# Uniform vorticity depletion and inviscid damping for periodic shear flows in
the high Reynolds number regime

Rajendra Beekie  Duke University<EMAIL_ADDRESS>,  Shan Chen
University of Minnesota<EMAIL_ADDRESS>and  Hao Jia  University of
Minnesota<EMAIL_ADDRESS>
###### Abstract.

We study the dynamics of the two dimensional Navier-Stokes equations
linearized around a shear flow on a (non-square) torus which possesses exactly
two non-degenerate critical points. We obtain linear inviscid damping and
vorticity depletion estimates for the linearized flow that are uniform with
respect to the viscosity, and enhanced dissipation type decay estimates. The
main task is to understand the associated Rayleigh and Orr-Sommerfeld
equations, under the natural assumption that the linearized operator around
the shear flow in the inviscid case has no discrete eigenvalues. The key
difficulty is to understand the behavior of the solution to Orr-Sommerfeld
equations in three distinct regimes depending on the spectral parameter: the
non-degenerate case when the spectral parameter is away from the critical
values, the intermediate case when the spectral parameter is close to but
still separated from the critical values, and the most singular case when the
spectral parameter is inside the viscous layer.

HJ and SC are supported in part by NSF grant DMS-1945179. RB is supported in
part by NSF grant DMS-2202974

###### Contents

  1. 1 Introduction and main results
  2. 2 Bounds on the Green’s function and modified Green’s function
  3. 3 Pointwise bounds on the kernel of generalized Airy operators and applications
  4. 4 The limiting absorption principle and main bounds on the spectral density function
  5. 5 Proof of the main theorem
  6. 6 Proof of the limiting absorption principle (I): non-degenerate region
  7. 7 Proof of the limiting absorption principle (II): the degenerate region
  8. 8 Proof of Proposition 7.1 \- Proposition 7.3

##  1\. Introduction and main results

Assume that $\nu\in(0,1)$ and $\mathfrak{p}\in(2\pi,\infty)$. Consider the
equation

$\left\\{\begin{array}[]{rl}\partial_{t}\omega-\nu\Delta\omega+b(y)\partial_{x}\omega-b^{\prime\prime}(y)\partial_{x}\psi=0,&\\\
\Delta\psi=\omega,&\end{array}\right.$ (1.1)

for $(x,y,t)\in\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}\times[0,\infty)$.

Equation (1.1) can be obtained as the linearization of the two dimensional
Navier Stokes equations on $\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}$ around
the meta-steady state $(b(y),0)$. Alternatively, one can view the shear flow
$(b(y),0)$ as an exact steady state to the Navier Stokes equations with a
small external forcing $F:=(-\nu
b^{\prime\prime}(y),0),y\in\mathbb{T}_{\mathfrak{p}}$. Equation (1.1) was
studied classically, in an attempt to explain the onset of turbulence. See for
example [20] for an interesting history on this problem. We refer also to [2,
15, 41] for more recent results.

We note that it is natural to study (1.1) on a non-square torus (for our
purposes), since on a square torus some of the most interesting shear flows
such as Kolmogorov flows are not stable [18]. To establish asymptotic
stability, a more natural object to study on a square torus seems to be the
dipole flow, see Remark 1.3 in [65].

Our main goal in this paper is to understand the precise dynamics of (1.1)
starting from relatively smooth initial data, and obtain bounds on the
solution that are uniform with respect to the viscosity $\nu\in(0,\nu_{0})$
for some small $\nu_{0}>0$. In particular we shall capture some of the more
refined dynamics associated with periodic shear flows, and prove uniform
vorticity depletion and linear inviscid damping results. Such problems are
well known and were discussed for example in section 11, chapter 6 of the book
[7] by Bedrossian and Vicol .

###  1.1. Review of recent results on inviscid damping and main problems

The study of stability problems in mathematical analysis of fluid dynamics has
a long and distinguished history, dating back to the work of Kelvin [42], Orr
[52] and Rayleigh [53] among many others, and continuing to the present day.
Despite many significant recent progresses, many important questions still
remain open. The main issue is that very often the most interesting physical
settings involve high Reynolds number (or equivalently the viscosity has to be
taken very small), where the Navier Stokes equations degenerate to Euler
equations and the continuous spectrum plays a dominant role. As a starting
point to a more complete and precise theory of hydrodynamic stability, it is
therefore important to understand mathematically the property of the
linearized flow in the high Reynolds number regime, including the limiting
inviscid case. Below we briefly review some recent works on the linearized
flow for both Euler and Navier Stokes equations with small viscosity, focusing
on two dimensional shear flows which is our main object of study in this
paper, and mentioning nonlinear inviscid damping and other extensions only
briefly.

####  1.1.1. The case of monotonic shear flow

For the Euler equations, there are significant recent progresses on the
asymptotic stability of monotonic shear flows and vortices, assuming spectral
stability, see for example [6, 27, 36, 39, 40, 47, 56, 62, 68, 69] for linear
results. The main mechanism of stabilization is the so called “inviscid
damping”, which refers to the transfer of energy of vorticity to higher and
higher frequencies leading to decay of the stream and velocity functions, as
$t\to\infty$. Extending the linearized stability analysis for inviscid fluid
equations to the full nonlinear setting is a challenging problem, and the only
available results are on spectrally stable monotonic shear flows [5, 33, 34,
50], and on point vortices [35]. We refer also to the recent review article
[37] for a more in-depth discussion of recent developments of both linear and
nonlinear inviscid damping.

In view of the results on linear inviscid damping for the Euler equation, it
is natural to expect that even for Navier Stokes equations with small
viscosity we should have the same inviscid damping results with explicit rates
of decay, in time, of the stream functions and velocity fields. There are
several results in this direction, all of which are for monotonic shear flows,
see for example [4, 8, 13, 48] for the Couette flow with full nonlinear
analysis or precise linear results, and [27] for the spectrally restricted
stream function for “mixing layer” type shear flows (but a description of the
full solution without spectral restrictions is not available). More recently,
Chen, Wei and Zhang [15] established uniform inviscid damping with
quantitative bounds for general spectrally stable monotonic shear flows in a
periodic channel. More precisely, they proved that the stream function decays
like $1/t$ as $t\to\infty$, uniformly in viscosity. We remark that it remains
an interesting question if the stream function decays as in the Euler equation
(with rate $1/t^{2}$). It seems that one needs to understand the boundary
layer more clearly to settle this quesiton. The third author [41] proved
uniform (in viscosity) Gevrey bounds on the profile of the vorticity, for
spectrally stable monotonic shear flows in $\mathbb{T}\times\mathbb{R}$ (thus
avoiding the boundary layer issue), which yields sharp bounds on the stream
function (and in particular $1/t^{2}$ decay rate).

####  1.1.2. The case of non-monotonic shear flows

Many physically important shear flows are not monotonic, such as Poiseuille
flow and Kolmogorov flows. For such flows on the linear inviscid level, there
is an additional significant physical phenomenon called “vorticity depletion”
which refers to the asymptotic vanishing of vorticity as $t\to\infty$ near the
critical points of the shear flow, first predicted in Bouchet and Morita [9],
and proved rigorously in Wei-Zhang-Zhao [63]. A similar phenomenon for the
case of vortices was observed earlier by Bassom and Gilbert [3] and
demonstrated in Bedrossian-Coti Zelati-Vicol [6]. In [36] Ionescu and the
third author obtained a refined description of the dynamics of the linearized
flow around monotonic vortices in Gevrey spaces, which is a critical step
towards proving nonlinear vortex symmetrization.

In [63] by Wei-Zhang-Zhao, sharp linear inviscid damping estimates and
quantitative depletion estimates were obtained for an important class of
“symmetric shear flows” in a periodic channel (see also [65] by Wei-Zhang-Zhao
for a similar result for Kolmogorov flow). When no symmetry is assumed,
Ionescu, Iyer and the third author proved optimal inviscid damping rates and
quantitative vorticity depletion estimates in [38].

As in the case of monotonic shear flows, it is also natural to consider the
linearized equation around non-monotonic shear flows for the Navier Stokes
equations and study the corresponding vorticity depletion and linear inviscid
damping estimates that are uniform with respect to the viscosity $0<\nu\ll 1$.
This problem is more complicated than the monotonic case and there are very
few results, since the corresponding Orr Sommerfeld equations are genuinely
fourth order singular ODEs and the estimates needed to establish optimal
vorticity depletion and linear inviscid damping are very strong. We note also
that the Orr-Sommerfeld equations have been studied classically, with a focus
on distribution of eigenvalues, see [28, 29] and references therein for recent
results in this direction. We also mention the essay [20] for an interesting
account on the history of research in this direction. Our main task here is
quite different, which is to establish very precise control for the resolvent
for the spectral parameter on the continuous spectrum (of the inviscid Euler
equation).

###  1.2. Main equations and assumptions

In this paper we address this problem and study the uniform vorticity
depletion and linear asymptotic stability around a periodic shear flow
$(b(y),0)$ on $\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}$, together with
enhanced dissipation estimates. The choice of the domain is motivated by our
goal of understanding the interaction of viscosity and critical points of
shear flows, and by the desire to avoid the boundary layers when there is a
boundary. We note that the issues of critical points and boundary layers are
somewhat decoupled, and for the sake of clarity of argument, it makes sense to
tackle these difficulties separately.

Strictly speaking, $(b(y),0)$ is not a steady state for the Navier Stokes
equations, and becomes steady state only with a small external force $F=(-\nu
b^{\prime\prime}(y),0)$. In our setting, we consider the viscosity $\nu$ to be
very small. The effect of diffusion on the background shear flow is
negligible, at least up to the diffusion time scale $T\ll\frac{1}{\nu}$. After
the diffusion time, due to the enhanced dissipation effect, the flow is
essentially dominated by a heat evolution, at least heuristically. Therefore
our analysis below still captures the main dynamics of the Navier Stokes
equations near shear flows, even when we do not add the external forcing.

####  1.2.1. Main spectral assumption on the background shear flow

Our main assumptions on the background flow
$(b(y),0),y\in\mathbb{T}_{\mathfrak{p}}$ are the following.

######  Assumption 1.1.

We assume that the shear flow (b(y),0), $y\in\mathbb{T}_{\mathfrak{p}}$
satisfies the following conditions:

  * •

$b\in C^{4}(\mathbb{T}_{\mathfrak{p}})$. The set
$S:=\\{y\in\mathbb{T}_{\mathfrak{p}}:b^{\prime}(y)=0\\}=\\{y_{1\ast},y_{2\ast}\\}$,
and for some $\varkappa\in(0,1)$, we have

$|b^{\prime\prime}(y)|\in[\varkappa,1/\varkappa]\,\,{\rm for\,\,}y\in
S,\quad{\rm and}\quad\|b\|_{C^{4}(\mathbb{T}_{\mathfrak{p}})}\leq
1/\varkappa.$ (1.2)

  * •

The linearized operator $L_{k}:L^{2}(\mathbb{T}_{\mathfrak{p}})\to
L^{2}(\mathbb{T}_{\mathfrak{p}})$ with $k\in\mathbb{Z}\backslash\\{0\\}$
defined for $g\in L^{2}(\mathbb{T}_{\mathfrak{p}})$ as

$L_{k}g(y):=b(y)g(y)-b^{\prime\prime}(y)\varphi,\quad{\rm
where}\,\,(-k^{2}+\partial_{y}^{2})\varphi(y)=g(y),\,\,y\in\mathbb{T}_{\mathfrak{p}},$
(1.3)

has no eigenvalues, nor singular limiting modes in
$\\{b(y),y\in\mathbb{T}_{\mathfrak{p}}\\}$.

In contrast to the spectral property of the linearized operator around
monotonic shear flows, the spectral property of $L_{k}$ is less understood,
especially on the generation of discrete eigenvalues and embedded eigenvalues.
From general spectral theory, we know that the spectrum of $L_{k}$ consists of
the continuous spectrum

$\Sigma:=\big{\\{}b(y):\,y\in\mathbb{T}_{\mathfrak{p}}\big{\\}},$ (1.4)

together with some discrete eigenvalues with nonzero imaginary part which can
only accumulate at the set of continuous spectrum $\Sigma$. Unlike the case of
monotonic shear flows where the discrete eigenvalues can accumulate only at
inflection points of the background shear flow, there appears no simple
characterization of the possible accumulation points for non-monotonic shear
flows.

Recall that $\lambda\in\Sigma$ is called an embedded eigenvalue if there
exists a nontrivial $g\in L^{2}(\mathbb{T}_{\mathfrak{p}})$, such that

$L_{k}g=\lambda g.$ (1.5)

For non-monotonic shear flows, this definition is too restrictive, as
accumulation points of discrete eigenvalues may no longer be embedded
eigenvalues. To capture the discrete eigenvalues, we recall the following
definition of “generalized embedded eigenvalues”, which can be found already
in [63], adapted to our setting.

######  Definition 1.1.

We call $\lambda\in\Sigma$ a generalized embedded eigenvalue, if one of the
following conditions is satisfied.

  * •

$\lambda$ is an embedded eigenvalue.

  * •

$\lambda\not\in\\{b(y_{1\ast}),b(y_{2\ast})\\}$ and there exists a nontrivial
$\psi\in
H^{1}(\mathbb{T}_{\mathfrak{p}}):\mathbb{T}_{\mathfrak{p}}\to\mathbb{C}$ such
that in the sense of distributions on $\mathbb{T}_{\mathfrak{p}}$,

$(k^{2}-\partial_{y}^{2})\psi(y)+{\rm
P.V.}\frac{b^{\prime\prime}(y)\psi(y)}{b(y)-\lambda}+i\pi\sum_{z\in\mathbb{T}_{\mathfrak{p}},\,b(z)=\lambda}\frac{b^{\prime\prime}(z)\psi(z)}{|b^{\prime}(z)|}\delta(y-z)=0.$
(1.6)

We remark that our assumption that the critical points $y_{1\ast},y_{2\ast}$
of $b(y)$ being non-degenerate implies that the sum in (1.6) is finite. The
spectral assumption 1.1 is satisfied by a large class of shear flows
$(b(y),0)$, including $\sin\frac{2\pi y}{\mathfrak{p}}$.

####  1.2.2. Main equations

Taking the Fourier transform in $x$ in the equation (1.1), we obtain for each
$k\in\mathbb{Z}$ that

$\left\\{\begin{array}[]{rl}\partial_{t}\omega_{k}+\nu(k^{2}-\partial_{y}^{2})\omega_{k}+ikb(y)\omega_{k}-ikb^{\prime\prime}(y)\psi_{k}=0,&\\\
(-k^{2}+\partial_{y}^{2})\psi_{k}=\omega_{k},&\end{array}\right.$ (1.7)

for $(y,t)\in\mathbb{T}_{\mathfrak{p}}\times[0,\infty)$. For
$k\in\mathbb{Z}\backslash\\{0\\}$, define the operator
$L_{k,\nu}:H^{2}(\mathbb{T}_{\mathfrak{p}})\to
L^{2}(\mathbb{T}_{\mathfrak{p}})$ as follows. For $g\in
H^{2}(\mathbb{T}_{\mathfrak{p}})$ and $y\in\mathbb{T}_{\mathfrak{p}}$,

$L_{k,\nu}g(y):=(\nu/k)\partial_{y}^{2}g-ib(y)g+ib^{\prime\prime}(y)\varphi,\quad{\rm
with}\,\,(-k^{2}+\partial_{y}^{2})\varphi=g.$ (1.8)

We can rewrite equation (1.7) more abstractly as

$\partial_{t}\omega_{k}^{\ast}=kL_{k,\nu}\omega_{k}^{\ast},\quad{\rm
for}\,\,t\geq 0,$ (1.9)

where we have set for $y\in\mathbb{T}_{\mathfrak{p}}$ and $t\geq 0$,

$\omega_{k}^{\ast}(t,y)=e^{\nu k^{2}t}\omega_{k}(t,y).$ (1.10)

By relatively standard spectral theory and our bounds below (see section 5 for
more details), for sufficiently small $\nu_{0}>0,\sigma_{\sharp}>0$ and
$\alpha\geq-\sigma_{\sharp}|\nu/k|^{1/2}$, (denoting $\iota={\rm sign}\,k$) we
have the following representation formula for $y\in\mathbb{T}_{\mathfrak{p}}$
and $t>0$,

$\begin{split}\omega_{k}^{\ast}(t,y)&=\frac{1}{2\pi
i}\int_{i\mathbb{R}+\iota\alpha}e^{\mu tk}\big{[}(\mu-
L_{k,\nu})^{-1}\omega_{0k}\big{]}(y)\,d\mu\\\
&=\frac{1}{2\pi}\int_{\mathbb{R}}e^{i\lambda
tk+\alpha|k|t}\big{[}(i\lambda+\iota\alpha-
L_{k,\nu})^{-1}\omega_{0k}\big{]}(y)\,d\lambda\\\
&=-\frac{1}{2\pi}e^{\alpha|k|t}\int_{\mathbb{R}}e^{-i\lambda
tk}\big{[}(i\lambda-\iota\alpha+L_{k,\nu})^{-1}\omega_{0k}\big{]}(y)d\lambda.\end{split}$
(1.11)

In the above, $\omega_{0k}$ denotes the $k-$th Fourier coefficient of the
initial data $\omega|_{t=0}$.

Define for $y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}$ and
$\iota={\rm sign}\,k$,

$\omega_{k,\nu}(y,\lambda+i\iota\alpha):=\big{[}(i\lambda-\iota\alpha+L_{k,\nu})^{-1}\omega_{0k}\big{]}(y).$
(1.12)

It follows from (1.12) that

$(i\lambda-\iota\alpha)\omega_{k,\nu}(y,\lambda+i\iota\alpha)+L_{k,\nu}\omega_{k,\nu}(y,\lambda)=\omega_{0k}(y),\quad{\rm
for}\,\,y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}.$ (1.13)

Therefore $\omega_{k,\nu}(y,\lambda)$ satisfies the equation for
$k\in\mathbb{Z}\backslash\\{0\\}$,
$y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}$, $\iota={\rm sign}\,k$,

$\left\\{\begin{array}[]{rl}\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\iota\alpha\Big{]}\omega_{k,\nu}(y,\lambda+i\iota\alpha)+ib^{\prime\prime}(y)\psi_{k,\nu}(y,\lambda+i\iota\alpha)=\omega_{0k}(y),&\\\
(-k^{2}+\partial_{y}^{2})\psi_{k,\nu}(y,\lambda+i\iota\alpha)=\omega_{k,\nu}(y,\lambda+i\iota\alpha).&\end{array}\right.$
(1.14)

We summarize our calculations in the following proposition.

######  Proposition 1.1.

Assume that $\nu\in(0,\nu_{0})$ with sufficiently small $\nu_{0}>0$ and
$k\in\mathbb{Z}\backslash\\{0\\}$. Suppose that $\omega_{k}(t,y)$ satisfies
the regularity condition $\omega_{k}(t,y)\in
C^{\infty}((0,\infty)\times\mathbb{T}_{\mathfrak{p}})$ and
$\omega_{k}(t,\cdot)\in C([0,\infty),L^{2}(\mathbb{T}_{\mathfrak{p}}))$ is the
solution to the system of equations

$\left\\{\begin{array}[]{rl}\partial_{t}\omega_{k}(t,y)+\nu(k^{2}-\partial_{y}^{2})\omega_{k}(t,y)+ikb(y)\omega_{k}(t,y)-ikb^{\prime\prime}(y)\psi_{k}(t,y)&=0,\\\
(-k^{2}+\partial_{y}^{2})\psi_{k}(t,y)&=\omega_{k}(t,y),\end{array}\right.$
(1.15)

for $(y,t)\in\mathbb{T}_{\mathfrak{p}}\times[0,\infty)$, with initial data
$\omega_{k}(0,y)=\omega_{0k}(y)\in C^{\infty}(\mathbb{T}_{\mathfrak{p}})$.
Define the operator $L_{k,\nu}:H^{2}(\mathbb{T}_{\mathfrak{p}})\to
L^{2}(\mathbb{T}_{\mathfrak{p}})$ as follows. For any $g\in
H^{2}(\mathbb{T}_{\mathfrak{p}})$,

$L_{k,\nu}g(y)=(\nu/k)\partial_{y}^{2}g-ib(y)g+ib^{\prime\prime}(y)\varphi,\quad{\rm
where\,\,}(-\partial_{y}^{2}+k^{2})\varphi=g\,\,{\rm
for}\,\,y\in\mathbb{T}_{\mathfrak{p}}.$ (1.16)

Set for $y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}$, $\iota={\rm
sign}\,k$, $\alpha\geq-\sigma_{\sharp}|\nu/k|^{1/2}$ with sufficiently small
$\sigma_{\sharp}\in(0,1/10)$,

$\omega_{k,\nu}(y,\lambda+i\iota\alpha)=\big{[}(i\lambda-\iota\alpha+L_{k,\nu})^{-1}\omega_{0k}\big{]}(y).$
(1.17)

Then $\omega_{k,\nu}(y,\Lambda)$ satisfies the equation for
$k\in\mathbb{Z}\backslash\\{0\\}$,
$y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}$,

$\left\\{\begin{array}[]{rl}\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\iota\alpha\Big{]}\omega_{k,\nu}(y,\lambda+i\iota\alpha)+ib^{\prime\prime}(y)\psi_{k,\nu}(y,\lambda+i\iota\alpha)=\omega_{0k}(y),&\\\
(-k^{2}+\partial_{y}^{2})\psi_{k,\nu}(y,\lambda+i\iota\alpha)=\omega_{k,\nu}(y,\lambda+i\iota\alpha).&\end{array}\right.$
(1.18)

We have the representation formulae for $y\in\mathbb{T}_{\mathfrak{p}},t>0$,

$\begin{split}\omega_{k}(t,y)&=-\frac{1}{2\pi}e^{\alpha|k|t}e^{-\nu
k^{2}t}\int_{\mathbb{R}}e^{-ik\lambda
t}\omega_{k,\nu}(y,\lambda+i\iota\alpha)\,d\lambda,\\\
\psi_{k}(t,y)&=-\frac{1}{2\pi}e^{\alpha|k|t}e^{-\nu
k^{2}t}\int_{\mathbb{R}}e^{-ik\lambda
t}\psi_{k,\nu}(y,\lambda+i\iota\alpha)\,d\lambda.\end{split}$ (1.19)

Strictly speaking the vorticity function $\omega_{k}(t,y)$ and stream function
$\psi_{k}(t,y)$ also depend on $\nu$. We omit this dependence from our
notations for the sake of simplicity, since there is no danger of confusion.

###  1.3. Main results

Our main result is the following theorem.

######  Theorem 1.1.

Assume the spectral condition 1.1. Fix $\gamma\in[15/8,2)$. There exist
$\sigma_{\sharp}\in(0,1)$ and $\nu_{0}\in(0,1/10)$ sufficiently small
depending on $\gamma$, such that the following statement holds for
$\nu\in(0,\nu_{0})$. Assume that

$\omega(t,x,y)\in
C([0,\infty),H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}))$

with the associated stream function $\psi(t,x,y)$ is the unique solution to
(1.1), with initial data $\omega_{0}\in
H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})$ satisfying for all
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\int_{\mathbb{T}}\omega_{0}(x,y)\,dx=0.$ (1.20)

Then we have the following bounds.

(i) Uniform inviscid damping estimates: for
$(x,y,t)\in\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}\times[0,\infty)$,

$|\psi(t,x,y)|\lesssim_{\gamma}\frac{|y-y_{1\ast}|^{\gamma-2}+|y-y_{2\ast}|^{\gamma-2}}{\langle
t\rangle^{2}}e^{-\sigma_{\sharp}\nu^{1/2}t}\|\omega_{0}\|_{H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})},$
(1.21) $\begin{split}&|u^{x}(t,x,y)|\lesssim_{\gamma}\frac{1}{\langle
t\rangle}e^{-\sigma_{\sharp}\nu^{1/2}t}\|\omega_{0}\|_{H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})},\\\
&|u^{y}(t,x,y)|\lesssim_{\gamma}\frac{|y-y_{1\ast}|^{\gamma-2}+|y-y_{2\ast}|^{\gamma-2}}{\langle
t\rangle^{2}}e^{-\sigma_{\sharp}\nu^{1/2}t}\|\omega_{0}\|_{H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(1.22)

(ii) Uniform vorticity depletion estimates: for $j\in\\{1,2\\}$ there exists a
decomposition

$\omega(t,x,y):=\omega^{j}_{\rm loc}(t,x,y)+\omega^{j}_{\rm nloc}(t,x,y),$
(1.23)

where for
$(x,y,t)\in\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}\times[0,\infty)$,

$\begin{split}&|\omega^{j}_{\rm
loc}(t,x,y)|\lesssim_{\gamma}\big{(}|y-y_{j\ast}|+\nu^{1/4}\big{)}^{\gamma}e^{-\sigma_{\sharp}\nu^{1/2}t}\|\omega_{0}\|_{H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})},\\\
&|\omega^{j}_{\rm nloc}(t,x,y)|\lesssim_{\gamma}\frac{1}{\langle
t\rangle^{\gamma/2}}e^{-\sigma_{\sharp}\nu^{1/2}t}\|\omega_{0}\|_{H^{3}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(1.24)

In the above, $\langle t\rangle:=\sqrt{1+t^{2}}$, and the implied constants
depend, in addition to $\gamma$, also on $\varkappa$ from (1.2) and the
structure constant $\kappa\in(0,1)$ from Proposition 4.1.

We have the following remarks on Theorem 1.1.

  1. (1)

The exponent $\gamma$ is almost sharp. The endpoint case $\gamma=2$ seems to
require new ideas to treat.

  2. (2)

The restriction that $\mathfrak{p}\in(2\pi,\infty)$ is only needed for the
spectral assumption 1.1 to hold. An important example that satisfies 1.1 is
the Kolmogorov flow $(\sin(2\pi y/\mathfrak{p}),0)$. See also Meshalkin and
Sinai [55] for an investigation of spectral stability of Kolmogorov flows on
general torus.

  3. (3)

The condition (1.20) removes the zero mode in $x$ of vorticity from our
results, since the evolution of the zero mode corresponding to $k=0$ in
equation (1.15) is trivial.

  4. (4)

The bounds (1.21)-(1.22) provide uniform-in-viscosity inviscid damping
estimates. We note that to get $1/t^{2}$ decay as $t\to\infty$, one needs to
use integration by parts in (1.19) twice with respect to $\lambda$, while the
decay bound on $u^{x}$ involves only one integration by parts in $\lambda$ and
one derivative in $y$. In the neighborhood of the critical points
$y_{1\ast},y_{2\ast}$, taking a derivative with respect to $\lambda$ is
roughly equivalent to $\frac{1}{b^{\prime}(y)+i\nu^{1/4}}\partial_{y}$, which
explains the unbounded factor
$|y-y_{1\ast}|^{\gamma-2}+|y-y_{2\ast}|^{\gamma-2}$ in the bounds for $\psi$
and $u^{y}$.

  5. (5)

The bounds (1.24) confirm the “vorticity depletion” phenomenon which predicts
the asymptotic vanishing of vorticity near the critical points $y_{1\ast}$ and
$y_{2\ast}$ as $t\to\infty$. In addition, we note that there is a cut-off
distance $\nu^{1/4}$ to the vorticity depletion effect in contrast to the
Euler equation case, which is natural since the viscous effect dominates in
small scales and the dissipation it causes offsets the vorticity depletion
over the viscous region.

  6. (6)

We comment on the regularity assumptions on the initial data. By a simple
dimension counting argument, we expect that (1.24) to be quite sharp. On the
other hand, in (1.21)-(1.22), one might be able to reduce the regularity
assumption to $\omega_{0}\in
H^{5/2}(\mathbb{T}\times\mathbb{T}_{\mathfrak{p}})$. For the sake of
conciseness we have refrained from optimization in this direction.

  7. (7)

It is a natural problem to also consider the problem in
$\mathbb{T}\times[0,1]$, to include a boundary. There are essential new
difficulties due to the presence of boundary layers. We refer to the recent
work [2, 15] and references therein for significant results on monotonic shear
flows in this direction. The boundary effect is completely decoupled from the
issue of critical points of $b(y)$ away from the boundary, and therefore can
be treated separately. We plan to address this issue in a future work.

###  1.4. Main ideas of the proof

In this section we briefly outline the main ideas of the proof. To present the
key elements in the simplest settings, we assume that $k=1$ and $\alpha=0$ in
(1.18).

The fact that we can take $\alpha$ as $-\sigma_{\sharp}|\nu/k|^{1/2}$ is
crucial to obtain the enhanced dissipation bound in our approach, which
requires that we show stability of our bounds with respect to moving from
$\alpha=0$ to $\alpha=-\sigma_{\sharp}|\nu/k|^{1/2}$. The higher $k$ cases
should, in principle, make the problem easier since the problem becomes more
“elliptic”. In practice, however, the analysis becomes considerably more
complicated as we need to track the dependence on $k$ of various quantities,
as optimally as possible. Nonetheless these are secondary issues that we
neglect here to emphasize the main points.

To further simplify the discussion, we assume also that $\omega_{0k}$ vanishes
in a neighborhood of $\\{y_{1\ast},y_{2\ast}\\}$. The depletion estimates are
still nontrivial and manifest as a repelling effect for the vorticity away
from the critical points. The general case can be reduced to a case similar to
this.

With the help of the representation formula (1.19), it suffices to study the
“spectral density functions” for the stream function $\psi(y,\lambda)$ and
$\omega(y,\lambda)$, see (1.18). For the sake of conciseness, we have
suppressed the dependence on $\nu$ (recall that we assumed $k=1$, $\alpha=0$).
Indeed, the uniform-in-viscosity inviscid damping bounds follow from
integration by parts in $\lambda$ using (1.19), and the vorticity depletion
bounds follow from weighted estimates on $\psi(y,\lambda)$ which imply
asymptotic vanishing of $\psi(y,\lambda)$ as $y\to\\{y_{1\ast},y_{2\ast}\\}$.

Roughly speaking, the bounds on $\psi(y,\lambda)$ are that

$\begin{split}&|\psi(y,\lambda)|\sim\big{(}|\lambda-b(y)|+\min\\{|\lambda-b(y_{1\ast})|,|\lambda-b(y_{2\ast})|\\}+|\nu|^{1/2}\big{)}^{\gamma/2},\\\
&|\partial_{y}\psi(y,\lambda)|\sim\big{(}|\lambda-b(y)|+\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}+|\nu|^{1/2}\big{)}^{\frac{\gamma-1}{2}}\sum_{j=1}^{2}\Big{\langle}\log|\frac{\lambda-b(y)}{\lambda-b(y_{j\ast})}\Big{|}\Big{\rangle},\\\
&|\partial^{2}_{y}\psi(y,\lambda)|\sim\big{(}|\lambda-b(y)|+\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}+|\nu|^{1/2}\big{)}^{\frac{\gamma-2}{2}}\sum_{j=1}^{2}\bigg{|}\frac{\lambda-b(y_{j\ast})}{\lambda-b(y)}\bigg{|}.\end{split}$
(1.25)

We also have similar bounds on $\partial_{\lambda}\psi(y,\lambda)$, with the
heuristic that in terms of size for $\beta\in\\{1,2\\}$,

$\partial^{\beta}_{\lambda}\psi(y,\lambda)\sim\big{(}|\lambda-b(y)|+\min\\{|\lambda-b(y_{1\ast})|,|\lambda-b(y_{2\ast})|\\}+|\nu|^{1/2}\big{)}^{-\beta/2}\partial^{\beta}_{y}\psi(y,\lambda).$
(1.26)

The bounds (1.25) show that $\psi(y,\lambda)$ becomes very small as $y$
approaches the critical points $\\{y_{1\ast},y_{2\ast}\\}$ when $\lambda$ is
close to one of the critical values $\\{b(y_{1*}),b(y_{2*})\\}$. These decay
bounds are crucial for obtaining the explicit depletion rate for the vorticity
functions, see (1.24). The bounds on $\partial_{y}^{2}\psi(y,\lambda)$ and
$\partial_{\lambda}^{2}\psi(y,\lambda)$ have to be refined near
$\\{y\in\mathbb{T}_{\mathfrak{p}}:b(y)-\lambda=0\\}$ so that they can be
integrated, see Lemma 4.1 below.

The first critical step in proving the bounds (1.25)-(1.26) is to reformulate
the equation (1.18) as

$(-1+\partial_{y}^{2})\psi(y,\lambda)+\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}\big{(}ib^{\prime\prime}\psi(\cdot,\lambda)\big{)}(y)=\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}\omega_{01}(y),$
(1.27)

and to justify the heuristic that

$\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}\approx\frac{1}{i(\lambda-b(y))+\ell_{\nu}(\lambda)},$
(1.28)

where

$\ell_{\nu}(\lambda):=\left\\{\begin{array}[]{ll}\nu^{1/3},&{\rm
if}\,\,\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\gtrsim 1,\\\
\min\limits_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}^{1/3}\nu^{1/3},&{\rm
if}\,\,\nu^{1/2}\ll\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\ll 1,\\\
\nu^{1/2},&{\rm
if}\,\,\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\lesssim\nu^{1/2}.\end{array}\right.$
(1.29)

To rigorously show (1.28) in various senses, we prove pointwise estimates for
the kernel of $\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}$, see
Proposition 3.1, as well as obtain refined characterizations of the
(quantitative) singular behavior near
$\\{y\in\mathbb{T}_{\mathfrak{p}}:b(y)-\lambda=0\\}$. The factor
$\ell_{\nu}(\lambda)$ in (1.29) is determined by the kernel estimates.

To study (1.27), we use (1.28) and obtain the following approximate equation
for $y\in\mathbb{T}_{\mathfrak{p}}$,

$(-1+\partial_{y}^{2})\psi(y,\lambda)+\frac{ib^{\prime\prime}(y)\psi(y,\lambda)}{i(\lambda-b(y))+\ell_{\nu}(\lambda)}\approx\frac{\omega_{01}(y)}{i(\lambda-b(y))+\ell_{\nu}(\lambda)}.$
(1.30)

As in [38], depending on how close $\lambda$ is to the critical values
$\\{b(y_{1\ast}),b(y_{2\ast})\\}$, the nonlocal term
$\frac{ib^{\prime\prime}(y)\psi(y,\lambda)}{i(\lambda-b(y))+\ell_{\nu}(\lambda)}$
plays very different roles.

  * •

Case I: non-degenerate region
$\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\gtrsim 1$. In this case, the
nonlocal term can be viewed as a compact perturbation of the main part
$-1+\partial_{y}^{2}$ of the equation and can be treated using our spectral
assumptions in 1.3.

  * •

Case II: the intermediate region
$\nu^{1/2}\ll\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\lesssim 1$. In
this case, as observed in [36] and [38], assuming for concreteness that
$\nu^{1/2}\ll|\lambda-b(y_{1\ast})|\ll 1$, we have by simple computations that
for $|\lambda-b(y_{1\ast})|^{1/2}\ll|y-y_{1\ast}|\ll 1$, the nonlocal term is

$\frac{ib^{\prime\prime}(y)}{i(\lambda-b(y))+\ell_{\nu}(\lambda)}\approx-\frac{2}{|y-y_{1\ast}|^{2}},$
(1.31)

which is both critically singular and favorable. This term can not be
considered as a perturbation. In section 2.2 we incorporate this term into the
main part (in addition to $-1+\partial_{y}^{2}$) and study the associated
Green’s functions which we call “modified Green’s function” as in [36, 38].
After taking off the non-perturbative part of the nonlocal term, the rest of
the terms can be treated perturbatively, using the spectral assumptions in
1.3.

  * •

Case III: the viscous region
$\min_{j\in\\{1,2\\}}\\{|\lambda-b(y_{j\ast})|\\}\lesssim\nu^{1/2}$. As is
expected, the viscosity dominates in small regions which in our setting limits
the depletion effect to a limit length scale. The scale $\ell_{\nu}(\lambda)$
is determined by the bounds we obtained for the kernel of
$\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}$. On the technical
level, in the viscous region, the analogous computation (1.31) is valid only
for $|y-y_{1\ast}|\gg|\nu|^{1/4}$, where we note that the lower bound is
independent of $\lambda$. This cutoff is ultimately responsible for the factor
$\nu^{1/4}$ that appeared in (1.24).

After identifying the main terms using the above analysis, we then show that
the resulting equations can be solved, by proving suitable limiting absorption
principle in weighted Sobolev spaces which are designed to capture both the
regularity and vanishing of the spectral density functions $\psi(y,\lambda)$.

In order to prove the limiting absorption principle, we reformulate the left
hand side of (1.30) as

$\psi(y,\lambda)+T_{\Theta}\psi(y,\lambda)$ (1.32)

where $T_{\Theta}$ is defined based on whether $\lambda$ is in the non-
degenerate or intermediate/viscous region (see (4.1) and (4.2)). Written in
this way, the limiting absorption principle can then be understood as the
statement that “$I+T_{\Theta}$ is invertible” in suitable weighted Sobolev
spaces. To establish this, we proceed via a contradiction argument after
showing that $T_{\Theta}$ is “small” when the frequency is high by proving
strong regularity estimates. The contradiction then follows as a consequence
of the spectral assumption 1.1.

An essential difficulty that arises when the viscosity $\nu>0$ in contrast to
the Euler equation case is that the singularities we need to handle are no
longer explicit. The complexity of the singularities can be seen from the
kernel of $\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}$, which
although on the level of regularity is comparable with the simpler term in
(1.28), behaves nonetheless rather differently. Indeed, the kernel is highly
oscillatory and does not admit a simple expression.

The lack of explicit formula for the operator
$\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-1}$ presents serious
technical challenges especially when estimating
$\partial_{\lambda}^{2}\psi(y,\lambda)$ since we need to take two derivatives
in $\lambda$ in equation (1.27) and the resulting singularities involve very
singular terms of the order
$\Big{[}\nu\partial_{y}^{2}+i(\lambda-b(y))\Big{]}^{-3}$. To resolve this
issue, instead of taking derivatives in $\lambda$, we use the good derivative
$D_{\lambda}$ defined in (4.9), which does not worsen the singularities. Using
the observation that $\partial_{y}^{\beta}\psi(y,\lambda)$ with
$\beta\in\\{1,2\\}$ can be understood using the equation (1.18) directly, we
can then obtain sufficient control on
$\partial_{\lambda}^{\beta}\psi(y,\lambda)$ for our purposes.

###  1.5. Notations and conventions

Here we summarize some important notations and conventions used throughout the
rest of the paper, for easy references.

In the sections below we assume that $k\in\mathbb{Z}\cap[1,\infty)$ as the
case of $k\in\mathbb{Z}\cap(-\infty,-1]$ follows by a complex conjugation.
Define for $k\in\mathbb{Z}\backslash\\{0\\},m\in\mathbb{Z}\cap[1,\infty)$, and
$h(y)\in H^{m}(\mathbb{T}_{\mathfrak{p}})$,

$\|h\|^{2}_{H^{m}_{k}(\mathbb{T}_{\mathfrak{p}})}:=\sum_{j=0}^{m}|k|^{2(m-j)}\|\partial_{y}^{j}h\|_{L^{2}(\mathbb{T}_{\mathfrak{p}})}^{2}.$
(1.33)

Below we list some notations that we reserve for important quantities and
briefly explain their roles.

  * •

Fundamental parameters $\lambda,\,\nu,\,k,\,\alpha,\,\epsilon,\,j$: These
parameters are directly given by the equations as in Proposition 1.1.
$\nu\in(0,1)$ is the viscosity and will be fixed throughout the paper.
$k\in\mathbb{Z}\cap[1,\infty)$ denotes the Fourier mode in $x$. We sometimes
also use $\epsilon:=\nu/k\in(0,1)$ to simplify notations. We will use
$\lambda\in\mathbb{R}$ to denote our main spectral parameter, and
$\alpha\geq-\sigma_{\sharp}|\nu/k|^{1/2}$ as an auxiliary spectral parameter
which are useful when we shift the contour of integral, as in Proposition 1.1,
where $\sigma_{\sharp}\in(0,1)$ will be specified below. $j\in\\{1,2\\}$ is
used to specify the the critical point $y_{j\ast}$ under consideration.

  * •

Derived parameters
$\delta_{0},\,\delta_{1}(\lambda),\,\delta_{2}(\lambda),\,\Sigma_{j,\delta_{0}},\,S^{j}_{d}$:
These parameters are introduced to simplify notations. It follows from
Assumption 1.1 that there exists a $\delta_{0}\in(0,1/8)$ such that

$\begin{split}&|y_{1\ast}-y_{2\ast}|>10\delta_{0},\sup_{y\in\cup_{y_{\ast}\in\\{y_{1\ast},y_{2\ast}\\}}(y_{\ast}-4\delta_{0},y_{\ast}+4\delta_{0})}|b^{\prime\prime\prime}(y)|\delta_{0}<|b^{\prime\prime}(y_{\ast})|/10.\end{split}$
(1.34)

Define

$\begin{split}\delta_{1}(\lambda):=&8\min_{y_{\ast}\in\\{y_{1\ast},y_{2\ast}\\}}\sqrt{|\lambda-b(y_{\ast})|/b^{\prime\prime}(y_{\ast})},\\\
\delta_{2}(\lambda):=&\frac{1}{8}\min_{y_{\ast}\in\\{y_{1\ast},y_{2\ast}\\}}\sqrt{|\lambda-b(y_{\ast})|/b^{\prime\prime}(y_{\ast})},\end{split}$
(1.35)

and for $j\in\\{1,2\\}$ the set

$\Sigma_{j,\delta_{0}}:=\Big{[}b(y_{j\ast})-|b^{\prime\prime}(y_{j\ast})|\delta_{0}^{2}/16,\,b(y_{j\ast})+|b^{\prime\prime}(y_{j\ast})|\delta_{0}^{2}/16\Big{]}.$
(1.36)

We introduce also for $j\in\\{1,2\\},d\in(0,1/10)$ the shorthand,

$S^{j}_{d}:=[y_{j\ast}-d,y_{j\ast}+d].$ (1.37)

  * •

Composite notations $\Lambda,\Theta$: Below we use the composite symbol
$\Lambda$ and $\Theta$ to indicate dependence of certain quantities on
$\Lambda:=(\lambda,\alpha,\epsilon)$ (or $\Lambda=(\lambda,\alpha,k,\nu)$
recalling that $\epsilon=\nu/k$). For example, $h(y;\Lambda)$ denotes a
function $h$ of $y$ that depends on the parameters $\lambda,\alpha$ and
$\epsilon$. We use $\Theta:=(\lambda,\alpha,\epsilon)$ (or
$\Theta=(\lambda,\alpha,k,\nu)$) for similar purposes. The motivation for
using both $\Lambda$ and $\Theta$ with identical purpose is purely aesthetic.

  * •

Weights: To capture the precise depletion phenomena, we need to prove various
weighted estimates. For this purpose, define

$\delta(\Lambda):=|\alpha|^{1/2}+C^{\dagger}|\epsilon|^{1/4}+8\min_{y_{\ast}\in\\{y_{1\ast},y_{2\ast}\\}}\sqrt{|\lambda-b(y_{\ast})|/b^{\prime\prime}(y_{\ast})}.$
(1.38)

The constant $C^{\dagger}\gg 1$ depends only on $\delta_{0}$ and will be
chosen below in Proposition 7.1. For $\lambda\in\mathbb{R}$ and
$j\in\\{1,2\\}$ with

$\delta(\Lambda)\leq\mathfrak{p}/8,$ (1.39)

we define the weights for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho_{j}(y;\Lambda):=|y-y_{j\ast}|+\delta(\Lambda),\quad\varrho_{j,k}(y;\Lambda):=\varrho_{j}(y;\Lambda)\wedge\frac{1}{k},\quad
d_{j,k}:=\delta(\Lambda)\wedge\frac{1}{k}.$ (1.40)

In the above, $\alpha\wedge\beta$ is the minimum of two real numbers
$\alpha,\beta$.

  * •

Weighted spaces: For $\mathfrak{M}:=(\lambda,\alpha,k,\nu,j)$ and
$\sigma_{1},\sigma_{2}\in\mathbb{R}$, we define the weighted space
$X^{\sigma_{1},\sigma_{2}}(\mathfrak{M})$ using the norm for any function $g$
on $\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}\|g\|_{X^{\sigma_{1},\sigma_{2}}(\mathfrak{M})}:=&\sum_{\beta\in\\{0,1\\}}(\delta(\Lambda))^{-1/2+\sigma_{1}}\Big{\|}\big{[}\delta(\Lambda)\wedge\frac{1}{k}\big{]}^{\sigma_{2}+\beta}\partial_{y}^{\beta}g\Big{\|}_{L^{2}(S^{j}_{\delta(\Lambda)})}\\\
&+\sum_{\beta\in\\{0,1\\}}\Big{\|}\varrho_{j}^{\sigma_{1}}(\cdot;\Lambda)\varrho_{j,k}^{\sigma_{2}+\beta}(\cdot;\Lambda)\partial^{\beta}_{y}g\Big{\|}_{L^{\infty}(\mathbb{T}_{\mathfrak{p}}\backslash
S^{j}_{\delta(\Lambda)})}.\end{split}$ (1.41)

Lastly we remark that all the implied constants are uniform with respect to
the parameters $\lambda,\alpha,k,\nu$ and only depend on the $\varkappa$ from
(1.2) and the structure constant $\kappa$ from Proposition 4.1 in subsection
4.2 and section 5.

###  1.6. Organization of the paper

The rest of the paper is organized as follows.

  * •

In section 2 we study the Green’s function and modified Green’s function for
the main elliptic operator from the Rayleigh equations;

  * •

In section 3 we prove pointwise bounds on the kernel of the inverse for the
generalized Airy operators that appear in the Orr-Sommerfeld equations, and
use them to bound the inverse of the generalized Airy operators in various
weighted spaces;

  * •

In section 4 we state the limiting absorption principle and use them to derive
the main bounds on the spectral density functions;

  * •

In section 5 we prove our main result, Theorem 1.1;

  * •

In sections 6 and 7 we give the proof of the limiting absorption principles
assuming various bounds on $T_{\Theta}$.

  * •

In section 8 we prove the technical propositions needed on $T_{\Theta}$ for
the proofs of the limiting absorption principles.

##  2\. Bounds on the Green’s function and modified Green’s function

In this section we recall and prove some useful estimates for Green’s
functions associated with Rayleigh equations, including the Green’s function
for the standard ellitpic operator $k^{2}-\partial_{y}^{2}$ on
$\mathbb{T}_{\mathfrak{p}}$ and for the elliptic operator
$k^{2}-\partial_{y}^{2}+V(y)$ on $\mathbb{T}_{\mathfrak{p}}$ where $V$ is a
potential term coming from the nonlocal term in the Rayleigh equations.
Throughout the section we assume that $k\in\mathbb{Z}\cap[1,\infty)$.

###  2.1. Elementary properties of the standard Green’s function

Recall that the Green’s function $G_{k}(y,z)$ solves for
$y,z\in\mathbb{T}_{\mathfrak{p}}$,

$-\frac{d^{2}}{dy^{2}}G_{k}(y,z)+k^{2}G_{k}(y,z)=\delta(y-z).$ (2.1)

$G_{k}$ has the symmetry

$G_{k}(y,z)=G_{k}(z,y),\qquad{\rm for}\,\,y,z\in\mathbb{T}_{\mathfrak{p}}.$
(2.2)

We note the following point-wise bounds for $G_{k}$ with
$y,z\in\mathbb{T}_{\mathfrak{p}}$,

$|G_{k}(y,z)|+|k|^{-1}|\partial_{y}G_{k}(y,z)|\lesssim|k|^{-1}e^{-|k||y-z|},$
(2.3)

and the integral bounds

$\begin{split}\sup_{y\in\mathbb{T}_{\mathfrak{p}},\beta\in\\{0,1\\}}\bigg{[}|k|^{3/2-\beta}\left\|\partial_{y,z}^{\beta}G_{k}(y,z)\right\|_{L^{2}(z\in\mathbb{T}_{\mathfrak{p}})}\bigg{]}\lesssim
1.\end{split}$ (2.4)

Define $F_{k}(y,z)$ for $y,z\in\mathbb{T}_{\mathfrak{p}}$ as

$F_{k}(y,z):=\partial_{z}\partial_{y}G_{k}(y,z)-\delta(y-z),\qquad{\rm
for}\,\,y,z\in\mathbb{T}_{\mathfrak{p}}.$ (2.5)

Then $F_{k}$ satisfies the bounds

$\begin{split}&\sup_{y\in\mathbb{T}_{\mathfrak{p}},\beta\in\\{0,1\\}}\bigg{[}|k|^{-1/2-\beta}\left\|\partial_{y,z}^{\beta}F_{k}(y,z)\right\|_{L^{2}(z\in\mathbb{T}_{\mathfrak{p}})}\bigg{]}\lesssim
1.\end{split}$ (2.6)

These estimates are well known and follow also from the argument below in
section 2.2.

###  2.2. Bounds on the modified Green’s function

For applications below, we need to study the “modified Green’s function”
$\mathcal{G}^{j}_{k}(y,z;\Lambda)$ defined for the parameters
$j\in\\{1,2\\},\lambda\in\Sigma_{j,\delta_{0}}$, and $\delta(\Lambda)\leq
1/8$, which satisfies for $y,z\in\mathbb{T}_{\mathfrak{p}},$

$(k^{2}-\partial_{y}^{2})\mathcal{G}^{j}_{k}(y,z;\Lambda)+\frac{b^{\prime\prime}(y)}{b(y)-\lambda-i\alpha}\Big{[}\varphi_{0}\big{(}\frac{y-y_{j\ast}}{\delta_{0}}\big{)}-\varphi_{0}\big{(}\frac{y-y_{j\ast}}{\delta(\Lambda)}\big{)}\Big{]}\mathcal{G}^{j}_{k}(y,z;\Lambda)=\delta(y-z).$
(2.7)

In the above, $\varphi_{0}\in C_{c}^{\infty}(-2,2)$ with $\varphi_{0}\equiv 1$
on $[-1,1]$.

The bounds we need for the modified Green’s function
$\mathcal{G}^{j}_{k}(y,z;\Lambda)$ are the following.

######  Lemma 2.1.

Let $\mathcal{G}^{j}_{k}(y,z;\Lambda)$ be defined as in (2.7) for
$j\in\\{1,2\\},\lambda\in\Sigma_{j,\delta_{0}}$ and $\delta(\Lambda)\leq 1/8.$
Then we have for $\beta\in\\{0,1\\}$ and $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho_{j,k}^{\beta}(y;\Lambda)\big{|}\partial_{y}^{\beta}\mathcal{G}^{j}_{k}(y,z;\Lambda)\big{|}\lesssim\varrho_{j,k}(z;\Lambda)\min\Big{\\{}e^{-|k||y-z|},\frac{\varrho_{j}^{2}(y;\Lambda)}{\varrho_{j}^{2}(z;\Lambda)},\,\frac{\varrho_{j}(z;\Lambda)}{\varrho_{j}(y;\Lambda)}\Big{\\}}.$
(2.8)

As a corollary of Lemma 2.1, we also have the following bounds.

######  Lemma 2.2.

Under the assumptions of Lemma 2.1, define for
$y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\mathcal{H}^{j}_{k}(y,z;\Lambda):=\Big{[}\partial_{z}+\varphi_{0}\big{(}\frac{y-y_{j\ast}}{10\delta(\Lambda)}\big{)}\partial_{y}\Big{]}\mathcal{G}^{j}_{k}(y,z;\Lambda).$
(2.9)

Then we have the bounds for $y\in\mathbb{T}_{\mathfrak{p}},z\in
S^{j}_{4\delta(\Lambda)}$ and $\beta\in\\{0,1\\}$,

$\begin{split}&\varrho_{j,k}^{\beta}(y;\Lambda)\big{|}\partial_{y}^{\beta}\mathcal{H}^{j}_{k}(y,z;\Lambda)\big{|}\lesssim\min\Big{\\{}e^{-|k||y-z|},\frac{\varrho_{j}^{2}(y;\Lambda)}{\varrho_{j}^{2}(z;\Lambda)},\,\frac{\varrho_{j}(z;\Lambda)}{\varrho_{j}(y;\Lambda)}\Big{\\}},\end{split}$
(2.10)

and for $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho_{j,k}^{\beta}(y;\Lambda)\big{|}\partial_{y}^{\beta}(\partial_{z}+\partial_{y})\mathcal{G}^{j}_{k}(y,z;\Lambda)\big{|}\lesssim\min\left\\{e^{-(|k|/2)|y-z|},\frac{\varrho_{j,k}(y;\Lambda)}{\varrho_{j}(z;\Lambda)},\frac{\varrho_{j,k}(z;\Lambda)}{\varrho_{j}(y;\Lambda)}\right\\}.$
(2.11)

The rest of the subsection is devoted to the proof of Lemma 2.1 and Lemma 2.2.

####  2.2.1. Proof of Lemma 2.1

For simplicity, we suppress the dependence of $\mathcal{G}_{k}^{j}$ on
$z,\Lambda$, $k$, $j$ which are fixed, and set for
$y\in\mathbb{T}_{\mathfrak{p}}$,

$h(y):=\mathcal{G}_{k}^{j}(y,z;\Lambda),\quad
V(y):=\frac{b^{\prime\prime}(y)}{b(y)-\lambda-i\alpha}\Big{[}\varphi_{0}\big{(}\frac{y-y_{j\ast}}{\delta_{0}}\big{)}-\varphi_{0}\big{(}\frac{y-y_{j\ast}}{\delta(\Lambda)}\big{)}\Big{]}.$

We also set for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho(y):=\varrho_{j}(y;\Lambda),\quad\varrho_{k}(y):=\varrho_{j,k}(y;\Lambda),\quad
y_{\ast}:=y_{j\ast}.$

We separate the proofs into three steps, corresponding to different
configurations of $y$ and $z$. With a slight abuse of notation we shall prove
the bounds (2.8) for
$y=y_{1}\in\mathbb{T}_{\mathfrak{p}},z\in\mathbb{T}_{\mathfrak{p}}$ and use
$y$ as a free variable.

Step 1: local estimates near $z\in\mathbb{T}_{\mathfrak{p}}$. Multiplying
$\overline{h}$ to (2.7) and integrating over $\mathbb{T}_{\mathfrak{p}}$, we
get that

$\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}h(y)|^{2}+|k|^{2}|h(y)|^{2}dy+\int_{\mathbb{T}_{\mathfrak{p}}}V(y)|h(y)|^{2}dy=\bar{h}(z).$
(2.12)

Notice that $\real V(y)\geq 0$ and for $y\in S^{j}_{\delta_{0}}$ satisfying
$|y-y_{*}|\gg\delta(\Lambda),$ that

$1+\real V(y)\gtrsim\frac{1}{\varrho^{2}(y)}.$ (2.13)

It follows from (2.12) that for large $C_{0}>1$,

$\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}h(y)|^{2}+|k|^{2}|h(y)|^{2}dy+\int_{y\in
S_{\delta_{0}},|y-y_{*}|>C_{0}\delta_{\ast}}\frac{1}{\varrho^{2}(y;\Lambda)}|h(y)|^{2}dy\lesssim|h(z)|.$
(2.14)

Using the inequality that

$\|h\|_{L^{\infty}(J)}\lesssim\|h\|_{L^{2}(J_{*})}|J|^{-\frac{1}{2}}+\|\partial_{y}h\|_{L^{2}(J)}|J|^{\frac{1}{2}},$
(2.15)

for any interval $J,\ J_{*}$ with $J_{*}\subseteq J$ and $|J_{*}|\gtrsim|J|$,
and choosing the interval

$J=\Big{\\{}y\in\mathbb{T}_{\mathfrak{p}}:|y-z|\leq
2C_{0}\min\\{1/|k|,\varrho(z)\\}\Big{\\}},$

we obtain that

$|h(z)|\lesssim\min\\{1/|k|,\varrho(z)\\}.$ (2.16)

It follows from Sobolev type inequality and (2.14) that

$\sup_{y,z\in\mathbb{T}_{\mathfrak{p}},\,|y-z|\leq\min\\{\varrho(z),1/|k|\\}}|\mathcal{G}^{j}_{k}(y,z;\lambda+i\alpha)|\lesssim\min\\{\varrho(z),1/|k|\\}.$
(2.17)

Then, by (2.7) we also obtain that

$\sup_{y,z\in\mathbb{T}_{\mathfrak{p}},\,|y-z|\leq\min\\{\varrho(z),1/|k|\\}}|\partial_{y}\mathcal{G}^{j}_{k}(y,z;\Lambda)|\lesssim
1.$ (2.18)

Step 2: Global bounds (I) large $k$ case. In this step we prove, under the
assumption that

$1/k\leq\varrho_{j}(z;\Lambda),$ (2.19)

for $\beta\in\\{0,1\\}$ and $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho_{j,k}^{\beta}(y;\Lambda)\big{|}\partial_{y}^{\beta}\mathcal{G}^{j}_{k}(y,z;\lambda+i\alpha)\big{|}\lesssim\varrho_{j,k}(z;\Lambda)e^{-|k||y-z|}.$
(2.20)

We will remove the assumption (2.19) at the end of the proof.

Let $\varphi\in C^{1}_{p}(\mathbb{T}_{\mathfrak{p}})$, the space of piecewise
$C^{1}$ functions, with $\varphi(z)=0$, to be chosen more precisely below. We
multiply $\varphi^{2}\overline{h}$ to (2.7) and integrate over
$\mathbb{T}_{\mathfrak{p}}$ to get that

$\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi(y)\partial_{y}h(y)|^{2}dy+\left(k^{2}+V(y)\right)|\varphi(y)h(y)|^{2}dy+\int_{\mathbb{T}_{\mathfrak{p}}}2\varphi^{\prime}(y)\varphi(y)\overline{h}(y)\partial_{y}h(y)dy=0.$
(2.21)

Taking the real part of (2.21), using Cauchy-Schwarz inequality and the fact
that $\real V(y)\geq 0$ we obtain that

$\int_{\mathbb{T}_{\mathfrak{p}}}\Big{[}|\varphi^{\prime}(y)|^{2}-|k|^{2}|\varphi(y)|^{2}\Big{]}|h(y)|^{2}dy\geq
0.$ (2.22)

We assume, without loss of generality that $y_{1}$ lies in the positive
direction to $z$ counterclockwise, and that the distance from $z$ to $y$ is
greater than $1/k$ (otherwise the desired bound (2.20) follows from (2.17)).
We now specify $\varphi$ as follows

$\begin{split}&\varphi(z)=0,\quad\varphi^{\prime\prime}(y)=0\quad\text{for}\
y\in\left[z,z+c/k\right]\backslash\\{0\\},\quad\varphi|_{y\in\\{z-c/k,z+c/k\\}}=1,\\\
&\varphi^{\prime}(y)=k\varphi(y)\quad\text{for}\
y\in\left[z+c/k,y_{1}-c/k\right],\quad\varphi^{\prime}(y)=0\quad\text{for}\
y\in\left[y_{1}-c/k,z+\mathfrak{p}/2\right].\end{split}$ (2.23)

In the above $0<c<1$ is a small constant. And then extend $\varphi$ to
$[z-\mathfrak{p}/2,z]$ by an even reflection across $z$. We denote
$I=[y_{1}-c/k,y_{1}+c/k]$, then $\varphi(y)\approx e^{|k||y_{1}-z|}$ for $y\in
I$ and

$\int_{I}|h(y)|^{2}dy\lesssim
e^{-2|k||y_{1}-z|}\int_{z-c/k}^{z+c/k}|h(y)|^{2}dy.$ (2.24)

By equation (2.7) and a local energy inequality, we obtain with
$I^{\prime}:=[y_{1}-c/(2k),y_{1}+c/(2k)]$,

$\int_{I^{\prime}}\varrho^{-2}(y)|h(y)|^{2}+|\partial_{y}h(y)|^{2}dy\lesssim
k^{2}\int_{I}|h(y)|^{2}dy.$ (2.25)

By (2.15), we obtain that for $y\in I^{\prime}$,

$|h(y)|\lesssim\varrho_{k}(z)e^{-|k||y_{1}-z|}.$ (2.26)

The rest of the bounds in (2.20) then follow from equation (2.7) and (2.26).

Step 3: Global estimates (II) large $V$ case. In this step we prove, under the
assumption that

$1/k\geq\varrho_{j}(z;\Lambda),$ (2.27)

the bounds for $\beta\in\\{0,1\\}$ and $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\varrho_{j,k}^{\beta}(y;\Lambda)\big{|}\partial_{y}^{\beta}\mathcal{G}^{j}_{k}(y,z;\Lambda)\big{|}\lesssim\varrho_{j,k}(z;\Lambda)\min\Big{\\{}\frac{\varrho_{j}^{2}(y;\Lambda)}{\varrho_{j}^{2}(z;\Lambda)},\,\frac{\varrho_{j}(z;\Lambda)}{\varrho_{j}(y;\Lambda)}\Big{\\}}.$
(2.28)

We will remove the assumption (2.27) at the end of the proof.

Without loss of generality, we assume the critical point $y_{j*}=0$, and
consider only the main case $y_{1},z\in[-\delta_{0},\delta_{0}]$ and
$z>\delta(\Lambda)$. In addition, we assume that $y_{1}<z$, the other case
being analogous.

Multiplying $\varphi^{2}\overline{h}$ to (2.7) and integrating over
$\mathbb{T}_{\mathfrak{p}}$, we have

$\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi(y)\partial_{y}h(y)|^{2}+\left(k^{2}+V(y)\right)|\varphi(y)h(y)|^{2}dy+\int_{\mathbb{T}_{\mathfrak{p}}}2\varphi^{\prime}(y)\varphi(y)\overline{h}(y)\partial_{y}h(y)\,dy=0.$
(2.29)

Set for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\displaystyle h(y)=r(y)h^{*}(y)$ (2.30)

where $r(y)$ is chosen to satisfy the following properties

$r(y)=\left\\{\begin{array}[]{lll}|y|^{1/2}&\text{on}\
[-\delta_{0},\delta_{0}]\backslash J,\
J:=[-\delta(\Lambda),\delta(\Lambda)],\\\
(\delta(\Lambda))^{\frac{1}{2}}A\cos(\frac{By}{\delta(\Lambda)})&\text{on}\
J,\\\ (\delta_{0})^{\frac{1}{2}}&\text{on}\,\
\mathbb{T}_{\mathfrak{p}}\backslash[-\delta_{0},\delta_{0}],\end{array}\right.$
(2.31)

where the coefficients $A,B$ satisfy

$\begin{split}&B\tan B=-\frac{1}{2},\quad\ B\in(\pi/2,\pi),\\\ &A\cos
B=1.\end{split}$

From (2.29) and (2.30) we obtain that

$\begin{split}&\int_{\mathbb{T}_{\mathfrak{p}}}\frac{1}{r^{2}(y)}\Big{[}(r^{2}(y)\varphi^{\prime}(y))^{2}-\left(r^{2}(y)|r^{\prime}(y)|^{2}+r^{4}(y)\real
V(y)\right)\varphi^{2}(y)\Big{]}|h^{\ast}(y)|^{2}dy\\\
&\geq-\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|h^{\ast}(y)|^{2}(r^{\prime}(y)r(y))^{\prime}dy.\end{split}$
(2.32)

By our construction of $r(y)$, we get that

$\int_{J}|r^{\prime}(y)|^{2}\varphi^{2}(y)|h^{\ast}(y)|^{2}dy\geq\int_{J}\varphi^{2}(y)|h^{\ast}(y)|^{2}(r^{\prime}(y)r(y))^{\prime}dy,$
(2.33)

and therefore,

$\int_{\mathbb{T}_{\mathfrak{p}}\backslash
J}\frac{1}{r^{2}(y)}\Big{[}(r^{2}(y)\varphi^{\prime}(y))^{2}-\left(r^{2}(y)|r^{\prime}(y)|^{2}+r^{4}(y)\real
V(y)\right)\varphi^{2}(y)\Big{]}|h^{*}(y)|^{2}dy\geq 0.$ (2.34)

We notice the pointwise bounds for $y\in[-\delta_{0},\delta_{0}]\backslash J$
and some $C_{2}>1$ that

$m^{2}(y):=r^{2}(y)|r^{\prime}(y)|^{2}+r^{4}(y)\real
V(y)\geq\max\left\\{\frac{1}{4},\frac{9}{4}-C_{2}\frac{\delta(\Lambda)^{2}}{y^{2}}-C_{2}|y|\right\\}.$
(2.35)

It is clear that there exists a $y_{1}^{\prime}\in\mathbb{T}_{\mathfrak{p}}$
such that

$\int_{[y_{1}+\varrho(y_{1})/8,z]}|m(y)|dy=\int_{[z,y_{1}^{\prime}-\varrho(y_{1}^{\prime})/8]}|m(y)|dy.$
(2.36)

In the above, we identify $\mathbb{T}_{\mathfrak{p}}$ with the circle of
circumference $\mathfrak{p}$, and for $a,b\in\mathbb{T}_{\mathfrak{p}}$ where
$b$ is in the positive direction of $a$ (counterclockwise), $[a,b]$ denotes
the interval from $a$ to $b$.

We now specify $\varphi\in C^{1}_{p}(\mathbb{T}_{\mathfrak{p}})$ as the
following

$\displaystyle\varphi(z)=0,\quad\varphi^{\prime\prime}(y)=0\quad
y\in[z-c\varrho(z),z+c\varrho(z)]\backslash\\{z\\},\quad\varphi|_{\\{z-c\varrho(z),z+c\varrho(z)\\}}=1;$
$\displaystyle
r(y)\varphi^{\prime}(y)=\Big{[}r^{2}(y)|r^{\prime}(y)|^{2}+r^{4}(y)\real
V(y)\Big{]}^{1/2}\varphi(y),\quad
y\in[z,y_{1}^{\prime}-\varrho(y_{1}^{\prime})/8];$ (2.37) $\displaystyle
r(y)\varphi^{\prime}(y)=-\Big{[}r^{2}(y)|r^{\prime}(y)|^{2}+r^{4}(y)\real
V(y)\Big{]}^{1/2}\varphi(y),\quad y\in[y_{1}+\varrho(y_{1})/8,z];$ (2.38)
$\displaystyle\varphi^{\prime}(y)=0,\
y\in[y_{1}^{\prime}-\varrho(y_{1}^{\prime})/8,y_{1}+\varrho(y_{1})/8].$ (2.39)

From the definition of $\varphi$, we obtain the bound for
$y\in[y_{1}-\varrho(y)/8,y_{1}+\varrho(y)/8]$ for a small $c>0$,

$\varphi(y)\geq c\Big{[}\varrho(z)/\varrho(y_{1})\Big{]}^{3/2}.$ (2.40)

Then the bound (2.34) and the definiton (2.2.1) imply that

$\begin{split}&\Big{[}\varrho(z)/\varrho(y_{1})\Big{]}^{3}\int_{[y_{1}-\varrho(y_{1})/8,y_{1}+\varrho(y_{1})/8]}\frac{1}{r^{2}(y)}|h^{\ast}(y)|^{2}dy\\\
&\lesssim\int_{[z-c\varrho(z),z+c\varrho(z)]}\varrho^{-2}(z)|h(y)|^{2}dy\lesssim\varrho(z).\end{split}$
(2.41)

The desired bounds (2.28) then follow from (2.41), equation (2.7) and local
energy inequality. We omit the routine details.

Step 4: Removing assumptions (2.19) and (2.27).  We consider first the
assumption (2.19). The key is to establish the bound

$k\int_{z-c/k}^{z+c/k}h^{2}(y)dy\lesssim\varrho^{2}(z),$ (2.42)

in the case that $1/k\geq\varrho(z)$. Then we can use the bound (2.28) and
(2.42) follows.

We now turn to the assumption (2.27). The key is to prove the bound

$\int_{[z-c\varrho(z),z+c\varrho(z)]}\varrho^{-1}(z)|h(y)|^{2}dy\lesssim
1/k^{2},$ (2.43)

in the case that $1/k\leq\varrho(z)$. In this case, we can use the bound
(2.20) and (2.43) follows.

Lemma 2.1 is now proved.

####  2.2.2. Proof of Lemma 2.2

This proof of (2.10) is almost the same as the one in [38]. We briefly outline
the main ideas below.

Denote for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\varphi^{{\dagger}}(y):=\varphi_{0}\Big{(}\frac{y-y_{j*}}{10\delta(\Lambda)}\Big{)}.$
(2.44)

Notice that if $y=z$ or $|y|\geq 20\delta(\Lambda)$,
$|\mathcal{H}_{k}^{j}(y,z;\Lambda)|\approx|\partial_{y}\mathcal{G}_{k}^{j}(y,z;\Lambda)|$.
The conclusion follows from (2.8). Thus for the rest, we consider the region
$y\neq z$ and $|y|<20\delta(\Lambda)$ where $\delta(y-z)\equiv 0$ and
$\varrho(y)\gtrsim|y|$. Under such assumptions,
$\mathcal{H}_{k}^{j}(y,z;\Lambda)$ satisfies

$\begin{split}&(k^{2}-\partial_{y}^{2})\mathcal{H}_{k}^{j}(y,z;\Lambda)+V(y)\mathcal{H}_{k}^{j}(y,z;\Lambda)\\\
&=-2\partial_{y}\varphi^{{\dagger}}(y)\partial_{y}^{2}\mathcal{G}^{j}_{k}(y,z;\Lambda)-\partial_{y}^{2}\varphi^{{\dagger}}(y)\partial_{y}\mathcal{G}^{j}_{k}(y,z;\Lambda)-\varphi^{{\dagger}}(y)\partial_{y}V(y)\cdot\mathcal{G}^{j}_{k}(y,z;\Lambda)\\\
&:=F(y,z;\Lambda).\end{split}$ (2.45)

Notice the fact that

$|F(y,z;\Lambda)|\lesssim\min\Big{\\{}|k|(\delta(\Lambda))^{-1}e^{-|k||y-z|},\varrho^{-2}(y)\Big{\\}},\quad\
\text{Supp}\,F\subseteq[-20\delta(\Lambda),20\delta(\Lambda)].$

Then the desired bound (2.10) follows from (2.8).

We now turn to the proof of (2.11). Notice that
$(\partial_{y}+\partial_{z})\mathcal{G}^{j}_{k}(y,z;\Lambda)$ satisfies the
equation for $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$\Big{[}k^{2}-\partial_{y}^{2}+V(y)\Big{]}(\partial_{y}+\partial_{z})\mathcal{G}^{j}_{k}(y,z;\Lambda)=-\partial_{y}V(y)\mathcal{G}^{j}_{k}(y,z;\Lambda).$
(2.46)

The desired bounds (2.11) then follow from (2.8).

##  3\. Pointwise bounds on the kernel of generalized Airy operators and
applications

In this section we consider the generalized Airy operator
$\epsilon\partial_{y}^{2}-\alpha+i(\lambda-b(y))$, where $b(y)$ satisfies
Assumption 1.2, $\epsilon\in(0,1),\lambda\in\mathbb{R}$ and
$\alpha\geq-\sigma_{0}\epsilon^{1/2}$ with a sufficiently small
$\sigma_{0}\in(0,1)$. Our main goal is to prove pointwise bounds for the
kernel of the generalized Airy operator and then obtain estimates on various
operators involving the generalized Airy operator.

###  3.1. Pointwise estimates on the kernel of generalized Airy operators

Depending on the value of $\lambda\in\mathbb{R}$, we divide into three
regions.

  1. (1)

the non-degenerate region:
$\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\gtrsim 1$;

  2. (2)

the intermediate region:
$\epsilon^{\frac{1}{2}}\ll\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\ll 1$;

  3. (3)

the viscous region:
$\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\lesssim\epsilon^{\frac{1}{2}}$.

######  Proposition 3.1.

Assume that $b(y)$ satisfies Assumption 1.2 and that $\epsilon\in(0,1)$. Then
there exist $\sigma_{0},c_{0}\in(0,1)$ such that the following statement
holds.

For $\lambda\in\mathbb{R}$, $y,z\in\mathbb{T}_{\mathfrak{p}}$, denote
$\Lambda:=(\epsilon,\lambda,\alpha)$ and let $k(y,z;\Lambda)$ be the
fundamental solution to the generalized Airy operator
$\epsilon\partial_{y}^{2}-\alpha+i(\lambda-b(y))$ on
$\mathbb{T}_{\mathfrak{p}}$, where $\alpha\geq-\sigma_{0}\epsilon^{1/2}$. More
precisely, for $y,z\in\mathbb{T}_{\mathfrak{p}}$, in the sense of
distributions,

$\epsilon\partial_{y}^{2}k(y,z;\Lambda)-\alpha
k(y,z;\Lambda)+i(\lambda-b(y))k(y,z;\Lambda)=\delta(y-z).$ (3.1)

We have the following bounds for $k(y,z;\Lambda)$ with $\lambda\in\mathbb{R}$,
$y,z\in\mathbb{T}_{\mathfrak{p}}$.

  1. (1)

if $\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\gtrsim 1$, then

$\begin{split}|k(y,z;\Lambda)|\lesssim&\frac{\epsilon^{-\frac{2}{3}}}{\langle\epsilon^{-\frac{1}{3}}\alpha,\epsilon^{-\frac{1}{3}}(b(z)-\lambda)\rangle^{\frac{1}{2}}}\\\
&\times\exp\left[-c_{0}\langle\epsilon^{-\frac{1}{3}}\alpha,\epsilon^{-\frac{1}{3}}(b(y)-\lambda),\epsilon^{-\frac{1}{3}}(b(z)-\lambda)\rangle^{\frac{1}{2}}|y-z|\epsilon^{-\frac{1}{3}}\right];\end{split}$
(3.2)

  2. (2)

if $\epsilon^{\frac{1}{2}}\ll\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\ll
1$, then

$\begin{split}|k(y,z;\Lambda)|\lesssim&\frac{\epsilon^{-\frac{2+\beta}{3}}}{\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha,\epsilon^{-\frac{1+2\beta}{3}}(b(z)-\lambda)\rangle^{\frac{1}{2}}}\\\
&\times\exp\left[-c_{0}\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha,(b(y)-\lambda)\epsilon^{-\frac{1+2\beta}{3}},(b(z)-\lambda)\epsilon^{-\frac{1+2\beta}{3}}\rangle^{\frac{1}{2}}|y-z|\epsilon^{-\frac{1-\beta}{3}}\right];\end{split}$
(3.3)

where $\beta$ is determined by
$\min_{i\in\\{1,2\\}}\\{|b(y_{i*})-\lambda|\\}\approx\epsilon^{2\beta}$,
$\beta\in[0,\frac{1}{4}]$.

  3. (3)

if
$\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}\lesssim\epsilon^{\frac{1}{2}}$,
then

$\begin{split}|k(y,z;\Lambda)|\lesssim&\frac{\epsilon^{-\frac{3}{4}}}{\langle\epsilon^{-\frac{1}{2}}\alpha,\epsilon^{-\frac{1}{2}}(b(z)-\lambda))\rangle^{\frac{1}{2}}}\\\
&\times\exp\left[-c_{0}\langle\epsilon^{-\frac{1}{2}}\alpha,\epsilon^{-\frac{1}{2}}(b(y)-\lambda),\epsilon^{-\frac{1}{2}}(b(z)-\lambda)\rangle^{\frac{1}{2}}|y-z|\epsilon^{-\frac{1}{4}}\right].\end{split}$
(3.4)

The rest of the subsection is devoted to the proof of Proposition 3.1. Without
loss of generality, below we only consider the case
$\min_{j\in\\{1,2\\}}\\{|b(y_{j*})-\lambda|\\}=|b(y_{1*})-\lambda|$, and
denote $y_{*}=y_{1*}$ for simplicity. Through the rest of this section, we
also denote $k(y,z)=k(y,z;\Lambda)$ for the ease of notations.

####  3.1.1. Global energy estimates

We first establish the following global energy estimates.

######  Lemma 3.1.

For all $z\in\mathbb{T}_{\mathfrak{p}}$, we have the following bounds.

  1. (1)

if $|b(y_{*})-\lambda|\gtrsim 1$,

$|k(z,z)|\lesssim\frac{\epsilon^{-\frac{2}{3}}}{\langle\epsilon^{-\frac{1}{3}}\alpha,\epsilon^{-\frac{1}{3}}(b(z)-\lambda)\rangle^{\frac{1}{2}}}.$
(3.5)

  2. (2)

if $\epsilon^{\frac{1}{2}}\ll|b(y_{*})-\lambda|\lesssim 1$,

$|k(z,z)|\lesssim\frac{\epsilon^{-\frac{2+\beta}{3}}}{\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha,\epsilon^{-\frac{1+2\beta}{3}}(b(z)-\lambda)\rangle^{\frac{1}{2}}}.$
(3.6)

  3. (3)

if $|b(y_{*})-\lambda|\lesssim\epsilon^{\frac{1}{2}}$,

$|k(z,z)|\lesssim\frac{\epsilon^{-\frac{3}{4}}}{\langle\epsilon^{-\frac{1}{2}}\alpha,\epsilon^{-\frac{1}{2}}(b(z)-\lambda)\rangle^{\frac{1}{2}}}.$
(3.7)

###### Proof.

We first prove the degenerate case (2). Multiplying (3.1) with
$\overline{k(y,z)}$ and $(\lambda-b(y))\overline{k(y,z)}$ respectively, and
integrating over $\mathbb{T}_{\mathfrak{p}}$, we get that

$-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}k(y,z)|^{2}dy-\alpha\int_{\mathbb{T}_{\mathfrak{p}}}|k(y,z)|^{2}dy+i\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))|k(y,z)|^{2}dy=\overline{k(z,z)};$
(3.8)
$\begin{split}-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))|\partial_{y}k(y,z)|^{2}dy+\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}b^{\prime}(y)\partial_{y}k(y,z)\overline{k(y,z)}dy-\alpha\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))|k(y,z)|^{2}\\\
+i\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))^{2}|k(y,z)|^{2}dy=(\lambda-b(z))\overline{k(z,z)}.\end{split}$
(3.9)

It follows from (3.8) that

$\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}k(y,z)|^{2}dy+\alpha\int_{\mathbb{T}_{\mathfrak{p}}}|k(y,z)|^{2}dy\leq|k(z,z)|.$
(3.10)

Similarly, it follows from (3.9) that

$\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))^{2}|k(y,z)|^{2}dy\leq|\lambda-b(z)||k(z,z)|+\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|b^{\prime}(y)||\partial_{y}k(y,z)||k(y,z)|dy$
(3.11)

Using the spectral gap inequality

$\int_{\mathbb{T}_{\mathfrak{p}}}|b^{\prime}(y)|^{2}|k(y,z)|^{2}dy\lesssim\epsilon^{-\frac{2-2\beta}{3}}\int_{\mathbb{T}_{\mathfrak{p}}}(\lambda-b(y))^{2}|k(y,z)|^{2}dy+\epsilon^{\frac{2+4\beta}{3}}\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}k(y,z)|^{2}dy$
(3.12)

we get that for any $\tau\in(0,1)$ and a corresponding $C(\tau)\in(1,\infty)$,

$\begin{split}&\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|b^{\prime}(y)|\cdot|\partial_{y}k(y,z)|\cdot|k(y,z)|dy\\\
&\leq\epsilon^{\frac{2-2\beta}{3}}\tau\int_{\mathbb{T}_{\mathfrak{p}}}|b^{\prime}(y)|^{2}|k(y,z)|^{2}dy+\epsilon^{\frac{4+2\beta}{3}}\frac{1}{\tau}\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}k(y,z)|^{2}dy\\\
&\lesssim\tau\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)^{2}|k(y,z)|^{2}dy+\epsilon^{\frac{4+2\beta}{3}}C(\tau)\int_{\mathbb{T}_{\mathfrak{p}}}|\partial_{y}k(y,z)|^{2}dy.\end{split}$
(3.13)

Thus combining (3.10), (3.11) and (3.13), we obtain that

$\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)^{2}|k(y,z)|^{2}dy\lesssim|b(z)-\lambda||k(z,z)|+\epsilon^{\frac{1+2\beta}{3}}|k(z,z)|,$
(3.14)

which further implies that

$\epsilon^{-\frac{1+2\beta}{3}}\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)^{2}|k(y,z)|^{2}dy\lesssim\langle\epsilon^{-\frac{1+2\beta}{3}}(b(z)-\lambda)\rangle|k(z,z)|.$
(3.15)

Then, in view of (3.10) and (3.14), we can get the desired bounds (3.6) for
$k(y,z)$ when $y$ close to $z$ using the bounds

$\|f\|_{L^{\infty}(I)}\lesssim\|f\|_{L^{2}(I)}|I|^{-\frac{1}{2}}+\|f^{\prime}\|_{L^{2}(I)}|I|^{\frac{1}{2}}$
(3.16)

for an interval $I$, as follows.

If $\alpha\lesssim\epsilon^{\frac{1+2\beta}{3}}$, we choose

$I=\left\\{y\in\mathbb{T}_{\mathfrak{p}}:\epsilon^{-\frac{1-\beta}{3}}|y-z|\leq\langle\epsilon^{-\frac{1+2\beta}{3}}(\lambda-b(z))\rangle^{-\frac{1}{2}}\right\\},\quad|I|\approx\epsilon^{\frac{1-\beta}{3}}\langle\epsilon^{-\frac{1+2\beta}{3}}(\lambda-b(z))\rangle^{-\frac{1}{2}}.$

If $\alpha\gg\epsilon^{\frac{1+2\beta}{3}}$, we choose
$I=\left\\{y\in\mathbb{T}_{\mathfrak{p}}:\epsilon^{-\frac{1-\beta}{3}}|y-z|\leq\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha\rangle^{-\frac{1}{2}}\right\\}$,
$|I|\approx\epsilon^{\frac{1-\beta}{3}}\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha\rangle^{-\frac{1}{2}}$.
This concludes the proof of case (2) of Lemma 3.1.

The cases (1) and (3) follow from similar arguments. Indeed the non-degenerate
case (1), we obtain (3.7) by just applying $\beta=0$ into (3.6), if
additionally $|\lambda|\lesssim 1$. If $|\lambda|\gg 1$, then there is no need
to use the spectral gap inequality (3.12), and (3.14) with $\beta=0$ follows
directly from (3.11).

For the viscous case, (3.5) follows from the same arguments as in the
degenerate case if we take $\beta=\frac{1}{4}$. The proof of Lemma 3.1 is now
complete. ∎

####  3.1.2. The entanglement inequality

The following lemma will be used frequently in our proof of Proposition 3.1.

######  Lemma 3.2.

For $\beta\in[0,1/4]$ and non-negative cutoff function $\varphi(y)\in
C^{1}(\mathbb{T}_{\mathfrak{p}})$ with $\varphi(z)=0$, there exists
$c_{0}\in(0,1)$ independent of $\beta$ and $\varphi$ such that

$\int_{\mathbb{T}_{\mathfrak{p}}}\left[|\varphi^{\prime}|^{2}-c_{0}^{2}\epsilon^{-\frac{2-2\beta}{3}}\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha,\epsilon^{-\frac{1+2\beta}{3}}(b(y)-\lambda)\rangle|\varphi|^{2}\right]|k(y,z)|^{2}dy\geq
0.$ (3.17)

######  Remark 3.1.

The statement of our entanglement inequality is the same as that in [41].
However, our proof is much more flexible since our method here does not
require the separation into regions where $\lambda-b(y)$ keeps its sign. In
particular, we no longer need to use the local to global energy inequality,
see section 3.2 of [41], which is essential in [41] but is somewhat
cumbersome.

###### Proof.

We provide the detailed proof only for the most complicated case when
$\lambda\in b(\mathbb{T}_{\mathfrak{p}})$. The proof of the case
$\lambda\in\mathbb{R}\backslash b(\mathbb{T}_{\mathfrak{p}})$ is similar and
easier.

We first introduce a real-valued test function $\phi(y)$ which is a standard
regularization of the jump function $\text{sgn}(b(y)-\lambda)$ on the scale of
$\epsilon^{\frac{1-\beta}{3}}$ which is the natural length scale at
$y\in\mathbb{T}_{\mathfrak{p}}$ with $\lambda=b(y)$, and satisfies the
following properties:

  1. (1)

$|\phi^{\prime}(y)|\lesssim\epsilon^{-\frac{1-\beta}{3}}$;

  2. (2)

$|\phi(y)-\text{sgn}(b(y)-\lambda)||b(y)-\lambda|\lesssim\epsilon^{\frac{1+2\beta}{3}}$.

Multiplying $\varphi^{2}(y)\overline{k(y,z)}$ to (3.1) and integrating over
$\mathbb{T}_{\mathfrak{p}}$, we have

$\begin{split}-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|\partial_{y}k(y,z)|^{2}dy-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}2\varphi^{\prime}(y)\varphi(y)\overline{k(y,z)}\partial_{y}k(y,z)dy-\alpha\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|k(y,z)|^{2}dy\\\
+i\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)\varphi^{2}(y)|k(y,z)|^{2}dy=0.\end{split}$
(3.18)

By taking the real part and applying the Hölder inequality, we get that

$\alpha\epsilon^{-1}\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|k(y,z)|^{2}dy+\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|\partial_{y}k(y,z)|^{2}dy\lesssim\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi^{\prime}(y)|^{2}|k(y,z)|^{2}dy.$
(3.19)

We also multiply $\varphi^{2}(y)\phi(y)\overline{k(y,z)}$ to (3.1) and
integrate over $\mathbb{T}_{\mathfrak{p}}$ to get that

$\begin{split}-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)\phi(y)|\partial_{y}k(y,z)|^{2}dy-\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\left(2\varphi(y)\varphi^{\prime}(y)\phi(y)+\varphi^{2}(y)\phi^{\prime}(y)\right)\overline{k(y,z)}\partial_{y}k(y,z)dy\\\
-\alpha\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)\phi(y)|k(y,z)|^{2}dy+i\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)\varphi^{2}(y)\phi(y)|k(y,z)|^{2}dy=0.\end{split}$
(3.20)

By taking the imaginary part, we get that

$\begin{split}&\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)\varphi^{2}(y)\phi(y)|k(y,z)|^{2}dy\\\
&\leq\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\Big{[}2|\varphi(y)||\varphi^{\prime}(y)||\phi(y)|+\varphi^{2}(y)\cdot|\phi^{\prime}(y)|\Big{]}\big{|}\overline{k(y,z)}\big{|}|\partial_{y}k(y,z)|dy\end{split}$
(3.21)

Thus by the construction of the function $\phi(y)$, we conclude that

$\begin{split}&\int_{\mathbb{T}_{\mathfrak{p}}}|b(y)-\lambda|\varphi^{2}(y)|k(y,z)|^{2}dy\\\
&=\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)\big{[}\text{sgn}(b(y)-\lambda)-\phi(y)\big{]}\varphi^{2}(y)|k(y,z)|^{2}dy+\int_{\mathbb{T}_{\mathfrak{p}}}(b(y)-\lambda)\phi(y)\varphi^{2}|k(y,z)|^{2}dy\\\
&\leq\int_{\mathbb{T}_{\mathfrak{p}}}\epsilon^{\frac{1+2\beta}{3}}\varphi^{2}(y)|k(y,z)|^{2}dy+\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\Big{[}2|\varphi(y)||\varphi^{\prime}(y)||\phi(y)|+\varphi^{2}(y)|\phi^{\prime}(y)|\Big{]}|k(y,z)||\partial_{y}k(y,z)|dy.\end{split}$
(3.22)

By the spectral gap inequality that for any $\tau\in(0,1)$, a corresponding
$C(\tau)\in(1,\infty)$, and any $h\in H^{1}(\mathbb{T}_{\mathfrak{p}})$,

$\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi^{2}||h(y)|^{2}dy\leq\tau\epsilon^{-\frac{1+2\beta}{3}}\int_{\mathbb{T}_{\mathfrak{p}}}|b(y)-\lambda||\varphi|^{2}|h(y)|^{2}dy+C(\tau)\epsilon^{\frac{2-2\beta}{3}}\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi^{\prime}|^{2}|h(y)|^{2}dy$
(3.23)

together with (3.22), we get that

$\begin{split}&\int_{\mathbb{T}_{\mathfrak{p}}}|b(y)-\lambda|\varphi^{2}(y)|k(y,z)|^{2}dy\\\
&\lesssim\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi^{\prime}|^{2}(y)|k(y,z)|^{2}+\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}\varphi^{2}(y)|\phi^{\prime}(y)|^{2}|k^{2}(y,z)|dy,\end{split}$
(3.24)

which in view of the property of $\phi$ implies that

$\begin{split}&\int_{\mathbb{T}_{\mathfrak{p}}}|b(y)-\lambda|\varphi^{2}(y)|k(y,z)|^{2}dy\lesssim\epsilon\int_{\mathbb{T}_{\mathfrak{p}}}|\varphi^{\prime}|^{2}(y)|k(y,z)|^{2}.\end{split}$
(3.25)

The desired inequality (3.17) follows from (3.19), (3.25) and the spectral gap
inequality (3.23).

∎

####  3.1.3. Point-wise bounds for the kernel

Finally, we prove our main conclusion Proposition 3.1.

We introduce the length scale $Q(x)$ for $x\in\mathbb{T}_{\mathfrak{p}}$ which
will be used frequently in our argument. Notice that there is a slight abuse
of notation and $x$ in this section is not to be confused with the $x$
variable for the non-square torus $\mathbb{T}\times\mathbb{T}_{\mathfrak{p}}$.

We first define for $x\in\mathbb{T}_{\mathfrak{p}},\beta\in[0,1/4]$,

$Q(x;\beta):=\frac{\epsilon^{\frac{1-\beta}{3}}}{\langle\epsilon^{-\frac{1+2\beta}{3}}\alpha,\epsilon^{-\frac{1+2\beta}{3}}(b(x)-\lambda)\rangle^{\frac{1}{2}}}.$

Since $\beta$ determined by $\lambda\in\mathbb{R}$ is fixed, we also use the
following shortened notation $Q(x)$. In the non-degenerate case,
$Q(x)=Q(x;0)$; in the degenerate case, $Q(x)=Q(x;\beta)$; and in the viscous
case, $Q(x)=Q(x;\frac{1}{4})$.

When $|y-z|\lesssim Q(z)$, notice here $\\{|y-z|\leq Q(z)\\}\subset I$ where
$I$ is taken from the proof of Lemma 3.1. Thus by Lemma 3.1,

$|k(y,z)|\lesssim\epsilon^{-1}Q(z)\cdot e^{0},$ (3.26)

which satisfies (3.2)-(3.4) naturally.

When $Q(z)\ll|z-y|$, we will apply the entanglement inequality to get the
bound. Without loss of generality, we assume $y\in\mathbb{T}_{\mathfrak{p}}$
is in the positive direction to $z$, counter-clockwisely.

Let $\Gamma_{x_{1},x_{2}}$ be the arc on $\mathbb{T}_{\mathfrak{p}}$ from
$x_{1}$ to $x_{2}$ in the counter-clockwise direction. We assume that

$\int_{\Gamma_{z+Q(z),y}}Q^{-1}(x)\,dx\geq\int_{\Gamma_{y,z-Q(z)}}Q^{-1}(x)\,dx.$
(3.27)

Then we can find $y_{\ast}\in[z+Q(z),y]$ such that

$m_{\ast}:=c_{0}\int_{\Gamma_{z+Q(z),y_{\ast}}}Q^{-1}(x)\,dx=c_{0}\int_{\Gamma_{y+Q(y),z-Q(z)}}Q^{-1}(x)\,dx.$
(3.28)

Consider the points

$\displaystyle A_{1}$ $\displaystyle=z-Q(z),$ $\displaystyle A_{2}$
$\displaystyle=z+Q(z),$ $\displaystyle A_{3}$ $\displaystyle=y_{\ast},$
$\displaystyle A_{4}$ $\displaystyle=y.$ (3.29)

Using (3.1.3) we choose $\varphi$ for the entanglement inequality (3.17)
satisfying the following conditions:

$\displaystyle\varphi^{\prime}(x)=c_{0}Q^{-1}(x)\varphi(x)\quad\text{ for
}x\in[A_{2},A_{3}];$
$\displaystyle\varphi^{\prime}(x)=-c_{0}Q^{-1}(x)\varphi(x)\quad\text{ for
}x\in[A_{4},A_{1}],\quad\varphi(A_{1})=\varphi(A_{2})=1;$
$\displaystyle\varphi^{\prime\prime}(x)\equiv 0,\quad\text{ for
}x\in[A_{3},A_{4}]\cup[A_{1},z)\cup(z,A_{2}],\quad\varphi(z)=0.$ (3.30)

We have from (3.1.3) that there exists $c_{1}\in(0,1)$, such that

$\varphi(A_{3})=e^{m_{\ast}}\gtrsim\exp\left[c_{1}\Big{\langle}\epsilon^{-\frac{1+2\beta}{3}}\alpha,\epsilon^{-\frac{1+2\beta}{3}}(b(y)-\lambda),\epsilon^{-\frac{1+2\beta}{3}}(b(z)-\lambda)\Big{\rangle}^{\frac{1}{2}}|y-z|\epsilon^{-\frac{1-\beta}{3}}\right].$
(3.31)

Using the entanglement inequality (3.17) we obtain that

$\int_{y}^{y+Q(y)}e^{2m_{\ast}}Q^{-2}(x)|k(x,z)|^{2}dx\lesssim\int_{z-Q(z)}^{z+Q(z)}Q^{-2}(x)|k(x,z)|^{2}\,dx,$
(3.32)

which together with equation (3.1) implies the desired point-wise estimates by
taking a smaller constant in the lower bound to $m_{*}$ to account for the
factor of $Q(y)$ instead of $Q(z)$ on the right-hand side. The proof of
Proposition 3.1 is now complete.

###  3.2. Applications of the kernel estimates

In this section we prove several bounds concerning the operator

$A_{\Theta}:=\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\alpha.$ (3.33)

In the above, we use the same assumptions and notations as in proposition 1.1
and recall that $k\in\mathbb{Z}\cap[1,\infty)$. We also used the composite
symbol $\Theta:=(k,\alpha,\nu,\lambda)$ for the ease of notations.

Assume that $j\in\\{1,2\\},\gamma\in[7/4,2)$. For
$\lambda\in\Sigma_{j,\delta_{0}},y\in\mathbb{T}_{\mathfrak{p}}$ and
$\epsilon:=\nu/k\in(0,1/8)$, we denote $\Lambda:=(\lambda,\alpha,\epsilon)$
and recall from (1.38) that

$d_{j,k}(\Lambda):=\delta(\Lambda)\wedge\frac{1}{|k|},\quad\varrho_{j,k}(y;\Lambda):=\varrho_{j}(y;\Lambda)\wedge\frac{1}{|k|}.$
(3.34)

Then we have the following bounds.

######  Lemma 3.3.

For any $\sigma_{1},\sigma_{2},\sigma_{3}\in\mathbb{R}$, there exists
$\epsilon_{1}=\epsilon_{1}(\sigma_{1},\sigma_{2},\sigma_{3})>0$ sufficiently
small, such that for $0<\nu<\epsilon_{1},k\in\mathbb{Z}\cap[1,\infty)$,
$\lambda\in\mathbb{R}$, and $1/4\geq\alpha\geq-\sigma_{0}|\nu/k|^{1/2}$ as in
Proposition 1.1, we have the following bounds for

$w=\varrho_{j}^{\sigma_{1}}(y;\Lambda)\varrho_{j,k}^{\sigma_{2}}(y;\Lambda)\big{(}i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/2}+|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}\big{)}^{\sigma_{3}}$
(3.35)

and $p\in[1,\infty]$, $h\in L^{p}(\mathbb{T}_{\mathfrak{p}})$.

(i) For $\lambda\in\Sigma_{j,\delta_{0}},m\in\mathbb{Z}\cap[0,1]$,
$j\in\\{1,2\\}$,

$\begin{split}&\left\|w(y)\partial_{y}^{m}A_{\Theta}^{-1}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim_{\sigma_{1},\sigma_{2},\sigma_{3}}\left\|\frac{w(y)}{(L_{j}(y;\Lambda))^{m}\big{(}i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/2}+|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}\big{)}}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})},\end{split}$
(3.36)

where for $y\in\mathbb{T}_{\mathfrak{p}}$ and
$|\lambda-b(y_{j\ast})|\geq|\nu/k|^{1/2}$,

$L_{j}(y;\Lambda):=\frac{|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{-1/6}}{\langle|\nu/k|^{-1/3}|\lambda-b(y_{j\ast})|^{-1/3}(\lambda-b(y)),|\nu/k|^{-1/3}|\lambda-b(y_{j\ast})|^{-1/3}\alpha\rangle^{1/2}},$
(3.37)

and for $y\in\mathbb{T}_{\mathfrak{p}}$,
$|\lambda-b(y_{j\ast})|<|\nu/k|^{1/2}$,

$L_{j}(y;\Lambda):=|\nu/k|^{1/4}\langle|\nu/k|^{-1/2}(\lambda-b(y)),|\nu/k|^{-1/2}\alpha\rangle^{-1/2};$
(3.38)

In addition, we have for $m\in\\{1,2\\}$ the bounds

$\begin{split}&\left\|w(y)\partial_{y}^{m}A_{\Theta}^{-1}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim_{\sigma_{1},\sigma_{2},m,p}\left\|\frac{w(y)}{(L_{j}(y;\Lambda))^{m-1}\big{(}i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/2}+|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}\big{)}}\partial_{y}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\qquad\qquad+\left\|\frac{w(y)}{(L_{j}(y;\Lambda))^{m-1}\big{(}i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/2}+|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}\big{)}^{3/2}}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(3.39)

In particular, if $h\equiv 0$ for $|y-y_{j\ast}|\leq\delta(\lambda)$ then we
have for $m\in\\{1,2\\}$,

$\begin{split}&\left\|\varrho_{j}^{\sigma_{1}}(y;\Lambda)\varrho_{j,k}^{\sigma_{2}}(y;\Lambda)\partial_{y}^{m}A_{\Theta}^{-1}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim_{\sigma_{1},\sigma_{2},m,p}\left\|\frac{\varrho_{j}^{\sigma_{1}-2}(y;\Lambda)\varrho_{j,k}^{\sigma_{2}}(y;\Lambda)}{(L_{j}(y;\Lambda))^{m-1}}\partial_{y}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}+\left\|\frac{\varrho_{j}^{\sigma_{1}-3}(y;\Lambda)\varrho_{j,k}^{\sigma_{2}}(y;\Lambda)}{(L_{j}(y;\Lambda))^{m-1}}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(3.40)

(ii) For
$\lambda\in\mathbb{R}\backslash\big{(}\cup_{j\in\\{1,2\\}}\Sigma_{j,\delta_{0}}\big{)}$,
$m\in\\{0,1\\}$,

$\left\|\partial_{y}^{m}A_{\Theta}^{-1}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\lesssim\left\|(L_{j}(y;\Lambda))^{-m}\big{(}i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/3}\big{)}^{-1}h\right\|_{L^{p}(\mathbb{T}_{\mathfrak{p}})},$
(3.41)

where

$L(y;\Lambda):=|\nu/k|^{1/3}\langle|\nu/k|^{-1/3}(\lambda-b(y)),|\nu/k|^{-1/3}\alpha\rangle^{-1/2}.$
(3.42)

###### Proof.

The main idea of the proof is to keep track of the correct length scales, and
to show that the kernel for $A_{\Theta}^{-1}$ always defines a smaller length
scale on which it decays when compared with the weights. Therefore we can
treat $A_{\Theta}^{-1}$ as a local operator when proving these weighted
estimates.

We focus only on the proof of (i), with (ii) being similar. Denote $N(y,z)$,
$y,z\in\mathbb{T}_{\mathfrak{p}}$ as the kernel of $A_{\Theta}^{-1}$, i.e. for
any $h\in L^{2}(\mathbb{T}_{\mathfrak{p}})$ and
$y\in\mathbb{T}_{\mathfrak{p}}$,

$A_{\Theta}^{-1}h(y)=\int_{\mathbb{T}_{\mathfrak{p}}}N(y,z)h(z)\,dz.$ (3.43)

It follows from Proposition 3.1 that for some $c_{0}\in(0,1)$,
$y,z\in\mathbb{T}_{\mathfrak{p}}$,

$|N(y,z)|\lesssim|\nu/k|^{-1}L_{j}(y;\Lambda)e^{-c_{0}\big{(}L^{-1}_{j}(y;\Lambda)+L^{-1}_{j}(z;\Lambda)\big{)}|y-z|}.$
(3.44)

Denote for $y\in\mathbb{T}_{\mathfrak{p}}$,

$w_{\ast}(y):=i(\lambda-b(y))+|\alpha|+|\nu/k|^{1/2}+|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}.$
(3.45)

We note that for

$w\in\\{\varrho_{j}^{\sigma_{1}},\varrho_{j,k}^{\sigma_{2}},L_{j}(\cdot;\Lambda),w_{\ast}\\}$
(3.46)

and any $\sigma>0$, $y,z\in\mathbb{T}_{\mathfrak{p}}$,

$w(y)\lesssim_{\sigma}w(z)\,e^{\sigma\big{(}L^{-1}_{j}(y;\Lambda)+L^{-1}_{j}(z;\Lambda)\big{)}|y-z|}.$
(3.47)

Using (3.47), we can bound for any $w$ as in (3.46),

$\begin{split}&\Big{\|}w(y)A_{\Theta}^{-1}h(y)\Big{\|}_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim\Big{\|}w(y)\int_{\mathbb{T}_{\mathfrak{p}}}N(y,z)h(z)dz\Big{\|}_{L^{p}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim_{\sigma_{1},\sigma_{2}}\Big{\|}|\nu/k|^{-1}L_{j}(y;\Lambda)w_{\ast}(y)\int_{\mathbb{T}_{\mathfrak{p}}}e^{-c_{1}\big{(}L^{-1}_{j}(y;\Lambda)+L^{-1}_{j}(z;\Lambda)\big{)}|y-z|}w_{\ast}^{-1}(z)w(z)|h(z)|dz\Big{\|}_{L^{p}(\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(3.48)

To complete the proof of (3.36), we divide $\mathbb{T}_{\mathfrak{p}}$ as the
union of intervals $\mathcal{I}_{i}$, $i\in\mathbb{Z}\cap[1,N]$ for some
integer $N\geq 1$, where the intervals $\mathcal{I}_{i}$ only intersects at
the endpoints with each other and $|\mathcal{I}_{i}|\approx
L_{j}(y;\Lambda)\approx\ell_{i}$ for $y\in\mathcal{I}_{i}$.

Then using (3.48) we can bound for suitable constants $c_{2},c_{3}\in(0,1)$
and each $i\in\mathbb{Z}\cap[1,N]$,

$\begin{split}&\Big{\|}w(y)A_{\Theta}^{-1}h(y)\Big{\|}_{L^{p}(\mathcal{I}_{i})}\\\
&\lesssim_{\sigma_{1},\sigma_{2}}\Big{\|}\sum_{i^{\prime}=1}^{N}\int_{\mathcal{I}_{i^{\prime}}}\ell_{i}^{-1}e^{-c_{2}(\ell_{i}^{-1}+\ell_{i^{\prime}}^{-1})|y-z|}w_{\ast}^{-1}(z)w(z)|h(z)|dz\Big{\|}_{L^{p}(\mathcal{I}_{i})}\\\
&\lesssim_{\sigma_{1},\sigma_{2}}\Big{\|}\sum_{i^{\prime}=1}^{N}\int_{\mathcal{I}_{i^{\prime}}}\ell_{i}^{-1}e^{-c_{3}|i-i^{\prime}|}e^{-c_{3}\ell_{i}^{-1}|y-z|}w_{\ast}^{-1}(z)w(z)|h(z)|dz\Big{\|}_{L^{p}(\mathcal{I}_{i})}\\\
&\lesssim_{\sigma_{1},\sigma_{2}}\sum_{i^{\prime}=1}^{N}e^{-c_{3}|i-i^{\prime}|}\big{\|}w_{\ast}^{-1}(z)w(z)h\big{\|}_{L^{p}(\mathcal{I}_{i^{\prime}})}.\end{split}$
(3.49)

The desired bound (3.36) follows from (3.49).

The desired bounds (3.40) follow from the bounds (3.36) and the equation for
$y\in\mathbb{T}_{\mathfrak{p}}$

$\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\alpha\Big{]}\partial_{y}u(y)=\partial_{y}h(y)+ib^{\prime}(y)u(y),$
(3.50)

where

$u=A_{\Theta}^{-1}h.$ (3.51)

∎

The above lemmas are very useful for treating $A_{\Theta}^{-1}$ away from the
singular region when $|\lambda-b(y)|\gtrsim\delta(\lambda)$. In the viscous
layer when $|\lambda-b(y_{j\ast})|\lesssim|\nu/k|^{1/2}$, we can use the
viscous term $(\nu/k)\partial_{y}^{2}$ to gain regularity without much loss.
To deal with the singular behavior when
$|\nu/k|^{1/2}\lesssim|\lambda-b(y)|\lesssim\delta(\lambda)$, we rely on the
following lemma.

######  Lemma 3.4.

For any $\sigma_{1},\sigma_{2}\in\mathbb{R}$, there exists
$\epsilon_{1}=\epsilon_{1}(\sigma_{1},\sigma_{2})>0$ sufficiently small, such
that for $0<\nu<\epsilon_{1},k\in\mathbb{Z}\cap[1,\infty)$,
$\lambda\in\mathbb{R}$, $j\in\\{1,2\\}$,
$-\sigma_{0}|\nu/k|^{1/2}\leq\alpha\ll|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}$,
where $\sigma_{0}$ is from Propostion 3.1, the following statement holds.

(i) Assume that $\lambda\in\Sigma_{j,\delta_{0}}\cap
b(\mathbb{T}_{\mathfrak{p}})$ and $|\lambda-b(y_{j\ast})|\gg|\nu/k|^{1/2}$. In
this case there exist $y^{+}_{\lambda},y^{-}_{\lambda}\in S^{j}_{3\delta}$,
arranged so that $y^{+}_{\lambda}$ is in the positive direction, counter-
clockwise, to $y^{-}_{\lambda}$, such that

$\lambda=b(y^{+}_{\lambda})=b(y^{-}_{\lambda}).$ (3.52)

Choose a cut-off function $\varphi\in C_{c}^{\infty}(-1,1)$ with
$\varphi\equiv 1$ on $[-1/2,1/2]$. Recall the definition (3.34) for $d_{j,k}$.
Assume that

${\rm supp}\,h\subseteq S^{j}_{9\delta}.$

Denote

$\epsilon:=\nu/k,\,\,M:=\|h\|_{L^{2}(S^{j}_{9\delta})}+d_{j,k}(\Lambda)\|\partial_{y}h\|_{L^{2}(S^{j}_{9\delta})}.$
(3.53)

There exists a decomposition for $y\in\mathbb{T}_{\mathfrak{p}}$,

$A_{\Theta}^{-1}h(y):=w_{1}(y,\lambda)+w_{2}(y,\lambda),$ (3.54)

satisfying the property that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$|w_{1}(y,\lambda)|\lesssim\sum_{\dagger\in\\{+,-\\}}\Big{[}d_{j,k}(\Lambda)|\lambda-b(y_{j\ast})|^{1/2}\Big{]}^{-1}|y-y^{\dagger}_{\lambda}|^{-1/2}M+\Big{[}\big{(}d_{j,k}(\Lambda)\big{)}^{1/2}|\lambda-b(y_{j\ast})|\Big{]}^{-1}M,$
(3.55)

and

$\begin{split}&\bigg{|}w_{2}(y,\lambda)-\sum_{\dagger\in\\{+,-\\}}\frac{h(y^{\dagger}_{\lambda})\varphi\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}}{ib^{\prime}(y^{\dagger}_{\lambda})(y-y^{\dagger}_{\lambda})+|\epsilon|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}}\bigg{|}\\\
&\lesssim\sum_{\dagger\in\\{+,-\\}}|\lambda-b(y_{j\ast})|^{-1/2}\frac{|\epsilon|^{-1/3}|\lambda-b(y_{j\ast})|^{1/6}}{1+(|\epsilon|^{1/3}|\lambda-b(y_{j\ast})|^{-1/6})^{-2}|y-y^{\dagger}_{\lambda}|^{2}}\big{(}d_{j,k}(\Lambda)\big{)}^{-1/2}M.\end{split}$
(3.56)

(ii) Assume that $\lambda\in
b(\mathbb{T}_{\mathfrak{p}})\backslash\Sigma_{j,\delta_{0}}$. In this case
there exist
$y^{+}_{\lambda},y^{-}_{\lambda}\in\mathbb{T}_{\mathfrak{p}}\backslash
S_{j,\delta_{0}}$, arranged so that $y^{+}_{\lambda}$ is in the positive
direction, counter-clockwise, to $y^{-}_{\lambda}$, such that

$\lambda=b(y^{+}_{\lambda})=b(y^{-}_{\lambda}).$ (3.57)

Denote

$\epsilon:=\nu/k,\,\,M:=\|h\|_{L^{2}(\mathbb{T}_{\mathfrak{p}})}+|k|^{-1}\|\partial_{y}h\|_{L^{2}(\mathbb{T}_{\mathfrak{p}})}.$
(3.58)

There exists a decomposition for $y\in\mathbb{T}_{\mathfrak{p}}$

$A_{\Theta}^{-1}h(y):=w_{1}(y,\lambda)+w_{2}(y,\lambda),$ (3.59)

satisfying the property that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$|w_{1}(y,\lambda)|\lesssim\sum_{\dagger\in\\{+,-\\}}|k||y-y^{\dagger}_{\lambda}|^{-1/2}M,$
(3.60)

and for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&\bigg{|}w_{2}(y,\lambda)-\sum_{\dagger\in\\{+,-\\}}\frac{h(y^{\dagger}_{\lambda})\varphi(y-y^{\dagger}_{\lambda})}{ib^{\prime}(y^{\dagger}_{\lambda})(y-y^{\dagger}_{\lambda})+|\epsilon|^{1/3}}\bigg{|}\lesssim\sum_{\dagger\in\\{+,-\\}}\frac{|\epsilon|^{-1/3}}{1+|\epsilon|^{-2/3}|y-y^{\dagger}_{\lambda}|^{2}}|k|^{1/2}M.\end{split}$
(3.61)

###### Proof.

Without loss of generality, we assume that $\epsilon>0$. We focus on the proof
of the case (i), with (ii) being similar and simpler. Define for
$y\in\mathbb{T}_{\mathfrak{p}}$,

$w(y,\lambda):=A_{\Theta}^{-1}h(y).$ (3.62)

It follows from Lemma 3.3, the bound (3.4) and Sobolev inequalities that for
$y\in\mathbb{T}_{\mathfrak{p}}$ with

$\min\\{|y-y_{\lambda}^{+}|,|y-y_{\lambda}^{-}|\\}\gtrsim|\lambda-b(y_{j\ast})|^{1/2}$

we have bound

$|w(y,\lambda)|\lesssim|\lambda-b(y_{j\ast})|^{-1}\|h\|_{L^{\infty}(\mathbb{T}_{\mathfrak{p}})}\lesssim\Big{[}\big{(}d_{j,k}(\Lambda)\big{)}^{1/2}|\lambda-b(y_{j\ast})|\Big{]}^{-1}M.$
(3.63)

Note that $w$ satisfies for $\dagger\in\\{+,-\\}$,
$|y-y^{\dagger}_{\lambda}|\ll|\lambda-b(y_{j\ast})|^{1/2}$,

$\begin{split}&\Big{[}\epsilon\partial_{y}^{2}+ib^{\prime}(y_{\lambda}^{\dagger})(y_{\lambda}^{\dagger}-y)-\alpha\Big{]}w(y,\lambda)\\\
&=h(y)+i\Big{[}b^{\prime}(y_{\lambda}^{\dagger})(y_{\lambda}^{\dagger}-y)+b(y)-b(y_{\lambda}^{\dagger})\Big{]}w(y,\lambda)\\\
&=h(y^{\dagger}_{\lambda})+h(y)-h(y^{\dagger}_{\lambda})+i\Big{[}b^{\prime}(y_{\lambda}^{\dagger})(y_{\lambda}^{\dagger}-y)+b(y)-b(y_{\lambda}^{\dagger})\Big{]}w(y,\lambda).\end{split}$
(3.64)

To extract the most singular part near $y=y^{\dagger}_{\lambda}$, we define
for $y\in\mathbb{R}$, $W(y,\Lambda)$ as the solution to

$\Big{[}\partial_{y}^{2}-iy-\epsilon^{-1/3}|b^{\prime}(y_{\lambda}^{\dagger})|^{-2/3}\alpha\Big{]}W(y,\Lambda)=1.$
(3.65)

It follows from Proposition 3.1 in [41] that $W(y,\Lambda)$ satisfies the
following bounds for $y\in\mathbb{R}$,

$\Big{|}W(y,\Lambda)-\frac{1}{1-iy}\Big{|}\lesssim\frac{1}{1+y^{2}}.$ (3.66)

The desired bounds then follow from Lemma 3.3 and the inequality (3.66) using
equation (3.64), by taking

$w_{2}(y,\lambda):=\sum_{\dagger\in\\{+,-\\}}\epsilon^{-1/3}|b^{\prime}(y_{\lambda}^{\dagger})|^{-2/3}h(y^{\dagger}_{\lambda})W\Big{(}(y-y^{\dagger}_{\lambda})\epsilon^{-1/3}|b^{\prime}(y_{\lambda}^{\dagger})|^{1/3},\Lambda\Big{)}\varphi\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}.$
(3.67)

∎

######  Remark 3.2.

Denote for $y\in\mathbb{T}_{\mathfrak{p}},\dagger\in\\{+,-\\}$,

$g(y):=\int_{\mathbb{T}_{\mathfrak{p}}}\mathcal{G}_{k}^{j}(y,z;\Lambda)A_{\Theta}^{-1}h(z)\,dz.$
(3.68)

It follows from (3.54) and (3.56), using integration by parts and the identity
for $\iota\in\mathbb{R}\backslash\\{0\\}$

$\frac{1}{y-y^{\dagger}+i\iota}=\partial_{y}\log\frac{y-y^{\dagger}+i\iota}{\delta},$
(3.69)

that

$\begin{split}&\left\|g\right\|_{L^{2}(S^{j}_{9\delta})}+d_{j,k}(\Lambda)\left\|\partial_{y}g\right\|_{L^{2}(S^{j}_{9\delta})}\lesssim\frac{d_{j,k}^{1/2}(\Lambda)}{\delta}M,\end{split}$
(3.70)

and for $y\in\mathbb{T}_{\mathfrak{p}}\backslash S^{j}_{9\delta}$,

$\begin{split}&|g(y)|+d_{j,k}(\Lambda)|\partial_{y}g(y)|\lesssim
d_{j,k}^{-1/2}\delta^{-3/4}\Big{[}\delta^{-1}\int_{|z-y_{j\ast}|<10\delta}\mathcal{G}_{k}^{j}(y,z;\Lambda)\,dz\Big{]}\,M.\end{split}$
(3.71)

##  4\. The limiting absorption principle and main bounds on the spectral
density function

In this section we state the limiting absorption principle, and derive bounds
on the spectral density functions using the limiting absorption principle and
the equations (1.18).

###  4.1. The limiting absorption principle for the Orr-Sommerfeld equation

For
$k\in\mathbb{Z}\cap[1,\infty),\nu\in(0,1/8),\lambda\in\mathbb{R},\alpha\geq-\sigma_{0}|\nu/k|^{1/2}$
with $\sigma_{0}\in(0,1/2)$ sufficiently small from Proposition 3.1, denoting
$\Theta:=(\lambda,\alpha,\nu,k)$, we define the operator
$T_{\Theta}:H^{1}(\mathbb{T}_{\mathfrak{p}})\to
H^{2}(\mathbb{T}_{\mathfrak{p}})$ as follows.

If
$\lambda\in\mathbb{R}\backslash\big{(}\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}\big{)}$
or $\alpha\geq\delta_{0}$, we define for any $h\in
H^{1}(\mathbb{T}_{\mathfrak{p}})$ and $y\in\mathbb{T}_{\mathfrak{p}}$,

$T_{\Theta}h(y):=-(k^{2}-\partial_{y}^{2})^{-1}A_{\Theta}^{-1}\big{(}ib^{\prime\prime}h\big{)}(y);$
(4.1)

If $\lambda\in\Sigma_{j,\delta_{0}}$ and
$-\sigma_{0}|\nu/k|^{1/2}\leq\alpha\leq\delta_{0}$, for some $j\in\\{1,2\\}$,
then we define for any $h\in H^{1}(\mathbb{T}_{\mathfrak{p}})$ and
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}T_{\Theta}h(y):=-\int_{\mathbb{T}_{\mathfrak{p}}}\mathcal{G}_{k}^{j}(y,z;\Lambda)&\Big{\\{}A_{\Theta}^{-1}\big{(}ib^{\prime\prime}h\big{)}(z)\\\
&\quad+\frac{b^{\prime\prime}(z)}{b(z)-\lambda-i\alpha}\Big{[}\varphi_{0}\big{(}\frac{z-y_{j\ast}}{\delta_{0}}\big{)}-\varphi_{0}\big{(}\frac{z-y_{j\ast}}{\delta(\Lambda)}\big{)}\Big{]}h(z)\Big{\\}}dz.\end{split}$
(4.2)

The main technical tool we need is the following limiting absorption
principle.

######  Proposition 4.1.

Fix $\gamma\in[7/4,2)$. Under the assumptions in Proposition 1.1, there exist
$\nu_{0},\kappa,\sigma_{\sharp}\in(0,1/10)$ sufficiently small depending on
$\gamma$, such that the following statement holds. Assume that

$(\sigma_{1},\sigma_{2})\in\big{\\{}(0,-\gamma),(1,-\gamma+1),(1-2(2-\gamma),-\gamma+2)\big{\\}}.$
(4.3)

For
$k\in\mathbb{Z}\cap[1,\infty),\nu\in(0,\nu_{0}),\alpha\geq-\sigma_{\sharp}|\nu/k|^{1/2}$
and $\lambda\in\mathbb{R}$, we have the following bounds.

(i) If $\lambda\in\Sigma_{j,\delta_{0}}$ for some $j\in\\{1,2\\}$, then for
any $h\in H^{2}(\mathbb{T}_{\mathfrak{p}})$ and $(\sigma_{1},\sigma_{2})$ as
in (4.3),

$\big{\|}h+T_{\Theta}h\big{\|}_{X^{\sigma_{1},\sigma_{2}}(\mathfrak{M})}\geq\kappa\|h\|_{X^{\sigma_{1},\sigma_{2}}(\mathfrak{M})};$
(4.4)

(ii) If
$\lambda\in\mathbb{R}\backslash\big{(}\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}\big{)}$,
then

$\big{\|}h+T_{\Theta}h\big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\geq\kappa\|h\|_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}.$
(4.5)

######  Remark 4.1.

We briefly explain the design of the exponents (4.3) and the weighted norm
(1.41), which will be used in section 4 to control the spectral density
functions and their derivatives in $\lambda$ up to two derivatives.

The weights used in section 4 are motivated by the decay of the modified
Green’s functions near the critical points $y_{1\ast},y_{2\ast}$. Assuming
$k=1$ and $\min\\{|z-y_{1\ast}|,|z-y_{2\ast}|\\}\gtrsim 1$ for simplicity, the
modified Green’s function
$\mathcal{G}_{k}^{j}(y,z;\Lambda)\sim(|y-y_{j\ast}|+\delta(\Lambda))^{2}$ for
$y$ close to $y_{j\ast}$. This asymptotic matches the weights used in (1.41)
for $(\sigma_{1},\sigma_{2})=(0,-\gamma)$. The use of $L^{2}$ and $L^{\infty}$
based spaces adapts to the singularity at $\lambda=b(y)$ that appears in
(4.2).

For $(\sigma_{1},\sigma_{2})=(0,-\gamma)$ and
$(\sigma_{1},\sigma_{2})=(1,-\gamma+1)$, the corresponding norms reflect the
heuristic that taking one derivative in $\lambda$ costs
$\varrho_{j}^{-1}\varrho_{j,k}^{-1}$ (in contrast to taking one derivative in
$y$ which costs only $\varrho_{j,k}^{-1}$). The extra adjustment for
$(\sigma_{1},\sigma_{2})=(1-2(2-\gamma),-\gamma+2)$, where the weight
$\varrho^{1-2(2-\gamma)}_{j}\varrho_{j,k}^{-\gamma+2+\alpha}$ was used in
favor of the seemingly more natural weight
$\varrho^{2}_{j}\varrho_{j,k}^{-\gamma+2+\alpha}$, is due to the requirement
that the weights need to vary more slowly than the modified Green’s function
to avoid nonlocal effects. In applications below (see (4.11)), we need to
incorporate an additional factor $\delta(\Lambda)^{1+2(2-\gamma)}$ to match
the two derivatives taken.

###  4.2. Main bounds on the spectral density function

Assume that $j\in\\{1,2\\},k\in\mathbb{Z}\cap[1,\infty)$,
$|\alpha|\leq\sigma_{\sharp}|\nu/k|^{1/2}$ and $\gamma\in[7/4,2)$.

We define the “modified spectral density function”
$\psi_{k,\nu}^{\ast}(y,\lambda)$ for $y\in\mathbb{T}_{\mathfrak{p}}$ as

$\psi_{k,\nu}^{\ast}(y,\Lambda):=\psi_{k,\nu}(y,\Lambda)-\sum_{j=1}^{2}\varphi_{0}\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\frac{\omega_{0k}(y)}{ib^{\prime\prime}(y)}.$
(4.6)

Then $\psi_{k,\nu}^{\ast}(y,\lambda)$ satisfies the equation for
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\left\\{\begin{array}[]{rl}\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\alpha\Big{]}\omega^{\ast}_{k,\nu}(y,\Lambda)+ib^{\prime\prime}(y)\psi^{\ast}_{k,\nu}(y,\Lambda)=f_{0k}(y,\Lambda),&\\\
(-k^{2}+\partial_{y}^{2})\psi^{\ast}_{k,\nu}(y,\Lambda)=\omega^{\ast}_{k,\nu}(y,\Lambda),&\end{array}\right.$
(4.7)

where

$\begin{split}f_{0k}(y,\Lambda):=&\,\,\omega_{0k}(y)-\sum_{j=1}^{2}\varphi_{0}\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\omega_{0k}(y)\\\
&-\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\alpha\Big{]}(\partial_{y}^{2}-k^{2})\Big{\\{}\sum_{j=1}^{2}\varphi_{0}\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\frac{\omega_{0k}(y)}{ib^{\prime\prime}(y)}\Big{\\}}.\end{split}$
(4.8)

The proof of our main theorems is based on the following proposition on the
regularity of the spectral density function.

######  Proposition 4.2.

Fix $\gamma\in[15/8,2)$ and $\nu_{0},\sigma_{\sharp}\in(0,1)$ from Proposition
4.1. Assume that $\nu\in(0,\nu_{0})$, $k\in\mathbb{Z}\cap[1,\infty)$,
$\epsilon:=\nu/k\in(0,1/8)$, $|\alpha|\leq\sigma_{\sharp}|\nu/k|^{1/2}$. Let
$\omega_{0k},\omega_{k,\nu},\psi_{k,\nu}$ be defined as in Proposition 1.1,
and let $\omega^{\ast}_{k,\nu},\psi^{\ast}_{k,\nu}$ be as in (4.7)-(4.8).

Define the “good derivative” $D_{\lambda}$ for
$\lambda\in\mathbb{R},y\in\mathbb{T}_{\mathfrak{p}}$ as follows. If
$\lambda\in b(\mathbb{T}_{\mathfrak{p}})$ and
$\min\\{|\lambda-b(y_{1\ast})|,|\lambda-b(y_{2\ast})|\\}\geq 10|\nu/k|^{1/2}$,
then

$D_{\lambda}:=\partial_{\lambda}+\Big{[}1-\varphi_{0}\big{(}\frac{y-y_{1\ast}}{\delta_{2}(\lambda)}\big{)}\Big{]}\Big{[}1-\varphi_{0}\big{(}\frac{y-y_{2\ast}}{\delta_{2}(\lambda)}\big{)}\Big{]}\frac{1}{b^{\prime}(y)}\partial_{y}.$
(4.9)

If $\lambda\not\in b(\mathbb{T}_{\mathfrak{p}})$ or
$\min\\{|\lambda-b(y_{1\ast})|,|\lambda-b(y_{2\ast})|\\}<10|\nu/k|^{1/2}$,
denote $D_{\lambda}:=\partial_{\lambda}$.

Then we have the following bounds.

(i) For
$\lambda\in\mathbb{R}\backslash\big{(}\cup_{j\in\\{1,2\\}}\Sigma_{j,\delta_{0}}\big{)},\sigma\in\\{0,1,2\\}$,

$\big{\|}D_{\lambda}^{\sigma}\,\psi_{k,\nu}(\cdot,\Lambda)\big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim|k|^{-5/2+\sigma}\frac{1}{\langle\lambda\rangle^{\sigma+1}}\big{\|}\omega_{0k}\big{\|}_{H^{3}_{k}(\mathbb{T}_{\mathfrak{p}})};$
(4.10)

(ii) For $j\in\\{1,2\\},\lambda\in\Sigma_{j,\delta_{0}}$,

$\begin{split}&\big{\|}\psi_{k,\nu}^{\ast}(\cdot,\Lambda)\big{\|}_{X^{0,-\gamma}(\mathfrak{M})}\lesssim|k|^{-1/2}\big{\|}\omega_{0k}\big{\|}_{H^{3}_{k}(\mathbb{T}_{\mathfrak{p}})},\\\
&\big{\|}D_{\lambda}\,\psi_{k,\nu}^{\ast}(\cdot,\Lambda)\big{\|}_{X^{1,-\gamma+1}(\mathfrak{M})}\lesssim|k|^{-1/2}\big{\|}\omega_{0k}\big{\|}_{H^{3}_{k}(\mathbb{T}_{\mathfrak{p}})},\\\
&\big{\|}D_{\lambda}^{2}\,\psi_{k,\nu}^{\ast}(\cdot,\Lambda)\big{\|}_{X^{1-2(2-\gamma),-\gamma+2}(\mathfrak{M})}\lesssim|k|^{-1/2}(\delta(\Lambda))^{-1-2(2-\gamma)}\big{\|}\omega_{0k}\big{\|}_{H^{3}_{k}(\mathbb{T}_{\mathfrak{p}})}.\end{split}$
(4.11)

As a corollary of Lemma 3.4 and Proposition 4.2, we also have the following
refined description of the singularities of
$\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)$ when $y$ approaches
the points $\\{y\in\mathbb{T}_{\mathfrak{p}}:b(y)=\lambda\\}$.

######  Corollary 4.1.

There exist $\nu_{0},\sigma_{\sharp}>0$ sufficiently small, such that for
$0<\nu<\nu_{0},k\in\mathbb{Z}\cap[1,\infty)$, $\lambda\in\mathbb{R}$,
$j\in\\{1,2\\}$, $|\alpha|\leq\sigma_{\sharp}|\nu/k|^{1/2}$, the following
statement holds. Denote

$M_{k}:=\|\omega_{0k}\|_{H^{3}_{k}(\mathbb{T}_{\mathfrak{p}})}.$ (4.12)

(i) Assume that $\lambda\in\Sigma_{j,\delta_{0}}\cap
b(\mathbb{T}_{\mathfrak{p}})$ and $|\lambda-b(y_{j\ast})|\gg|\nu/k|^{1/2}$.
Then we have the bounds for $|\lambda-b(y)|\ll|b(y)-b(y_{j\ast})|$,

$|\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)|\lesssim\big{(}|y-y_{j\ast}|\wedge\frac{1}{|k|}\big{)}^{\gamma-2}|y-y_{j\ast}|^{-1}\bigg{(}1+\bigg{|}\log{\bigg{|}\frac{\lambda-b(y)}{b(y)-b(y_{j\ast})}\bigg{|}}\bigg{|}\bigg{)}|k|^{-1/2}M_{k},$
(4.13)

and

$\begin{split}&\bigg{|}\omega^{\ast}_{k,\nu}(y,\Lambda)+\frac{ib^{\prime\prime}(y)\psi^{\ast}_{k,\nu}(y,\Lambda)}{i(\lambda-b(y))+|\nu/k|^{1/3}|y-y_{j\ast}|^{2/3}}\bigg{|}\\\
&\lesssim\varrho_{j,k}^{\gamma-1/2}(y,\Lambda)|y-y_{j\ast}|^{-1/2}\frac{|\nu/k|^{-1/3}|y-y_{j\ast}|^{1/3}}{1+(|\nu/k|^{1/3}|y-y_{j\ast}|^{2/3})^{-2}|\lambda-b(y)|^{2}}|k|^{-1/2}M_{k}.\end{split}$
(4.14)

(ii) If $\lambda\in
b(\mathbb{T}_{\mathfrak{p}})\backslash\big{(}\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}\big{)}$,
we have the bounds for $y\in\mathbb{T}_{\mathfrak{p}}$ with $|\lambda-b(y)|\ll
1$,

$|\partial_{y}D_{\lambda}\psi_{k,\nu}(y,\Lambda)|\lesssim\big{(}1+|\log{|\lambda-b(y)|}|\big{)}|k|^{-1/2}M_{k},$
(4.15)

and

$\begin{split}&\bigg{|}\omega_{k,\nu}(y,\Lambda)-\frac{\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda)}{i(\lambda-b(y))+|\nu/k|^{1/3}}\bigg{|}\\\
&\lesssim\frac{|\nu/k|^{-1/3}}{1+|\nu/k|^{-2/3}|\lambda-b(y)|^{2}}|k|^{-5/2}M_{k}.\end{split}$
(4.16)

The proof of Corollary 4.1 on the refined properties of the functions
$\partial_{y}\psi^{\ast}_{k,\nu}$,
$\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}$ and $\omega_{k,\nu}^{\ast}$
follows from equations (4.7), (4.22), and (4.35) using Proposition 4.2 and
Lemmas 3.3 and 2.1. These refined bounds are useful below in the proof of the
main theorem. The rest of the section is devoted to the proofs of Proposition
4.2. We organize the proofs into different sections as the details of the
argument are somewhat lengthy.

###  4.3. Proof for statement (i) of Proposition 4.2

We divide the proof into the cases $\sigma\in\\{0,1,2\\}$, organized in
subsections.

####  4.3.1. The case $\sigma=0$ for statement (i) of Proposition 4.2

To establish (4.2) for $\sigma=0$, by the limiting absorption principle (4.5),
it suffices to show that

$\big{\|}(k^{2}-\partial_{y}^{2})^{-1}A_{\Theta}^{-1}\omega_{0k}\big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim|k|^{-5/2}\langle\lambda\rangle^{-1}M_{k}.$
(4.17)

If $\lambda\not\in b(\mathbb{T}_{\mathfrak{p}})$, the desired bounds (4.17)
follow from Lemma 3.3 and standard elliptic estimates, since in this case
$A_{\Theta}^{-1}$ does not involve any singularities.

We now consider the case $\lambda\in b(\mathbb{T}_{\mathfrak{p}})$. We note
first the inequality that

$\Big{\|}\omega_{0k}\Big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim|k|^{-2}M_{k}.$
(4.18)

To prove (4.17), we use (4.18), Lemma 3.3 and Lemma 3.4, by considering the
regions (I) $|\lambda-b(y)|\gtrsim 1$ and (II) $|\lambda-b(y)|\ll 1$
respectively. In the region (I), the desired bound (4.17) follows from (4.18)
and Lemma 3.3. In region (II), we use Lemma 3.4. After peeling off the easier
bounds for $\dagger\in\\{+,-\\}$ (using the notation as in Lemma 3.4)

$\bigg{\|}\int_{\mathbb{T}_{\mathfrak{p}}}G_{k}(y,z)\frac{|\nu/k|^{-1/3}}{1+|\nu/k|^{-2/3}|z-y_{\lambda}^{\dagger}|^{2}}dz\bigg{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim|k|^{-1/2},$
(4.19)

it suffices to notice the inequality that for $\dagger\in\\{+,-\\}$ (with the
notation that
$\epsilon^{\prime}:=|\nu/k|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}/b^{\prime}(y_{\lambda}^{\dagger})$),

$\bigg{\|}|k|^{-1}\int_{\mathbb{R}}e^{-|k||y-z|}\frac{\varphi(z-y_{\lambda}^{\dagger})}{i(z-y_{\lambda}^{\dagger})+\epsilon^{\prime}}dz\bigg{\|}_{H^{1}_{k}(\mathbb{R})}\lesssim|k|^{-1/2}.$
(4.20)

This completes the proof of (4.10) for $\sigma=0$.

####  4.3.2. The case $\sigma=1$ for statement (i) of Proposition 4.2

We now turn to the case $\sigma=1$. Set for
$y\in\mathbb{T}_{\mathfrak{p}},\lambda\in\mathbb{R}$,

$a(y,\lambda):=\Big{[}1-\varphi_{0}\big{(}\frac{y-y_{1\ast}}{\delta_{2}(\lambda)}\big{)}\Big{]}\Big{[}1-\varphi_{0}\big{(}\frac{y-y_{2\ast}}{\delta_{2}(\lambda)}\big{)}\Big{]}\frac{1}{b^{\prime}(y)}.$
(4.21)

Taking the derivative $D_{\lambda}$, by a straightforward but somewhat lengthy
calculation, we obtain that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda)+A_{\Theta}^{-1}\big{(}ib^{\prime\prime}(\cdot)D_{\lambda}\psi_{k,\nu}(\cdot,\Lambda)\big{)}(y)\\\
&=A_{\Theta}^{-1}\big{(}g^{\ast}_{0k}(\cdot,\Lambda)\big{)}(y)+\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}\Big{]}(y,\Lambda),\end{split}$
(4.22)

where for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}g^{\ast}_{0k}(y,\Lambda):=&\,\,D_{\lambda}\omega_{0k}(y)+\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega_{k,\nu}(y,\Lambda)\\\
&+2\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}(y,\Lambda)-ib^{\prime\prime\prime}(y)a(y,\lambda)\psi_{k,\nu}(y,\Lambda)\\\
&+i\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega_{k,\nu}(y,\Lambda).\end{split}$
(4.23)

The calculations to get (4.22)-(4.23) are similar to those in subsection 4.4.2
which are slightly more complicated. For the sake of conciseness, we omit the
repetitive details here. By Proposition 4.1, it suffices to show for

$\begin{split}h\in
A_{\Theta}^{-1}\Big{\\{}&D_{\lambda}\omega_{0k},\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega_{k,\nu}(y,\Lambda),\\\
&2\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}(y,\Lambda),b^{\prime\prime\prime}(y)a(y,\lambda)\psi_{k,\nu}(y,\Lambda),\\\
&\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega_{k,\nu}(y,\Lambda)\Big{\\}},\end{split}$
(4.24)

and

$\begin{split}h\in\Big{\\{}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu},\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}\Big{\\}},\end{split}$
(4.25)

the following bounds hold

$\big{\|}(k^{2}-\partial_{y}^{2})^{-1}h\big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim\langle\lambda\rangle^{-2}|k|^{-3/2}M_{k}.$
(4.26)

If
$\lambda\in\mathbb{R}\backslash\Big{[}b(\mathbb{T}_{\mathfrak{p}})\cup\big{(}\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}\big{)}\Big{]}$,
then the desired bounds (4.26) follow from (4.10) for the case $\sigma=0$
easily since in this case $A_{\Theta}^{-1}$ does not involve strong
singularities. Below we assume that $\lambda\in
b(\mathbb{T}_{\mathfrak{p}})\backslash\big{(}\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}\big{)}$.
The desired bounds for

$\begin{split}h\in\Big{\\{}&A_{\Theta}^{-1}D_{\lambda}\omega_{0k},A_{\Theta}^{-1}\big{(}b^{\prime\prime\prime}a(y,\lambda)\psi_{k,\nu}),\\\
&A_{\Theta}^{-1}\big{[}(b^{\prime}(y)a(y,\lambda)-1)\omega_{k,\nu}\big{]},\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu}\Big{\\}},\end{split}$
(4.27)

follow from the case $\sigma=0$, using the inequalities (4.19)-(4.20) and
Lemma 3.4. We note that $b^{\prime}a-1\equiv 0$ in a neighborhood of
$\\{y\in\mathbb{T}_{\mathfrak{p}}:b(y)-\lambda=0\\}$. Hence
$(b^{\prime}(y)a(y,\lambda)-1)\omega_{k,\nu}$ is not singular.

To handle the case $h=\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}$,
we use the identity for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\partial^{2}_{y}\psi_{k,\nu}(y,\Lambda)=k^{2}\psi_{k,\nu}(y,\Lambda)+\omega_{k,\nu}(y,\Lambda).$
(4.28)

The desired bounds (4.26) for
$h=\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}$ then follow from
analogous argument as in the proof of (4.17). It remains to prove (4.26) for

$\begin{split}h\in
A_{\Theta}^{-1}\Big{\\{}\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega_{k,\nu}(y,\Lambda),\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}(y,\Lambda)\Big{\\}},\end{split}$
(4.29)

We consider only the harder case when
$h=A_{\Theta}^{-1}\Big{[}\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}\Big{]}$
which is more singular.

It follows from equation (1.18) that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\alpha\Big{]}\omega_{k,\nu}(y,\Lambda)=\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda).$
(4.30)

Thus for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&\Big{[}\frac{\nu}{k}\partial_{y}^{2}+i(\lambda-b(y))-\iota\alpha\Big{]}\partial_{y}\omega_{k,\nu}(y,\Lambda)\\\
&=\partial_{y}\big{[}\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda)\big{]}+ib^{\prime}(y)\omega_{k,\nu}(y,\Lambda).\end{split}$
(4.31)

Using the bounds

$\big{\|}\partial_{y}\big{[}\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda)\big{]}\big{\|}_{L^{2}(\mathbb{T}_{\mathfrak{p}})}\lesssim|k|^{-2}M_{k},$
(4.32)

we obtain from Lemma 3.3 that

$\begin{split}&\Big{\|}\frac{\nu}{k}A_{\Theta}^{-1}\Big{\\{}\partial_{y}a\,\partial_{y}A_{\Theta}^{-1}\partial_{y}\big{[}\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda)\big{]}\Big{\\}}\Big{\|}_{L^{2}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim\Big{\|}(i(\lambda-b(y))+|\nu/k|^{1/3})^{-2}\frac{\nu}{k}L^{-1}(y;\Lambda)\partial_{y}\big{[}\omega_{0k}(y)-ib^{\prime\prime}(y)\psi_{k,\nu}(y,\Lambda)\big{]}\Big{\|}_{L^{2}(\mathbb{T}_{\mathfrak{p}})}\\\
&\lesssim|k|^{-2}M_{k},\end{split}$ (4.33)

and for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&\Big{|}\frac{\nu}{k}A_{\Theta}^{-1}\Big{\\{}\partial_{y}a\,(y,\lambda)\partial_{y}A_{\Theta}^{-1}\big{(}b^{\prime}\omega_{k,\nu}\big{)}(y,\Lambda)\Big{\\}}\Big{|}\\\
&\lesssim\frac{1}{(|\lambda-b(y)|+|\nu/k|^{1/3})^{3}}|\nu/k|\frac{|k|^{-2}M_{k}}{|\nu/k|^{1/2}(|\nu/k|^{1/3}+|\lambda-b(y)|)^{-1/2}}\\\
&\lesssim\frac{|\nu/k|^{-1/3}}{(1+|\nu/k|^{-1/3}|\lambda-b(y)|)^{5/2}}|k|^{-2}M_{k}.\end{split}$
(4.34)

The desired bounds (4.26) for
$h=A_{\Theta}^{-1}\Big{[}\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}\Big{]}$
follow from (4.31), and (4.33)-(4.34). This completes the proof of (4.26) and
therefore also the bound (4.11) for $\sigma=1$.

####  4.3.3. The case $\sigma=2$ for statement (i) of Proposition 4.2

Similar calculations to (4.68)-(4.71) below (which we again omit to avoid
repetitive details) show that

$\begin{split}&(\partial_{y}^{2}-k^{2})D^{2}_{\lambda}\psi_{k,\nu}(y,\Lambda)+A_{\Theta}^{-1}\big{(}ib^{\prime\prime}(\cdot)D^{2}_{\lambda}\psi_{k,\nu}(\cdot,\Lambda)\big{)}(y)\\\
&=A_{\Theta}^{-1}g_{0k}^{\ast\ast}(y,\Lambda)+\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi_{k,\nu}\Big{]}(y,\Lambda),\end{split}$
(4.35)

where we have defined

$\begin{split}g_{0k}^{\ast\ast}(y,\Lambda):=&D_{\lambda}\Big{\\{}g_{0k}^{\ast}+A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}\Big{]}(y,\Lambda)\Big{\\}}\\\
&+\Big{[}\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}+2\frac{\nu}{k}\partial_{y}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}^{2}\Big{]}D_{\lambda}\psi_{k,\nu}(y,\Lambda)\\\
&+i\Big{[}a(y,\lambda)b^{\prime}(y)-1\Big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda)-ia(y,\lambda)b^{\prime\prime\prime}(y)D_{\lambda}\psi_{k,\nu}(y,\Lambda).\end{split}$
(4.36)

As in the case of $\sigma=1$, it suffices to show that for

$\begin{split}&h\in
A_{\Theta}^{-1}\Big{\\{}D_{\lambda}g_{0k}^{\ast},D_{\lambda}A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}\Big{]},\\\
&\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}D_{\lambda}\psi_{k,\nu}(y,\Lambda),\frac{\nu}{k}\partial_{y}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}^{2}D_{\lambda}\psi_{k,\nu}(y,\Lambda),\\\
&\Big{[}a(y,\lambda)b^{\prime}(y)-1\Big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda),a(y,\lambda)b^{\prime\prime\prime}(y)D_{\lambda}\psi_{k,\nu}(y,\Lambda)\Big{\\}},\end{split}$
(4.37)

and for

$h\in\Big{\\{}\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi_{k,\nu},\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi_{k,\nu}\Big{\\}},$
(4.38)

we have the bound

$\big{\|}(k^{2}-\partial_{y}^{2})^{-1}h\big{\|}_{H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})}\lesssim\langle\lambda\rangle^{-3}|k|^{-1/2}M_{k}.$
(4.39)

To prove the bounds (4.39) we shall use the following commutator relations for
any $q(x,\lambda)$ with sufficient regularity,

$\begin{split}\big{[}A_{\Theta},D_{\lambda}\big{]}&=i\big{(}a(y,\lambda)b^{\prime}(y)-1\big{)}+\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}+2\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2},\\\
\big{[}D_{\lambda},q(x,\lambda)\big{]}&=D_{\lambda}q(x,\lambda),\,\,\big{[}D_{\lambda},\partial_{y}\big{]}=-\partial_{y}a(y,\lambda)\partial_{y}.\end{split}$
(4.40)

We observe from (4.40) that the commutator terms behave better than the
uncommutated terms, by either gaining a factor of $\nu/k$ or one derivative.

Using equation (4.22), the commutator relations (4.40) and the identity

$\partial^{2}_{y}D_{\lambda}\psi_{k,\nu}(y,\Lambda)=\big{(}\partial^{2}_{y}-k^{2}\big{)}D_{\lambda}\psi_{k,\nu}(y,\Lambda)+k^{2}D_{\lambda}\psi_{k,\nu}(y,\Lambda),$
(4.41)

the desired bounds for

$h\in\Big{\\{}\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi_{k,\nu},\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi_{k,\nu}\Big{\\}}$
(4.42)

and

$\begin{split}&h\in
A_{\Theta}^{-1}\Big{\\{}D_{\lambda}A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi_{k,\nu}\Big{]},\\\
&\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}D_{\lambda}\psi_{k,\nu}(y,\Lambda),\frac{\nu}{k}\partial_{y}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}^{2}D_{\lambda}\psi_{k,\nu}(y,\Lambda),\\\
&\Big{[}a(y,\lambda)b^{\prime}(y)-1\Big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda),a(y,\lambda)b^{\prime\prime\prime}(y)D_{\lambda}\psi_{k,\nu}(y,\Lambda)\Big{\\}},\end{split}$
(4.43)

follow from the bounds (4.10) for $\sigma\in\\{0,1\\}$ and the arguments as in
proof of the cases $\sigma\in\\{0,1\\}$, after peeling off the easier
commutator terms.

It remains to prove (4.39) for $h=A_{\Theta}^{-1}D_{\lambda}g_{0k}^{\ast}$.
Using the definition (4.23), it suffices to prove (4.39) for

$\begin{split}h\in
A_{\Theta}^{-1}D_{\lambda}\Big{\\{}&\,D_{\lambda}\omega_{0k}(y),\,\,\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega_{k,\nu}(y,\Lambda),\\\
&\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}(y,\Lambda),\,\,b^{\prime\prime\prime}(y)a(y,\lambda)\psi_{k,\nu}(y,\Lambda),\\\
&\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega_{k,\nu}(y,\Lambda)\Big{\\}}.\end{split}$
(4.44)

Following similar arguments as in the proof for $h$ as in (4.42)-(4.43) and
considering only the more singular terms, it suffices to prove (4.39) for

$\begin{split}h\in
A_{\Theta}^{-1}D_{\lambda}\Big{\\{}&\,\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega_{k,\nu}(y,\Lambda),\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega_{k,\nu}(y,\Lambda)\Big{\\}}.\end{split}$
(4.45)

After peeling off the easier commutator terms using (4.40), we reduce to prove
(4.39) for

$\begin{split}h\in
A_{\Theta}^{-1}\Big{\\{}&\,\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda),\\\
&\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}(y,\Lambda)\Big{\\}}.\end{split}$
(4.46)

The desired bounds (4.39) for
$h=A_{\Theta}^{-1}\big{\\{}\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}\big{\\}}(y,\Lambda)$
follows from Lemma 3.4 noting that $b^{\prime}a-1$ vanishes in a neighborhood
of $\\{y\in\mathbb{T}_{\mathfrak{p}}:\,b(y)=\lambda\\}$. The proof of (4.39)
for
$A_{\Theta}^{-1}\big{\\{}\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi_{k,\nu}\big{\\}}(y,\Lambda)$
is similar to (4.30)-(4.34).

###  4.4. Proof of statement (ii) in Proposition 4.2

We choose $\gamma_{0},\gamma_{1},\gamma_{2}\in(1,2)$ with
$\gamma<\gamma_{2}<\gamma_{1}<\gamma_{0}<2$, and $\nu_{0}$ sufficiently small
such that Proposition 4.2 holds for $\gamma_{0},\gamma_{1},\gamma_{2}$.

Fix $j\in\\{1,2\\}$. We provide the detailed proof only for $\lambda\in
b(\mathbb{T}_{\mathfrak{p}})\cap\Sigma_{j,\delta_{0}}$ with
$|\lambda-b(y_{j\ast})|\gtrsim|\nu/k|^{1/2}$. The case when
$|\lambda-b(y_{j\ast})|\ll|\nu/k|^{1/2}$ or
$\lambda\in\Sigma_{j,\delta_{0}}\backslash b(\mathbb{T}_{\mathfrak{p}})$ is
easier since the singular behavior of $A_{\Theta}^{-1}$ for
$|y-y_{j\ast}|\lesssim|\nu/k|^{1/4}$ is compensated by the decay property of
the modified Green’s function and we can use Lemma 3.4 to get effective
bounds.

The following weighted bounds are very useful to treat various nonlocal
contributions below.

######  Lemma 4.1.

We have the following pointwise inequality for
$\sigma_{1}\in(-1,1),\sigma_{2}\in\mathbb{R}$ with
$\sigma_{1}+\sigma_{2}\in(-1,2)$, any $h\in
L^{\infty}(\mathbb{T}_{\mathfrak{p}})$ and $y\in\mathbb{T}_{\mathfrak{p}}$,

$\int_{\mathbb{T}_{\mathfrak{p}}}\Big{|}\mathcal{G}_{k}^{j}(y,z;\Lambda)\varrho_{j}^{\sigma_{1}-1}\varrho_{j,k}^{\sigma_{2}-1}(z;\Lambda)h(z)\Big{|}dz\lesssim_{\sigma_{1},\sigma_{2}}\varrho_{j}^{\sigma_{1}-1}\varrho_{j,k}^{\sigma_{2}+1}(y;\Lambda)\|h\|_{L^{\infty}(\mathbb{T}_{\mathfrak{p}})}.$
(4.47)

The proof follows from direct calculations using the bounds on the modified
Green’s function in Lemma (2.2).

The inequality (4.47) is useful when
$|\lambda-b(y)|\gtrsim|\lambda-b(y_{j\ast})|$ so that $A^{-1}_{\Theta}$ is not
singular. To handle the case when $|\lambda-b(y)|\ll|\lambda-b(y_{j\ast})|$,
in view of Lemma 3.4 and adopting the notation there, we use the following
bounds. Denote $\epsilon:=\nu/k$ and define for
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\mathcal{I}_{k,\nu}(y,\lambda):=\int_{\mathbb{T}_{\mathfrak{p}}}\mathcal{G}_{k}^{j}(y,z;\Lambda)\sum_{\dagger\in\\{+,-\\}}\frac{\varphi_{0}\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}}{ib^{\prime}(y^{\dagger}_{\lambda})(y-y^{\dagger}_{\lambda})+|\epsilon|^{1/3}|\lambda-b(y_{j\ast})|^{1/3}}dz.$
(4.48)

Then we have,

$\begin{split}&d^{-1/2}_{j,k}d^{-1}_{j}(\Lambda)\sum_{\beta\in\\{0,1\\}}\big{\|}d^{\beta}_{j,k}(\Lambda)\partial^{\beta}_{y}\mathcal{I}_{k,\nu}(y,\lambda)\big{\|}_{L^{2}(S^{j}_{8\delta})}\\\
&+d^{-2}_{j,k}(\Lambda)\sum_{\beta\in\\{0,1\\}}\big{\|}\varrho_{j,k}^{1+\beta}(y,\Lambda)\partial^{\beta}_{y}\mathcal{I}_{k,\nu}(y,\lambda)\big{\|}_{L^{\infty}(\mathbb{T}_{\mathfrak{p}}\backslash
S^{j}_{8\delta})}\lesssim|\lambda-b(y_{j\ast})|^{-1/2}.\end{split}$ (4.49)

We prove the bounds for
$D^{\sigma}_{\lambda}\,\psi_{k,\nu}^{\ast},\sigma\in\\{0,1,2\\}$,
respectively, organized as subsections.

####  4.4.1. Proof of the bounds for
$D^{\sigma}_{\lambda}\,\psi_{k,\nu}^{\ast},\sigma=0$ in statement (ii) in
Proposition 4.2

Using the bounds (4.4) of the limiting absorption principle and the equation
(4.7)-(4.8), it suffices to show that

$\big{\|}\mathcal{G}_{k}^{j}(\Lambda)A_{k,\nu}^{-1}(\Lambda)f_{0k}(\cdot,\lambda,\nu,\alpha)\big{\|}_{X^{0,-\gamma_{0}}(\mathfrak{M})}\lesssim|k|^{-1/2}M_{k}.$
(4.50)

In the above, with a slight abuse of notation, we denote for
$k\in\mathbb{Z}\cap[1,\infty),\nu\in(0,1/18),\lambda\in\cup_{j=1}^{2}\Sigma_{j,\delta_{0}}$,
$\mathcal{G}_{k}^{j}(\Lambda):H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})\to
H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})$ as follows. For any $h\in
H^{1}_{k}(\mathbb{T}_{\mathfrak{p}})$,

$\mathcal{G}_{k}^{j}(\Lambda)h(y):=\int_{\mathbb{T}_{\mathfrak{p}}}\mathcal{G}_{k}^{j}(y,z;\Lambda)h(z)dz.$
(4.51)

Notice that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}A_{\Theta}^{-1}f_{0k}(\cdot,\lambda,\nu,\alpha)=&\,A_{\Theta}^{-1}\Big{[}\omega_{0k}(y)-\sum_{j=1}^{2}\varphi_{0}\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\omega_{0k}(y)\Big{]}\\\
&-(\partial_{y}^{2}-k^{2})\Big{[}\sum_{j=1}^{2}\varphi_{0}\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\frac{\omega_{0k}(y)}{ib^{\prime\prime}(y)}\Big{]}.\end{split}$
(4.52)

The desired bounds (4.50) follow from Lemma 3.4 and the bounds on the modified
Green’s function.

####  4.4.2. Proof of the bounds for
$D^{\sigma}_{\lambda}\,\psi_{k,\nu}^{\ast},\sigma=1$ in statement (ii) in
Proposition 4.2

We note that $\partial_{\lambda}\psi^{\ast}_{k,\nu}$ satisfies the equation

$\begin{split}A_{\Theta}\partial_{\lambda}\omega^{\ast}_{k,\nu}(y,\Lambda)+ib^{\prime\prime}(y)\partial_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)&=-i\omega^{\ast}_{k,\nu}(y,\Lambda)\\\
&\quad+\partial_{\lambda}f_{0k}(y,\Lambda),\\\
(-k^{2}+\partial_{y}^{2})\partial_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)&=\partial_{\lambda}\omega^{\ast}_{k,\nu}(y,\Lambda).\end{split}$
(4.53)

Direct computation shows that $a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}$
satisfies the equation for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&A_{\Theta}(\partial_{y}^{2}-k^{2})\Big{[}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}(y,\Lambda)\Big{]}+ib^{\prime\prime}(y)a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&=a(y,\lambda)\partial_{y}f_{0k}(y,\Lambda)+A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda)\\\
&\,+\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega^{\ast}_{k,\nu}(y,\Lambda)+2\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega^{\ast}_{k,\nu}(y,\Lambda)\\\
&\,-ib^{\prime\prime\prime}(y)a(y,\lambda)\psi^{\ast}_{k,\nu}(y,\Lambda)+ib^{\prime}(y)a(y,\lambda)\omega^{\ast}_{k,\nu}(y,\Lambda).\end{split}$
(4.54)

It follows from equations (4.53) and (4.54) that
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&(\partial_{y}^{2}-k^{2})D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)+A_{\Theta}^{-1}\big{(}ib^{\prime\prime}(\cdot)D_{\lambda}\psi^{\ast}_{k,\nu}(\cdot,\Lambda)\big{)}(y)\\\
&=A_{\Theta}^{-1}\big{(}f_{0k}^{\ast}(\cdot,\Lambda)\big{)}(y)+\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda),\end{split}$
(4.55)

where for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}f_{0k}^{\ast}(y,\Lambda):=&\,\,D_{\lambda}f_{0k}(y,\Lambda)+\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)\partial_{y}\omega^{\ast}_{k,\nu}(y,\Lambda)\\\
&+2\frac{\nu}{k}\partial_{y}a(y,\lambda)\partial_{y}^{2}\omega^{\ast}_{k,\nu}(y,\Lambda)-ib^{\prime\prime\prime}(y)a(y,\lambda)\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&+i\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega^{\ast}_{k,\nu}(y,\Lambda).\end{split}$
(4.56)

Using the bounds (4.4) of the limiting absorption principle and the equation
(4.55)-(4.56), it suffices to show that

$\big{\|}\mathcal{G}_{k}^{j}(\Lambda)h\big{\|}_{X^{1,-\gamma+1}(\mathfrak{M})}\lesssim|k|^{-1/2}M_{k},$
(4.57)

where

$h\in\Big{\\{}A_{\Theta}^{-1}\big{(}f_{0k}^{\ast}(\cdot,\Lambda)\big{)}(y),\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda)\Big{\\}}.$
(4.58)

To prove (4.57), we divide into the singular region for
$y\in\mathbb{T}_{\mathfrak{p}}$ with $|\lambda-b(y)|\ll|\lambda-b(y_{j\ast})|$
and the non-singular region for $y\in\mathbb{T}_{\mathfrak{p}}$ with
$|\lambda-b(y)|\gtrsim|\lambda-b(y_{j\ast})|$. More precisely, using the
notations in Lemma 3.4, it suffices to prove that for $h$ as in (4.58),

$\Big{\|}\mathcal{G}_{k}^{j}(\Lambda)\Big{[}1-\sum_{\dagger\in\\{+,-\\}}\varphi_{0}\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}\Big{]}h\Big{\|}_{X^{1,-\gamma+1}(\mathfrak{M})}\lesssim|k|^{-1/2}M_{k},$
(4.59)

and for $\dagger\in\\{+,-\\}$

$\big{\|}\mathcal{G}_{k}^{j}(\Lambda)\varphi_{0}\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}h\big{\|}_{X^{1,-\gamma+1}(\mathfrak{M})}\lesssim|k|^{-1/2}M_{k}.$
(4.60)

To establish (4.59), we notice the bounds that for
$y\in\mathbb{T}_{\mathfrak{p}},\dagger\in\\{+,-\\}$ and $h$ as in (4.58),

$\Big{|}\Big{[}1-\sum_{\dagger\in\\{+,-\\}}\varphi_{0}\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}\Big{]}h(y)\Big{|}\lesssim\varrho_{j,k}^{\gamma_{0}-2}\varrho_{j}^{-2}(y;\Lambda)\lesssim\varrho_{j,k}^{\gamma_{1}-2}\varrho_{j}^{\gamma_{0}-\gamma_{1}-2}(y;\Lambda).$
(4.61)

The desired bounds (4.59) then follow from the estimates (4.51).

To establish (4.60), we use similar arguments as in section 4.3.2, since the
weights are constant over the support of
$\varphi\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}$
and can be taken out of the estimates when appropriately accounted for. We
consider the terms

$h\in
A_{\Theta}^{-1}\Big{\\{}D_{\lambda}f_{0k},\frac{\nu}{k}\partial_{y}^{2}a\partial_{y}\omega^{\ast}_{k,\nu},\frac{\nu}{k}\partial_{y}a\partial_{y}^{2}\omega^{\ast}_{k,\nu},b^{\prime\prime\prime}(y)a(y,\lambda)\psi^{\ast}_{k,\nu},\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega^{\ast}_{k,\nu}\Big{\\}}$
(4.62)

and

$h\in\Big{\\{}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}(y,\Lambda),\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}(y,\Lambda)\Big{\\}}.$
(4.63)

Using (4.8) and setting for $y\in\mathbb{T}_{\mathfrak{p}}$,

$h=(\partial_{y}^{2}-k^{2})\Big{[}\sum_{j=1}^{2}\varphi\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\omega_{0k}(y)\Big{]},$

by direct calculations we obtain that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}D_{\lambda}f_{0k}=&a\partial_{y}\Big{[}\omega_{0k}(y)-\sum_{j=1}^{2}\varphi\big{(}(y-y_{j\ast})/\delta_{0}\big{)}\omega_{0k}(y)\Big{]}+i(ab^{\prime}-1)h\\\
&+A_{\Theta}\partial_{y}\Big{[}a\,h\Big{]}+\frac{\nu}{k}\Big{[}-\partial_{y}^{3}a\,h+3\partial_{y}(\partial_{y}^{2}a\,h)-3\partial_{y}^{2}(\partial_{y}a\,h)\Big{]},\end{split}$
(4.64)

To prove the desired bound (4.60) for $h=A_{\Theta}^{-1}D_{\lambda}f_{0k}$, by
(4.64), noting the support property of the first and second terms on the right
hand side of (4.64) and using integration by parts in the third term after
cancelling $A_{\Theta}$, it suffices to consider the term

$h=A_{\Theta}^{-1}\frac{\nu}{k}\Big{[}-\partial_{y}^{3}a\,h+3\partial_{y}(\partial_{y}^{2}a\,h)-3\partial_{y}^{2}(\partial_{y}a\,h)\Big{]},$
(4.65)

which follows from Lemma 3.3.

The desired bounds (4.60) for

$h\in
A_{\Theta}^{-1}\Big{\\{}\frac{\nu}{k}\partial_{y}^{2}a\partial_{y}\omega^{\ast}_{k,\nu},\frac{\nu}{k}\partial_{y}a\partial_{y}^{2}\omega^{\ast}_{k,\nu},\big{[}b^{\prime}(y)a(y,\lambda)-1\big{]}\omega^{\ast}_{k,\nu}\Big{\\}}$
(4.66)

follow from Lemma 3.3 and equation (4.7) noting the support property of
$b^{\prime}a-1$. The desired bounds for

$h\in\Big{\\{}b^{\prime\prime\prime}(y)a(y,\lambda)\psi^{\ast}_{k,\nu},\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}(y,\Lambda),\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}(y,\Lambda)\Big{\\}}$
(4.67)

follow from equation (4.7) and Lemma (3.4), as in (4.28). We omit the lengthy
but straightforward calculations and refer to (4.29)-(4.34) for details.

####  4.4.3. Proof of the bounds for
$D^{\sigma}_{\lambda}\,\psi_{k,\nu}^{\ast},\sigma=2$ in statement (ii) in
Proposition 4.2

Similar calculations show that for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&A_{\Theta}(\partial_{y}^{2}-k^{2})\partial_{\lambda}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)+ib^{\prime\prime}(y)\partial_{\lambda}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&=\partial_{\lambda}f_{0k}^{\ast}(y,\Lambda)+\partial_{\lambda}\Big{\\{}A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda)\Big{\\}}\\\
&\quad-i(\partial_{y}^{2}-k^{2})D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda),\end{split}$
(4.68)

and

$\begin{split}&A_{\Theta}(\partial_{y}^{2}-k^{2})\Big{(}a(y,\lambda)\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\Big{)}+ib^{\prime\prime}(y)a(y,\lambda)\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&=a(y,\lambda)\partial_{y}\Big{\\{}f_{0k}^{\ast}+A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda)\Big{\\}}\\\
&\quad+ia(y,\lambda)b^{\prime}(y)(\partial_{y}^{2}-k^{2})D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)-ia(y,\lambda)b^{\prime\prime\prime}(y)D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&\quad+\Big{[}\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}+2\frac{\nu}{k}\partial_{y}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}^{2}\Big{]}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&\quad+A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda).\end{split}$
(4.69)

Therefore, $D_{\lambda}^{2}\psi^{\ast}_{k,\nu}$ satisfies the equation for
$y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}&(\partial_{y}^{2}-k^{2})D^{2}_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)+A_{\Theta}^{-1}\big{(}ib^{\prime\prime}(\cdot)D^{2}_{\lambda}\psi^{\ast}_{k,\nu}(\cdot,\Lambda)\big{)}(y)\\\
&=A_{\Theta}^{-1}f_{0k}^{\ast\ast}(y,\Lambda)+\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda),\end{split}$
(4.70)

where we have defined for $y\in\mathbb{T}_{\mathfrak{p}}$,

$\begin{split}f_{0k}^{\ast\ast}(y,\Lambda):=&D_{\lambda}\Big{\\{}f_{0k}^{\ast}+A_{\Theta}\Big{[}\partial_{y}^{2}a(y,\lambda)\partial_{y}\psi^{\ast}_{k,\nu}+2\partial_{y}a(y,\lambda)\partial^{2}_{y}\psi^{\ast}_{k,\nu}\Big{]}(y,\Lambda)\Big{\\}}\\\
&+\Big{[}\frac{\nu}{k}\partial_{y}^{2}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}+2\frac{\nu}{k}\partial_{y}a(y,\lambda)(\partial_{y}^{2}-k^{2})\partial_{y}^{2}\Big{]}D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)\\\
&+i\Big{[}a(y,\lambda)b^{\prime}(y)-1\Big{]}(\partial_{y}^{2}-k^{2})D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda)-ia(y,\lambda)b^{\prime\prime\prime}(y)D_{\lambda}\psi^{\ast}_{k,\nu}(y,\Lambda).\end{split}$
(4.71)

Using the bounds (4.4) of the limiting absorption principle and the equation
(4.55)-(4.56), it suffices to show that

$\big{\|}\mathcal{G}_{k}^{j}(\Lambda)h\big{\|}_{X^{1-2(2-\gamma),-\gamma+2}(\mathfrak{M})}\lesssim|k|^{-1/2}(\delta(\Lambda))^{-1-2(2-\gamma)}M_{k},$
(4.72)

where

$h\in\Big{\\{}A_{\Theta}^{-1}\big{(}f_{0k}^{\ast\ast}(\cdot,\Lambda)\big{)}(y),\partial_{y}^{2}a(y,\lambda)\partial_{y}D_{\lambda}\psi^{\ast}_{k,\nu},\partial_{y}a(y,\lambda)\partial^{2}_{y}D_{\lambda}\psi^{\ast}_{k,\nu}\Big{\\}}.$
(4.73)

To prove the bounds (4.72)-(4.73), as in section 4.4.2, we divide we divide
into the singular regions for $y\in\mathbb{T}_{\mathfrak{p}}$ with
$|\lambda-b(y)|\ll|\lambda-b(y_{j\ast})|$ and non-singular regions for
$y\in\mathbb{T}_{\mathfrak{p}}$ with
$|\lambda-b(y)|\gtrsim|\lambda-b(y_{j\ast})|$. More precisely, using the
notation as in Lemma 3.4, it suffices to prove that for $h$ as in (4.73),

$\Big{\|}\mathcal{G}_{k}^{j}(\Lambda)\Big{[}1-\sum_{\dagger\in\\{+,-\\}}\varphi\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}\Big{]}h\Big{\|}_{X^{1-2(2-\gamma),-\gamma+2}(\mathfrak{M})}\lesssim|k|^{-1/2}(\delta(\Lambda))^{-1-2(2-\gamma)}M_{k},$
(4.74)

and for $\dagger\in\\{+,-\\}$,

$\big{\|}\mathcal{G}_{k}^{j}(\Lambda)\varphi\big{(}(y-y^{\dagger}_{\lambda})/|\lambda-b(y_{j\ast})|^{1/2}\big{)}h\big{\|}_{X^{1-2(2-\gamma),-\gamma+2}(\mathfrak{M})}\lesssim|k|^{-1/2}(\delta(\Lambda))^{-1-2(2-\gamma)}M_{k}.$
(4.75)

To establish (4.74), using Lemma (3.3), (4.11) with $\sigma\in\\{0,1\\}$ and
equation (4.7) and (4.55)-(4.56), we obtain the bounds that for
$y\in\mathbb{T}_{\mathfrak{p}},\dagger\in\\{+,-\\}$ and $h$ as in (4.73),