#  The $L^{p}$-continuity of wave operators for higher order Schrödinger
operators

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

We consider the higher order Schrödinger operator $H=(-\Delta)^{m}+V(x)$ in
$n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in\mathbb{N}$,
$m>1$. When $n$ is odd, we prove that the wave operators extend to bounded
operators on $L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$ under $n$ and
$m$ dependent conditions on the potential analogous to the case when $m=1$.
Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave
operators are bounded on the same range for even $n$. We further show that if
the smallness assumption is removed in even dimensions the wave operators
remain bounded in the range $1<p<\infty$.

The first author is partially supported by Simons Foundation Grant 634269. The
second author is partially supported by Simons Foundation Grant 511825.

##  1\. Introduction

We consider the higher order Schrödinger equation

$\displaystyle i\psi_{t}=(-\Delta)^{m}\psi+V\psi,\qquad
x\in\mathbb{R}^{n},\qquad m>1,\quad m\in\mathbb{N}.$

We restrict our focus to the case when the spatial dimension $n>2m$. Here $V$
is a real-valued, decaying potential. We denote the free higher order
Schrödinger operator by $H_{0}=(-\Delta)^{m}$ and the perturbed operator by
$H=(-\Delta)^{m}+V(x)$. We study the $L^{p}$ boundedness of the wave
operators, which are defined by

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

For the classes of potentials $V$ we consider, the wave operators exist and
are asymptotically complete, see the work of Agmon, [1], Hörmander, [15] and
Schechter, [23, 22].

We use the notation $\langle x\rangle$ to denote $(1+|x|^{2})^{\frac{1}{2}}$,
$\mathcal{F}(f)$ or $\widehat{f}$ to denote the Fourier transform of $f$. We
write $A\lesssim B$ to say that there exists a constant $C$ with $A\leq CB$,
and write $a-:=a-\epsilon$ and $a+:=a+\epsilon$ for some $\epsilon>0$
throughout the paper. We use the norm
$\|f\|_{H^{\delta}}=\|\langle\cdot\rangle^{\delta}\widehat{f}(\cdot)\|_{2}$.
We first state a small potential result that is valid in all dimensions
$n>2m$.

######  Theorem 1.1.

Let $n>2m$. Assume that the $V$ is a real-valued potential on $\mathbb{R}^{n}$
and fix $0<\delta\ll 1$. Then $\exists C=C(\delta,n,m)>0$ so that the wave
operators extend to bounded operators on $L^{p}(\mathbb{R}^{n})$ for all
$1\leq p\leq\infty$, provided that

  1. i)

$\big{\|}\langle\cdot\rangle^{\frac{4m+1-n}{2}+\delta}V(\cdot)\big{\|}_{2}<C$
when $2m<n<4m-1$,

  2. ii)

$\big{\|}\langle\cdot\rangle^{1+\delta}V(\cdot)\big{\|}_{H^{\delta}}<C$ when
$n=4m-1$,

  3. iii)

$\big{\|}\mathcal{F}(\langle\cdot\rangle^{\sigma}V(\cdot))\big{\|}_{L^{\frac{n-1-\delta}{n-2m-\delta}}}<C$
for some $\sigma>\frac{2n-4m}{n-1-\delta}+\delta$ when $n>4m-1$.

For boundedness on $L^{p}$ when $1<p<\infty$, we may remove the smallness
assumption above provided $V$ decays sufficiently at spatial infinity. We
define zero energy to be regular if there are no non-trivial distributional
solutions to $H\psi=0$ with $\langle x\rangle^{\frac{n}{2}-2m-}\psi(x)\in
L^{2}$. We show

######  Theorem 1.2.

Let $n>2m$. Assume that the $V$ is a real-valued potential on $\mathbb{R}^{n}$
so that

  1. i)

$|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n+3$ when $n$ is odd
and for some $\beta>n+4$ when $n$ is even

  2. ii)

$\|\langle\cdot\rangle^{1+}V(\cdot)\|_{H^{0+}}<\infty$ when $n=4m-1$,

  3. iii)

for some $0<\delta\ll 1$ and $\sigma>\frac{2n-4m}{n-1-\delta}$,
$\|\mathcal{F}(\langle\cdot\rangle^{\sigma}V(\cdot))\|_{L^{\frac{n-1-\delta}{n-2m-\delta}}}<\infty$
when $n>4m-1$,

  4. iv)

$H=(-\Delta)^{m}+V(x)$ has no positive eigenvalues and zero energy is regular.

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

Finally, with slightly more decay on the potential we recover the endpoints
$p=1,\infty$ in odd dimensions:

######  Theorem 1.3.

Let $n>2m$ be odd. Assume that $V$ satisfies the hypothesis of Theorem 1.2 and
in addition $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n+5$.
Then, the wave operators extend to bounded operators on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$.

In even dimensions, we lose the boundedness on the endpoints of $p=1,\infty$
due to the low energy. In particular, the energies away from zero are bounded
on the full range including $p=1,\infty$, see Proposition 6.5 below. We hope
to address the cases of $p=1,\infty$ when $n>2m$ even and the case when there
are threshold obstructions in a future work.

We note that the norm used when $n>4m-1$ is finite when $\langle
x\rangle^{\sigma}V(x)$ has more than $\frac{n}{n-2m}(\frac{n-4m+1}{2})$
derivatives in $L^{2}(\mathbb{R}^{n})$. In all cases above, we also note that

$\|V\|_{L^{2}(B(x,1))}\lesssim\langle x\rangle^{-1-},\quad
x\in\mathbb{R}^{n}.$

This suffices to imply, [22, 1, 23], the existence, asymptotic completeness,
and intertwining identity for the wave operators. In particular, we have

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

Here $P_{ac}(H)$ is the projection onto the absolutely continuous spectral
subspace of $H$, and $f$ is any Borel function. Using (1) one may obtain
$L^{p}$-based mapping properties for the more complicated, perturbed operator
$f(H)P_{ac}(H)$ from the simpler free operator $f((-\Delta)^{m})$. The
boundedness of the wave operators on $L^{p}(\mathbb{R}^{n})$ for any choice of
$p\geq 2$ with the function $f(\cdot)=e^{-it(\cdot)}$ yield the dispersive
estimate

(2) $\displaystyle\|e^{-itH}P_{ac}(H)\|_{L^{p^{\prime}}\to
L^{p}}\lesssim|t|^{-\frac{n}{2m}+\frac{n}{2p}},$

where $p^{\prime}$ is the Hölder conjugate of $p$. In particular in all odd
dimensions $n>2m$, under the hypothesis of Theorem 1.3, we have

$\|e^{-itH}P_{ac}(H)\|_{L^{1}\to L^{\infty}}\lesssim|t|^{-\frac{n}{2m}}.$

Our work is inspired by recent work by Feng, Soffer, Wu and Yao on weighted
$L^{2}$-based “local dispersive estimates” for higher order Schrödinger
operators considered in [9], as well as the recent work on the
$L^{p}(\mathbb{R}^{3})$ boundedness of the wave operators for the fourth order
($m=2$) Schrödinger operators by Goldberg and the second author [12], and the
extensive works of Yajima, [24, 25, 26, 27, 28], in the case of $m=1$. The
wave operators for the usual Schrödinger operator $-\Delta+V$, when $m=1$ are
well-studied, see for example [24, 25, 26, 17, 18, 5, 21] in all dimensions
$n\geq 1$. On $\mathbb{R}^{3}$, Beceanu and Schlag obtained detailed structure
formulas for the wave operators, [2, 3, 4]. The $L^{2}$ existence and other
properties of the higher order wave operators have been studied by many
authors, including Agmon [1], Kuroda [19, 20], Hörmander [15], and Schechter,
[22, 23]. We note that the only result on the $L^{p}$ boundedness of the wave
operators for higher order Schrödinger operators is the case of $m=2$ and
$n=3$ by Goldberg and the second author, [12]. There appears to be three
regimes in the analysis of $L^{p}$ boundedness of the wave operators: $n<2m$,
$n=2m$, and $n>2m$. In the case $n<2m$, as in [12], zero energy is not regular
for the free operator and the main difficulty in the analysis is the small
energies. However the large energy argument is more straightforward since the
resolvent decays in the spectral parameter $\lambda$. In the range $n>2m$ the
zero energy is regular for the free operator and the resolvent remains bounded
as $\lambda\to 0$. However, the large energies, and in particular the Born
series terms, are not easy to deal with. When $n>4m-1$ one needs a smoothness
requirement on the potential $V$ as in the case $m=1$ and $n>3$, [24, 13], due
to the growth of the resolvents as the spectral variable goes to infinity. The
case $n=2m$ is challenging in both the low and high energy regimes.

Similar to the usual second order Schrödinger operator, for the types of
potentials we consider there is a Weyl criterion and
$\sigma_{ac}(H)=\sigma_{ac}(H_{0})=[0,\infty)$. In contrast, decay of the
potential is not sufficient to ensure the lack of eigenvalues embedded in the
continuous spectrum for the higher order operators, [9]. Even perturbing with
compactly supported, smooth potentials may induce embedded eigenvalues. We
leave this as an overarching assumption and note that there are conditions
that ensure the lack of embedded eigenvalues, see Theorem 1.11 in [9].

To prove Theorem 1.2 we use a time-independent representation of the wave
operators based on resolvent operators. We have the splitting identity for
$z\in\mathbb{C}\setminus[0,\infty)$, (c.f. [9])

(3)
$\mathcal{R}_{0}(z)(x,y):=((-\Delta)^{m}-z)^{-1}(x,y)=\frac{1}{mz^{1-\frac{1}{m}}}\sum_{\ell=0}^{m-1}\omega_{\ell}R_{0}(\omega_{\ell}z^{\frac{1}{m}})(x,y)$

where $\omega_{\ell}=\exp(i2\pi\ell/m)$ are the $m^{th}$ roots of unity,
$R_{0}(z)=(-\Delta-z)^{-1}$ is the usual ($2^{nd}$ order) Schrödinger
resolvent. Using the change of variables $z=\lambda^{2m}$ with $\lambda$
restricted to the sector in the complex plane with $0<\arg(\lambda)<\pi/m$,

(4)
$\mathcal{R}_{0}(\lambda^{2m})(x,y):=((-\Delta)^{m}-\lambda^{2m})^{-1}(x,y)=\frac{1}{m\lambda^{2m-2}}\sum_{\ell=0}^{m-1}\omega_{\ell}R_{0}(\omega_{\ell}\lambda^{2})(x,y).$

By the well-known Bessel function expansions, for $n>3$ odd we have

(5)
$R_{0}(z^{2})(x,y)=\frac{e^{iz|x-y|}}{|x-y|^{n-2}}\sum_{j=0}^{\frac{n-3}{2}}c_{n,j}|x-y|^{j}z^{j},\qquad\Im(z)>0.$

Even dimensions are more complicated due to the appearance of logarithmic
terms.

Our usual starting point to study the wave operators is the stationary
representation

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

where $\mathcal{R}_{V}(\lambda)=((-\Delta)^{m}+V-\lambda)^{-1}$, where the ‘+’
and ‘-’ denote the usual limiting values as $\lambda$ approaches the positive
real line from above and below, [9]. Since the identity operator is bounded on
$L^{p}$, we need only bound the second term involving the integral. It is
convenient to make the change of variables $\lambda\mapsto\lambda^{2m}$ and
consider the integral kernel of the operator

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

Our result in Theorem 1.1 follows by using resolvent identities to expand
$\mathcal{R}_{V}^{+}$ in an infinite series and directly summing the series.
To remove the smallness assumption to show that the operator defined in (6)
extends to a bounded operator on $L^{p}$ requires different strategies in the
low ($0<\lambda\ll 1$) and high ($\lambda\gtrsim 1$) energy regimes. To
delineate these cases, we use the even, smooth cut-off function $\chi$ with
$\chi(\lambda)=1$ for $|\lambda|<\lambda_{0}$ for some sufficiently small
$\lambda_{0}\ll 1$, and $\chi(\lambda)=0$ for $|\lambda|>2\lambda_{0}$, as
well as the complimentary cut-off $\widetilde{\chi}(\lambda)=1-\chi(\lambda)$.

We note that the different assumptions on the potential we impose based on the
size of $n$ versus $m$ are natural. When $n\leq 2m$ the low energy expansions
of the resolvent $\mathcal{R}_{0}$ are singular as the spectral parameter
$\lambda\to 0$. This complication necessitates a different strategy to invert
certain operators and develop expansions for both the free and perturbed
resolvents, see [14, 7] for the case when $m=2$ and $n=4,3$ respectively.
Smoothness of the potential is required for the second order Schrödinger
operator in dimensions $n>3$ since the kernel free resolvent
$R_{0}^{\pm}(\lambda^{2})$ grows like $\lambda^{\frac{n-3}{2}}$ as the
spectral parameter $\lambda\to\infty$. This causes the $L^{1}\to L^{\infty}$
dispersive estimates to fail in dimensions greater than three without some
smoothness assumptions on the potential, see the counterexample constructed by
Goldberg and Visan [13]. The higher order Schrödinger resolvent,
$\mathcal{R}_{0}(\lambda^{2m})$ grows like $\lambda^{\frac{n+1}{2}-2m}$ when
$n>4m-1$, which necessitates a control over derivatives of the potential which
we measure in terms of the $\mathcal{F}L^{r}$ norm similar to the conditions
for the second order Schrödinger established by Yajima, [24]. Our
$\epsilon$-smoothness requirement in the case $n=4m-1$ could be an artifact of
our methods.

The paper is organized as follows. We first control the Born series terms that
arise by iterating the resolvent identity for the perturbed resolvent in the
stationary representation, (6), of the wave operator in Section 2. Next, we
prove Theorem 1.2 and Theorem 1.3. First in odd dimensions, in Section 3 and
Section 4, we control the remainder in the low energy regime, when the
spectral parameter $\lambda$ is in a neighborhood of zero. In Section 5 we
control the remainder in the high energy regime, when $\lambda\gtrsim 1$ in
odd dimensions. In Section 6 we show how the arguments in Sections 3 and 5 may
be adapted to the even dimensional case. Finally, in Section 7 we provide
integral estimates that are used throughout the paper.

##  2\. Born Series

By iterating the resolvent identity, one has the expansion

(7)
$\displaystyle\mathcal{R}_{V}(z)=\sum_{J=0}^{2\ell}\big{[}\mathcal{R}_{0}(z)(-V\mathcal{R}_{0}(z))^{J}\big{]}-(\mathcal{R}_{0}(z)V)^{\ell}\mathcal{R}_{V}(z)(V\mathcal{R}_{0}(z))^{\ell}.$

Consider the contribution of an arbitrary summand in the Born series to (6),

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

In this section by modifying the proof of Yajima in [24] to control the Born
series terms for the second order Schrödinger, we prove that $W_{J}$ extends
to a bounded operator on $L^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$:

######  Theorem 2.1.

Fix $1\leq p\leq\infty$ and $0<\delta\ll 1$. Then $\exists C=C(\delta,n,m)>0$
so that for $2m<n<4m-1$, we have

$\|W_{J}\|_{L^{p}\to L^{p}}\leq
C^{J}\|\langle\cdot\rangle^{\frac{4m+1-n}{2}+\delta}V(\cdot)\|_{L^{2}}^{J},$

for $n=4m-1$, we have

$\|W_{J}\|_{L^{p}\to L^{p}}\leq C^{J}\|\langle
x\rangle^{1+\delta}V\|_{H^{\delta}}^{J},$

for $n>4m-1$, we have

$\|W_{J}\|_{L^{p}\to L^{p}}\leq C^{J}\|\mathcal{F}(\langle
x\rangle^{\frac{2n-4m}{n-1-\delta}+\delta}V)\|_{L^{\frac{n-1-\delta}{n-2m-\delta}}}^{J}.$

In what follows we will ignore most implicit constants; their affect on the
final inequality is of the form $C^{J}$, where $C$ depends on $n,m$ and the
actual value of the implicit small constants in the hypothesis above. Theorem
1.1 follows from this result.

Our approach is inspired by the paper [24], in which Yajima proved the result
in the case of $m=1$. We will bound the adjoint operator $Z_{J}=W_{J}^{*}$.
Fix $f\in\mathcal{S}$ and let

(8) $Z_{J}f(x)=\lim_{\epsilon_{1}\to 0^{+}}\cdots\lim_{\epsilon_{J}\to
0^{+}}\lim_{\epsilon_{0}\to 0^{+}}Z_{J,\vec{\epsilon},\epsilon_{0}}f(x),$

where

$Z_{J,\vec{\epsilon},\epsilon_{0}}f(x):=\frac{1}{2\pi
i}\int_{\mathbb{R}}\big{[}\mathcal{R}_{0}(\lambda-i\epsilon_{0})V\mathcal{R}_{0}(\lambda+i\epsilon_{1})\cdots
V\mathcal{R}_{0}(\lambda+i\epsilon_{J})f\big{]}(x)d\lambda.$

The main result of this sections is to show this operator is bounded on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$. As in [24], it suffices
to prove that the limit above exists in $L^{p}$ and the bounds stated in the
theorem hold for $f\in\mathcal{S}$ and $\widehat{V}\in C_{0}^{\infty}$.

Taking the Fourier transform in $x$ yields, up to constants,

$\displaystyle\mathcal{F}(Z_{J,\vec{\epsilon},\epsilon_{0}}f)(\xi)=\int_{\mathbb{R}}\int_{\mathbb{R}^{Jn}}\frac{1}{\xi^{2m}-\lambda+i\epsilon_{0}}\prod_{j=1}^{J}\frac{\widehat{V}(k_{j})}{(\xi-\sum_{\ell=1}^{j}k_{\ell})^{2m}-\lambda-i\epsilon_{j}}\widehat{f}(\xi-\sum_{j=1}^{J}k_{j})d\lambda.$

Applying Cauchy’s integral formula to the $\lambda$ integral in the definition
of $Z_{J}$ and taking $\epsilon_{0}\to 0^{+}$ yield

$\mathcal{F}(Z_{J,\vec{\epsilon}}f)(\xi)=\int_{\mathbb{R}^{Jn}}\bigg{[}\prod_{j=1}^{J}\frac{\widehat{V}(k_{j})}{(|\xi-\sum_{\ell=1}^{j}k_{\ell}|^{2m}-|\xi|^{2m}-i\epsilon_{j})}\bigg{]}\widehat{f}(\xi-\sum_{j=1}^{J}k_{j})dk_{1}\cdots
dk_{J}.$

Now, we utilize the change of variables $\sum_{\ell=1}^{j}k_{\ell}\mapsto
k_{j}$ for $j=1,\ldots,J$ and define $k_{0}=0$ to obtain

$\mathcal{F}(Z_{J,\vec{\epsilon}}f)(\xi)=\int_{\mathbb{R}^{Jn}}\bigg{[}\prod_{j=1}^{J}\frac{\widehat{V}(k_{j}-k_{j-1})}{(|\xi-
k_{j}|^{2m}-|\xi|^{2m}-i\epsilon_{j})}\bigg{]}\widehat{f}(\xi-
k_{J})dk_{1}\cdots dk_{J}.$

We define the multiplier operator $T_{k,\epsilon}^{m}$ by

(9)
$T_{k,\epsilon}^{m}f=\mathcal{F}^{-1}\bigg{(}\frac{\widehat{f}(\xi)}{|\xi-k|^{2m}-|\xi|^{2m}-i\epsilon}\bigg{)}.$

Let $K_{J}(k_{1},k_{2},\dots,k_{J})=\prod_{j=1}^{J}\widehat{V}(k_{j}-k_{j-1})$
and $f_{k_{J}}(x)=e^{ik_{J}\cdot x}f(x)$. Then, we have

(10) $Z_{J}f(x)\\\ =\lim_{\epsilon_{1}\to 0^{+}}\cdots\lim_{\epsilon_{J}\to
0^{+}}\int_{\mathbb{R}^{n}}T_{k_{1},\epsilon_{1}}^{m}\bigg{\\{}\int_{\mathbb{R}^{n}}T_{k_{2},\epsilon}^{m}\bigg{\\{}\cdots\int_{\mathbb{R}^{n}}K_{J}(k_{1},k_{2},\dots,k_{J})T_{k_{J},\epsilon_{J}}^{m}f_{k_{J}}\,dk_{J}\bigg{\\}}\cdots\bigg{\\}}dk_{2}\bigg{\\}}dk_{1},$

Now, we need to study the operators $T^{m}_{k,\epsilon}$ in some detail. We
note the algebraic identity

$|\xi-k|^{2m}-|\xi|^{2m}-i\epsilon=(|\xi-k|^{2}-|\xi|^{2})\big{(}\sum_{\ell=0}^{m-1}|\xi-k|^{2\ell}|\xi|^{2m-2-2\ell}\big{)}-i\epsilon\\\
=2i\frac{|k|^{2m-1}}{p_{\omega}(\xi/|k|)}\big{(}-\frac{i|k|}{2}+i\omega\cdot\xi-\frac{\epsilon
p_{\omega}(\xi/|k|)}{2|k|^{2m-1}}\big{)},$

where

(11) $\omega=\frac{k}{|k|}\in S^{n-1},\,\,\,\text{ and
}\,\,p_{\omega}(\xi)=\frac{1}{\sum_{\ell=0}^{m-1}|\omega-\xi|^{2\ell}|\xi|^{2m-2-2\ell}}.$

We therefore have

$T_{k,\epsilon}^{m}f=\frac{1}{2i|k|^{2m-1}}\mathcal{F}^{-1}\bigg{(}\frac{p_{\omega}(\xi/|k|)\widehat{f}(\xi)}{-\frac{i|k|}{2}+i\omega\cdot\xi-\frac{\epsilon
p_{\omega}(\xi/|k|)}{2|k|^{2m-1}}}\bigg{)}.$

Writing (note that $p_{\omega}(\xi)>0$)

$\frac{1}{-\frac{i|k|}{2}+i\omega\cdot\xi-\frac{\epsilon
p_{\omega}(\xi/|k|)}{2|k|^{2m-1}}}=-\int_{0}^{\infty}e^{-\frac{i|k|t}{2}+it\omega\cdot\xi}e^{-\frac{\epsilon
p_{\omega}(\xi/|k|)}{2|k|^{2m-1}}t}dt,$

we obtain

$\mathcal{F}^{-1}\bigg{(}\frac{p_{\omega}(\xi/|k|)\widehat{f}(\xi)}{-\frac{i|k|}{2}+i\omega\cdot\xi-\frac{\epsilon
p_{\omega}(\xi/|k|)}{2|k|^{2m-1}}}\bigg{)}(x)=-\int_{0}^{\infty}e^{-\frac{i|k|t}{2}}h_{k,\frac{\epsilon
t}{2|k|^{2m-1}}}*f(x+t\omega)dt,$

where $*$ denotes convolution and

$h_{k,\epsilon}=\mathcal{F}^{-1}\Big{(}p_{\omega}(\xi/|k|)e^{-\epsilon
p_{\omega}(\xi/|k|)}\Big{)}.$

######  Lemma 2.2.

We have the following bounds (with $k=s\omega,s>0,\omega\in S^{n-1}$)

$\big{\|}\sup_{\epsilon>0}h_{k,\epsilon}\big{\|}_{L^{1}}\lesssim 1,$
$\big{\|}\sup_{\epsilon>0}|\partial_{s}^{j}h_{s\omega,\frac{\epsilon}{s^{2m-1}}}|\big{\|}_{L^{1}}\lesssim
s^{-j},\,\,\,j=1,2,\ldots$

Furthermore, $h_{k,\epsilon}$ converges to $h_{k}:=h_{k,0}$ and
$\partial_{s}^{j}h_{s\omega,\frac{\epsilon}{s^{2m-1}}}$ converges to
$\partial_{s}^{j}h_{k}$ as $\epsilon\to 0$ a.e. and in $L^{1}$, and $h_{k}$
satisfies the same bounds above.

###### Proof.

We first prove the claims for $h_{k}$. Note that

$\|h_{s\omega}\|_{L^{1}}=\Big{\|}\mathcal{F}^{-1}(p_{\omega})\Big{\|}_{L^{1}}.$

A simple calculation shows that

$\big{|}\nabla_{\xi}^{N}p_{\omega}(\xi)\big{|}\lesssim\frac{1}{\langle\xi\rangle^{2m-2+N}},\qquad
N=0,1,2,\dots$

This is seen by considering cases based on the size of $|\xi|$ and
$|\omega|=1$ in (11). Therefore, for $N\geq n-2m+3$,
$|x|^{N}\mathcal{F}^{-1}(p_{\omega})(x)$ is a bounded continuous function, and
hence

$\mathcal{F}^{-1}(p_{\omega})(x)=u+O\big{(}\min(|x|^{-n-1},|x|^{-n+1})\big{)},$

where $u$ is a distribution supported at $0$. Since $p_{\omega}(\xi)\to 0$ as
$|\xi|\to\infty$, we conclude that $u=0$, which yields the claim for $j=0$.
For $j>0$, note that

$\partial_{s}\mathcal{F}^{-1}p_{\omega}(sx)=x\cdot[\nabla\mathcal{F}^{-1}p_{\omega}](xs)=\frac{1}{s}\mathcal{F}^{-1}(\nabla\cdot\xi\,p_{\omega}(\xi))(xs).$

Similarly,
$\partial_{s}^{\ell}\mathcal{F}^{-1}p_{\omega}(sx)=s^{-\ell}\mathcal{F}^{-1}((\nabla\cdot\xi)^{\ell}p_{\omega}(\xi))(xs)$.
Therefore,

$|\partial_{s}^{j}h_{s\omega}(x)|\lesssim\sum_{\ell=0}^{j}s^{n+\ell-j}s^{-\ell}|\mathcal{F}^{-1}((\nabla\cdot\xi)^{\ell}p_{\omega}(\xi))(xs)|.$

The claim follows from this as above since
$(\nabla\cdot\xi)^{\ell}p_{\omega}(\xi)$ satisfies the same bounds as
$p_{\omega}(\xi)$.

Now, we consider $h_{k,\epsilon}$. Let
$H_{\omega}(\epsilon,x)=\mathcal{F}^{-1}\Big{(}p_{\omega}e^{-\epsilon
p_{\omega}}\Big{)}(x)$. Using the bounds on the derivatives of $p_{\omega}$,
and noting that $p_{\omega}(\xi)\approx\langle\xi\rangle^{2-2m}$ and that
$\sup_{\alpha>0}\alpha e^{-\alpha}\lesssim 1$, we conclude that

$\big{|}\nabla_{\xi}^{N}[p_{\omega}(\xi)e^{-\epsilon
p_{\omega}(\xi)}]\big{|}\lesssim\frac{1}{\langle\xi\rangle^{2m-2+N}},\qquad
N=0,1,2,\dots$

Therefore we have

$|\mathcal{F}^{-1}(p_{\omega}e^{-\epsilon
p_{\omega}})(x)|\lesssim\min(|x|^{-n-1},|x|^{-n+1}),$

uniformly in $\epsilon>0$. This yields the claim for $j=0$ since
$h_{k,\epsilon}=s^{n}H_{\omega}(\epsilon,sx)$.

Similarly, note that

$\big{|}\nabla_{\xi}^{N}[p_{\omega}(\xi)(e^{-\epsilon
p_{\omega}(\xi)}-1)]\big{|}\lesssim\frac{\epsilon}{\langle\xi\rangle^{4m-4+N}},\qquad
N=0,1,2,\dots.$

This implies the a.e. and $L^{1}$ convergence of $h_{k,\epsilon}$ to $h_{k}$.

For the $j$th derivative of $h_{k,\epsilon}$, by chain rule and scaling as
above, it suffices to prove that the $L^{1}$ norms of
$\sup_{\epsilon}\epsilon^{j_{1}}\partial_{\epsilon}^{j_{1}}(x\cdot\nabla_{x})^{j_{2}}\mathcal{F}^{-1}[p_{\omega}e^{-\epsilon
p_{\omega}}](x)$ are $\lesssim 1$ for $j_{1},j_{2}\geq 0$. Note that

$\nabla_{\xi}^{N}\epsilon^{j_{1}}\partial_{\epsilon}^{j_{1}}(\nabla_{\xi}\cdot\xi)^{j_{2}}p_{\omega}(\xi)e^{-\epsilon
p_{\omega}(\xi)}\in L^{1}$

for $N\geq n-2m+3$. The claim follows as above. Convergence of the $s$
derivatives of $h_{k,\epsilon}$ follow similarly. ∎

We conclude that for $f\in\mathcal{S}$

$T_{k,\epsilon}^{m}f(x)=\frac{i}{2|k|^{2m-1}}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{-i|k|t/2}h_{k,\frac{\epsilon
t}{2|k|^{2m-1}}}(y)f(x-y+t\omega)\,dy\,dt,$

and for all $x\in\mathbb{R}^{n}$

$\lim_{\epsilon\to
0^{+}}T_{k,\epsilon}^{m}f(x)=\frac{i}{2|k|^{2m-1}}\int_{0}^{\infty}e^{-it|k|/2}\int_{\mathbb{R}^{n}}h_{k}(y)f(x-y+t\omega)\,dy\,dt:=T_{k}^{m}f(x).$

Following the notation of [24], for $\epsilon>0$, let

$G_{\epsilon}f=\int_{\mathbb{R}^{n}}T_{k,\epsilon}^{m}f(k,\cdot)dk,\qquad
G_{0}f=\int_{\mathbb{R}^{n}}T_{k}^{m}f(k,\cdot)dk,$

Note that

(12)
$G_{\epsilon}f(x)=\int_{\mathbb{R}^{n}}\frac{i}{2|k|^{2m-1}}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{-i|k|t/2}h_{k,\frac{\epsilon
t}{2|k|^{2m-1}}}(y)f(k,x-y+t\omega)\,dy\,dt\,dk.$

Passing to polar coordinates, $k=s\omega$, and changing the order of
integration, we have

$G_{\epsilon}f(x)=\frac{i}{2}\int_{S^{n-1}}\int_{0}^{\infty}F_{\epsilon}(t,\omega,x)\,dt\,d\omega,$

where

$F_{\epsilon}(t,\omega,x)=\int_{0}^{\infty}e^{-ist/2}s^{n-2m}h_{s\omega,\frac{\epsilon
t}{2s^{2m-1}}}*f(s\omega,\cdot)(x+t\omega)\,ds.$

Also note that $G_{0}f$ satisfies the same formula with $F_{0}$ replacing
$F_{\epsilon}$.

######  Lemma 2.3.

Let $\epsilon>0$ and
$f(k,x)\in\mathcal{S}(\mathbb{R}^{n}_{k},\mathcal{S}(\mathbb{R}^{n}_{x}))$.
For all $n>2m+1$ and $1\leq p\leq\infty$, we have

$\|G_{\epsilon}f\|_{L^{p}}\leq C_{n,m}\int_{\mathbb{R}^{n}}\langle
k\rangle^{n-2m}\sum_{j=0}^{2}\|D_{k}^{j}f(k,\cdot)\|_{L^{p}}\frac{dk}{|k|^{n-1}}.$

For $n=2m+1$, we have

$\|G_{\epsilon}f\|_{L^{p}}\leq C_{n,m}\int_{\mathbb{R}^{n}}\langle
k\rangle\min(1,|k|)^{-\frac{1}{2}}\sum_{j=0}^{3}\|D_{k}^{j}f(k,\cdot)\|_{L^{p}}\frac{dk}{|k|^{n-1}}.$

Moreover, $G_{\epsilon}f\to G_{0}f$ in $L^{p}$ as $\epsilon\to 0^{+}$.

###### Proof.

Note that

$\big{\|}F_{\epsilon}(t,\omega,x)\big{\|}_{L^{p}_{x}}\lesssim\int_{0}^{\infty}s^{n-2m}\|\sup_{\epsilon}h_{s\omega,\epsilon}\|_{L^{1}}\|f(s\omega,\cdot)\|_{L^{p}}\,ds\\\
\lesssim\int_{0}^{\infty}s^{n-2m}\|f(s\omega,\cdot)\|_{L^{p}}\,ds.$

For $t>1$, and $n>2m+1$, we integrate by parts twice in the $s$ integral to
obtain

$|F_{\epsilon}(t,\omega,x)|\lesssim\frac{1}{t^{2}}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\big{|}\partial_{s}^{2}\big{(}s^{n-2m}h_{s\omega,\frac{\epsilon
t}{2s^{2m-1}}}(y)f(s\omega,x-y+t\omega)\big{)}\big{|}\,ds\,dy.$

Let
$H_{s\omega}(y)=|\sup_{\epsilon>0,j=0,1,2}s^{j}\partial_{s}^{j}h_{s\omega,\frac{\epsilon
t}{2s^{2m-1}}}(y)|$. Using this we obtain the bound

$|F_{\epsilon}(t,\omega,x)|\lesssim\frac{1}{t^{2}}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\langle
s\rangle^{2}s^{n-2m-2}H_{s\omega}(y)\sum_{j=0}^{2}\big{|}\partial_{s}^{j}f(s\omega,x-y+t\omega)\big{|}\,ds\,dy\\\
\lesssim\frac{1}{t^{2}}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}H_{s\omega}(y)\langle
s\rangle^{n-2m}\sum_{j=0}^{2}\big{|}\partial_{s}^{j}f(s\omega,x-y+t\omega)\big{|}\,ds\,dy.$

By Lemma 2.2, $\|H_{s\omega}\|_{L^{1}}\lesssim 1$, therefore uniformly in $t$
and $\omega$, we have

$\big{\|}F_{\epsilon}(t,\omega,x)\big{\|}_{L^{p}_{x}}\lesssim\frac{1}{\langle
t\rangle^{2}}\int_{0}^{\infty}\langle
s\rangle^{n-2m}\sum_{j=0}^{2}\big{\|}\partial_{s}^{j}f(s\omega,\cdot)\big{\|}_{L^{p}}\,ds,$

which implies the claim for $G_{\epsilon}f$ when $n>2m+1$. The convergence of
$G_{\epsilon}f$ to $G_{0}f$ in $L^{p}$ also follows by applying the same
argument with $h_{s\omega,\frac{\epsilon t}{2s^{2m-1}}}-h_{s\omega}$ replacing
$h_{s\omega,\frac{\epsilon t}{2s^{2m-1}}}$ and using dominated convergence
theorem.

We now consider the case $n=2m+1$. For $t\gg 1$, after an integration by
parts, we have

$F_{\epsilon}(t,\omega,x)=-\frac{2i}{t}\int_{0}^{\infty}e^{-ist/2}\partial_{s}[sh_{s\omega,\frac{\epsilon
t}{2s^{2m-1}}}*f(s\omega,\cdot)(x+t\omega)]\,ds.$

We cannot integrate by parts again to gain another power of $t$ in this case.
Therefore we utilize the identity (with
$K(s)=\partial_{s}[sh_{s\omega,\frac{\epsilon
t}{2s^{2m-1}}}*f(s\omega,\cdot)(x+t\omega)]$)

$\int_{0}^{\infty}e^{-ist/2}K(s)ds=\frac{1}{2}\int_{0}^{2\pi/t}e^{-ist/2}K(s)ds+\frac{1}{2}\int_{0}^{\infty}e^{-i(s+2\pi/t)t/2}[K(s+2\pi/t)-K(s)]ds.$

This implies that

$\Big{\|}\int_{0}^{\infty}e^{-ist/2}K(s)ds\Big{\|}_{L^{p}_{x}}\lesssim\\\
\int_{0}^{2\pi/t}\|K(s)\|_{L^{p}_{x}}ds+\int_{0}^{\infty}(\|K(s+2\pi/t)\|_{L^{p}_{x}}+\|K(s)\|_{L^{p}_{x}})^{\frac{1}{2}}\Big{(}\int_{s}^{s+2\pi/t}\big{\|}\partial_{\rho}K(\rho)\big{\|}_{L^{p}_{x}}d\rho\Big{)}^{\frac{1}{2}}ds\\\
\lesssim
t^{-\frac{1}{2}}\sup_{0<s<1}\|K(s)\|_{L^{p}_{x}}+t^{-\frac{1}{2}}\int_{0}^{\infty}\big{[}\sup_{s<\rho<s+1}\|K(\rho)\|_{L^{p}}\big{]}^{1/2}\big{[}\sup_{s<\rho<s+1}\|\partial_{\rho}K(\rho)\|_{L^{p}}\big{]}^{1/2}ds.$

Note that

$\|K(\rho)\|_{L^{p}_{x}}\lesssim\langle\rho\rangle\big{(}\|f(\rho\omega,\cdot)\|_{L^{p}}+\|\partial_{\rho}f(\rho\omega,\cdot)\|_{L^{p}}\big{)}$
$\big{\|}\partial_{\rho}K(\rho)\big{\|}_{L^{p}_{x}}\lesssim\langle\rho\rangle\min(1,\rho)^{-1}\big{(}\|f(\rho\omega,\cdot)\|_{L^{p}}+\|\partial_{\rho}f(\rho\omega,\cdot)\|_{L^{p}}+\|\partial^{2}_{\rho}f(\rho\omega,\cdot)\|_{L^{p}}\big{)}.$

Therefore,

$\Big{\|}\int_{0}^{\infty}e^{-ist/2}K(s)ds\Big{\|}_{L^{p}_{x}}\lesssim
t^{-\frac{1}{2}}\int_{0}^{\infty}\langle
s\rangle\min(1,s)^{-\frac{1}{2}}\sup_{s<\rho<s+1}\sum_{j=0}^{2}\|\partial_{\rho}^{j}f(\rho\omega,\cdot)\|_{L^{p}}ds.$

Noting that, for $s<\rho<s+1$

$\sum_{j=0}^{2}\|\partial_{\rho}^{j}f(\rho\omega,\cdot)\|_{L^{p}}\leq\sum_{j=0}^{2}\|\partial_{s}^{j}f(s\omega,\cdot)\|_{L^{p}}+\int_{s}^{s+1}\sum_{j=0}^{3}\|\partial_{\rho}^{j}f(\rho\omega,\cdot)\|_{L^{p}},$

and applying Fubini’s theorem yield the claim bounding $G_{\epsilon}$ in
$L^{p}$. Convergence in $L^{p}$ follows similarly. ∎

We now return to the operator $Z_{J}$ defined in (10). For fixed $k_{1},\ldots
k_{J-1}$, the inner most integral is $G_{\epsilon_{J}}\widetilde{f}_{k_{J}}$
where $\widetilde{f}_{k_{J}}(k_{J},x)=e^{ik_{J}\cdot
x}K_{J}(k_{1},k_{2},\dots,k_{J})f(x)$. By Lemma 2.3, it converges to
$G_{0}\widetilde{f}_{k_{J}}$ in $L^{p}$ for $f\in\mathcal{S}$. Using Lemma
2.3, we also take $\epsilon_{J-1},\ldots,\epsilon_{1}\to 0^{+}$ to obtain

(13)
$Z_{J}f(x)=\int_{\mathbb{R}^{n}}T_{k_{1}}^{m}\bigg{\\{}\int_{\mathbb{R}^{n}}T_{k_{2}}^{m}\bigg{\\{}\cdots\int_{\mathbb{R}^{n}}T_{k_{J}}^{m}\widetilde{f}_{k_{J}}\,dk_{J}\bigg{\\}}\cdots\bigg{\\}}dk_{2}\bigg{\\}}dk_{1}.$

We rewrite the inner most integral using (12) (with $\epsilon=0$) as

(14) $G_{0}\widetilde{f}_{k_{J}}(x)\\\
=\int_{\mathbb{R}^{n}}\frac{i}{2|k_{J}|^{2m-1}}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{-i\frac{|k_{J}|t_{J}}{2}}h_{k_{J}}(y_{J})e^{ik_{J}\cdot(x-y_{J}+t_{J}\omega_{J})}K_{J}(k_{1},\dots,k_{J})f(x-y_{J}+t_{J}\omega_{J})\,dy_{J}\,dt_{J}\,dk_{J}\\\
=\int_{0}^{\infty}\int_{S^{n-1}}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\frac{is_{J}^{n-2m}}{2}e^{i\frac{s_{J}t_{J}}{2}+is\omega\cdot(x-y)}h_{s_{J}\omega_{J}}(y_{J})K_{J}(k_{1},\dots,s_{J}\omega_{J})f(x-y_{J}+t_{J}\omega_{J})\,ds_{J}\,dt_{J}\,dy_{J}\,d\omega_{J}.$

Letting $t_{J}+2\omega_{J}\cdot(x-y_{J})\to-t_{J}$, we have

$\eqref{eqn:G0
fkx}=\frac{i}{2}\int_{S^{n-1}}\int_{\mathbb{R}^{n}}\int_{-\infty}^{-2\omega_{J}\cdot(x-y_{J})}F_{J}(k_{1},\ldots,k_{J-1},t_{J},y_{J},\omega_{J})f(\overline{x}-\overline{y_{J}}+t_{J}\omega_{J})\,dt_{J}\,dy_{J}\,d\omega_{J},$

where $\overline{x}=x-2\omega_{J}(x\cdot\omega_{J})$ and

$F_{J}(k_{1},\ldots,k_{J-1},t_{J},y_{J},\omega_{J})=\int_{0}^{\infty}s_{J}^{n-2m}e^{i\frac{s_{J}t_{J}}{2}}h_{s_{J}\omega_{J}}(y_{J})K_{J}(k_{1},\dots,k_{J-1},s_{J}\omega_{J})\,ds_{J}.$

Now, using (12) (with $\epsilon=0$) we rewrite the integral in $k_{J-1}$ in
(13) to obtain

$\eqref{eqn:G0
fkx}=\bigg{(}\frac{i}{2}\bigg{)}^{2}\int_{S^{n-1}\times\mathbb{R}^{n}\times(0,\infty)}\int_{S^{n-1}\times\mathbb{R}^{n}\times(-\infty,\sigma_{J-1})}F_{J-1}f(\overline{x}-\gamma_{J-1})\,dt_{J}dy_{J}d\omega_{J}dt_{J-1}dy_{J-1}d\omega_{J-1},$

where for $j=1,\dots,J-1$,

$\gamma_{j}:=\overline{y_{J}}-t_{J}\omega_{J}+\sum_{\ell=j}^{J-1}\overline{y_{\ell}-t_{\ell}\omega_{\ell}},\qquad\sigma_{j}=-2\omega_{J}\cdot(x-y_{J}-\sum_{\ell=j}^{J-1}(y_{\ell}+t_{\ell}\omega_{\ell})),$

and

$F_{J-1}=F_{J-1}(k_{1},\dots,k_{J-2},t_{J-1},\omega_{J-1},y_{J-1},t_{J},\omega_{J},y_{J})\\\
:=\int_{(0,\infty)^{2}}\prod_{j=J-1}^{J}\big{[}s_{j}^{n-2m}e^{-i\frac{s_{j}t_{j}}{2}}h_{s_{j}\omega_{j}}(y_{j})\big{]}K_{J}(k_{1},\dots,k_{J-2},s_{J-1}\omega_{J-1},s_{J}\omega_{J})\,ds_{J}\,ds_{J-1}.$

Continuing in this manner we have

$Z_{J}f(x)=\big{(}\frac{i}{2}\big{)}^{J}\int_{(S^{n-1}\times\mathbb{R}^{n}\times(0,\infty))^{J-1}}\int_{S^{n-1}\times\mathbb{R}^{n}\times(-\infty,\sigma_{1})}F\,f(\overline{x}-\gamma_{1})dt_{J}dy_{J}d\omega_{J}\cdots
dt_{1}dy_{1}d\omega_{1},$

where

$F=F(t_{1},\omega_{1},y_{1},\dots,t_{J},\omega_{J},y_{J})\\\
:=\int_{(0,\infty)^{J}}\prod_{j=1}^{J}\big{[}s_{j}^{n-2m}e^{-i\frac{s_{j}t_{j}}{2}}h_{s_{j}\omega_{j}}(y_{j})\big{]}K_{J}(s_{1}\omega_{1},\dots,s_{J}\omega_{J})\,ds_{J}\cdots
ds_{1}.$

Taking the absolute values and then extending the integrals in $t_{j}$,
$j=1,2,\dots,J$ to $\mathbb{R}$, we have

$|Z_{J}f(x)|\lesssim\int_{(S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R})^{J}}|F(t_{1},\omega_{1},y_{1},\dots,t_{J},\omega_{J},y_{J})||f(\overline{x}-\gamma_{1})|dt_{J}dy_{J}d\omega_{J}\cdots
dt_{1}dy_{1}d\omega_{1}.$

Therefore, by Minkowski’s integral inequality and noting that
$x\to\overline{x}$ is an isometry), we have

$\|Z_{J}f\|_{L^{p}}\lesssim\|F\|_{L^{1}((S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R})^{J})}\|f\|_{L^{p}}.$

The following lemma finishes the proof of $L^{p}$ boundedness of $Z_{J}$.

######  Lemma 2.4.

For $2m<n<4m-1$, we have

$\|F\|_{L^{1}((S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R})^{J})}\leq
C^{J}\|\langle\cdot\rangle^{\frac{4m+1-n}{2}+}V(\cdot)\|_{L^{2}}^{J},$

for $n=4m-1$, we have

$\|F\|_{L^{1}((S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R})^{J})}\leq
C^{J}\|\langle x\rangle^{1+}V\|_{H^{0+}}^{J},$

for $n>4m-1$ and $\sigma>\frac{n-2m}{n-1}$, we have

$\|F\|_{L^{1}((S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R})^{J})}\leq
C^{J}\|\mathcal{F}(\langle
x\rangle^{2\sigma}V)\|_{L^{\frac{n-1}{n-2m}-}}^{J}.$

Here $C$ depends on $n,m$ and the actual values of $\pm$ signs.

###### Proof.

We write $F$ as a sum of $2^{J}$ operators of the form (for each subset
$\mathcal{J}$ of $\\{1,2,...,J\\}$)

$F_{\mathcal{J}}(t_{1},\omega_{1},y_{1},\dots,t_{J},\omega_{J},y_{J})=F(t_{1},\omega_{1},y_{1},\dots,t_{J},\omega_{J},y_{J})\big{[}\prod_{j\in\mathcal{J}}\chi(y_{j})\big{]}\big{[}\prod_{j\not\in\mathcal{J}}\widetilde{\chi}(y_{j})\big{]}.$

It suffices to prove that each $F_{\mathcal{J}}$ satisfies the claim.

Fix $r\geq 2$ and $\frac{1}{q}+\frac{1}{r}=1$. By Hausdorff-Young inequality,
we have

$\|F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\int_{(S^{n-1}\times\mathbb{R}^{n})^{J}}\Big{[}\int_{(0,\infty)^{J}}\big{[}\prod_{j=1}^{J}s_{j}^{n-2m}h_{s_{j}\omega_{j}}(y_{j})\big{]}^{q}\times\\\
|K_{J}(s_{1}\omega_{1},\dots,s_{J}\omega_{J})|^{q}ds_{1}\dots
ds_{J}\Big{]}^{1/q}\big{[}\prod_{j\in\mathcal{J}}\chi(y_{j})\big{]}\big{[}\prod_{j\not\in\mathcal{J}}\widetilde{\chi}(y_{j})\big{]}d\vec{y}d\vec{\omega}.$

Note that, by the proof of Lemma 2.2 above (for $0<\delta\leq 1$)

$|h_{s\omega}(y)|\lesssim
s^{n}\min((s|y|)^{-n-\delta},(s|y|)^{-n+\delta})\lesssim\chi(y)|y|^{-n+\delta}s^{\delta}+\widetilde{\chi}(y)|y|^{-n-\delta}s^{-\delta}.$

Since $\chi(y)|y|^{-n+\delta}\in L^{1}$ and
$\widetilde{\chi}(y)|y|^{-n-\delta}\in L^{1}$ for any $\delta>0$, we can bound
the norm above by

$\int_{(S^{n-1})^{J}}\Big{[}\int_{(0,\infty)^{J}}\big{[}\prod_{j\in\mathcal{J}}^{J}s_{j}^{(n-2m+\delta)q}\big{]}\big{[}\prod_{j\not\in\mathcal{J}}^{J}s_{j}^{(n-2m-\delta)q}\big{]}|K_{J}(s_{1}\omega_{1},\dots,s_{J}\omega_{J})|^{q}d\vec{s}\Big{]}^{1/q}d\vec{\omega}.$

By Holder in $\omega_{j}$ integrals we conclude that

(15)
$\|F\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\Big{[}\int_{\mathbb{R}^{nJ}}\big{[}\prod_{j\in\mathcal{J}}^{J}|k_{j}|^{(n-2m+\delta)q-n+1}\big{]}\big{[}\prod_{j\not\in\mathcal{J}}^{J}|k_{j}|^{(n-2m-\delta)q-n+1}\big{]}\times\\\
|K_{J}(k_{1},\dots,k_{J})|^{q}dk_{1}\dots dk_{J}\Big{]}^{1/q}.$

Similarly, (here $\alpha_{j}=0$ or $1$ independently)

$\|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\\\
\int_{(S^{n-1}\times\mathbb{R}^{n})^{J}}\Big{[}\int_{(0,\infty)^{J}}\Big{|}\partial_{s_{1}}^{\alpha_{1}}\dots\partial_{s_{J}}^{\alpha_{J}}\prod_{j=1}^{J}\big{(}s_{j}^{n-2m}h_{s_{j}\omega_{j}}(y_{j})\big{)}\times\\\
|K_{J}(s_{1}\omega_{1},\dots,s_{J}\omega_{J})|^{q}ds_{1}\dots
ds_{J}\Big{]}^{1/q}\big{[}\prod_{j\in\mathcal{J}}\chi(y_{j})\big{]}\big{[}\prod_{j\not\in\mathcal{J}}\widetilde{\chi}(y_{j})\big{]}d\vec{y}d\vec{\omega}.$

Since $\partial_{s}h_{s\omega}$ satisfies the same bounds as
$\frac{1}{s}h_{s\omega}$, proceeding as above, we obtain the estimate

$\|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\Big{[}\int_{\mathbb{R}^{nJ}}\big{[}\prod_{j\in\mathcal{J}}|k_{j}|^{(n-2m+\delta)q-n+1}\big{]}\big{[}\prod_{j\not\in\mathcal{J}}|k_{j}|^{(n-2m-\delta)q-n+1}\big{]}\times\\\
\Big{|}\prod_{j=1}^{J}(\nabla_{k_{j}}^{\alpha_{j}}+|k_{j}|^{-\alpha_{j}})K_{J}(k_{1},\dots,k_{J})\Big{|}^{q}dk_{1}\dots
dk_{J}\Big{]}^{1/q}.$

Using Hardy’s inequality, this implies that

(16) $\|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\Big{[}\int_{\mathbb{R}^{nJ}}\big{[}\prod_{j\in\mathcal{J}}|k_{j}|^{(n-2m+\delta)q-n+1}\big{]}\big{[}\prod_{j\not\in\mathcal{J}}|k_{j}|^{(n-2m-\delta)q-n+1}\big{]}\times\\\
\Big{|}\prod_{j=1}^{J}\nabla_{k_{j}}^{\alpha_{j}}K_{J}(k_{1},\dots,k_{J})\Big{|}^{q}dk_{1}\dots
dk_{J}\Big{]}^{1/q}.$

Let $2m<n<4m-1$. Applying (15) with $0<\delta\ll 1$ and $q=r=2$, we obtain

$\|F_{\mathcal{J}}\|^{2}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2}(\mathbb{R}^{J})}\lesssim\int_{\mathbb{R}^{nJ}}\big{[}\prod_{j\in\mathcal{J}}|k_{j}|^{n-4m+1+2\delta}\big{]}\big{[}\prod_{j\not\in\mathcal{J}}|k_{j}|^{n-4m+1-2\delta}\big{]}|K_{J}(k_{1},\dots,k_{J})|^{2}d\vec{k}.$

Note that by Hardy’s inequality the integral in $k_{J}$ is bounded by

$\int||D_{k_{J}}|^{\frac{4m-1-n}{2}\pm\delta}\widehat{V}(k_{J-1}-k_{J})|^{2}dk_{J}\lesssim\|\langle\cdot\rangle^{\frac{4m-1-n}{2}\pm\delta}V(\cdot)\|_{L^{2}}^{2}\lesssim\|\langle\cdot\rangle^{\frac{4m-1-n}{2}+\delta}V(\cdot)\|_{L^{2}}^{2}.$

Repeated application of this inequality yields

$\|F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2}(\mathbb{R}^{J})}\lesssim\|\langle\cdot\rangle^{\frac{4m-1-n}{2}+\delta}V(\cdot)\|_{L^{2}}^{J}.$

Similarly, applying (16) with $r=q=2$ and $0<\delta\ll 1$ yield

$\|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2}(\mathbb{R}^{J})}\lesssim\|\langle\cdot\rangle^{2+\frac{4m-1-n}{2}+\delta}V(\cdot)\|_{L^{2}}^{J}.$

Writing

$\prod_{j=1}^{J}(1+|t_{j}|)=\sum_{\alpha_{1},\dots,\alpha_{J}\in\\{0,1\\}}|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}|,$

these inequalities imply with that

$\big{\|}\prod_{j=1}^{J}\langle t_{j}\rangle
F\big{\|}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2}(\mathbb{R}^{J})}\lesssim\|\langle\cdot\rangle^{2+\frac{4m-1-n}{2}+\delta}V(\cdot)\|_{L^{2}}^{J},$

which by multilinear complex interpolation leads to

$\big{\|}\prod_{j=1}^{J}\langle
t_{j}\rangle^{\frac{1}{2}+}F_{\mathcal{J}}\big{\|}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2}(\mathbb{R}^{J})}\lesssim\|\langle\cdot\rangle^{1+\frac{4m-1-n}{2}+\delta+}V(\cdot)\|_{L^{2}}^{J}.$

This proves the claim for $n<4m-1$ by Cauchy-Schwarz in $t$ integrals.

For $n=4m-1$, with $q=2-$, $r=2+$, (15) implies

$\|F_{\mathcal{J}}\|^{2-}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2+}(\mathbb{R}^{J})}\lesssim\int_{\mathbb{R}^{nJ}}\big{[}\prod_{j\not\in\mathcal{J}}^{J}|k_{j}|^{0-}\big{]}|K_{J}(k_{1},\dots,k_{J})|^{2-}dk_{1}\dots
dk_{J}.$

By Hardy’s inequality, the integral in $k_{J}$ is

$\lesssim\int\big{|}|D_{k_{J}}|^{0+}\mathcal{F}(V(\cdot)e^{ik_{J-1}\cdot})(k_{J})\big{|}^{2-}dk_{J}\lesssim\int\big{|}\mathcal{F}(\langle\cdot\rangle^{0+}V(\cdot)e^{ik_{J-1}\cdot})(k_{J})\big{|}^{2-}dk_{J}\\\
\lesssim\int\big{|}\mathcal{F}(\langle\cdot\rangle^{0+}V(\cdot))(k_{J})\big{|}^{2-}dk_{J}\lesssim\Big{[}\int\langle
k_{J}\rangle^{0+}\big{|}\mathcal{F}(\langle\cdot\rangle^{0+}V(\cdot))(k_{J})\big{|}^{2}dk_{J}\Big{]}^{\frac{2-}{2}}\lesssim\|\langle\cdot\rangle^{0+}V(\cdot)\|_{H^{0+}}^{2-}.$

Repeating the same argument in the remaining variables yield

$\|F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{2+}(\mathbb{R}^{J})}\lesssim\|\langle\cdot\rangle^{0+}V(\cdot)\|_{H^{0+}}^{J}.$

Similar modifications in the other inequalities imply the claim in this case.

When $n>4m-1$, we apply the inequalities with $0<\delta\ll 1$ and
$q=\frac{n-1-\delta}{n-2m}$, $r=\frac{n-1-\delta}{2m-1-\delta}$ to obtain

$\|F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\Big{[}\int_{\mathbb{R}^{nJ}}\prod_{j\not\in\mathcal{J}}|k_{j}|^{0-}|K_{J}(k_{1},\dots,k_{J})|^{q}dk_{1}\dots
dk_{J}\Big{]}^{1/q}\lesssim\|\mathcal{F}(\langle\cdot\rangle^{0+}V(\cdot))\|_{L^{q}}^{J}.$

Similarly, we obtain

$\|t_{1}^{\alpha_{1}}\dots
t_{J}^{\alpha_{J}}F_{\mathcal{J}}\|_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{r}(\mathbb{R}^{J})}\lesssim\|\mathcal{F}(\langle\cdot\rangle^{2+}V(\cdot))\|_{L^{q}}^{J},$

which implies that

$\big{\|}\prod_{j=1}^{J}\langle t_{j}\rangle
F_{\mathcal{J}}\big{\|}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{\frac{n-1-\delta}{2m-1-\delta}}(\mathbb{R}^{J})}\lesssim\|\mathcal{F}(\langle
x\rangle^{2+}V)\|_{L^{\frac{n-1-\delta}{n-2m}}}^{J}.$

Interpolating the two bounds we obtain (with $\sigma>\frac{n-2m}{n-1-\delta}$)

$\big{\|}\prod_{j=1}^{J}\langle
t_{j}\rangle^{\sigma}F_{\mathcal{J}}\big{\|}_{L^{1}(S^{n-1}\times\mathbb{R}^{n})^{J}L^{\frac{n-1-\delta}{2m-1-\delta}}(\mathbb{R}^{J})}\lesssim\|\mathcal{F}(\langle
x\rangle^{2\sigma}V)\|_{L^{\frac{n-1-\delta}{n-2m}}}^{J},$

which implies the claim by Holder’s inequality in $t$ integrals.

∎

Keeping track of the relationship between $q,r,\sigma$ and $\delta$ in the
proof above leads to the statement in Theorem 2.1.

##  3\. Low Energies: Odd dimensions

Throughout this section we consider odd dimensions $n$, as the Schrödinger
resolvent has a closed form representation, (5), that is entire. We prove that
the wave operators are bounded on the range $1<p<\infty$ for odd $n$. We show
in Section 6 how to adapt the arguments here to account for the logarithmic
singularities present in even dimensions. Further, in Section 4 we show that
for odd $n$ it is possible to capture boundedness on the endpoints of
$p=1,\infty$.

Having controlled the contribution of the Born series terms to (6), to
establish the claim of Theorem 1.2 we need to show the boundedness of the tail
of the Born series in (7). Noting that spectral localization, multiplying by
the cut-off $\chi(\lambda)$ in (6) is bounded on $L^{p}$, we need only control
the contribution of

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

With $v=|V|^{\frac{1}{2}}$, $U(x)=1$ if $V(x)\geq 0$ and $U(x)=-1$ if
$V(x)<0$, we define $M^{+}(\lambda)=U+v\mathcal{R}_{0}^{+}(\lambda^{2m})v$. We
also define $w(x)=U(x)v(x)$. Using the symmetric resolvent identity, one has

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

which is valid in a sufficiently small neighborhood of $\lambda=0$. We show

######  Proposition 3.1.

Let $n>2m$ be odd. If $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some
$\beta>n+2$, then the operator defined by

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

extends to a bounded operator on $L^{p}(\mathbb{R}^{n})$ for all $1<p<\infty$.

We utilize the representation of the $m^{th}$ order resolvent frequently.

######  Lemma 3.2.

Let $n>2m$ be odd. Then, we have the following representation of the free
resolvent

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

Here $|F^{(N)}(r)|\lesssim\langle r\rangle^{\frac{n+1}{2}-2m-N}$,
$N=0,1,2,...$

###### Proof.

By the splitting identity, and (5) we have

$\displaystyle\mathcal{R}_{0}^{+}(\lambda^{2m})(y,u)$
$\displaystyle=\frac{1}{m\lambda^{2m-2}}\big{[}R_{0}^{+}(\lambda^{2})(y,u)+\sum_{\ell=1}^{m-1}\omega_{\ell}R_{0}^{+}(\omega_{\ell}\lambda^{2})(y,u)\big{]}$
$\displaystyle=\frac{e^{i\lambda|y-u|}}{m\lambda^{2m-2}|y-u|^{n-2}}\big{[}P_{\frac{n-3}{2}}(\lambda|y-u|)+\sum_{\ell=1}^{m-1}\omega_{\ell}e^{i(\omega_{\ell}^{\frac{1}{2}}-1)\lambda|y-u|}P_{\frac{n-3}{2}}(\omega_{\ell}^{\frac{1}{2}}\lambda|y-u|)\big{]}$

Here $P_{k}(s)$ indicates a polynomial of degree $k$ in $s$, the exact
coefficients are not important. Therefore,

$F(r)=\frac{1}{mr^{2m-2}}\big{[}P_{\frac{n-3}{2}}(r)+\sum_{\ell=1}^{m-1}\omega_{\ell}e^{i(\omega_{\ell}^{\frac{1}{2}}-1)r}P_{\frac{n-3}{2}}(\omega_{\ell}^{\frac{1}{2}}r)\big{]}=:\frac{g(r)}{r^{2m-2}}.$

Note that $g$ is entire and bounded by a constant multiple of $\langle
r\rangle^{\frac{n-3}{2}}$ on the positive real line. Moreover,
$|\partial_{r}^{N}g(r)|\lesssim\langle r\rangle^{\frac{n-3}{2}-N}$ for each
$N\in\mathbb{N}$ and $r>0$. By a Taylor series expansion, see for example
Proposition 2.4 in [9], the resolvent is bounded in $\xi$ as $|\xi|\to 0$ and
has a series expansion in $|\xi|\,|y-u|$ near $\xi=0$. This implies that $g$
has a zero of degree $\geq 2m-2$ at $0$, which implies the first claim. ∎

To prove Proposition 3.1, we need to understand the operator
$M^{+}(\lambda)^{-1}$. By the assumption that zero energy is regular,
$M^{+}(\lambda)^{-1}$ is a bounded operator. To show this, we use the
following low energy bounds on the resolvent.

######  Lemma 3.3.

Let $n>2m$ be odd. We have the following bounds on the derivatives of the
resolvent. For $k=1,2,\dots,$ we have

$\sup_{0<\lambda<1}|\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)|\lesssim|x-y|^{2m+1-n}+|x-y|^{k-(\frac{n-1}{2})}.$

###### Proof.

In all cases we use the expansions in Lemma 3.2. By the product and chain
rules, we have

$\big{|}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)\big{|}=\bigg{|}\frac{1}{|x-y|^{n-2m}}\sum_{\ell=0}^{k}\binom{k}{\ell}\big{(}\partial_{\lambda}^{k-\ell}e^{i\lambda|x-y|}\big{)}\big{(}\partial_{\lambda}^{\ell}F(\lambda|x-y|)\big{)}\bigg{|}\\\
\lesssim|x-y|^{2m-n+k}\sum_{\ell=0}^{k}\langle\lambda|x-y|\rangle^{\frac{n+1}{2}-2m-\ell}\lesssim|x-y|^{2m-n+k}\langle\lambda|x-y|\rangle^{\frac{n+1}{2}-2m}.$

From here, it follows that

$\displaystyle\big{|}\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)\big{|}\lesssim\lambda^{k-1}|x-y|^{k+2m-n}\langle\lambda|x-y|\rangle^{\frac{n+1}{2}-2m}.$

When $\lambda|x-y|\leq 1$, we cannot use the terms in the bracket, but instead
rearrange to see

$\displaystyle\chi(\lambda|x-y|)\big{|}\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)\big{|}\lesssim\chi(\lambda|x-y|)(\lambda|x-y|)^{k-1}|x-y|^{2m+1-n}\lesssim|x-y|^{2m+1-n}.$

Here we used that $k-1\geq 0$. When $\lambda|x-y|\geq 1$, we have

$\displaystyle\widetilde{\chi}(\lambda|x-y|)\big{|}\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)\big{|}\lesssim\widetilde{\chi}(\lambda|x-y|)(\lambda|x-y|)^{k-1-2m+\frac{n+1}{2}}|x-y|^{2m+1-n}.$

Here we consider cases, either $k-1-2m+\frac{n+1}{2}<0$ hence the first term
is bounded by one and we have the bound $|x-y|^{1+2m-n}$. On the other hand,
if $k-1-2m+\frac{n+1}{2}\geq 0$ we bound by

$\widetilde{\chi}(\lambda|x-y|)\lambda^{k-1-2m+\frac{n+1}{2}}|x-y|^{k-(\frac{n-1}{2})}.$

Since the exponent on $\lambda$ is non-negative, taking the supremum on
$0<\lambda<1$ yields the bound of $|x-y|^{k-(\frac{n-1}{2})}$. ∎

To control the low energy, we define the following terms. First, we define an
operator $T:L^{2}\to L^{2}$ with integral kernel $T(\cdot,\cdot)$ to be
absolutely bounded if the operator with kernel $|T(\cdot,\cdot)|$ is also
bounded on $L^{2}$. Further, we define the operator

$T_{0}:=U+v\mathcal{R}_{0}^{+}(0)v=M^{+}(0).$

Here $v=|V|^{\frac{1}{2}}$ and $V=vw$, recall that $|w|=v$. By the assumption
that zero energy is regular, $T_{0}$ is invertible.

The bounds in Lemma 3.3 imply that the operator $R_{k}$ with kernel

(17)
$R_{k}(x,y):=v(x)v(y)\sup_{0<\lambda<1}|\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)|$

is bounded on $L^{2}(\mathbb{R}^{2})$ for $1\leq k\leq\frac{n+1}{2}$ provided
that $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some $\beta>n+2$. We note
that when $n$ is large compared to $m$, we identify $|x-y|^{2m+1-n}$ as a
multiple of the fractional integral operator $I_{2m+1}:L^{2,\sigma}\to
L^{2,-\sigma}$, see Propositions 3.2 and 3.3 in [13] for example. Using the
decay of $v(x)v(y)$ suffices when identifying
$\sigma=\sigma^{\prime}=\frac{\beta}{2}$, to apply the Propositions in [13]
and establish boundedness on $L^{2}$.

Note that by a Neumann series expansion and the invertibility of $T_{0}$ we
have

$[M^{+}(\lambda)]^{-1}=\sum_{k=0}^{\infty}(-1)^{k}T_{0}^{-1}(E(\lambda)T_{0}^{-1})^{k},$

where
$E(\lambda)=v[\mathcal{R}^{+}_{0}(\lambda^{2m})-\mathcal{R}^{+}_{0}(0)]v$ for
$0<\lambda<\lambda_{0}$. By (17) and the mean value theorem we have

$E_{0}(x,y):=\sup_{0<\lambda<\lambda_{0}}|E(\lambda)(x,y)|\lesssim\lambda_{0}R_{1}(x,y)$

is a bounded operator on $L^{2}$ with norm $\lesssim\lambda_{0}$. Therefore,

$\Gamma_{0}(x,y):=\sup_{0<\lambda<\lambda_{0}}|[M^{+}(\lambda)]^{-1}(x,y)|$

is bounded on $L^{2}$ for sufficiently small $\lambda_{0}$.

Similarly, note that by the resolvent identity the operator
$\lambda^{N}\partial_{\lambda}^{N}[M^{+}(\lambda)]^{-1}$ is a linear
combination of operators of the form

$[M^{+}(\lambda)]^{-1}\prod_{j=1}^{J}\big{[}v\big{(}\lambda^{k_{j}}\partial_{\lambda}^{k_{j}}\mathcal{R}_{0}^{+}(\lambda^{2m})\big{)}v[M^{+}(\lambda)]^{-1}\big{]},$

where $\sum k_{j}=N$ and each $k_{j}\geq 1$. Therefore using (17) we see that

(18)
$\Gamma_{N}(x,y):=\sup_{0<\lambda<\lambda_{0}}\lambda^{N}|\partial_{\lambda}^{N}[M^{+}(\lambda)]^{-1}(x,y)|$

is bounded in $L^{2}$ for $N=0,1,\ldots,\frac{n+1}{2}$ provided that
$\beta>n+2$. Further, for $N\geq 1$ we may replace $\lambda^{N}$ with
$\lambda^{N-1}$, and the operator remains bounded on $L^{2}$. This bound
suffices for $n<4m$, odd. However, for odd $n>4m$ we need to modify the
approach to account for the fact that $|x-\cdot|^{2m-n}$ is no longer locally
$L^{2}(\mathbb{R}^{n})$. We iterate the Born series further and utilize the
following

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

By repeated iterations of Lemma 7.2 using the representation of Lemma 3.2,
each iteration of the resolvent smooths out $2m$ power of the singularity.
Selecting $\kappa$ large enough ensures that $A$ is bounded. That is, we have

######  Lemma 3.4.

Fix odd $n>4m$. If $\kappa\in\mathbb{N}$ is sufficiently large depending on
$n,m$ and $|V(x)|\lesssim\langle x\rangle^{-\frac{n+3}{2}-}$, then

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

for $0\leq\ell\leq\frac{n+1}{2}$.

We will prove this lemma at the end of this section. We say an operator $K$ is
admissible if its integral kernel $K(x,y)$ satisfies

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

By the Schur test, an admissible operator is bounded on $L^{p}$ for all $1\leq
p\leq\infty$.

By iterating the Born series sufficiently many times it suffices to prove that
the operator with kernel

$\displaystyle\int_{0}^{\infty}\lambda^{2m-1}\chi(\lambda)[\mathcal{R}_{0}^{+}(\lambda^{2m})VA(\lambda)vM^{-1}(\lambda)vA(\lambda)V\mathcal{R}_{0}^{\mp}(\lambda^{2m})](x,y)\,d\lambda.$

is bounded on $L^{p}$, $1<p<\infty$. Letting (recall that $|w|=v$)

$\Gamma=wA(\lambda)vM^{-1}(\lambda)vA(\lambda)w,$

and using Lemma 3.4 and (18) we see that $\Gamma$ satisfies

(20) $\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\sup_{0\leq
k\leq\frac{n+1}{2}}\big{|}\lambda^{k}\partial_{\lambda}^{k}\Gamma(\lambda)(x,y)\big{|}\lesssim\langle
x\rangle^{-\frac{n}{2}-}\langle y\rangle^{-\frac{n}{2}-},$

provided that $\beta>n+2$. Hence, Proposition 3.1 is a consequence of the
following bound.

######  Lemma 3.5.

Fix $n$ odd and let $\Gamma$ be a $\lambda$ dependent absolutely bounded
operator. Let

$\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\sup_{0\leq
k\leq\frac{n+1}{2}}\big{|}\lambda^{k}\partial_{\lambda}^{k}\Gamma(\lambda)(x,y)\big{|}.$

For $2m<n<4m$ assume that $\widetilde{\Gamma}$ is bounded on $L^{2}$, and for
$n>4m$ assume that $\widetilde{\Gamma}$ satisfies (20). Then the operator with
kernel

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

is bounded on $L^{p}$ for $1<p<\infty$ provided that $\beta>n$.

###### Proof.

Using the representation in Lemma 3.2 with $r_{1}=|x-z_{1}|$ and
$r_{2}:=|z_{2}-y|$ we have

(21)
$K(x,y)=\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})}{r_{1}^{n-2m}r_{2}^{n-2m}}\int_{0}^{\infty}e^{i\lambda(r_{1}-r_{2})}\chi(\lambda)\lambda^{2m-1}\Gamma(\lambda)(z_{1},z_{2})F(\lambda
r_{1})F(\lambda r_{2})d\lambda dz_{1}dz_{2}.$

Using the bounds in Lemma 3.2, (20), and the assumption $\beta>n$, we bound
the $\lambda$-integral above by

(22)
$\displaystyle\widetilde{\Gamma}(z_{1},z_{2})\int_{0}^{1}\frac{\lambda^{2m-1}}{\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}}\langle\lambda
r_{2}\rangle^{2m-\frac{n+1}{2}}}d\lambda.$

Also note that by integrating by parts $N\leq\frac{n+1}{2}$ times in $\lambda$
when $\lambda|r_{1}-r_{2}|>1$ and using (22) when $\lambda|r_{1}-r_{2}|<1$,
and recalling the bounds for the derivatives of $F$, we obtain

(23)
$\displaystyle|K(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})}{r_{1}^{n-2m}r_{2}^{n-2m}}\int_{0}^{1}\frac{\lambda^{2m-1}}{\langle\lambda(r_{1}-r_{2})\rangle^{N}\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}}\langle\lambda
r_{2}\rangle^{2m-\frac{n+1}{2}}}d\lambda dz_{1}dz_{2}.$

Note that there are no boundary terms here since we include the cutoff
$\widetilde{\chi}(\lambda(r_{1}-r_{2}))$ in the integration by parts argument
above. Also note that we can choose $N$ depending on $z_{1},z_{2}$. We write

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

where the integrand in $K_{1}$ is restricted to the set $r_{1},r_{2}\lesssim
1$, in $K_{2}$ to the set $r_{1}\approx r_{2}\gg 1$, in $K_{3}$ to the set
$r_{2}\gg\langle r_{1}\rangle$, in $K_{4}$ to the set $r_{1}\gg\langle
r_{2}\rangle$.

Note that $K_{1}$ is admissible using (23) with $N=0$: For $n<4m$ we have

$\int|K_{1}(x,y)|dx\lesssim\int_{\mathbb{R}^{2n}}\int_{r_{1}<1}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})}{r_{1}^{n-2m}r_{2}^{n-2m}}dxdz_{1}dz_{2}\lesssim\|v(\cdot)|y-\cdot|^{2m-n}\|_{L^{2}}\|\widetilde{\Gamma}\|_{L^{2}\to
L^{2}}\|v\|_{L^{2}}\lesssim 1,$

provided that $\beta>n$. For $n>4m$, we instead have

$\int|K_{1}(x,y)|dx\lesssim\int_{\mathbb{R}^{2n}}\int_{r_{1}<1}\frac{\langle
z_{1}\rangle^{-n-}\langle
z_{2}\rangle^{-n-}}{r_{1}^{n-2m}r_{2}^{n-2m}}dxdz_{1}dz_{2}\\\
\lesssim\big{\|}|\cdot|^{2m-n}\big{\|}_{L^{1}(B(0,1))}\big{\|}\langle\cdot\rangle^{-n-}|y-\cdot|^{2m-n}\big{\|}_{L^{1}}\big{\|}\langle\cdot\rangle^{-n-}\big{\|}_{L^{1}}\lesssim
1,$

uniformly in $y$. The $y$-integrals can be estimated similarly.

Similarly $K_{2}$ is admissible using (23) with $N=2$: For $n<4m$ we have

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

provided that $\beta>n$. When $n>4m$ we bound the last integral by

$\int_{\mathbb{R}^{2n}}\langle z_{1}\rangle^{-n-}\langle
z_{2}\rangle^{-n-}\int_{0}^{1}\int_{\eta\approx\lambda
r_{2}\gg\lambda}\frac{\lambda^{n-2m-1}\eta^{4m-n-1}}{\langle\eta-\lambda
r_{2}\rangle^{2}\langle\eta\rangle^{4m-n-1}}d\eta d\lambda dz_{1}dz_{2}\\\
\lesssim\int_{\mathbb{R}^{2n}}\langle z_{1}\rangle^{-n-}\langle
z_{2}\rangle^{-n-}\int_{0}^{1}\Big{[}\int_{1}^{\infty}\frac{\lambda^{n-2m-1}d\eta}{\langle\eta-\lambda
r_{2}\rangle^{2}}+\int_{1>\eta\approx\lambda
r_{2}\gg\lambda}\frac{\lambda^{2m-2}d\eta}{\langle\eta-\lambda
r_{2}\rangle^{2}}\Big{]}d\lambda dz_{1}dz_{2}\lesssim 1.$

The $y$-integrals can be estimated similarly.

We will prove that $K_{3}$ and $K_{4}$ are bounded in $L^{p}$ for
$1<p<\infty$. By symmetry we will only consider $K_{3}$. By using (23) with
$N=\frac{n+1}{2}$ we have the bound

(24)
$|K_{3}(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})}{r_{1}^{n-2m}r_{2}^{n-2m}}\int_{0}^{1}\frac{\lambda^{2m-1}\langle\lambda
r_{1}\rangle^{\frac{n+1}{2}-2m}}{\langle\lambda r_{2}\rangle^{2m}}d\lambda
dz_{1}dz_{2}.$

When $n<4m$, we bound this by

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

By Hölder we have

$\Big{\|}\int
K_{3}(x,y)f(y)dy\Big{\|}_{L^{p}}\lesssim\|f\|_{L^{p}}\Big{\|}\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})}{|x-z_{1}|^{n-2m}\langle
x-z_{1}\rangle^{\frac{n}{p}-}}dz_{1}dz_{2}\Big{\|}_{L^{p}}.$

When $|x-z_{1}|>1$ the bound is easy by Minkowski integral inequality.
Similarly, when $|x-z_{1}|<1$ and $p<\frac{n}{n-2m}$. When $|x-z_{1}|<1$ and
$p\geq\frac{n}{n-2m}$, we estimate the integral by

$\langle
x\rangle^{-\beta/2}\int_{\mathbb{R}^{2n}}\frac{\widetilde{\Gamma}(z_{1},z_{2})\chi_{|x-z_{1}|<1}v(z_{2})}{|x-z_{1}|^{n-2m}}dz_{1}dz_{2}\\\
\lesssim\|\widetilde{\Gamma}\|_{L^{2}\to
L^{2}}\|v\|_{L^{2}}\||z_{1}|^{2m-n}\|_{L^{2}_{B(0,1)}}\langle
x\rangle^{-\beta/2}\lesssim\langle x\rangle^{-\beta/2}\in L^{p},$

provided that $\beta/2>\frac{n}{p}$, which holds if $\beta>n$.

For $n>4m$, we bound the $\lambda$-integral in (24) by

$\int_{0}^{1}\frac{\lambda^{2m-1}}{\langle\lambda
r_{2}\rangle^{2m}}d\lambda+\int_{0}^{1}\frac{\lambda^{\frac{n-1}{2}-2m}r_{1}^{\frac{n+1}{2}-2m}}{r_{2}^{2m}}d\lambda\lesssim\frac{\log(r_{2})}{r_{2}^{2m}}+\frac{r_{1}^{\frac{n+1}{2}-2m}}{r_{2}^{2m}}.$

Therefore,

$|K_{3}(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\langle z_{1}\rangle^{-n-}\langle
z_{2}\rangle^{-n-}\Big{[}\frac{\log(r_{2})}{r_{1}^{n-2m}r_{2}^{n}}+\frac{1}{r_{1}^{\frac{n-1}{2}}r_{2}^{n}}\Big{]}dz_{1}dz_{2}.$

This can be bounded as above considering the cases $|x-z_{1}|>1$ and
$|x-z_{1}|<1$ separately. ∎

We now complete the proof of Proposition 3.1 by proving Lemma 3.4:

###### Proof of Lemma 3.4.

Using Lemma 3.2, we note that when $\lambda<1$ we have (with $u_{0}=z_{1}$ and
$u_{\kappa+1}=z_{2}$)

(25)
$\bigg{|}\partial_{\lambda}^{\ell}\bigg{(}\prod_{j=1}^{\kappa}\mathcal{R}_{0}(\lambda^{2m})(u_{j-1},u_{j})V(u_{j})\mathcal{R}_{0}(\lambda^{2m})(u_{\kappa},z_{2})\bigg{)}\bigg{|}\\\
=\bigg{|}\partial_{\lambda}^{\ell}\bigg{(}e^{i\lambda\sum_{j=1}^{\kappa+1}|u_{j-1}-u_{j}|}\prod_{j=1}^{\kappa}\frac{F(\lambda|u_{j-1}-u_{j}|)V(u_{j})}{|u_{j-1}-u_{j}|^{n-2m}}\frac{F(\lambda|u_{\kappa}-u_{\kappa+1}|)}{|u_{\kappa}-u_{\kappa+1}|^{n-2m}}\bigg{)}\bigg{|}\\\
\lesssim(\sum_{j=1}^{\kappa+1}|u_{j-1}-u_{j}|^{\ell})\prod_{j=1}^{\kappa}\frac{\langle
u_{j-1}-u_{j}\rangle^{\frac{n+1}{2}-2m}|V(u_{j})|}{|u_{j-1}-u_{j}|^{n-2m}}\frac{\langle
u_{\kappa}-u_{\kappa+1}\rangle^{\frac{n+1}{2}-2m}}{|u_{\kappa}-u_{\kappa+1}|^{n-2m}}.$

We only consider the case when $j=\kappa+1$ in the first sum above; the other
cases boils down to this case. We need to bound

$\int\prod_{j=1}^{\kappa}\frac{\langle
u_{j-1}-u_{j}\rangle^{\frac{n+1}{2}-2m}|V(u_{j})|}{|u_{j-1}-u_{j}|^{n-2m}}\frac{\langle
u_{\kappa}-u_{\kappa+1}\rangle^{\frac{n+1}{2}-2m}}{|u_{\kappa}-u_{\kappa+1}|^{n-2m-\ell}}du_{1}\dots
du_{\kappa}.$

Note that for $a=1,...,\lfloor n/2m\rfloor-1$, we have

$\int\frac{\langle
u_{0}-u\rangle^{\frac{n+1}{2}-2ma}}{|u_{0}-u|^{n-2ma}}\langle
u\rangle^{-\frac{n+1}{2}-}\frac{\langle
u-u_{1}\rangle^{\frac{n+1}{2}-2m}}{|u-u_{1}|^{n-2m}}du\lesssim\frac{\langle
u_{0}-u_{1}\rangle^{\frac{n+1}{2}-2m(a+1)}}{|u_{0}-u_{1}|^{n-2m(a+1)}},$

namely the power of the singularity decreases by $2m$ but the decay rate does
not change. To see this inequality consider the cases $|u_{0}-u|<1$,
$|u_{0}-u|>1$ separately and same for $|u-u_{1}|$. Also note that if
$a\geq\lfloor n/2m\rfloor$ , then the bound is $\langle
u_{0}-u_{1}\rangle^{-\frac{n-1}{2}}$.

Using this bound in $u_{1},\dots,u_{\kappa-1}$ integrals, and assuming
$\kappa$ is large, we obtain the bound

$\int\langle u_{0}-u_{\kappa}\rangle^{-\frac{n-1}{2}}\langle
u_{\kappa}\rangle^{-\frac{n+3}{2}-}\frac{\langle
u_{\kappa}-u_{\kappa+1}\rangle^{\frac{n+1}{2}-2m}}{|u_{\kappa}-u_{\kappa+1}|^{n-2m-\ell}}du_{\kappa}.$

This is $\lesssim 1$ if $\ell=0,1,\dots,\frac{n-1}{2}$. If
$\ell=\frac{n+1}{2}$, then the bound is $\langle u_{\kappa+1}\rangle$.

If $j\neq 1,\kappa+1$, we start integrating from the farther end to the $j$th
term and obtain the bound $\lesssim 1$. ∎

##  4\. Low Energy: Endpoint estimates in odd dimensions

In this section we prove that the low energy portion of the wave operators in
odd dimensions is bounded at the endpoint values of $p=1,\infty$. The proof
relies on the explicit closed form representation of the odd dimensional
resolvents. Namely, we show

######  Proposition 4.1.

Let $n>2m$ be odd. If $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some
$\beta>n+4$, then the operator defined by

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

extends to a bounded operator on $L^{p}(\mathbb{R}^{n})$ for all $1\leq
p\leq\infty$.

Unlike Proposition 3.1, this proposition relies on a detailed analysis of the
cancellation in $\mathcal{R}_{0}^{+}-\mathcal{R}_{0}^{-}$. We start with the
following

######  Lemma 4.2.

Let $n>2m$ be odd. We have

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

where $\widetilde{F}$ is an entire function satisfying

$|\partial_{r}^{j}\widetilde{F}(r)|\lesssim\langle
r\rangle^{\frac{1-n}{2}-j},\,\,\,r\in\mathbb{R}.$

###### Proof.

By the splitting identity (4) and the explicit form of the odd dimensional
Schrödinger resolvent, we may write:

$\displaystyle[\mathcal{R}_{0}^{+}-\mathcal{R}_{0}^{-}](\lambda^{2m})(y,u)$
$\displaystyle=\frac{1}{m\lambda^{2m-2}}[R_{0}^{+}-R_{0}^{-}](\lambda^{2})(y,u)$
$\displaystyle=\lambda^{n-2}\frac{e^{i\lambda|y-u|}P_{\frac{n-3}{2}}(\lambda|y-u|)-e^{-i\lambda|y-u|}P_{\frac{n-3}{2}}(-\lambda|y-u|)}{(\lambda|y-u|)^{n-2}}$

Here $P_{\frac{n-3}{2}}(r)$ is a polynomial of order $\frac{n-3}{2}$ whose
coefficients may be computed exactly. We identify

$\widetilde{F}(r)=\frac{e^{ir}P_{\frac{n-3}{2}}(r)-e^{-ir}P_{\frac{n-3}{2}}(-r)}{r^{n-2}}.$

For $r>1$ the bounds and are clear. For $0<r<1$, a careful Taylor series
expansion as in [16, 10] shows that for $c_{j}\in\mathbb{R}$,

$\frac{R_{0}^{\pm}(\lambda^{2})(y,u)}{(\lambda|y-u|)^{n-2}}=c_{0}+c_{1}(\lambda|y-u|)+c_{2}(\lambda|y-u|)^{2}+\dots+c_{n-3}(\lambda|y-u|)^{n-3}\\\
+\sum_{j=\frac{n-2}{2}}^{\infty}(c_{2j}(\pm
i\lambda|y-u|)^{2j}+c_{2j+1}(\lambda|y-u|)^{2j+1}).$

From which we deduce, for $0<r<1$,

$\widetilde{F}(r)=\sum_{j=0}^{\infty}2ic_{2j+n-2}r^{2j},$

which suffices to prove the claim.

∎

As in the previous section, the proposition follows from the following

######  Lemma 4.3.

Fix $n$ odd and let $\Gamma$ be a $\lambda$ dependent absolutely bounded
operator. Let

$\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\Big{[}\big{|}\Gamma(\lambda)(x,y)\big{|}+\big{|}\partial_{\lambda}\Gamma(\lambda)(x,y)\big{|}+\sup_{2\leq
k\leq\frac{n+3}{2}}\big{|}\lambda^{k-2}\partial_{\lambda}^{k}\Gamma(\lambda)(x,y)\big{|}\Big{]}.$

For $2m<n<4m$ assume that $\widetilde{\Gamma}$ is bounded on $L^{2}$, and for
$n>4m$ assume that $\widetilde{\Gamma}$ satisfies (20). Then the operator with
kernel

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

is admissible, and hence it is bounded on $L^{p}$ for $1\leq p\leq\infty$
provided that $\beta>n$.

Note that the assumption on $\Gamma$ is stronger than the one in Lemma 3.5. By
a straightforward modification of Lemma 3.3 and Lemma 3.4, which requires
$\beta>n+4$, the operator $\Gamma=vM^{+}(\lambda)^{-1}v$ satisfies the
assumption for $2m<n<4m$. When $n>4m$ the operator
$\Gamma=wA(\lambda)vM^{-1}(\lambda)vA(\lambda)w,$ satisfies the hypotheses for
sufficiently large $\kappa$ when $n>4m$.

###### Proof of Lemma 4.3.

We define $K_{1},...,K_{4}$ as in the proof of Lemma 3.5 and use the notation
$r_{1}=|x-z_{1}|$, $r_{2}=|y-z_{2}|$. Since we already proved the
admissibility of $K_{1}$ and $K_{2}$, it remains to consider $K_{3}$
restricted to the region $r_{2}\gg\langle r_{1}\rangle$ and $K_{4}$ restricted
to the region $r_{1}\gg\langle r_{2}\rangle$. We first consider $K_{4}$. Using
the splitting identity for the resolvent on the left and Lemma 4.2 on the
right, we write the kernel of $K_{4}$ as follows (ignoring constants):

(26) $K_{4}(x,y)=\sum_{\ell=0}^{m-1}\omega_{\ell}\times\\\
\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{1}^{n-2}}\int_{0}^{\infty}\frac{e^{i\omega_{\ell}^{1/2}\lambda
r_{1}}P_{\frac{n-3}{2}}(\omega_{\ell}^{1/2}\lambda
r_{1})}{\lambda^{2m-2}}\chi(\lambda)\lambda^{2m-1}\Gamma(\lambda)(z_{1},z_{2})\lambda^{n-2m}\widetilde{F}(\lambda
r_{2})d\lambda dz_{1}dz_{2}.$

Here $i\omega_{\ell}^{1/2}$ has nonpositive real part and $P_{\frac{n-3}{2}}$
is a polynomial of degree $\frac{n-3}{2}$. Therefore it suffices to prove the
admissibility of operators with kernel

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

for $j=0,1,\ldots,\frac{n-3}{2}$, $|c|=1,\Re(c)\leq 0$. This suffices to
control all the terms that arise in the polynomial and for different choices
of $\ell$ in (26). Integrating by parts in $\lambda$ integral $j+3$ times we
rewrite the lambda integral as

$-\sum_{\ell=0}^{j+2}\Big{(}\frac{-1}{cr_{1}}\Big{)}^{\ell+1}\partial_{\lambda}^{\ell}\big{[}\chi(\lambda)\lambda^{n+j+1-2m}\Gamma(\lambda)\widetilde{F}(\lambda
r_{2})\big{]}\big{|}_{\lambda=0}\\\
+\Big{(}\frac{-1}{cr_{1}}\Big{)}^{j+3}\int_{0}^{\infty}e^{c\lambda
r_{1}}\partial_{\lambda}^{j+3}\big{[}\chi(\lambda)\lambda^{n+j+1-2m}\Gamma(\lambda)\widetilde{F}(\lambda
r_{2})\big{]}d\lambda.$

Note that the boundary terms are zero when $\ell<n+j+1-2m$. Since
$n+j+1-2m\geq j+2$, there is a nonzero boundary term, $\ell=j+2$, only when
$n=2m+1$, and it is a constant multiple of
$r_{1}^{-j-3}\Gamma(0)(z_{1},z_{2})$. The contribution of this to $K_{4,j}$ is
of the form

$\int_{\mathbb{R}^{2n}}\frac{v(z_{1})|\Gamma(0)(z_{1},z_{2})|v(z_{2})\chi_{r_{1}\gg\langle
r_{2}\rangle}}{r_{1}^{n+1}}dz_{1}dz_{2},$

which is admissible. We now consider the integral term. Ignoring the cases
when the derivative hits the smooth cutoff and using the bound for
$\widetilde{F}$ in Lemma 4.2, we bound the integral by

$\sum_{j_{1}+j_{2}+j_{3}=j+3}\int_{0}^{1}\frac{r_{2}^{j_{3}}}{r_{1}^{j+3}}\frac{\lambda^{n+j+1-2m-j_{1}}|\Gamma^{(j_{2})}(\lambda)|}{\langle\lambda
r_{2}\rangle^{\frac{n-1}{2}+j_{3}}}d\lambda.$

Here, $j_{1},j_{2},j_{3}\geq 0$ and $j_{1}\leq n+j+1-2m$. Note that the
condition on $j_{1}$ is relevant only when $n=2m+1$. Assume first that $n\geq
2m+3$, so that $n+1-2m+j\geq j+4$. We bound the integral by

$\widetilde{\Gamma}(z_{1},z_{2})\sum_{j_{1}+j_{2}+j_{3}=j+3}\int_{0}^{1}\frac{1}{r_{1}^{j+3}}\lambda^{j+4-j_{1}-j_{2}-j_{3}}d\lambda\lesssim\widetilde{\Gamma}(z_{1},z_{2})r_{1}^{-j-3}$

whose contribution to $K_{4,j}$ is admissible. When $n=2m+1$, either
$j_{2}\geq 1$ or $j_{3}\geq 1$. In both cases we can bound the integral by

$\widetilde{\Gamma}(z_{1},z_{2})\sum_{j_{1}+j_{2}+j_{3}=j+3}\int_{0}^{1}\frac{\langle
r_{2}\rangle}{r_{1}^{j+3}}\lambda^{j+2-j_{1}-(j_{2}+j_{3}-1)}d\lambda\lesssim\widetilde{\Gamma}(z_{1},z_{2})\frac{\langle
r_{2}\rangle}{r_{1}^{j+3}},$

which has admissible contribution to $K_{4,j}$. Hence, we conclude that the
operator $K_{4}$ is admissible.

We now consider $K_{3}$. Writing

$[\mathcal{R}_{0}^{+}(\lambda^{2m})-\mathcal{R}_{0}^{-}(\lambda^{2m})](z_{2},y)=\frac{1}{\lambda^{2m-2}}[R_{0}^{+}(\lambda^{2})-R_{0}^{-}(\lambda^{2})](z_{2},y)\\\
=\frac{1}{\lambda^{2m-2}r_{2}^{n-2}}\big{[}e^{i\lambda
r_{2}}P_{\frac{n-3}{2}}(\lambda r_{2})-e^{-i\lambda
r_{2}}P_{\frac{n-3}{2}}(-\lambda r_{2})\big{]},$

and using Lemma 3.2 for the resolvent on the left, it suffices to prove the
admissibility of kernels of the form

$K_{3,j}(x,y)=\\\
\int_{\mathbb{R}^{2n}}\frac{v(z_{1})v(z_{2})\chi_{r_{2}\gg\langle
r_{1}\rangle}}{r_{1}^{n-2m}r_{2}^{n-2-j}}\int_{0}^{\infty}\Big{[}e^{i\lambda(r_{1}+r_{2})}-(-1)^{j}e^{i\lambda(r_{1}-r_{2})}\Big{]}\chi(\lambda)\lambda^{1+j}\Gamma(\lambda)F(\lambda
r_{1})d\lambda dz_{1}dz_{2},$

for $j=0,1,\ldots,\frac{n-3}{2}$. In contrast to $K_{4,j}$ there is decay in
both $r_{1}$ and $r_{2}$ present. Integrating by parts in $\lambda$ integral
$j+3$ times we rewrite the lambda integral as

$\sum_{\ell=0}^{j+2}(-1)^{\ell}\Big{[}\Big{(}\frac{i}{r_{1}+r_{2}}\Big{)}^{\ell+1}-(-1)^{j}\Big{(}\frac{i}{r_{1}-r_{2}}\Big{)}^{\ell+1}\Big{]}\partial_{\lambda}^{\ell}\big{[}\chi(\lambda)\lambda^{j+1}\Gamma(\lambda)F(\lambda
r_{1})\big{]}\big{|}_{\lambda=0}\\\
+(-1)^{j+3}\int_{0}^{\infty}\Big{[}\Big{(}\frac{i}{r_{1}+r_{2}}\Big{)}^{j+3}e^{i\lambda(r_{1}+r_{2})}-(-1)^{j}\Big{(}\frac{i}{r_{1}-r_{2}}\Big{)}^{j+3}e^{i\lambda(r_{1}-r_{2})}\Big{]}\partial_{\lambda}^{j+3}\big{[}\chi(\lambda)\lambda^{j+1}\Gamma(\lambda)F(\lambda
r_{1})\big{]}d\lambda.$

Once again, many of the boundary terms are zero. The only nonzero boundary
terms occur when $\ell=j+1$ or $j+2$. When $\ell=j+2$, it is of the form

$\Big{[}\frac{1}{(r_{1}+r_{2})^{j+3}}-(-1)^{j}\frac{1}{(r_{1}-r_{2})^{j+3}}\Big{]}\partial_{\lambda}\big{[}\chi(\lambda)\Gamma(\lambda)F(\lambda
r_{1})\big{]}\big{|}_{\lambda=0}.$

We can bound the magnitude of this by $r_{2}^{-j-3}\langle
r_{1}\rangle\widetilde{\Gamma}(z_{1},z_{2}),$ whose contribution to $K_{3,j}$
is admissible as before. On the other hand, we need to utilize cancellation
for $\ell=j+1$ to see

$\Big{|}\frac{1}{(r_{1}+r_{2})^{j+2}}-(-1)^{j}\frac{1}{(r_{1}-r_{2})^{j+2}}\Big{|}=\frac{1}{r_{2}^{j+2}}\Big{|}\frac{1}{(1+\frac{r_{1}}{r_{2}})^{j+2}}-\frac{(-1)^{2j+2}}{(1-\frac{r_{1}}{r_{2}})^{j+2}}\Big{|}\lesssim\frac{r_{1}}{r_{2}^{j+3}}.$

Hence we may bound it’s contribution by $r_{2}^{-j-3}\langle
r_{1}\rangle\widetilde{\Gamma}(z_{1},z_{2})$ as well.

Using the bounds for $F$ in Lemma 3.2, we bound the integral term by (ignoring
the cases when the derivative hits the cutoff)

$\sum_{j_{1}+j_{2}+j_{3}=j+3}\frac{r_{1}^{j_{3}}}{r_{2}^{j+3}}\int_{0}^{1}\lambda^{j+1-j_{1}}\Gamma^{(j_{2})}(\lambda)\frac{1}{\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}+j_{3}}}d\lambda.$

Here $j_{1},j_{2},j_{3}\geq 0$ and $j_{1}\leq j+1$. We consider the cases
$j_{2}=0,1$ and $j_{2}\geq 2$ seperately. In the former case, we bound the sum
by

$\widetilde{\Gamma}(z_{1},z_{2})\sum_{j+2\leq j_{1}+j_{3}\leq
j+3}\frac{r_{1}^{j_{3}}}{r_{2}^{j+3}}\int_{0}^{1}\frac{\lambda^{j+1-j_{1}}}{\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}+j_{3}}}d\lambda.$

When $r_{1}\lesssim 1$, this is bounded by
$r_{2}^{-j-3}\widetilde{\Gamma}(z_{1},z_{2})$ whose contribution to $K_{3,j}$
is admissible. When $r_{1}\gg 1$, it is bounded by

$\widetilde{\Gamma}(z_{1},z_{2})\sum_{j+2\leq j_{1}+j_{3}\leq
j+3}\frac{r_{1}^{j_{3}-j-2+j_{1}}}{r_{2}^{j+3}}\int_{0}^{r_{1}}\frac{\eta^{j+1-j_{1}}}{\langle\eta\rangle^{2m-\frac{n+1}{2}+j_{3}}}d\eta\lesssim\widetilde{\Gamma}(z_{1},z_{2})\frac{1+r_{1}^{\frac{n+1}{2}-2m}}{r_{2}^{j+3}},$

here using the power of $r_{1}^{2m-n}$ in $K_{3,j}$, this contribution to
$K_{3,j}$ is admissible. In the latter case, we have the bound

$\widetilde{\Gamma}(z_{1},z_{2})\sum_{j_{2}=2}^{j+3}\sum_{j_{1}+j_{3}=j+3-j_{2}}\frac{r_{1}^{j_{3}}}{r_{2}^{j+3}}\int_{0}^{1}\lambda^{j+1-j_{1}-j_{2}+2}\frac{1}{\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}+j_{3}}}d\lambda\\\
\lesssim\widetilde{\Gamma}(z_{1},z_{2})\sum_{j_{2}=2}^{j+3}\sum_{j_{1}+j_{3}=j+3-j_{2}}\frac{1}{r_{2}^{j+3}}\int_{0}^{1}\lambda^{j+3-j_{1}-j_{2}-j_{3}}d\lambda\lesssim\widetilde{\Gamma}(z_{1},z_{2})r_{2}^{-j-3},$

which has admissible contribution. ∎

##  5\. High Energy: Odd dimensions

Since we can control the contribution of the Born series to arbitrary length,
we need only consider the tail of the series in (7) and show that

$\widetilde{\chi}(\lambda)[(\mathcal{R}_{0}^{+}V)^{\ell}V\mathcal{R}_{V}^{+}(V\mathcal{R}_{0}^{+})^{\ell}V\mathcal{R}_{0}^{\pm}](\lambda)$

extend to bounded operators on $L^{p}(\mathbb{R}^{n})$ provided $\ell$ is
sufficiently large. To do this, we invoke the limiting absorption principle
established in [9]. In all cases we assume there are no positive eigenvalues
of $H$.

######  Theorem 5.1 (Theorem 3.9 in [9]).

For $k=0,1,2,3\dots,$ let $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some
$\beta>2+2k$, then for $s,s^{\prime}>k+\frac{1}{2}$
$\mathcal{R}_{V}^{(k)}(z)\in B(s,-s^{\prime})$ is continuous for $z>0$.
Furthermore, we have

$\big{\|}\mathcal{R}_{V}^{(k)}(z)\big{\|}_{L^{2,s}\to
L^{2,-s^{\prime}}}\lesssim|z|^{\frac{1-2m}{2m}(1+k)}.$

Note that, in particular, these bounds hold for the free resolvent. We now
collect some useful bounds on the free resolvent on high energy, when
$\lambda\gtrsim 1$. To do, we define

$\displaystyle\mathcal{G}_{x}^{\pm}(\lambda,z)=e^{\mp
i\lambda|x|}\mathcal{R}_{0}(\lambda^{2m})(x,z)=\frac{e^{\pm
i\lambda(|x-z|-|x|)}}{|x-z|^{n-2m}}F(\lambda|x-y|)$

Following the bounds of Lemma 3.2 and using $\lambda\gtrsim 1$, we see that

(27)
$\displaystyle|\partial_{\lambda}^{\ell}[\widetilde{\chi}(\lambda)G_{x}^{\pm}(\lambda,z)]|$
$\displaystyle\lesssim\lambda^{\frac{n+1}{2}-2m}\langle
z_{1}\rangle^{\ell}\bigg{(}\frac{1}{|x-z|^{n-2m}}+\frac{1}{|x-z|^{\frac{n-1}{2}}}\bigg{)},$
(28)
$\displaystyle|\partial_{\lambda}^{\ell}[\widetilde{\chi}(\lambda)\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,y)]|$
$\displaystyle\lesssim\lambda^{\frac{n+1}{2}-2m}\bigg{(}\frac{1}{|x-y|^{n-2m-\ell}}+\frac{1}{|x-y|^{\frac{n-1}{2}-\ell}}\bigg{)}$

We utilize the following fact. It may be viewed as an extension of Lemma 3.1
in [28] and Lemma 2.1 in [12] to higher dimensions.

######  Lemma 5.2.

Suppose that $K$ is an integral operator whose kernel obeys the pointwise
bounds

(29) $\displaystyle|K(x,y)|\lesssim\frac{1}{\langle
x\rangle^{\frac{n-1}{2}}\langle
y\rangle^{\frac{n-1}{2}}\langle|x|-|y|\rangle^{\frac{n+1}{2}+\epsilon}}.$

Then $K$ is a bounded operator on $L^{p}(\mathbb{R}^{n})$ for $1\leq
p\leq\infty$ if $\epsilon>0$, and on $1<p<\infty$ if $\epsilon=0$.

######  Proposition 5.3.

We have the bound

(30)
$\bigg{|}\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell+1}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,y)\,d\lambda\bigg{|}\\\
\lesssim\frac{1}{\langle|x|-|y|\rangle^{\frac{n+3}{2}}\langle
x\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}},$

provided $\ell$ is sufficiently large, and $|V(x)|\lesssim\langle
x\rangle^{-\beta}$ for some $\beta>n+5$. In particular, this kernel is
admissible and hence the tail extends to a bounded operator on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$.

###### Proof.

We first establish the boundedness of the integral. We note that for
$\sigma>\frac{1}{2}$ and $\ell_{1}=\lfloor\frac{n}{4m}\rfloor+1$ we have

(31)
$\displaystyle\|(V\mathcal{R}_{0}^{+})^{\ell_{1}-1}V\mathcal{R}_{0}^{\pm}(\lambda^{2m})(\cdot,y)\|_{L^{2,\sigma}}\lesssim\frac{\lambda^{\ell_{1}(\frac{n+1}{2}-2m)}}{\langle
y\rangle^{\frac{n-1}{2}}}.$

This follows using the representations (28) with $\ell=0$ and Lemma 7.2
repeatedly as in Lemma 3.4. After $\ell_{1}=\lfloor\frac{n}{4m}\rfloor+1$
iterations we arrive at a bound dominated by $|x-z_{j}|^{-(\frac{n-1}{2})}$
which is locally $L^{2}(\mathbb{R}^{n})$ in $z_{j}$. By Lemma 7.1

$\|\langle
z_{j}\rangle^{-\beta}|y-z_{j}|^{-(\frac{n-1}{2})}\|_{L^{2,\sigma}}\lesssim\frac{1}{\langle
y\rangle^{\frac{n-1}{2}}},$

provided that $\beta>\sigma+\frac{n}{2}$. Similarly,

(32)
$\displaystyle\|\mathcal{(}\mathcal{R}_{0}^{+}V)^{\ell_{1}}(x,\cdot)\|_{L^{2,-\sigma}}\lesssim\frac{\lambda^{\ell_{1}(\frac{n+1}{2}-2m)}}{\langle
x\rangle^{\frac{n-1}{2}}}.$

By repeated uses of Theorem 5.1, we see that

(33)
$\displaystyle\|(\mathcal{R}_{0}^{+}V)^{\ell_{2}}\mathcal{R}_{V}^{+}(V\mathcal{R}_{0}^{+})^{\ell_{2}}\|_{L^{2,\sigma}\to
L^{2,-\sigma}}\lesssim\lambda^{(2\ell_{2}+1)(1-2m)}.$

Let $\ell=\ell_{1}+\ell_{2}$, then combining (31), (32) and (33) we see that

(34)
$\bigg{|}\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell+1}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,y)\,d\lambda\bigg{|}\\\
=\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}\|(\mathcal{R}_{0}^{+}V)^{\ell_{1}}(x,\cdot)\|_{L^{2,-\frac{1}{2}-}}\|(\mathcal{R}_{0}^{+}V)^{\ell_{2}}\mathcal{R}_{V}^{+}(V\mathcal{R}_{0}^{+})^{\ell_{2}}\|_{L^{2,\frac{1}{2}+}\to
L^{2,-\frac{1}{2}-}}\\\
\times\|(V\mathcal{R}_{0}^{+})^{\ell_{1}}V\mathcal{R}_{0}^{\pm}(\lambda^{2m})(\cdot,y)\|_{L^{2,\frac{1}{2}+}}\,d\lambda\\\
\lesssim\frac{1}{\langle x\rangle^{\frac{n-1}{2}}\langle
y\rangle^{\frac{n-1}{2}}}\int_{1}^{\infty}\lambda^{\ell_{1}(n+1-4m)+(2\ell_{2}+1)(1-2m)}\,d\lambda\lesssim\frac{1}{\langle
x\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}}.$

By selecting $\ell_{2}$ large enough, the $\lambda$ integral converges. To
complete the proof, we use the functions $\mathcal{G}^{\pm}$ and integrate by
parts $\frac{n+3}{2}$ times. That is,

$\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell+1}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}R_{0}^{\pm}(\lambda^{2m})(x,y)\,d\lambda\\\
=\int_{0}^{\infty}e^{-i\lambda(|x|\pm|y|)}\widetilde{\chi}(\lambda)\lambda^{2m-1}\mathcal{G}_{x}^{+}(\lambda,z_{1})(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{G}^{\pm}_{y}(\lambda,z_{2\ell+1})\,d\lambda\\\
=\bigg{(}\frac{-1}{i(|x|\pm|y|)}\bigg{)}^{\frac{n+3}{2}}\int_{0}^{\infty}e^{-i\lambda(|x|\pm|y|)}\partial_{\lambda}^{\frac{n+3}{2}}\bigg{(}\widetilde{\chi}(\lambda)\lambda^{3-2m}\mathcal{G}^{+}_{x}(\lambda,\cdot)(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\\\
\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{G}^{\pm}_{y}(\lambda,\cdot)\bigg{)}\,d\lambda.$

By the limiting absorption principle and the support of
$\widetilde{\chi}(\lambda)$, there are no boundary terms when integrating by
parts. To complete the argument, let $k_{j}\in\mathbb{N}\cup\\{0\\}$ be such
that $\sum k_{j}=\frac{n+3}{2}$, then the contribution will be bounded by

$\frac{1}{|\,|x|-|y|\,|^{\frac{n+3}{2}}}\int_{0}^{\infty}|\widetilde{\chi}(\lambda)\lambda^{3-2m-k_{1}}|\partial_{\lambda}^{k_{2}}\mathcal{G}^{+}_{x}(\lambda,\cdot)V|\,|\partial_{\lambda}^{k_{3}}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell_{1}}|\\\
|\partial_{\lambda}^{k_{4}}[\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell_{2}}\mathcal{R}_{V}^{+}(\lambda^{2m})(V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{\ell_{2}}]|\,|\partial_{\lambda}^{k_{5}}(V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{\ell_{1}}V||\partial_{\lambda}^{k_{6}}\mathcal{G}^{\pm}_{y}(\lambda,\cdot)|\,|\,d\lambda.$

Invoking the bounds in (28) and an argument similar to the first case shows
that we have the bound

$\frac{1}{|\,|x|-|y|\,|^{\frac{n+3}{2}}}\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1-k_{1}}\||\partial_{\lambda}^{k_{2}}\mathcal{G}^{+}_{x}(\lambda,\cdot)V|\,|\partial_{\lambda}^{k_{3}}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell_{1}}\|_{L^{2,\frac{1}{2}+k_{4}+}}\\\
\|\partial_{\lambda}^{k_{4}}[\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell_{2}}\mathcal{R}_{V}^{+}(\lambda^{2m})(V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{\ell_{2}}]\|_{L^{2,-\frac{1}{2}-k_{4}-}\to
L^{2,-\frac{1}{2}-k_{4}-}}\\\
\||\partial_{\lambda}^{k_{5}}((V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{\ell_{1}}V||\partial_{\lambda}^{k_{6}}\mathcal{G}^{\pm}_{y}(\lambda,\cdot)|\,|\|_{L^{2,\frac{1}{2}+k_{4}+}}\,d\lambda\\\
\lesssim\frac{1}{|\,|x|-|y|\,|^{\frac{n+3}{2}}\langle
x\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}}.$

We note that the decay rate of $|V(z)|\lesssim\langle z\rangle^{-(n+5)-}$ is
necessitated when all derivatives act on $\mathcal{R}_{V}$ to apply the
limiting absorption principle, Theorem 5.1. This suffices to control the other
extreme cases, when all derivatives act on a single free resolvent, then by
(27), (28) and 7.1 this decay rate on $V$ suffices to push forward decay in
$x$ or $y$ respectively. Combining this with (34) establishes the desired
bound. Invoking Lemma 5.2 establishes the claim on $L^{p}$ boundedness.

∎

By integrating by parts one less time, one obtains the following which
requires less decay of the potential but fails to capture the endpoints.

######  Corollary 5.4.

We have the bound

(35)
$\bigg{|}\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell+1}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,y)\,d\lambda\bigg{|}\\\
\lesssim\frac{1}{\langle|x|-|y|\rangle^{\frac{n+1}{2}}\langle
x\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}},$

provided $\ell$ is sufficiently large and $|V(x)|\lesssim\langle
x\rangle^{-\beta}$ for some $\beta>n+3$. In particular, this kernel extends to
a bounded operator on $L^{p}(\mathbb{R}^{n})$ for all $1<p<\infty$.

##  6\. Even dimensions

In this section we show how the low and high energy results for the tail of
the Born series in odd dimensions proven in Sections 3 and 5 may be applied to
even dimensions. One requires minor modifications to account for the
logarithmic singularities of the resolvent. After developing an appropriate
representation of the free resolvent in Lemma 6.2, the arguments may be easily
adapted.

First we sketch the argument for low energies. We will prove

######  Proposition 6.1.

Let $n>2m$ be even. If $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some
$\beta>n+3$, then the operator defined by

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

extends to a bounded operator on $L^{p}(\mathbb{R}^{n})$ for all $1<p<\infty$.

We have the following representation for the even dimensional free resolvent.

######  Lemma 6.2.

Let $n>2m$ be even. Then, we have the following representation of the free
resolvent

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

Here $|F^{(N)}(r)|\lesssim\langle r\rangle^{\frac{n+1}{2}-2m-N}$,
$N=0,1,2,...,2m-1$, and is valid for any $N$ when $r\gtrsim 1$, while when
$r\ll 1$ we have $|F^{(2m)}(r)|\lesssim\log(r)$ and $|F^{(N)}(r)|\lesssim
r^{2m-N}$ for $N>2m$.

###### Proof.

To prove this we consider cases when $\lambda|y-u|\ll 1$ and
$\lambda|y-u|\gtrsim 1$. We consider first the second-order Schrödinger
resolvent, which may be expressed in terms of the Bessel functions

$R_{0}^{+}(\lambda^{2})(y,u)=\frac{i}{4}\bigg{(}\frac{\lambda}{2\pi|y-u|}\bigg{)}^{\frac{n-2}{2}}H^{(1)}_{\frac{n-2}{2}}(\lambda|y-u|).$

Unlike in odd dimensions, we do not have a closed form representation for the
Hankel function of the first kind $H^{(1)}_{\frac{n-2}{2}}(\cdot)$. Following
the approach in [11], see also [16], for $\lambda|y-u|\ll 1$, we have a series
of the form

$R_{0}^{+}(\lambda^{2m})(y,u)=\frac{1}{|y-u|^{n-2}}\sum_{j=0}^{\infty}\sum_{k=0}^{1}c_{j}(\lambda|y-u|)^{2j}(a_{j}\log(\lambda|y-u|)+b_{j})^{k}.$

The constants $a_{j},b_{j},c_{j}$ may be computed explicitly. Of particular
importance is that $a_{j}=0$ for $j\leq\frac{n}{2}-2$. Combining this with the
splitting identity (4), we have

$\mathcal{R}_{0}^{+}(\lambda^{2m})(y,u)=\frac{1}{m|y-u|^{n-2}\lambda^{2m-2}}\sum_{j=0}^{\infty}\sum_{k=0}^{1}\sum_{\ell=0}^{m-1}c_{j}\omega_{\ell}^{j+1}(\lambda|y-u|)^{2j}(a_{j}\log(\lambda|y-u|)+a_{j}\log(\omega_{\ell})+b_{j})^{k}.$

Using the fact that

$\sum_{\ell=0}^{m-1}\omega_{\ell}^{j+1}\neq 0\quad\text{ if and only if }\quad
j=km-1,\quad k=1,2,\dots,$

we may write (for $\lambda|y-u|\ll 1$)

$\mathcal{R}_{0}^{+}(\lambda^{2m})(y,u)=\frac{1}{|y-u|^{n-2m}}\bigg{(}\sum_{j=0}^{m-1}d_{j}(\lambda|y-u|)^{2j}+\sum_{j=m}^{\infty}d_{j}(\lambda|y-u|)^{2j}(1+d_{j,l}\log(\lambda|y-u|))\bigg{)}$

In particular, the first logarithm occurs at the term $\lambda^{2m}$. The
claim on $F$ for $r\ll 1$ follows since we may write, for any choice of $N$

$F(r)=e^{-ir}\bigg{(}\sum_{j=0}^{m-1}d_{j}r^{2j}+\sum_{j=m}^{N}d_{j}(r^{2j}(1+d_{j,l}\log(r))\bigg{)}+O(r^{N-})$

where the remainder may be differentiated arbitrarily many times.

The large argument expansion of the resolvent is the same from the Bessel
functions, one has for $\lambda|y-u|\gtrsim 1$ that

$\lambda^{2-2m}R_{0}^{+}(\lambda^{2})(y,u)=e^{i\lambda|y-u|}\frac{(\lambda|y-u|)^{\frac{n+2}{2}-2m}}{|y-u|^{n-2m}}\omega_{+}(\lambda|y-u|),$

where $|\partial_{r}^{k}\omega_{+}(r)|\lesssim r^{-\frac{1}{2}-k}$. The
splitting identity (4) along with the exponential decay of the other
resolvents suffices to establish the claim.

∎

######  Lemma 6.3.

Let $n>2m$ be even. We have the following bounds on the derivatives of the
resolvent. For $k=1,2,\dots,$ we have

$\sup_{0<\lambda<1}|\lambda^{k-1}\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)|\lesssim|x-y|^{2m+1-n}+|x-y|^{k-(\frac{n-1}{2})}.$

###### Proof.

Since $|\partial_{r}^{k}F(r)|$ satisfies the same bounds as in the odd case
for $k\leq 2m-1$ and for $r>1$, we can assume that $k\geq 2m$ and
$\lambda|x-y|<1$. We have

$\lambda^{k-1}|\partial_{\lambda}^{k}\mathcal{R}_{0}(\lambda^{2m})(x,y)|\lesssim\lambda^{k-1}|x-y|^{k+2m-n}\sum_{\ell=0}^{k}|F^{(\ell)}(\lambda|x-y|)|\\\
\lesssim\lambda^{k-1}|x-y|^{k+2m-n}\big{[}1+|\log(\lambda|x-y|)|+(\lambda|x-y|)^{2m-k}\big{]}\\\
\lesssim|x-y|^{1+2m-n}[\lambda|x-y|]^{k}\big{[}(\lambda|x-y|)^{0-}+(\lambda|x-y|)^{2m-k}\big{]}\lesssim|x-y|^{1+2m-n}.$

∎

With this the invertibility of $M(\lambda)$ and the bounds on its derivatives
follow by similar arguments to the odd dimensional case, namely

(36)
$\Gamma_{N}(x,y):=\sup_{0<\lambda<\lambda_{0}}\lambda^{N}|\partial_{\lambda}^{N}[M^{+}(\lambda)]^{-1}(x,y)|$

is bounded in $L^{2}$ for $N=0,1,\ldots,\frac{n+2}{2}$, provided that
$\beta>n+3$. This will suffice for $n<4m$ even. For $n\geq 4m$ even, we
iterate the Born series and note that $A(\lambda,z_{1},z_{2})$ defined via
(19) satisfies a slightly modified version of the claim of Lemma 3.4:

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

for $0\leq\ell\leq\frac{n+2}{2}$. The inclusion of $\lambda^{\ell}$ power
takes care of the singularity arising from the logarithm in Lemma 6.2 as in
Lemma 6.3. Therefore letting

$\Gamma=wA(\lambda)vM^{-1}(\lambda)vA(\lambda)w,$

as above, we see that

(37) $\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\sup_{0\leq
k\leq\frac{n+2}{2}}\big{|}\lambda^{k}\partial_{\lambda}^{k}\Gamma(\lambda)(x,y)\big{|}\lesssim\langle
x\rangle^{-\frac{n}{2}-}\langle y\rangle^{-\frac{n}{2}-},$

provided that $\beta>n+3$. The following variant of Lemma 3.5 finishes the
proof:

######  Lemma 6.4.

Fix $n>2m$ even and let $\Gamma$ be a $\lambda$ dependent absolutely bounded
operator. Assume that

$\widetilde{\Gamma}(x,y):=\sup_{0<\lambda<\lambda_{0}}\sup_{0\leq
k\leq\frac{n+2}{2}}\big{|}\lambda^{k}\partial_{\lambda}^{k}\Gamma(\lambda)(x,y)\big{|}$

is bounded on $L^{2}$ for $2m<n<4m$ and satisfies (37) for $n\geq 4m$. Then
the operator with kernel

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

is bounded on $L^{p}$ for $1<p<\infty$ provided that $\beta>n$.

###### Proof.

Writing

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

we see that the $\lambda$ integral satisfies the bound (22):

$\widetilde{\Gamma}(z_{1},z_{2})\int_{0}^{1}\frac{\lambda^{2m-1}}{\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}}\langle\lambda
r_{2}\rangle^{2m-\frac{n+1}{2}}}d\lambda.$

We will use this for $\lambda|r_{1}-r_{2}|<1$ and integrate by parts
$N\leq\frac{n+2}{2}$ times otherwise. Note that by Lemma 6.2, when $\lambda
r>1$ or when $\ell\leq 2m-1$, we have
$|\lambda^{\ell}\partial_{\lambda}^{\ell}F(\lambda r)|\lesssim\langle
r\rangle^{\frac{n+1}{2}-2m}$. When $\lambda r<1$ and $\ell\geq 2m$, we once
again have

$|\lambda^{\ell}\partial_{\lambda}^{\ell}F(\lambda r)|\lesssim(\lambda
r_{1})^{\ell}\big{(}|\log(\lambda r)|+(\lambda r)^{2m-\ell}\big{)}\lesssim
1\lesssim\langle\lambda r\rangle^{\frac{n+1}{2}-2m}.$

Therefore, we obtain the following bound essentially identical to (23):

(38)
$|K(x,y)|\lesssim\int_{\mathbb{R}^{2n}}\frac{v(z_{1})\widetilde{\Gamma}(z_{1},z_{2})v(z_{2})}{r_{1}^{n-2m}r_{2}^{n-2m}}\int_{0}^{1}\frac{\lambda^{2m-1}}{\langle\lambda(r_{1}-r_{2})\rangle^{N}\langle\lambda
r_{1}\rangle^{2m-\frac{n+1}{2}}\langle\lambda
r_{2}\rangle^{2m-\frac{n+1}{2}}}d\lambda dz_{1}dz_{2},$

for all $0\leq N\leq\frac{n+2}{2}$, noting $N$ need not be an integer. The
rest of the proof is identical to the proof of Lemma 3.5 using (38) with $N=0$
for $K_{1}$ with $N=2$ for $K_{2}$ and $N=\frac{n+1}{2}$ for $K_{3}$ and
$K_{4}$. ∎

######  Proposition 6.5.

We have the bound

$\bigg{|}\int_{0}^{\infty}\widetilde{\chi}(\lambda)\lambda^{2m-1}(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell+1}\mathcal{R}_{V}^{+}(\lambda^{2m})V(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,y)\,d\lambda\bigg{|}\\\
\lesssim\frac{1}{\langle|x|-|y|\rangle^{\frac{n+2}{2}}\langle
x\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}},$

provided $\ell$ is sufficiently large, and $|V(x)|\lesssim\langle
x\rangle^{-\beta}$ for some $\beta>n+4$. In particular, this kernel is
admissible and hence the tail extends to a bounded operator on
$L^{p}(\mathbb{R}^{n})$ for all $1\leq p\leq\infty$.

This proof is essentially identical to the proof of Proposition 5.3 in the odd
dimensional case. Here, by Lemma 6.2, the bounds (27) and (28) hold, hence the
proposition follows by integrating by parts $\frac{n+2}{2}$ times to invoke
Lemma 5.2.

##  7\. Integral estimates and Proofs of Technical Lemmas

We now present the proofs of some technical lemmas that are used throughout
the paper. For completeness we provide a proof of Lemma 5.2.

###### Proof of Lemma 5.2.

We first consider the case when $\epsilon=0$, we decompose the integral into
three regions according to whether $|x|>2|y|$, $|x|<\frac{1}{2}|y|$, or
$\frac{1}{2}|y|\leq|x|\leq 2|y|$. In the region where $|x|\approx|y|$,
switching to polar coordinates we see that

$\int_{|x|\approx|y|}|K(x,y)|dx\lesssim\frac{1}{\langle
y\rangle^{n-1}}\int_{|y|/2}^{2|y|}\frac{r^{2}}{\langle
r-|y|\rangle^{\frac{n+1}{2}}}\,dr\lesssim\frac{|y|^{n-1}}{\langle
y\rangle^{n-1}}\int_{|y|/2}^{2|y|}\frac{1}{\langle
r-|y|\rangle^{\frac{n+1}{2}}}\,dr\lesssim 1,$

uniformly in $y$. By symmetry in $x$ and $y$, this part of the operator has an
admissible kernel and is bounded for any $1\leq p\leq\infty$.

On the second region (using that $|\,|x|-|y|\,|\approx|y|$ when
$|x|<\frac{1}{2}|y|$) we see

$\int_{|x|<\frac{1}{2}|y|}\frac{1}{\langle x\rangle^{\frac{n-1}{2}p}\langle
y\rangle^{\frac{n-1}{2}p}\langle|x|-|y|\rangle^{\frac{n+1}{2}p}}dx\lesssim\int_{|x|<\frac{1}{2}|y|}\frac{1}{\langle
x\rangle^{\frac{n-1}{2}p}\langle y\rangle^{np}}dx\lesssim\langle
y\rangle^{\max(n-\frac{3n-1}{2}p,-np)},$

and (using that $|\,|x|-|y|\,|\approx|x|$ when $|x|>2|y|$)

$\int_{|x|>2|y|}\frac{1}{\langle x\rangle^{\frac{n-1}{2}p}\langle
y\rangle^{\frac{n-1}{2}p}\langle|x|-|y|\rangle^{\frac{n+1}{2}p}}dx\lesssim\int_{|x|>2|y|}\frac{1}{\langle
x\rangle^{np}\langle y\rangle^{\frac{n-1}{2}p}}dx\lesssim\langle
y\rangle^{n-\frac{3n-1}{2}p}\text{ when }p>1.$

The constraint on the range of $p$ occurs when $|x|$ is large. Noting that
$\langle y\rangle^{\max(n/p-\frac{3n-1}{2},-n)}=\langle
y\rangle^{\max(-n/p^{\prime}-(n-1)/2,-n)}$ belongs to
$L^{p^{\prime}}(\mathbb{R}^{n})$ for any $1<p<\infty$ so these parts of the
operator $K(x,y)$ are bounded on $L^{p}(\mathbb{R}^{n})$ as long as
$1<p<\infty$.

When $\epsilon>0$ using polar coordinates we see that

$\sup_{y\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{1}{\langle
x\rangle^{\frac{n-1}{2}}\langle
y\rangle^{\frac{n-1}{2}}\langle|x|-|y|\rangle^{\frac{n+1}{2}+\epsilon}}\,dx=\sup_{y\in\mathbb{R}^{n}}C_{n}\int_{0}^{\infty}\frac{r^{n-1}}{\langle
r\rangle^{\frac{n-1}{2}}\langle y\rangle^{\frac{n-1}{2}}\langle
r-|y|\rangle^{\frac{n+1}{2}+\epsilon}}dr\lesssim 1.$

The last inequality follows by breaking up into regions based on whether
$r\leq\frac{1}{2}|y|$, $r\approx|y|$ or $r\geq 2|y|$. Similar to the previous
case, integrability for large $r$ requires $\epsilon>0$. By symmetry in $x$
and $y$, $K$ has an admissible kernel and is bounded for $1\leq p\leq\infty$.

∎

Finally, the following elementary integral estimates are used throughout the
paper.

######  Lemma 7.1 (Lemma 3.8 in [13]).

Let $k,\beta$ be such that $k<n$ and $n<\beta+k$. Then

$\int_{\mathbb{R}^{n}}\frac{du}{\langle
u\rangle^{\beta}|x-u|^{k}}\lesssim\left\\{\begin{array}[]{ll}\langle
x\rangle^{n-\beta-k}&\beta<n\\\ \langle
x\rangle^{-k}&\beta>n\end{array}\right..$

######  Lemma 7.2 (Lemma 6.3 in [6]).

Fix $u_{1},u_{2}\in\mathbb{R}^{n}$, and let $0\leq k,\ell<n$, $\beta>0$,
$k+\ell+\beta\geq n$, $k+\ell\neq n$. We have

$\displaystyle\int_{\mathbb{R}^{n}}\frac{\langle
z\rangle^{-\beta-}dz}{|z-u_{1}|^{k}|z-u_{2}|^{\ell}}\lesssim\left\\{\begin{array}[]{lc}\big{(}\frac{1}{|u_{1}-u_{2}|}\big{)}^{\max(0,k+\ell-n)}&|u_{1}-u_{2}|\leq
1\\\
\big{(}\frac{1}{|u_{1}-u_{2}|}\big{)}^{\min(k,\ell,k+\ell+\beta-n)}&|u_{1}-u_{2}|>1\end{array}\right..$

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