2}|)}\right).$

Let $\Phi^{\pm}_{j}$ be defined by 4.7 and

$\displaystyle b^{\pm}_{j}(\lambda,x,y)$
$\displaystyle=\int_{{\mathbb{R}}^{2}}e^{\lambda\Phi^{\pm}_{j}(x,y,u_{1},u_{2},1,1)}\widetilde{\Gamma}(\lambda,u_{1},u_{2})du_{1}du_{2},$
$\displaystyle K^{\pm}_{b_{1}}(x,y)$
$\displaystyle=-\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\lambda\chi(\lambda)b_{1}^{\pm}(\lambda,x,y)d\lambda,$
$\displaystyle K^{\pm}_{b_{2}}(x,y)$
$\displaystyle=-i\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\lambda\chi(\lambda)b_{2}^{\pm}(\lambda,x,y)d\lambda.$

Then, as before,
$K^{0}_{4}=K^{+}_{b_{1}}+K^{-}_{b_{1}}+K^{+}_{b_{2}}+K^{-}_{b_{2}}$. By virtue
of 2.12, the bound $|v(x)|\lesssim{\langle}x{\rangle}^{-\mu/2}$ with
$\mu>15$ and 4.10, $b^{\pm}_{j}(\lambda,x,y)$ satisfy

$\displaystyle|\partial_{\lambda}^{\ell}b^{\pm}_{1}(\lambda,x,y)|+e^{-\lambda|x|}|\partial_{\lambda}^{\ell}b^{\pm}_{2}(\lambda,x,y)|\lesssim{\|{\langle}x{\rangle}^{2\ell}V\|}_{L^{1}}\lambda^{-\ell}$
( 4.14)

for $\lambda>0$, $x,y\in{\mathbb{R}}$ and $\ell=0,1,2$. To deal with a
possible singularity of $\partial_{\lambda}b_{j}^{\pm}$ in $\lambda\ll 1$, we
decompose $\chi$ by using the dyadic partition of unity $\\{\varphi_{N}\\}$
defined in Subsection 1.8, as

$\chi(\lambda)=\sum_{N=-\infty}^{N_{0}}\widetilde{\chi}_{N}(\lambda),\quad\widetilde{\chi}_{N}(\lambda):=\chi(\lambda)\varphi_{N}(\lambda),\quad\lambda>0,$

where $N_{0}\lesssim|\log\lambda_{0}|\lesssim 1$ since
$\mathop{\mathrm{supp}}\nolimits\chi\subset[0,\lambda_{0}]$. Note that
$\mathop{\mathrm{supp}}\nolimits\widetilde{\chi}_{N}\subset[2^{N-2},2^{N}]$
and

$\displaystyle|\partial_{\lambda}^{\ell}\widetilde{\chi}_{N}(\lambda)|\leq
C_{\ell}2^{-N\ell}$ ( 4.15)

for all $\ell$. Let $K^{\pm}_{b_{j},N}$ is given by $K^{\pm}_{b_{j}}$ with
$\chi$ replaced by $\widetilde{\chi}_{N}$ and decompose $K^{\pm}_{b_{j}}$ as

$K^{\pm}_{b_{j}}=\sum_{N\leq N_{0}}K^{\pm}_{b_{j},N}.$

Since $\lambda\sim 2^{N}$ on
$\mathop{\mathrm{supp}}\nolimits\widetilde{\chi}_{N}$, we know by 4.14 that

$|K^{\pm}_{b_{j},N}(x,y)|\lesssim
2^{N}\int_{\mathop{\mathrm{supp}}\nolimits\widetilde{\chi}_{N}}d\lambda\lesssim
2^{2N},\quad x,y\in{\mathbb{R}}.$

In particular, if $||x|-|y||\leq 1$ then

$|K^{\pm}_{b_{j},N}(x,y)|\lesssim 2^{2N}{\langle}|x|-|y|{\rangle}^{-2}.$

On the other hand, when $||x|-|y||>1$, we obtain by integrating by parts twice
that

$\displaystyle
K^{\pm}_{b_{1},N}(x,y)=-\frac{1}{(|x|\pm|y|)^{2}}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\Big{[}2\partial_{\lambda}(\widetilde{\chi}_{N}b^{\pm}_{1})+\lambda\partial_{\lambda}^{2}(\widetilde{\chi}_{N}b^{\pm}_{1})\Big{]}d\lambda$

since $\widetilde{\chi}_{N}(0)=0$. Then 4.14, 4.15 and the support
property of $\widetilde{\chi}_{N}$ imply

$\displaystyle|K^{\pm}_{b_{1},N}(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-2}2^{-N}\int_{2^{N-2}}^{2^{N}}d\lambda\lesssim{\langle}|x|-|y|{\rangle}^{-2}$

if $||x|-|y||>1$. Therefore, $K^{\pm}_{b_{1},N}(x,y)$ satisfies

$|K^{\pm}_{b_{1},N}(x,y)|\lesssim\min\\{2^{2N},{\langle}|x|-|y|{\rangle}^{-2}\\}\lesssim
2^{2N(1-\theta)}{\langle}|x|-|y|{\rangle}^{-2\theta},\quad\theta\in[0,1],$

uniformly in $N\leq N_{0}$, $x,y\in{\mathbb{R}}$. In particular, taking
$\theta=3/4$ for instance, we obtain

$\displaystyle|K^{\pm}_{b_{1}}(x,y)|\lesssim{\langle}|x|-|y|{\rangle}^{-3/2}\sum_{N\leq
N_{0}}2^{N/2}\lesssim{\langle}|x|-|y|{\rangle}^{-3/2}.$

It follows similarly from 4.14, 4.15 and the support property of
$\widetilde{\chi}_{N}$ that

$\displaystyle|K^{\pm}_{b_{2}}(x,y)|\lesssim{\langle}|x|+|y|{\rangle}^{-3/2}.$

Therefore, $K^{0}_{4}(x,y)=O({\langle}|x|-|y|{\rangle}^{-3/2})$ and the result
follows by Lemmas 3.1 and 3.2. ∎

Next we deal with the class (II), namely $T_{K^{0}_{1}}$ and $T_{K^{0}_{33}}$.
We begin with $T_{K^{0}_{33}}$.

######  Proposition 4.5.

If $1<p<\infty$, $w_{p}\in A_{p}$ and $w_{1}\in A_{1}$ then $T_{K^{0}_{33}}$
and $T_{K^{0}_{33}}^{*}$ satisfy the same bounds as 3.1 and 3.2. Moreover,
$T_{K^{0}_{33}},T_{K^{0}_{33}}^{*}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$.

###### Proof.

We shall show that $K^{0}_{33}$ is written in the form

$\displaystyle
K^{0}_{33}(x,y)=\frac{-1+i}{8}g_{1}^{+}(x,y)+O({\langle}|x|-|y|{\rangle}^{-2}),$
( 4.16)

with $g_{1}^{+}=g_{1,1,1}^{+}$ defined in Lemma 3.5 with the choice of
$a=b=1$. Then Lemmas 3.1–3.5 apply to $T_{K^{0}_{33}}$, yielding the desired
assertion. As before, using 2.1 and Lemma 2.7, we have

$\displaystyle K^{0}_{33}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{6}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})v\widetilde{P}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda$
$\displaystyle=\frac{1}{16}\int_{0}^{\infty}\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}}(v\widetilde{P}v)(u_{1},u_{2})f_{00}(\lambda,x-u_{1},y-u_{2})du_{1}du_{2}\right)d\lambda$
$\displaystyle=K^{+}_{33,1}(x,y)+K^{-}_{33,1}(x,y)+K^{+}_{33,2}(x,y)+K^{-}_{33,2}(x,y),$

where, using $\Phi^{\pm}_{j}$ defined by 4.8, we set

$\displaystyle c_{j}^{\pm}(\lambda,x,y)$
$\displaystyle=\frac{1}{16}\int_{{\mathbb{R}}^{2}}e^{\lambda\Phi^{\pm}_{j}(x,y,u_{1},u_{2},1,1)}(v\widetilde{P}v)(u_{1},u_{2})du_{1}du_{2},$
$\displaystyle K^{\pm}_{33,1}(x,y)$
$\displaystyle=-\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\chi(\lambda)c_{1}^{\pm}(\lambda,x,y)d\lambda,$
$\displaystyle K^{\pm}_{33,2}(x,y)$
$\displaystyle=-i\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\chi(\lambda)c_{2}^{\pm}(\lambda,x,y)d\lambda.$

By 4.2 and 4.10, $c_{j}^{\pm}$ satisfy a similar estimates as that for
$a_{j}^{\pm}$: for $x,y\in{\mathbb{R}}$, $\lambda\geq 0$,

$\displaystyle|\partial_{\lambda}^{\ell}c_{1}^{\pm}(\lambda,x,y)|+e^{-\lambda|x|}|\partial_{\lambda}^{\ell}c_{2}^{\pm}(\lambda,x,y)|$
$\displaystyle\lesssim{\|{\langle}x{\rangle}^{2\ell}V\|}_{L^{1}}\lesssim
1,\quad\ell=0,1,2.$

Hence, since $\chi\in C_{0}^{\infty}$, $K^{0}_{33}$ is bounded on
${\mathbb{R}}^{2}$. Let $\psi(||x|\pm|y||^{2})$ be defined in Lemma 3.3 such
that $\psi(||x|\pm|y||^{2})$ is supported away from the region
$\\{|x|\pm|y|\leq 1\\}$. We decompose

$K^{\pm}_{33,1}=\psi_{\pm}K^{\pm}_{33,1}+\left(1-\psi_{\pm}\right)K^{\pm}_{33,1},$

where $\psi_{\pm}:=\psi(||x|\pm|y||^{2})$ for short. The second part of the
right hand side satisfies

$|\left(1-\psi_{\pm}\right)K^{\pm}_{33,1}(x,y)|\lesssim
1\lesssim{\langle}|x|-|y|{\rangle}^{-2},\quad x,y\in{\mathbb{R}}.$

To estimate the first term, we recall that
$\widetilde{P}=\frac{-2(1+i)}{{\|V\|}_{L^{1}}^{2}}{\langle}\cdot,v{\rangle}v$
and hence, for all $x,y,j$,

$c^{\pm}_{j}(0,x,y)=\frac{1}{16}\int_{{\mathbb{R}}^{2}}(v\widetilde{P}v)(u_{1},u_{2})du_{1}du_{2}=-\frac{1+i}{8}.$

Then we obtain by integration by parts twice that

$\displaystyle\psi_{\pm}K^{\pm}_{33,1}(x,y)$
$\displaystyle=\frac{-1+i}{8}\frac{\psi_{\pm}}{|x|\pm|y|}+\frac{\psi_{\pm}}{i(|x|\pm|y|)}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\partial_{\lambda}(\chi
c_{1}^{\pm})(\lambda,x,y)d\lambda$
$\displaystyle=\frac{-1+i}{8}\frac{\psi_{\pm}}{|x|\pm|y|}+\frac{\psi_{\pm}\partial_{\lambda}(\chi
c_{1}^{\pm})(0,x,y)}{(|x|\pm|y|)^{2}}$
$\displaystyle\quad\quad-\frac{\psi_{\pm}}{(|x|\pm|y|)^{2}}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\partial_{\lambda}^{2}(\chi
c_{1}^{\pm})(\lambda,x,y)d\lambda$
$\displaystyle=\frac{-1+i}{8}\frac{\psi_{\pm}}{|x|\pm|y|}+O({\langle}|x|-|y|{\rangle}^{-2}).$

Decomposing $K^{\pm}_{33,2}$ as
$K^{\pm}_{33,2}=\psi_{-}K^{\pm}_{33,2}+(1-\psi_{-})K^{\pm}_{33,2},$ we
similarly have

$K^{\pm}_{33,2}(x,y)=\frac{-1+i}{8}\frac{\psi_{-}}{|x|\pm
i|y|}+O({\langle}|x|-|y|{\rangle}^{-2}).$

Therefore, 4.16 follows. This completes the proof. ∎

It remains to deal with the most technical and delicate term $T_{K^{0}_{1}}$.

######  Proposition 4.6.

For any $1<p<\infty$ and $w_{p}\in A_{p}$, $T_{K^{0}_{1}}$ and
$T_{K^{0}_{1}}^{*}$ satisfy the same bound as 3.1. Moreover, $T_{K^{0}_{1}}$
satisfies the following statements:

  * (1)

If $V$ is compactly supported, then
$T_{K^{0}_{1}},T_{K^{0}_{1}}^{*}\in\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$;

  * (2)

If $Q_{1}A_{1}^{0}Q_{1}$ is finite rank, then $T_{K^{0}_{1}}$ and
$T_{K^{0}_{1}}^{*}$ satisfy the same bound as 3.2;

  * (3)

$T_{K^{0}_{1}},T_{K^{0}_{1}}^{*}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$.

###### Proof of Proposition 4.6.

The proof is divided into five steps.

Step 1. We first derive a useful asymptotic formula of $K^{0}_{1}$. Using
Lemma 2.5, we compute

$\displaystyle K^{0}_{1}(x,y)$
$\displaystyle=\int_{0}^{\infty}\lambda^{4}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{1}A_{1}^{0}Q_{1}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda$
$\displaystyle=\int_{0}^{\infty}\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{11}(X_{1},Y_{2},\Theta)f_{11}(\lambda,X_{1},Y_{2})d\Theta\right)d\lambda,$
( 4.17)

where
$\Theta=(u_{1},u_{2},\theta_{2},\theta_{2})\in{\mathbb{R}}^{2}\times[0,1]^{2}$,
$X_{1}=x-\theta_{1}u_{1}$, $Y_{2}=y-\theta_{2}u_{2},$ and $M_{11}$ is given by
4.6 with $B=A^{0}_{1}$. Since $f_{11}$ is bounded on
${\mathbb{R}}_{+}\times{\mathbb{R}}^{2}$, $\chi\in C_{0}^{\infty}$ and
$M_{\alpha\beta}$ satisfies 4.9, $K^{0}_{1}$ is absolutely convergent and
bounded on ${\mathbb{R}}^{2}$. Fubini’s theorem then yields

$\displaystyle K^{0}_{1}(x,y)$
$\displaystyle=\int_{{\mathbb{R}}\times[0,1]}\left(\int_{0}^{\infty}\chi(\lambda)\left(\int_{{\mathbb{R}}\times[0,1]}M_{11}(X_{1},Y_{2},\Theta)f_{11}(\lambda,X_{1},Y_{2})du_{2}d\theta_{2}\right)d\lambda\right)du_{1}d\theta_{1}$
$\displaystyle=\int_{{\mathbb{R}}\times[0,1]}\widetilde{K}^{0}_{1}(x,y;\theta_{1},u_{1})du_{1}d\theta_{1},$
( 4.18)

where

$\displaystyle\widetilde{K}^{0}_{1}(x,y;\theta_{1},u_{1})=\int_{0}^{\infty}\chi(\lambda)\left(\int_{{\mathbb{R}}\times[0,1]}M_{11}(X_{1},Y_{2},\Theta)f_{11}(\lambda,X_{1},Y_{2})du_{2}d\theta_{2}\right)d\lambda.$
( 4.19)

Now we shall show that $\widetilde{K}^{0}_{1}$ is of the form

$\displaystyle\widetilde{K}^{0}_{1}(x,y;\theta_{1},u_{1})=\mathop{\mathrm{sgn}}\nolimits(X_{1})g_{1}^{-}(x,y)m_{1}(y,u_{1},\theta_{1})+O\left({\langle}|x|-|y|{\rangle}^{-2}\rho_{3}(u_{1})\right),$
( 4.20)

where $g_{1}^{-}=g_{1,i,1}^{-}$ is given in Lemma 3.5 (with $a=i,b=1$) and
$\widetilde{m}_{1},\rho_{\ell}$ are given by

$\displaystyle m_{1}(y,u_{1},\theta_{1})$
$\displaystyle:=\int_{{\mathbb{R}}\times[0,1]}\frac{M_{11}(X_{1},Y_{2},\Theta)}{\mathop{\mathrm{sgn}}\nolimits
X_{1}}du_{2}d\theta_{2}$
$\displaystyle=\frac{1}{16}\int_{{\mathbb{R}}\times[0,1]}(\mathop{\mathrm{sgn}}\nolimits
Y_{2})u_{1}u_{2}(vQ_{1}A_{1}^{0}Q_{1}v)(u_{1},u_{2})du_{2}d\theta_{2},$

and, for $\ell=0,1,2,...$,

$\rho_{\ell}(u_{1}):=\frac{1}{16}{\langle}u_{1}{\rangle}^{\ell}\int_{{\mathbb{R}}}|(vQ_{1}A_{1}^{0}Q_{1}v)(u_{1},u_{2})|{\langle}u_{2}{\rangle}^{\ell}du_{2}.$

Note that $|m_{1}(y,u_{1},\theta_{1})|\leq\rho_{1}(u_{1})$. To prove 4.20,
we set

$\displaystyle d^{\pm}_{j}(\lambda,x,y;u_{1},\theta_{1})$
$\displaystyle=\int_{{\mathbb{R}}\times[0,1]}e^{\lambda\Phi^{\pm}_{j}(x,y,\Theta)}M_{11}(X_{1},Y_{2},\Theta)du_{2}d\theta_{2},$
$\displaystyle K^{\pm}_{1,1}(x,y;u_{1},\theta_{1})$
$\displaystyle=\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\chi(\lambda)d^{\pm}_{1}(\lambda,x,y;u_{1},\theta_{1})d\lambda,$
$\displaystyle K^{\pm}_{1,2}(x,y;u_{1},\theta_{1})$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda(|x|\pm
i|y|)}\chi(\lambda)d^{\pm}_{2}(\lambda,x,y;u_{1},\theta_{1})d\lambda,$

where $\Phi^{\pm}_{j}$ are defined by 4.7. It follows from 4.19 and Lemma
2.7 with $\alpha=\beta=1$ that

$\displaystyle\widetilde{K}^{0}_{1}(x,y;u_{1},\theta_{1})=K^{+}_{1,1}-K^{-}_{1,1}+K^{+}_{1,2}-K^{-}_{1,2}.$

Moreover, since
$d^{\pm}_{j}(0,x,y;u_{1},\theta_{1})=\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}(y,u_{1},\theta_{1})$,
4.10 and 4.2 imply

$\displaystyle|\partial_{\lambda}^{\ell}d^{\pm}_{1}(\lambda,x,y;u_{1},\theta_{1})|+e^{-\lambda|x|}|\partial_{\lambda}^{\ell}d^{\pm}_{2}(\lambda,x,y;u_{1},\theta_{1})|\lesssim\rho_{\ell+1}(u_{1})$
( 4.21)

uniformly in $\lambda\geq 0$, $x,y\in{\mathbb{R}}$, $u_{1}\in{\mathbb{R}}$ and
$\theta_{1}\in[0,1]$. Hence $K^{\pm}_{1,j}$ satisfy

$\displaystyle|K^{\pm}_{1,j}(x,y;u_{1},\theta_{1})|\lesssim\rho_{1}(u_{1})$ (
4.22)

uniformly in $x,y,u_{1}$ and $\theta_{1}$. We next let
$\psi_{\pm}=\psi(||x|\pm|y||^{2})$ be as in Lemma 3.3 and apply integration by
parts twice to $\psi_{\pm}K^{\pm}_{1,1}$ as in the previous case, obtaining

$\displaystyle\psi_{\pm}K^{\pm}_{1,1}$
$\displaystyle=-\frac{\psi_{\pm}\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}}{i(|x|\pm|y|)}-\frac{\psi_{\pm}}{i(|x|\pm|y|)}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\partial_{\lambda}(\chi
d^{\pm}_{1})d\lambda$
$\displaystyle=-\frac{\psi_{\pm}\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}}{i(|x|\pm|y|)}-\frac{\psi_{\pm}\partial_{\lambda}(\chi
d^{\pm}_{1})|_{\lambda=0}}{(|x|\pm|y|)^{2}}-\frac{\psi_{\pm}}{(|x|\pm|y|^{2})}\int_{0}^{\infty}e^{i\lambda(|x|\pm|y|)}\partial_{\lambda}^{2}(\chi
d^{\pm}_{1})d\lambda$
$\displaystyle=-\frac{\psi_{\pm}\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}}{i(|x|\pm|y|)}+O({\langle}|x|-|y|{\rangle}^{-2}\rho_{3}(u_{1})).$

The same calculation and 4.21 also yield

$\displaystyle\psi_{-}K^{\pm}_{1,2}=\frac{\psi_{-}\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}}{|x|\pm
i|y|}+O({\langle}|x|-|y|{\rangle}^{-2}\rho_{3}(u_{1})).$

Moreover, since $1-\psi_{\pm}$ is supported in $\\{|x|\pm|y|\leq 1\\}$, we
know by 4.22 that $(1-\psi_{\pm})K^{\pm}_{1,1}$ and
$(1-\psi_{-})K^{\pm}_{1,2}$ are dominated by
${\langle}|x|-|y|{\rangle}^{-2}\rho_{3}(u_{1})$. Therefore, we have

$\displaystyle\widetilde{K}^{0}_{1}$
$\displaystyle\equiv\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}(y,u_{1},\theta_{1})\left(-i\frac{\psi_{+}}{(|x|+|y|)}+i\frac{\psi_{-}}{(|x|-|y|)}-\frac{\psi_{-}}{(|x|+i|y|)}+\frac{\psi_{-}}{(|x|-i|y|)}\right)$
$\displaystyle\equiv\mathop{\mathrm{sgn}}\nolimits(X_{1})m_{1}(y,u_{1},\theta_{1})g_{1}^{-}(x,y)$

modulo the error term $O({\langle}|x|-|y|{\rangle}^{-2}\rho_{3}(u_{1}))$ and
4.20 thus follows.

Step 2: Proof of 3.1. Let
$T_{u_{1},\theta_{1}}=T_{\widetilde{K}^{0}_{1}(\cdot,\cdot,u_{1},\theta_{1})}$
be the integral operator with kernel
$\widetilde{K}^{0}_{1}(x,y,u_{1},\theta_{1})$, where $u_{1},\theta_{1}$ are
considered as parameters. We apply Fubini’s theorem and Minkowski’s integral
inequality (which holds for any $\sigma$-finite measures) to 4.18, obtaining

$\displaystyle{\|T_{K^{0}_{1}}f\|}_{L^{p}(w_{p})}\lesssim\int_{{\mathbb{R}}\times[0,1]}{\|T_{u_{1},\theta_{1}}f\|}_{L^{p}(w_{p})}du_{1}d\theta_{1},\quad
1\leq p<\infty.$ ( 4.23)

Thanks to 4.20, the main term of $T_{u_{1},\theta_{1}}$ is the composition
$\mathop{\mathrm{sgn}}\nolimits(X_{1})T_{g_{1}^{-}}m_{1}$. Moreover, since
$|m_{1}(y,u_{1},\theta_{1})|\leq\rho_{1}(u_{1})$, the multiplication by
$m_{1}$ is bounded on $L^{p}(w_{p})$ for any $1\leq p<\infty$ with the
operator norm at most $\rho_{1}(u_{1})$. It thus follows from Lemmas 3.2 and
3.3 that

$\displaystyle{\|T_{u_{1},\theta_{1}}f\|}_{L^{p}(w_{p})}$
$\displaystyle\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}}\rho_{3}(u_{1})\left({\|f\|}_{L^{p}(w_{p})}+{\|\tau
f\|}_{L^{p}(w_{p})}\right),\quad 1<p<\infty,$

where $\tau f(x)=f(-x)$. Since $\rho_{3}(u_{1})\in L^{1}({\mathbb{R}})$ by the
assumption on $V$ and 4.2, the desired the bound 3.1 for $T_{K^{0}_{1}}$
follow by applying this bound to 4.23. By the same argument, we also obtain
the same the bounds 3.1 for its adjoint $T_{K^{0}_{1}}^{*}$.

Step 3: Proof of (1). Suppose $\mathop{\mathrm{supp}}\nolimits
V\subset\\{|x|\leq r\\}$ with some $r>0$. Integrating 4.20 over
$(u_{1},\theta_{1})\in{\mathbb{R}}\times[0,1]$ and using 4.18, we have

$\displaystyle
K^{0}_{1}(x,y)=g_{1}^{-}(x,y)\widetilde{m}_{1}(x,y)+O({\langle}|x|-|y|{\rangle}^{-2}),$
( 4.24)

where $\widetilde{m}_{1}$ is given by

$\displaystyle\widetilde{m}_{1}(x,y)=\frac{1}{16}\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}(\mathop{\mathrm{sgn}}\nolimits
X_{1})(\mathop{\mathrm{sgn}}\nolimits
Y_{2})u_{1}u_{2}(vQ_{1}A_{1}^{0}Q_{1}v)(u_{1},u_{2})du_{1}du_{2}d\theta_{1}d\theta_{2}.$
( 4.25)

Hence it is enough to prove
$T_{\widetilde{m}_{1}g_{1}^{-}}\in\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$.
We decompose it as

$T_{\widetilde{m}_{1}g_{1}^{-}}=\mathds{1}_{\\{|x|\geq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}+\mathds{1}_{\\{|x|\leq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}.$

For the first part, since $|\theta_{1}u_{1}|\leq r$ for
$u_{1}\in\mathop{\mathrm{supp}}\nolimits v=\mathop{\mathrm{supp}}\nolimits V$
and $0\leq\theta_{1}\leq 1$, we have $\mathop{\mathrm{sgn}}\nolimits
X_{1}=\mathop{\mathrm{sgn}}\nolimits(x-\theta_{1}u_{1})=\mathop{\mathrm{sgn}}\nolimits
x$ if $|x|\geq r+1$, and hence

$\mathds{1}_{\\{|x|\geq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}=\mathds{1}_{\\{|x|\geq
r+1\\}}\cdot\mathop{\mathrm{sgn}}\nolimits x\cdot
T_{g_{1}^{-}}\cdot\widetilde{m}_{2},$

where $\widetilde{m}_{2}(y):=(\mathop{\mathrm{sgn}}\nolimits
x)^{-1}\widetilde{m}_{1}(x,y)$ depends only on $y$ and is bounded on
${\mathbb{R}}$. Hence, we obtain $\mathds{1}_{\\{|x|\geq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}\in\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$
by Lemma 3.5. In fact, we also obtain the same weighted weak-type bound as
3.2 for $\mathds{1}_{\\{|x|\geq r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}$. For
the second term, we set

$E_{\lambda}=\\{x\in{\mathbb{R}}\ |\ |\mathds{1}_{\\{|x|\leq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}f(x)|>\lambda\\}$

for $f\in L^{1}({\mathbb{R}})$. Since $\widetilde{m}_{1}g_{1}^{-}$ is bounded
on ${\mathbb{R}}$, we obtain

$|\mathds{1}_{\\{|x|\leq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}f(x)|\lesssim\|f\|_{L^{1}({\mathbb{R}})}$

We also have $|E_{\lambda}|\lesssim r$ thanks to the restriction
$\mathds{1}_{\\{|x|\leq r+1\\}}$. Thus,

$\|\mathds{1}_{\\{|x|\leq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}f\|_{L^{1,\infty}({\mathbb{R}})}\lesssim\sup_{\lambda>0}\lambda|E_{\lambda}|\lesssim\|f\|_{L^{1}({\mathbb{R}})}.$

This completes the proof of the item (1).

Step 4: Proof of (2). Suppose $Q_{1}A_{1}^{0}Q_{1}$ is finite rank. The proof
for this case is almost analogous to that for $\mathds{1}_{\\{|x|\geq
r+1\\}}T_{\widetilde{m}_{1}g_{1}^{-}}$. Indeed, we can write

$(Q_{1}A_{1}^{0}Q_{1})(u_{1},u_{2})=\sum_{i,j=1}^{N}a_{ij}\varphi_{i}(u_{1})\overline{\varphi_{j}(u_{2})}$

with some $\varphi_{j}\in L^{2}({\mathbb{R}})$, $a_{ij}\in{\mathbb{C}}$ and
$N<\infty$. With this expression, we can apply Fubini’s theorem in 4.25 to
compute the $(u_{1},\theta_{1})$-integral and $(u_{2},\theta_{2})$-integral
separately, and obtain

$\widetilde{m}_{1}(x,y)=\sum_{i,j=1}^{N}a_{ij}c_{i}(x)\overline{c_{j}(y)},\quad
c_{i}(x)=\frac{1}{4}\int_{{\mathbb{R}}\times[0,1]}(\mathop{\mathrm{sgn}}\nolimits
X_{1})u_{1}v(u_{1})\varphi_{i}(u_{1})du_{1}d\theta_{1}\in
L^{\infty}({\mathbb{R}}).$

Hence, the same argument as above yields the bound 3.2 for $T_{K_{0}^{1}}$
and $T_{K_{0}^{1}}^{*}$.

Step 5: Proof of (3). In order to prove the item (3), it is enough to show
$T_{K^{0}_{1}},T_{K^{0}_{1}}^{*}\in\mathbb{B}({\mathcal{H}}^{1},L^{1})$ by the
duality. For that purpose, 4.20 is not useful since the multiplication by
$m_{1}$ does not leave ${\mathcal{H}}^{1}$ invariant since
$f\in{\mathcal{H}}^{1}$ must satisfy $\int_{\mathbb{R}}f(x)dx=0$. Instead, we
use a simple trick based on the translation invariance. Let

$k^{0}_{1}(x,y)=\mathop{\mathrm{sgn}}\nolimits(x)\mathop{\mathrm{sgn}}\nolimits(y)\int_{0}^{\infty}\chi(\lambda)f_{11}(\lambda,x,y)d\lambda.$

Then, recalling the formula 4.1, we have

$\displaystyle
K^{0}_{1}(x,y)=\frac{1}{16}\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}u_{1}u_{2}(vQ_{1}A^{0}_{1}Q_{1}v)(u_{1},u_{2})k^{0}_{1}(x-\theta_{1}u_{1},y-\theta_{2}u_{2})d\Theta.$
( 4.26)

Since the $L^{1}$-norm and ${\mathcal{H}}^{1}$-norm are invariant under the
translation $f\mapsto f(\cdot-u)$, assuming
$T_{k^{0}_{1}}\in\mathbb{B}({\mathcal{H}}^{1},L^{1})$, we obtain by the change
of variables $x\mapsto x+\theta_{1}u_{1}$ and $y\mapsto y+\theta_{2}u_{2}$
that

$\displaystyle{\|T_{K^{0}_{1}}f\|}_{L^{1}}$
$\displaystyle\leq\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}{\langle}u_{1}{\rangle}{\langle}u_{2}{\rangle}|(vQ_{1}A^{0}_{1}Q_{1}v)(u_{1},u_{2})|{\|T_{k^{0}_{1}(\cdot-\theta_{1}u_{1},\cdot-\theta_{2}u_{2})}f\|}_{L^{1}}d\Theta$
$\displaystyle\lesssim\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}{\langle}u_{1}{\rangle}{\langle}u_{2}{\rangle}|(vQ_{1}A^{0}_{1}Q_{1}v)(u_{1},u_{2})|{\|f(\cdot+\theta_{2}u_{2}\|}_{{\mathcal{H}}^{1}}d\Theta$
$\displaystyle\lesssim{\|f\|}_{{\mathcal{H}}^{1}}.$

The same argument also applies to $T_{K^{0}_{1}}^{*}$. It remains to show
$T_{k^{0}_{1}},T_{k^{0}_{1}}^{*}\in\mathbb{B}({\mathcal{H}}^{1},L^{1})$. By a
similar argument as in the Step 1 based on Lemma 2.7, one can obtain

$\displaystyle
k^{0}_{1}(x,y)=g_{4}^{-}(x,y)+O({\langle}|x|-|y|{\rangle}^{-2}),$ ( 4.27)

where $g_{4}^{-}=g_{4,i,1}^{-}$ are defined in Lemma 3.5 with the choice of
$a=i$ and $b=1$. Moreover, the kernel of $T_{k^{0}_{1}}^{*}$ is given by

$\overline{k^{0}_{1}(y,x)}=\overline{g_{4,i,1}^{-}(y,x)}+O({\langle}|x|-|y|{\rangle}^{-2})=g_{4,-i,i}^{+}(x,y)+O({\langle}|x|-|y|{\rangle}^{-2}).$

Therefore, Lemmas 3.1 and 3.5 imply
$T_{k^{0}_{1}},T_{k^{0}_{1}}^{*}\in\mathbb{B}({\mathcal{H}}^{1},L^{1})$. This
completes the proof. ∎

By Propositions 4.3–4.6 and 4.1, we have obtained for the regular case and
$1<p<\infty$,

$W_{-}^{L},(W_{-}^{L})^{*}\in\mathbb{B}(L^{p}({\mathbb{R}}))\cap\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$

as well as the weighted estimate

${\|W_{-}^{L}f\|}_{L^{p}(w_{p})}+{\|(W_{-}^{L})^{*}f\|}_{L^{p}(w_{p})}\lesssim[w_{p}]_{A_{p}}^{\max\\{1,1/(p-1)\\}}\left({\|f\|}_{L^{p}(w_{p})}+{\|\tau
f\|}_{L^{p}(w_{p})}\right)$

We have also proved
$W_{\pm},W_{\pm}^{*}\in\mathbb{B}(L^{1}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$
if $V$ is compactly supported, and

${\|W_{-}^{L}f\|}_{L^{1,\infty}(w_{1})}+{\|(W_{-}^{L})^{*}f\|}_{L^{1,\infty}(w_{1})}\lesssim[w_{1}]_{A_{1}}(1+\log[w]_{A_{1}})({\|f\|}_{L^{1}(w_{1})}+{\|\tau
f\|}_{L^{1}(w_{1})})$

if $Q_{1}A_{1}^{0}Q_{1}$ is finite rank. This completes the proof of Theorem
4.1 for the regular case.

###  4.2. First kind resonant case

Next we consider the case when zero is a first kind resonance of $H$ and
$|V(x)|\lesssim{\langle}x{\rangle}^{-\mu}$ with $\mu>21$. By 2.7 and (ii),
$W^{L}_{-}$ is of the form

$\displaystyle
W^{L}_{-}=T_{K^{1}_{-1}}+\sum_{j=1}^{2}T_{K^{1}_{0j}}+\sum_{j=1}^{3}T_{K^{1}_{1j}}+\sum_{j=1}^{2}T_{K^{1}_{2j}}+\sum_{j=1}^{3}T_{K^{1}_{3j}}+T_{K^{1}_{4}},$
( 4.28)

where

$\displaystyle K^{1}_{-1}(x,y)$
$\displaystyle:=\int_{0}^{\infty}\lambda^{2}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{2}^{0}A_{-1}^{1}Q_{2}^{0}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
$\displaystyle K^{1}_{kj}(x,y)$
$\displaystyle:=\int_{0}^{\infty}\lambda^{3+k}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vB_{kj}^{1}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
$\displaystyle K^{1}_{4}(x,y)$
$\displaystyle:=\int_{0}^{\infty}\lambda^{3}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})v\Gamma_{4}^{1}(\lambda)v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda$

with $k=0,1,2$ and

  * •

$B_{01}^{1}=Q_{2}A_{01}^{1}Q_{1}$ and $B_{02}^{1}=Q_{1}A_{02}^{1}Q_{2}$;

  * •

$B_{1}^{1}=Q_{1}A_{11}^{1}Q_{1}$, $B_{12}^{1}=Q_{2}A_{12}^{1}$ and
$B_{13}^{1}=A_{13}^{1}Q_{2}$;

  * •

$B_{21}^{1}=Q_{1}A_{21}^{1}$ and $B_{22}^{1}=A_{22}^{1}Q_{1}$;

  * •

$B_{31}^{1}=Q_{1}A_{31}^{1}$, $B_{32}^{1}=A_{32}^{1}Q_{1}$ and
$B_{33}^{1}=\widetilde{P}$.

For any $B\in\\{Q_{2}^{0}A_{-1}^{1}Q_{2}^{0},B_{kj}^{1},\Gamma_{4}^{1}\\}$ and
$k\leq 8$, $vBv$ is an integral operator satisfying the bounds 4.2. As in
the regular case, Theorem 4.1 for the second kind resonant case follows from
the following proposition.

######  Proposition 4.7.

Let $1<p<\infty$, $w_{p}\in A_{p}$ and $w_{1}\in A_{1}$. Then all the integral
operators in 4.28 satisfy the same bound as 3.1 and belong to
$\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$.
Moreover, we have:

  * •

these operators also belong to
$\mathbb{B}(L^{\infty}({\mathbb{R}}),L^{1,\infty}({\mathbb{R}}))$ if $V$ is
compactly supported;

  * •

$T_{K_{31}^{1}},T_{K_{32}^{1}}$ and $T_{K_{4}^{1}}$ in fact belong to
$\mathbb{B}(L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}))$.

###### Proof.

The proof is essentially same as that of the regular case, so we only give a
brief outline. Recall that we do not distinguish $Q_{2},Q_{2}^{0}$ and use the
same notation $Q_{2}$ to denote them.

At first, $T_{K_{33}^{1}}=T_{K_{33}^{0}}$. Moreover, $K_{31}^{1}$ and
$K_{32}^{1}$ are written in the form 4.3 with some $B\in\mathbb{B}(L^{2})$
such that $Q_{\alpha}BQ_{\beta}$ is absolutely bounded, and
$(\alpha,\beta)=(1,0)$ for $K_{31}^{1}$ and $(\alpha,\beta)=(0,1)$ for
$K_{32}^{1}$. Hence the proofs for $K_{31}^{1}$ and $K_{32}^{1}$ are
completely same as that of Proposition 4.3. The proof for $T_{K_{4}^{1}}$ is
also completely same as that for $T_{K_{4}^{0}}$ since $\Gamma_{4}^{1}$
satisfies the same estimates as $\Gamma_{4}^{0}$ (see 2.12).

Next, for the other cases, precisely for the operators
$T_{K^{1}_{-1}},T_{K^{1}_{0j}},T_{K^{1}_{1j}},T_{K^{1}_{2j}}$, the
corresponding kernel is written in the following form:

$\displaystyle\int_{0}^{\infty}\lambda^{6-\alpha-\beta}\chi(\lambda)\Big{(}R_{0}^{+}(\lambda^{4})vQ_{\alpha}BQ_{\beta}v[R_{0}^{+}-R_{0}^{-}](\lambda^{4})\Big{)}(x,y)d\lambda,$
( 4.29)

where $Q_{\alpha}BQ_{\beta}$ is absolutely bounded and

$\displaystyle(\alpha,\beta)=\begin{cases}(2,2)&\text{for}\quad
K=K^{1}_{-1},\\\ (2,1)&\text{for}\quad K=K^{1}_{01},\\\ (1,2)&\text{for}\quad
K=K^{1}_{02},\\\ (1,1)&\text{for}\quad
K=K^{1}_{11},\end{cases}\quad(\alpha,\beta)=\begin{cases}(2,0)&\text{for}\quad
K=K^{1}_{12},\\\ (0,2)&\text{for}\quad K=K^{1}_{13},\\\ (1,0)&\text{for}\quad
K=K^{1}_{21},\\\ (0,1)&\text{for}\quad K=K^{1}_{22}.\end{cases}$ ( 4.30)

Recall that $X_{1}=x-\theta_{1}u_{1}$, $Y_{2}=y-\theta_{2}u_{2}$,
$\Theta=(u_{1},u_{2},\theta_{1},\theta_{2})$,
$\Theta_{j}=(u_{1},u_{2},\theta_{j})$. Let
$M_{\alpha\beta}(X_{1},Y_{2},\Theta)$, $M_{\alpha 0}(X_{1},\Theta_{1})$ and
$M_{0\beta}(Y_{2},\Theta_{2})$ be as in 4.6, 4.12 and 4.13,
respectively. For simplicity, without any confusion, we shall use the same
notation $M_{\alpha\beta}(X_{1},Y_{2},\Theta)$ to denote $M_{\alpha
0}(X_{1},\Theta_{1})$ and $M_{0\beta}(Y_{2},\Theta_{2})$ by regarding
$M_{\alpha 0}(X_{1},\Theta_{1})$ (resp. $M_{0\beta}(Y_{2},\Theta_{2}))$ as a
constant function of $y,\theta_{2}$ (resp. $x,\theta_{1}$). Let
$G^{1}_{\alpha\beta}(x,y)$ be the function given by 4.29. Using
$f_{\alpha\beta}$ defined in Lemma 2.7, we have

$\displaystyle
G^{1}_{\alpha\beta}(x,y)=\int_{0}^{\infty}\chi(\lambda)\left(\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{\alpha\beta}(X_{1},Y_{2},\Theta)f_{\alpha\beta}(\lambda,X_{1},Y_{2})d\Theta\right)d\lambda.$
( 4.31)

We consider the two cases (i) $(\alpha,\beta)\neq(1,1)$ and (ii)
$(\alpha,\beta)=(1,1)$, separately. It will be seen that the proof for the
case case (i) (resp. (ii)) is similar to that for $T_{K^{0}_{33}}$ (resp.
$T_{K^{0}_{1}}$). Case (i). Let $(\alpha,\beta)\neq(1,1)$. Then the same
argument as in the proof of Proposition 4.5 based on Lemma 2.7 yields that
$G^{1}_{\alpha\beta}(x,y)$ is of the form

$C_{\alpha\beta}(x,y)\Big{(}a_{\alpha\beta}\left(k_{1}^{+}+(-1)^{\beta}k_{1}^{-}\right)+b_{\alpha\beta}\left(k_{2}^{+}+(-1)^{\beta}k_{2}^{-}\right)\Big{)}(x,y)+O({\langle}|x|-|y|{\rangle}^{-2}),$

where $k^{\pm}_{j}$ are defined in Lemma 3.3,
$a_{\alpha\beta}=i^{\alpha+\beta+1}$,
$b_{\alpha\beta}=(-1)^{\alpha+\beta}i^{\beta+1}$ and

$C_{\alpha\beta}(x,y)=\int_{{\mathbb{R}}^{2}\times[0,1]^{2}}M_{\alpha\beta}(X_{1},Y_{2},\Theta)d\Theta.$

An important feature is that only one of $\mathop{\mathrm{sgn}}\nolimits
X_{1}$ or $\mathop{\mathrm{sgn}}\nolimits Y_{2}$ appears in the integrand of
$C_{\alpha\beta}$ since one of $\alpha,\beta$ is even in 4.30, except for
$(\alpha,\beta)=(1,1)$. In particular, $C_{\alpha\beta}(x,y)$ is of the form
$C^{1}_{\alpha}(x)C^{2}_{\beta}(y)$ with some bounded functions
$C^{1}_{\alpha}$ and $C^{2}_{\beta}$ (see 4.6, 4.12 and 4.13 and recall
the convention $(\mathop{\mathrm{sgn}}\nolimits x)^{2}=1$). Hence,
$T_{G^{1}_{\alpha\beta}}$ is a sum of the composition
$C^{1}_{\alpha}T_{g_{\alpha\beta}}C^{2}_{\beta}$ and an error term satisfying
the condition of Lemma 3.1, where

$\displaystyle
g_{\alpha\beta}=a_{\alpha\beta}\left(k_{1}^{+}+(-1)^{\beta}k_{1}^{-}\right)+b_{\alpha\beta}\left(k_{2}^{+}+(-1)^{\beta}k_{2}^{-}\right).$
( 4.32)

Since the multiplication operator by bounded function is bounded on
$L^{p}(w_{p})$ for any $1\leq p<\infty$ and on $L^{1,\infty}(w_{1})$, we can
apply Lemma 3.3 to obtain that the same bounds as 3.1 holds for
$T_{G^{1}_{\alpha\beta}},T_{G^{1}_{\alpha\beta}}^{*}$ and hence for all
$T_{K^{1}_{-1}},T_{K^{1}_{0j}},T_{K^{1}_{1j}},T_{K^{1}_{2j}}$ and their
adjoint operators.

To obtain
$T_{G^{1}_{\alpha\beta}}\in\mathbb{B}({\mathcal{H}}^{1}({\mathbb{R}}),L^{1}({\mathbb{R}}))\cap\mathbb{B}(L^{\infty}({\mathbb{R}}),{\mathop{\mathrm{BMO}}}({\mathbb{R}}))$,
we use the same trick as in Proposition 4.6 based on the translation