Im}(\varphi_{m,\varepsilon}^{\pm})+\frac{1}{4((y-y_{0})^{2}+\varepsilon^{2})^{2}}\left(((y-y_{0})^{2}-\varepsilon^{2}){\rm
Im}(\varphi_{m,\varepsilon}^{\pm})\mp 2\varepsilon(y-y_{0}){\rm
Re}(\varphi_{m,\varepsilon}^{\pm})\right)={\rm Im}(F_{m,\varepsilon}^{\pm}).$

Cross multiplying the equations by ${\rm Im}(\varphi_{m,\varepsilon}^{\pm})$
and ${\rm Re}(\varphi_{m,\varepsilon}^{\pm})$, respectively, subtracting them
and integrating, we obtain

$\pm\int_{0}^{1}\frac{\varepsilon(y-y_{0})}{2((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}y=\int_{0}^{1}{\rm
Im}(\varphi_{m,\varepsilon}^{\pm}){\rm Re}(F_{m,\varepsilon}^{\pm})-{\rm
Re}(\varphi_{m,\varepsilon}^{\pm}){\rm
Im}(F_{m,\varepsilon}^{\pm})\mathrm{d}y$

so that

$\int_{0}^{1}\frac{\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}y\lesssim\int_{0}^{1}|\varphi_{m,\varepsilon}^{\pm}|^{2}+|F_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}y\lesssim\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\|\rho_{m}^{0}\|^{2}_{H^{2}}.$

The third statement of the proposition follows from the density formula
(2.10), the Hardy-type inequality and the uniform bounds from the first
statement of the proposition. The proof is finished. ∎

From the arguments of the proof, one can directly obtain the following result.

######  Corollary 6.14.

Let $y_{0}\leq 0$ or $y_{0}\geq 1$. Then, $y_{0}+ic$ is not an eigenvalue of
$L_{m}$, for any $c\in{\mathbb{R}}$.

With the $\varepsilon$-uniform $H^{1}_{0}$ for
$\psi_{m,\varepsilon}^{\pm}(y,y_{0})$ at hand we are now able to prove Theorem
4.

######  Proposition 6.15.

We have that

$\lim_{\varepsilon\rightarrow 0}\left\|\int_{-\frac{\beta}{m}}^{0}{\rm
e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon_{*}t}\psi_{m,\varepsilon_{*}}^{-}(y,y_{0})-{\rm
e}^{-m\varepsilon_{*}t}\psi_{m,\varepsilon_{*}}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}=0$

and

$\lim_{\varepsilon\rightarrow 0}\left\|\int_{-\frac{\beta}{m}}^{0}{\rm
e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon_{*}t}\rho_{m,\varepsilon_{*}}^{-}(y,y_{0})-{\rm
e}^{-m\varepsilon_{*}t}\rho_{m,\varepsilon_{*}}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}=0.$

###### Proof.

Let us denote
$\psi_{m,\varepsilon}(y,y_{0})=\psi_{m,\varepsilon}^{+}(y,y_{0})-\psi_{m,\varepsilon}^{-}(y,y_{0})$.
Using (2.9), we have

$\psi_{m,\varepsilon}(y,y_{0})=\frac{2i\varepsilon}{\beta^{2}}{\omega}_{m}^{0}(y)+\varphi_{m,\varepsilon}^{+}(y,y_{0})-\varphi_{m,\varepsilon}^{-}(y,y_{0})$

and we further denote
$\varphi_{m,\varepsilon}(y,y_{0})=\varphi_{m,\varepsilon}^{+}(y,y_{0})-\varphi_{m,\varepsilon}^{-}(y,y_{0})$,
which solves

$\left(\partial_{y}^{2}-m^{2}+\beta^{2}\frac{1}{(y-y_{0}+i\varepsilon)^{2}}\right)\varphi_{m,\varepsilon}(y,y_{0})=\beta^{2}\frac{4i\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}\varphi_{m,\varepsilon}^{-}(y,y_{0})-\frac{2i\varepsilon}{\beta^{2}}{\Delta}_{m}{\omega}_{m}^{0}(y).$

Multiplying by $\overline{\varphi_{m,\varepsilon}(y,y_{0})}$ integrating by
parts and proceeding as before, we see

$(1-4\beta^{2})\int_{0}^{1}|\partial_{y}\varphi_{m,\varepsilon}|^{2}\mathrm{d}y+m^{2}\int_{0}^{1}|\varphi_{m,\varepsilon}|^{2}\mathrm{d}y\lesssim\frac{2\varepsilon}{\beta^{2}}\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\beta^{2}\int_{0}^{1}\frac{4\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}||\varphi_{m,\varepsilon}|\mathrm{d}y.$

Moreover, using Young’s inequality we can bound

$\displaystyle\beta^{2}\int_{0}^{1}\frac{4\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}||\varphi_{m,\varepsilon}|\mathrm{d}y$
$\displaystyle\leq\beta^{2}\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}\left(c_{0}^{2}|\varphi_{m,\varepsilon}|^{2}+\frac{1}{c_{0}^{2}}|\varphi_{m,\varepsilon}^{-}|^{2}\right)\mathrm{d}y$
$\displaystyle\leq
c_{0}^{2}\beta^{2}\int_{0}^{1}\frac{1}{y^{2}}||\varphi_{m,\varepsilon}|^{2}\mathrm{d}y+\frac{\beta^{2}}{c_{0}^{2}}\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}|^{2}\mathrm{d}y$
$\displaystyle\leq
4c_{0}^{2}\beta^{2}\int_{0}^{1}|\partial_{y}\varphi_{m,\varepsilon}|^{2}\mathrm{d}y+\frac{\beta^{2}}{c_{0}^{2}}\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}|^{2}\mathrm{d}y.$

Therefore, absorbing the derivative term in the left hand side for some
$c_{0}$ small enough, we obtain

$\int_{0}^{1}|\partial_{y}\varphi_{m,\varepsilon}|^{2}\mathrm{d}y+m^{2}\int_{0}^{1}|\varphi_{m,\varepsilon}|^{2}\mathrm{d}y\lesssim\varepsilon\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}|^{2}\mathrm{d}y.$
(6.6)

Given the uniform bounds in $\varepsilon>0$ from Proposition 6.13, we have
that

$\displaystyle\lim_{\varepsilon\rightarrow 0}$
$\displaystyle\left\|\int_{-\frac{\beta}{m}}^{0}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon t}\psi_{m,\varepsilon}^{-}(y,y_{0})-{\rm e}^{-m\varepsilon
t}\psi_{m,\varepsilon}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}$
$\displaystyle=\lim_{\varepsilon\rightarrow
0}\left\|\int_{-\frac{\beta}{m}}^{0}{\rm e}^{-imy_{0}t}{\rm e}^{m\varepsilon
t}\big{(}\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}$
$\displaystyle\leq\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left(\frac{2\varepsilon}{\beta}\|{\omega}_{m}^{0}\|_{L^{2}_{y}}+\left\|\varphi_{m,\varepsilon}(y,y_{0})\right\|_{L^{2}_{y}}\right)\mathrm{d}y_{0}$
$\displaystyle=\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left\|\varphi_{m,\varepsilon}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}.$

Now, note that with (6.6) we can estimate

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left\|\varphi_{m,\varepsilon}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}$
$\displaystyle\lesssim\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left(\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{-}|^{2}\mathrm{d}y\right)^{\frac{1}{2}}\mathrm{d}y_{0}$
$\displaystyle\lesssim\lim_{\varepsilon\rightarrow
0}\varepsilon^{\frac{1}{2}}\int_{-\frac{\beta}{m}}^{0}\left(\int_{0}^{1}\frac{1}{(y-y_{0})^{2}}\mathrm{d}y\right)^{\frac{1}{2}}\mathrm{d}y_{0},$

where we have used the pointwise bound
$|\varphi_{m,\varepsilon}^{-}(y,y_{0})|^{2}\lesssim y$, obtained from
Proposition 6.13. The conclusion follows.

For the density statement, we recall that

$\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})=\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}+\frac{2i\varepsilon}{(y-y_{0})^{2}+\varepsilon^{2}}\varphi_{m,\varepsilon}^{+}(y,y_{0}),$

from which, together with Proposition 6.13, we deduce that

$\displaystyle\lim_{\varepsilon\rightarrow 0}$
$\displaystyle\left\|\int_{-\frac{\beta}{m}}^{0}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon_{*}t}\rho_{m,\varepsilon_{*}}^{-}(y,y_{0})-{\rm
e}^{-m\varepsilon_{*}t}\rho_{m,\varepsilon_{*}}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}$
$\displaystyle\leq\lim_{\varepsilon\rightarrow
0}\left\|\int_{-\frac{\beta}{m}}^{0}{\rm e}^{-imy_{0}t}{\rm
e}^{m\varepsilon_{*}t}\big{(}\rho_{m,\varepsilon_{*}}^{-}(y,y_{0})-\rho_{m,\varepsilon_{*}}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}$
$\displaystyle\lesssim\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left(\left\|\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}\right\|_{L^{2}_{y}}+\left\|\frac{2i\varepsilon\varphi_{m,\varepsilon}^{+}(y,y_{0})}{(y-y_{0})^{2}+\varepsilon^{2}}\right\|_{L^{2}_{y}}\right)\mathrm{d}y_{0}$

Using the Hardy inequality (6.4), the estimates from (6.6) and the above
arguments, we have

$\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left\|\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\lesssim\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left\|\partial_{y}\varphi_{m,\varepsilon}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}=0.$

On the other hand, thanks to the bounds from Proposition 6.13, we also have

$\lim_{\varepsilon\rightarrow
0}\int_{-\frac{\beta}{m}}^{0}\left\|\frac{2i\varepsilon\varphi_{m,\varepsilon}^{+}(y,y_{0})}{(y-y_{0})^{2}+\varepsilon^{2}}\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\leq\lim_{\varepsilon\rightarrow
0}\varepsilon^{\frac{1}{2}}\int_{-\frac{\beta}{m}}^{0}\left\|\frac{2\varepsilon^{\frac{1}{2}}(y-y_{0})^{\frac{1}{2}}\varphi_{m,\varepsilon}^{+}(y,y_{0})}{(y-y_{0})^{2}+\varepsilon^{2}}\right\|_{L^{2}_{y}}\frac{1}{(-y_{0})^{\frac{1}{2}}}\mathrm{d}y_{0}=0.$

With this, the proof is finished. ∎

We next show that the contribution from the vertical boundaries of the contour
integral is also negligible.

######  Proposition 6.16.

Let $y_{0}\in\left[-\frac{\beta}{m},0\right]$. We have that

$\lim_{\varepsilon\rightarrow 0}\left\|\int_{0}^{\varepsilon}{\rm
e}^{-imy_{0}t}\big{(}{\rm e}^{mst}\psi_{m,s}^{-}(y,y_{0})+{\rm
e}^{-ms_{*}t}\psi_{m,s}^{+}(y,y_{0})\big{)}\mathrm{d}s\right\|_{L^{2}_{y}}=0$

and

$\lim_{\varepsilon\rightarrow 0}\left\|\int_{0}^{\varepsilon}{\rm
e}^{-imy_{0}t}\big{(}{\rm e}^{mst}\rho_{m,\varepsilon_{*}}^{-}(y,y_{0})+{\rm
e}^{-mst}\rho_{m,s}^{+}(y,y_{0})\big{)}\mathrm{d}s\right\|_{L^{2}_{y}}=0.$

###### Proof.

The statement follows from Minkowski inequality and the fact that

$\lim_{\varepsilon\rightarrow
0}\int_{0}^{\varepsilon}\left\|\psi_{m,s}^{-}(y,y_{0})\right\|_{L^{2}_{y}}+\left\|\psi_{m,s}^{+}(y,y_{0})\right\|_{L^{2}_{y}}+\left\|\rho_{m,s}^{-}(y,y_{0})\right\|_{L^{2}_{y}}+\left\|\rho_{m,s}^{+}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}s=0,$

due to the uniform bounds in $s\in[0,\varepsilon]$ of these quantities from
Proposition 6.13. ∎

We are now in position to carry out the proof of Proposition 6.1 for the case
$\beta^{2}<1/4$.

###### Proof of Proposition 6.1 .

Since the resolvent operator $\mathcal{R}(c,L_{m})$ is invertible for all
$c\in{\mathbb{C}}$ with ${\rm Re}(c)\leq 0$ and $|c|\geq c_{0}$ for some
$c_{0}>0$, confer Proposition 6.13, we can reduce the contour integral to the
boundary of the set
$R_{\varepsilon}:=\left\\{c=y_{0}+is\in{\mathbb{C}}:y_{0}\in\left[-\beta/m,0\right],\,s\in[-\varepsilon,\varepsilon]\right\\}$.
Now, Proposition 6.15 and Proposition 6.16 show that the integral along
$\partial R_{\varepsilon}$ is negligible as $\varepsilon\rightarrow 0$. The
Proposition follows. ∎

We finish the subsection with the proof of Theorem 4 for $b^{2}<1/4$, which is
a direct consequence of the following Lemma.

######  Lemma 6.17.

Let $y,z\in[0,1]$. There exists $\varepsilon_{0}>0$ such that

$\sup_{y,z\in[0,1]}\left|{\mathcal{G}}_{m,\varepsilon}^{-}(y,y_{0},z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,y_{0},z)\right|\lesssim\varepsilon^{2\mu}+\varepsilon^{\frac{1}{2}},\qquad
y_{0}\in\\{0,1\\},$

for all $\varepsilon\leq\varepsilon_{0}$.

###### Proof.

We take $y_{0}=0$, the other case is analogous. The argument is similar to the
one presented for Lemma 6.11. As before, we need to understand

${\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,0,z)=2i{\rm
Im}\left({\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)\right)$

For $y\leq z$, we have

$\displaystyle 2i{\rm
Im}\left({\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)\right)=\frac{2i}{|{{\mathcal{W}}_{m,\varepsilon}^{-}}(0)|^{2}}{\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right).$

Due to the explicit solutions of the Taylor-Goldstein equation, we can find
that

$\displaystyle{\phi_{l,m,\varepsilon}^{-}(y,0)}$
$\displaystyle{{\mathcal{W}}_{m,\varepsilon}^{+}(0)}$ $\displaystyle=4\mu
m\Big{(}M_{+}(-i\varepsilon)M_{-}(y)M_{+}(i\varepsilon)M_{-}(1)+M_{-}(-i\varepsilon)M_{+}(y)M_{-}(i\varepsilon)M_{+}(1)\Big{.}$
$\displaystyle\qquad\qquad-\Big{.}M_{+}(-i\varepsilon)M_{-}(y)M_{-}(i\varepsilon)M_{+}(1)-\Big{.}M_{-}(-i\varepsilon)M_{+}(y)M_{+}(i\varepsilon)M_{-}(1)+R_{1}(\varepsilon)\Big{)},$

where
$|R_{1}(\varepsilon)|\lesssim_{m,\mu}\varepsilon^{\frac{1}{2}-\mu}|M_{-}(i\varepsilon)|^{2}$.
Moreover, since now $\overline{M_{\pm}(\zeta)}=M_{\pm}(\overline{\zeta})$ for
all $\zeta\in{\mathbb{C}}$, we can write

$\displaystyle{\phi_{l,m,\varepsilon}^{-}(y,0)}{{\mathcal{W}}_{m,\varepsilon}^{+}(0)}$
$\displaystyle=4\mu
m\Big{(}|M_{+}(i\varepsilon)|^{2}M_{-}(y)M_{-}(1)+|M_{-}(i\varepsilon)|^{2}M_{+}(y)M_{+}(1)\Big{.}$
$\displaystyle\quad-\Big{.}M_{+}(-i\varepsilon)M_{-}(i\varepsilon)M_{-}(y)M_{+}(1)-M_{+}(i\varepsilon)M_{-}(-i\varepsilon)M_{+}(y)M_{-}(1)+R_{1}(\varepsilon)\Big{)}.$

On the other hand,

$\displaystyle\phi_{u,m,\varepsilon}^{-}(z,0)$
$\displaystyle=M_{+}(1)M_{-}(z)-M_{-}(1)M_{+}(z)+R_{2}(\varepsilon)=\phi_{u}(z)+R_{2}(\varepsilon)$

with $|R_{2}(\varepsilon)|\lesssim\varepsilon^{\frac{1}{2}-\mu}$. Thus,

$\displaystyle\phi_{l,m,\varepsilon}^{-}(y,0)$
$\displaystyle\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)$
$\displaystyle=-4\mu
m\phi_{u}(z)\Big{(}M_{+}(-i\varepsilon)M_{-}(i\varepsilon)M_{-}(y)M_{+}(1)+M_{+}(i\varepsilon)M_{-}(-i\varepsilon)M_{+}(y)M_{-}(1)\Big{)}$
$\displaystyle\quad+4\mu
m\phi_{u}(z)\Big{(}|M_{+}(i\varepsilon)|^{2}M_{-}(y)M_{-}(1)+|M_{-}(i\varepsilon)|^{2}M_{+}(y)M_{+}(1)\Big{)}+R_{3}(\varepsilon),$

where $|R_{3}(\varepsilon)|\lesssim\mu
m\varepsilon^{\frac{1}{2}-\mu}|M_{-}(i\varepsilon)|^{2}$, uniformly in
$y,z\in[0,1]$. In particular, since $M_{\pm}(y)\in{\mathbb{R}}$, for all
$y\in[0,1]$, we have

${\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)=-4\mu
m\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(M_{+}(-i\varepsilon)M_{-}(i\varepsilon)\right)+{\rm
Im}\left(R_{3}(\varepsilon)\right).$

Moreover, due to the symmetry of the Green’s function with respect to $y$ and
$z$, we also have that

${\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(z,0)\phi_{u,m,\varepsilon}^{-}(y,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)=-4\mu
m\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(M_{+}(-i\varepsilon)M_{-}(i\varepsilon)\right)+{\rm
Im}\left(\widetilde{R}_{3}(\varepsilon)\right).$

For the Wronskian, we have from (3.6) that

$\displaystyle\left|{\mathcal{W}}_{m,\varepsilon}^{+}(0)\right|=4\mu
m|M_{-}(i\varepsilon)||M_{+}(1)|\left|1-\frac{M_{+}(i\varepsilon)}{M_{-}(i\varepsilon)}\frac{M_{-}(1)}{M_{+}(1)}+R_{4}(\varepsilon)\right|.$

where $|R_{4}(\varepsilon)|\lesssim\varepsilon^{\frac{1}{2}-\mu}$. In
particular, for $\varepsilon\leq\varepsilon_{0}$ small enough we have from
Lemma A.16 that

$\displaystyle\left|{\mathcal{W}}_{m,\varepsilon}^{+}(0)\right|\geq 2\mu
m|M_{-}(i\varepsilon)||M_{+}(1)|.$

Therefore,

$\displaystyle\left|{\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,0,z)\right|$
$\displaystyle=\frac{2}{|{\mathcal{W}}_{m,\varepsilon}^{+}(0)|^{2}}\left|{\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)\right|$
$\displaystyle\leq\frac{2}{\mu m}\frac{\left|\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(M_{+}(-i\varepsilon)M_{-}(i\varepsilon)\right)\right|}{|M_{-}(i\varepsilon)|^{2}|M_{+}(1)|^{2}}+R_{5}(\varepsilon)$
$\displaystyle\lesssim\varepsilon^{2\mu}+\varepsilon^{\frac{1}{2}-\mu},$

and the lemma follows. ∎

###  6.3. Integral reduction for $\beta^{2}=1/4$

The special case in which $\beta^{2}=1/4$ is critical in the sense that the
Hardy inequality (6.4) may saturate and thus the derivative bounds in
Proposition 6.13 are no longer uniform in $\varepsilon>0$. Still, we are able
to prove the following result.

######  Proposition 6.18.

Let $y_{0}\leq 0$ and $0<\varepsilon\leq 1$. Then,

$\frac{\varepsilon^{2}}{1+\varepsilon^{2}}\|\partial_{y}\psi_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}}^{2}+m^{2}\|\psi_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}}^{2}\lesssim\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\|\rho_{m}^{0}\|_{H^{2}}^{2}.$

Moreover,

$\displaystyle\int_{0}^{1}\frac{\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}|\varphi_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}y\lesssim\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\|\rho_{m}^{0}\|_{H^{2}}^{2}.$

If we further assume that $|-y_{0}\pm i\varepsilon|\geq c_{0}$, for some
$c_{0}>0$, then

$\frac{c_{0}^{2}}{1+c_{0}^{2}}\|\partial_{y}\psi_{m,\varepsilon}^{\pm}(y,y_{0})\|_{L^{2}}^{2}+m^{2}\|\psi_{m,\varepsilon}^{\pm}(y,y_{0})\|_{L^{2}}^{2}\lesssim\frac{1}{c_{0}^{2}}\|{\omega}_{m}^{0}\|_{L^{2}}^{2}+\frac{1}{c_{0}^{4}}\|\rho_{m}^{0}\|_{L^{2}}^{2}.$

In particular, $c=-y_{0}\pm i\varepsilon$ belongs to the resolvent set of
$L_{m}$.

###### Proof.

The proof is similar to the one for Proposition 6.13. Here, since
$\beta^{2}=1/4$ we estimate

$\frac{1}{4}\int_{0}^{1}\frac{|\psi_{m,\varepsilon}^{\pm}|^{2}}{|y-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}y\leq\frac{1}{1+c_{0}^{2}}\int_{0}^{1}\frac{|\psi_{m,\varepsilon}^{\pm}|^{2}}{4y^{2}}\mathrm{d}y\leq\frac{1}{1+c_{0}^{2}}\int_{0}^{1}|\partial_{y}\psi_{m,\varepsilon}|^{2}\mathrm{d}y,$

which can be absorbed by
$\int_{0}^{1}|\partial_{y}\psi_{m,\varepsilon}|^{2}\mathrm{d}y$, thus
producing the desired $H^{1}$ estimates. ∎

The estimate on the $L^{2}$ norm of the derivative degenerates as
$\varepsilon$ becomes small. We may lose pointwise bounds on the solution, and
for this reason we investigate more thoroughly the Green’s function
${\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)$ when $-1\ll y_{0}\leq 0$. In
particular, we have that

######  Proposition 6.19.

Let $y,z\in[0,1]$. There exists $\delta>0$ such that

$|{\mathcal{G}}_{m,\varepsilon}^{+}(y,y_{0},z)|\lesssim|y-y_{0}+i\varepsilon|^{\frac{1}{2}}|z-y_{0}+i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log
m|y-y_{0}+i\varepsilon|\big{|}\right)\left(1+\big{|}\log
m|z-y_{0}+i\varepsilon|\big{|}\right)$

and

$|\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{+}(y,y_{0},z)|\lesssim|y-y_{0}+i\varepsilon|^{-\frac{1}{2}}|z-y_{0}+i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log
m|y-y_{0}+i\varepsilon|\big{|}\right)\left(1+\big{|}\log
m|z-y_{0}+i\varepsilon|\big{|}\right)$

for all $y_{0}<0$ and $\varepsilon>0$ with $|-y_{0}\pm
i\varepsilon|\leq\delta$.

We remark that the hidden implicit constant may depend on $m$, but for our
purposes this is unimportant.

###### Proof.

The proof follows the same steps as the one for Proposition 5.1. We shall
obtain suitable estimates on the Wrosnkian. Now, since $y_{0}<0$, we recall

${\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0}):=\frac{2m}{\sqrt{\pi}}\Big{(}W_{0}(-y_{0}\pm
i\varepsilon)M_{0}(1-y_{0}\pm i\varepsilon)-M_{0}(-y_{0}\pm
i\varepsilon)W_{0}(1-y_{0}\pm i\varepsilon)\Big{)}.$

Using Lemma A.11 and Lemma A.12, there exists $C>0$ and $\delta>0$ such that

$\left|\frac{W_{0}(1-y_{0}\pm i\varepsilon)}{M_{0}(1-y_{0}\pm
i\varepsilon)}\right|\leq C,\quad\left|\frac{M_{0}(-y_{0}\pm
i\varepsilon)}{W_{0}(-y_{0}\pm i\varepsilon)}\right|\leq\frac{1}{2C},$

for all $|-y_{0}\pm i\varepsilon|\leq\delta$. Hence, we can lower bound

$\left|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})\right|\geq\frac{m}{\sqrt{\pi}}\left|W_{0}(-y_{0}\pm
i\varepsilon)\right|\left|M_{0}(1-y_{0}\pm i\varepsilon)\right|$

and the proposition follows from the asymptotic expansions of the homogeneous
solutions that conform the Green’s function. ∎

With the above asymptotics at hand, we are now able to prove the following
result.

######  Proposition 6.20.

Let $\delta>0$ be given by Proposition 6.19 and let $y_{0}<0$ such that
$|y_{0}|\leq\frac{\delta}{2}$. We have that

$\left\|\int_{-\frac{\delta}{2}}^{0}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon t}\psi_{m,\varepsilon}^{-}(y,y_{0})-{\rm e}^{-m\varepsilon
t}\psi_{m,\varepsilon}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}\lesssim\varepsilon^{\frac{1}{2}}$

and also

$\left\|\int_{-\frac{\delta}{2}}^{0}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{m\varepsilon t}\rho_{m,\varepsilon}^{-}(y,y_{0})-{\rm e}^{-m\varepsilon
t}\rho_{m,\varepsilon}^{+}(y,y_{0})\big{)}\mathrm{d}y_{0}\right\|_{L^{2}_{y}}\lesssim\varepsilon^{\frac{1}{2}},$

for all $\varepsilon>0$ such that $|-y_{0}+i\varepsilon|\leq\delta$.

###### Proof.

Following the same strategy as in the proof of Proposition 6.15, we see that
$\varphi_{m,\varepsilon}(y,y_{0})$ satisfies

$m^{2}\|\varphi_{m,\varepsilon}\|_{L^{2}}^{2}\lesssim\varepsilon\|{\omega}_{m}^{0}\|_{H^{2}}^{2}+\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}\left(|\varphi_{m,\varepsilon}^{-}|^{2}+|\varphi_{m,\varepsilon}^{+}|^{2}\right)\mathrm{d}y.$

In particular, using the asymptotic bounds from Proposition 6.19 we can
estimate

$\displaystyle\int_{0}^{1}\frac{2\varepsilon(y-y_{0})}{((y-y_{0})^{2}+\varepsilon^{2})^{2}}\left(|\varphi_{m,\varepsilon}^{-}|^{2}+|\varphi_{m,\varepsilon}^{+}|^{2}\right)\mathrm{d}y$
$\displaystyle\lesssim\int_{0}^{1}\frac{\varepsilon(y-y_{0})}{|y-y_{0}+i\varepsilon|^{3}}\left(1+\left|\log|y-y_{0}+i\varepsilon|\right|\right)^{2}\mathrm{d}y$
$\displaystyle\lesssim\int_{0}^{1}\frac{\varepsilon(y-y_{0})}{|y-y_{0}+i\varepsilon|^{\frac{7}{2}}}\mathrm{d}y$
$\displaystyle\lesssim\varepsilon\int_{0}^{1}\frac{1}{(y-y_{0})^{\frac{5}{2}}}\mathrm{d}y$
$\displaystyle\lesssim\varepsilon\left(1+(-y_{0})^{-\frac{3}{2}}\right).$

We conclude the first part of the proof upon noting that

$\int_{-\frac{\delta}{2}}^{0}\left\|\varphi_{m,\varepsilon}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\lesssim\varepsilon^{\frac{1}{2}}\|{\omega}_{m}^{0}\|_{H^{2}}+\varepsilon^{\frac{1}{2}}\int_{-\frac{\delta}{2}}^{0}\left(1+(-y_{0})^{-\frac{3}{2}}\right)^{\frac{1}{2}}\mathrm{d}y_{0}\lesssim\varepsilon^{\frac{1}{2}}.$

For the second part of the proposition, from (2.10) we have

$\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})=\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}+\frac{2i\varepsilon}{(y-y_{0})^{2}+\varepsilon^{2}}\varphi_{m,\varepsilon}^{+}(y,y_{0})$

and we write

$\varphi_{m,\varepsilon}(y,y_{0})=\int_{0}^{1}{\mathcal{G}}_{m,\varepsilon}^{-}(y,y_{0},z)\left(\frac{i\varepsilon(z-y_{0})}{((z-y_{0})^{2}+\varepsilon^{2})^{2}}\varphi_{m,\varepsilon}^{-}(z,y_{0})-8i\varepsilon{\Delta}_{m}{\omega}_{m}^{0}(z)\right)\mathrm{d}z.$

In particular, using Proposition 6.18 and Proposition 6.19 we estimate

$\displaystyle\left|\int_{0}^{1}{\mathcal{G}}_{m,\varepsilon}^{-}(y,y_{0},z)\frac{i\varepsilon(z-y_{0})}{((z-y_{0})^{2}+\varepsilon^{2})^{2}}\varphi_{m,\varepsilon}^{-}(z,y_{0})\mathrm{d}z\right|$
$\displaystyle\qquad\lesssim\varepsilon^{\frac{1}{2}}\left(\int_{0}^{1}\frac{(z-y_{0})}{((z-y_{0})^{2}+\varepsilon^{2})^{2}}|{\mathcal{G}}_{m,\varepsilon}^{-}(y,y_{0},z)|^{2}\mathrm{d}z\right)^{\frac{1}{2}}$
$\displaystyle\qquad\lesssim\varepsilon^{\frac{1}{2}}|y-y_{0}-i\varepsilon|^{\frac{1}{2}}(1+\left|\log|y-y_{0}-i\varepsilon|\right|)\left(\int_{0}^{1}\frac{(z-y_{0})}{|z-y_{0}-i\varepsilon|^{3}}(1+\left|\log|z-y_{0}-i\varepsilon|\right|)^{2}\mathrm{d}z\right)^{\frac{1}{2}}$
$\displaystyle\qquad\lesssim\varepsilon^{\frac{1}{2}}|y-y_{0}-i\varepsilon|^{\frac{1}{2}}(1+\left|\log|y-y_{0}-i\varepsilon|\right|)\left(\int_{0}^{1}\frac{1}{(z-y_{0})^{2+\frac{1}{4}}}\mathrm{d}z\right)^{\frac{1}{2}}$
$\displaystyle\qquad\lesssim\varepsilon^{\frac{1}{2}}|y-y_{0}-i\varepsilon|^{\frac{1}{2}}(1+\left|\log|y-y_{0}-i\varepsilon|\right|)\left(1+(-y_{0})^{-\frac{1}{2}-\frac{1}{8}}\right).$

With this pointwise bound, we obtain

$\displaystyle\left\|\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}\right\|_{L^{2}_{y}}$
$\displaystyle\lesssim\varepsilon^{\frac{1}{2}}\left(1+(-y_{0})^{-\frac{1}{2}-\frac{1}{8}}\right)\left(\int_{0}^{1}|y-y_{0}-i\varepsilon|^{-1}(1+\left|\log|y-y_{0}-i\varepsilon|\right|)^{2}\mathrm{d}y\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\varepsilon^{\frac{1}{2}}\left(1+(-y_{0})^{-\frac{1}{2}-\frac{1}{8}}\right)\left(\int_{0}^{1}|y-y_{0}-i\varepsilon|^{-1-\frac{1}{4}}\mathrm{d}y\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\varepsilon^{\frac{1}{2}}\left(1+(-y_{0})^{-\frac{3}{4}}\right)$

and thus

$\int_{-\frac{\delta}{2}}^{0}\left\|\frac{\varphi_{m,\varepsilon}(y,y_{0})}{y-y_{0}-i\varepsilon}\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\lesssim\varepsilon^{\frac{1}{2}}.$

On the other hand, from the bounds obtained in Proposition 6.18, we have

$\int_{-\frac{\delta}{2}}^{0}\left\|\frac{2i\varepsilon}{(y-y_{0})^{2}+\varepsilon^{2}}\varphi_{m,\varepsilon}^{+}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\lesssim\varepsilon^{\frac{1}{2}}\int_{-\frac{\delta}{2}}^{0}\frac{1}{(-y_{0})^{\frac{1}{2}}}\left\|\frac{\varepsilon^{\frac{1}{2}}(y-y_{0})^{\frac{1}{2}}}{(y-y_{0})^{2}+\varepsilon^{2}}\varphi_{m,\varepsilon}^{+}(y,y_{0})\right\|_{L^{2}_{y}}\mathrm{d}y_{0}\lesssim\varepsilon^{\frac{1}{2}},$

and the proof is concluded. ∎

Similarly, the contribution from the resolvent integral along the vertical
boundaries of the contour is also negligible.

######  Proposition 6.21.

Let $y_{0}\in\left[-\beta/m,0\right]$. We have that

$\left\|\int_{0}^{\varepsilon}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{mst}\psi_{m,s}^{-}(y,y_{0})+{\rm
e}^{-ms_{*}t}\psi_{m,s}^{+}(y,y_{0})\big{)}\mathrm{d}s\right\|_{L^{2}_{y}}\lesssim\varepsilon$

and

$\left\|\int_{0}^{\varepsilon}{\rm e}^{-imy_{0}t}\big{(}{\rm
e}^{mst}\rho_{m,s}^{-}(y,y_{0})+{\rm
e}^{-mst}\rho_{m,s}^{+}(y,y_{0})\big{)}\mathrm{d}s\right\|_{L^{2}_{y}}\lesssim\varepsilon^{\frac{1}{4}}.$

###### Proof.

The first part concerning the stream-functions
$\psi_{m,\varepsilon}^{\pm}(y,y_{0})$ is a direct consequence of the uniform
$L^{2}$ bounds of $\psi_{m,\varepsilon}^{\pm}(y,y_{0})$ obtained in
Proposition 6.18. As for the density statement, we use (2.10); thanks to the
asymptotic bounds from Proposition 6.19 we further observe that

$\displaystyle\int_{0}^{\varepsilon}\left\|\frac{\varphi_{m,\varepsilon}^{\pm}(y,y_{0})}{y-y_{0}\pm
is}\right\|_{L^{2}_{y}}\mathrm{d}s$
$\displaystyle\lesssim\int_{0}^{\varepsilon}\left(\int_{0}^{1}|y-y_{0}\pm
is|^{-1}(1+\left|\log|y-y_{0}-i\varepsilon|\right|)^{2}\mathrm{d}y\right)^{\frac{1}{2}}\mathrm{d}s$
$\displaystyle\lesssim\int_{0}^{\varepsilon}\left(\int_{0}^{1}|y-y_{0}\pm
is|^{-\frac{3}{2}}\mathrm{d}y\right)^{\frac{1}{2}}\mathrm{d}s\lesssim\int_{0}^{\varepsilon}|s|^{-\frac{3}{4}}\mathrm{d}s.$

With the above estimate, the bound follows swiftly. ∎

We are now in position to prove Proposition 6.1 for the special case
$\beta^{2}=1/4$.

###### Proof of Proposition 6.1.

Let $\delta>0$ be given by Proposition 6.19. For all
$\varepsilon<\frac{\delta}{2}$, we introduce the rectangular region
$R_{\varepsilon}:=\left\\{c=y_{0}+is\in{\mathbb{C}}:y_{0}\in\left[-\delta/2,0\right],\,s\in[-\varepsilon,\varepsilon]\right\\}$.
From Proposition 6.20 and Proposition 6.21 we conclude that

$\left\|\int_{\partial R_{\varepsilon}}{\rm
e}^{-imct}\mathcal{R}(c,L_{m})\mathrm{d}c\right\|_{L^{2}_{y}}\lesssim\varepsilon^{\frac{1}{4}},\qquad\left\|\int_{\partial(R\setminus
R_{\varepsilon})}{\rm
e}^{-imct}\mathcal{R}(c,L_{m})\mathrm{d}c\right\|_{L^{2}_{y}}=0,$

because any $c\in R\setminus R_{\varepsilon}$ belongs to the resolvent set of
the operator $L_{m}$. Indeed, any $c\in R\setminus R_{\varepsilon}$ is such
that ${\rm Re}(c)\leq 0$, $|c|\geq\frac{\delta}{2}$, and we can see from
Proposition 6.18 that $\mathcal{R}(c,L_{m})$ is invertible. ∎

Finally, in order to prove Theorem 4 for $\beta^{2}=1/4$, we state and prove
the following key Lemma, from which the Theorem easily follows.

######  Lemma 6.22.

Let $y_{0}=0$ and $y,z\in[0,1]$. Then, there exists $\varepsilon_{0}>0$ such
that

$\sup_{y,z\in[0,1]}\left|{\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,0,z)\right|\lesssim\frac{1}{\log\left(\frac{4}{\varepsilon}\right)}+\varepsilon^{\frac{1}{4}},$

for all $\varepsilon\leq\varepsilon_{0}$.

###### Proof.

We have
${\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,0,z)=2i{\rm
Im}\left({\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)\right)$ and for $y\leq z$,

$2i{\rm
Im}\left({\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)\right)=\frac{2i}{|{{\mathcal{W}}_{m,\varepsilon}^{-}}(0)|^{2}}{\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right).$

Now, using Proposition 3.3, Lemma A.4 and Lemma A.11,

$\displaystyle{\phi_{l,m,\varepsilon}^{-}(y,0)}$
$\displaystyle{{\mathcal{W}}_{m,\varepsilon}^{+}(0)}$
$\displaystyle=\frac{2m}{\sqrt{\pi}}\Big{(}|W_{0}(i\varepsilon)|^{2}M_{0}(y)M_{0}(1)+|M_{0}(i\varepsilon)|^{2}W_{0}(y)W_{0}(1)\Big{.}$
$\displaystyle\qquad\qquad-\Big{.}W_{0}(i\varepsilon)M_{0}(-i\varepsilon)W_{0}(y)M_{0}(1)-W_{0}(-i\varepsilon)M_{0}(i\varepsilon)M_{0}(y)W_{0}(1)+R_{1}(\varepsilon)\Big{)},$

where
$|R_{1}(\varepsilon)|\lesssim_{m,\mu}\varepsilon^{\frac{1}{4}}|W_{0}(i\varepsilon)|^{2}$.
Similarly,

$\displaystyle\phi_{u,m,\varepsilon}^{-}(z,0)$
$\displaystyle=W_{0}(1)M_{0}(z)-M_{0}(1)W_{0}(z)+R_{2}(\varepsilon)=:\phi_{u,m}(z)+R_{2}(\varepsilon)$

with $|R_{2}(\varepsilon)|\lesssim\varepsilon^{\frac{1}{4}}$. Thus,

$\displaystyle\phi_{l,m,\varepsilon}^{-}(y,0)$
$\displaystyle\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)$
$\displaystyle=-\frac{2m}{\sqrt{\pi}}\phi_{u,m}(z)\Big{(}W_{0}(-i\varepsilon)M_{0}(i\varepsilon)M_{0}(y)W_{0}(1)+W_{0}(i\varepsilon)M_{0}(-i\varepsilon)W_{0}(y)M_{0}(1)\Big{)}$
$\displaystyle\quad+\frac{2m}{\sqrt{\pi}}\phi_{u,m}(z)\Big{(}|W_{0}(i\varepsilon)|^{2}M_{0}(y)M_{0}(1)+|M_{0}(i\varepsilon)|^{2}W_{0}(y)W_{0}(1)\Big{)}+R_{3}(\varepsilon),$

where $|R_{3}(\varepsilon)|\lesssim
m\varepsilon^{\frac{1}{4}}|W_{0}(i\varepsilon)|^{2}$, uniformly in
$y,z\in[0,1]$. In particular, since $M_{0}(y)\in{\mathbb{R}}$ and
$W_{0}(y)\in{\mathbb{R}}$, for all $y\in[0,1]$, we have

${\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)=-\frac{2m}{\sqrt{\pi}}\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(W_{0}(-i\varepsilon)M_{0}(i\varepsilon)\right)+{\rm
Im}\left(R_{3}(\varepsilon)\right).$

Due to symmetry, we also have

${\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(z,0)\phi_{u,m,\varepsilon}^{-}(y,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)=-\frac{2m}{\sqrt{\pi}}\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(W_{0}(-i\varepsilon)M_{0}(i\varepsilon)\right)+{\rm
Im}\left(\widetilde{R}_{3}(\varepsilon)\right).$

For the Wronskian, we have from (3.14) that

$\displaystyle\left|{\mathcal{W}}_{m,\varepsilon}^{+}(0)\right|=\frac{2m}{\sqrt{\pi}}|W_{0}(i\varepsilon)||M_{0}(1)|\left|1-\frac{M_{0}(i\varepsilon)}{W_{0}(i\varepsilon)}\frac{W_{0}(1)}{M_{0}(1)}+R_{4}(\varepsilon)\right|.$

where $|R_{4}(\varepsilon)|\lesssim\varepsilon$. In particular, for
$\varepsilon\leq\varepsilon_{0}$ small enough we have from Lemma A.11 that

$\displaystyle\left|{\mathcal{W}}_{m,\varepsilon}^{+}(0)\right|\geq\frac{m}{\sqrt{\pi}}|W_{0}(i\varepsilon)||M_{0}(1)|.$

Therefore,

$\displaystyle\left|{\mathcal{G}}_{m,\varepsilon}^{-}(y,0,z)-{\mathcal{G}}_{m,\varepsilon}^{+}(y,0,z)\right|$
$\displaystyle=\frac{2}{|{\mathcal{W}}_{m,\varepsilon}^{+}(0)|^{2}}\left|{\rm
Im}\left(\phi_{l,m,\varepsilon}^{-}(y,0)\phi_{u,m,\varepsilon}^{-}(z,0){\mathcal{W}}_{m,\varepsilon}^{+}(0)\right)\right|$
$\displaystyle\leq\frac{4\sqrt{\pi}}{m}\frac{\left|\phi_{u,m}(z)\phi_{u,m}(y){\rm
Im}\left(W_{0}(-i\varepsilon)M_{0}(i\varepsilon)\right)\right|}{|W_{0}(i\varepsilon)|^{2}|M_{0}(1)|^{2}}+R_{5}(\varepsilon)$
$\displaystyle\lesssim\left|\frac{M_{0}(i\varepsilon)}{W_{0}(i\varepsilon)}\right|+\varepsilon^{\frac{1}{4}},$

and the conclusion follows from Lemma A.11. ∎

##  7\. Bounds on solutions to the inhomogeneous Taylor-Goldstein equation

This section provides bounds for solutions $\Phi_{m,\varepsilon}$ to the
inhomogeneous Taylor-Goldstein equation (TGf) with boundary conditions
$\Phi_{m,\varepsilon}(0,y_{0})=\Phi_{m,\varepsilon}(1,y_{0})=0$. The following
lemma relates regions of the interval $(0,1)$ that are far away from a fixed
$y_{0}\in[0,1]$ to nearby regions of $y_{0}$.

######  Lemma 7.1.

Let $y_{0}\in[0,1]$, $n\geq 1$ and $\Phi_{m,\varepsilon}$ be the solution to
(TGf). Then, we have that

$\|\partial_{y}\Phi_{m,\varepsilon}\|_{L^{2}_{y}(J_{3}^{c})}^{2}+m^{2}\|\Phi_{m,\varepsilon}\|_{L^{2}_{y}(J_{3}^{c})}^{2}\lesssim
m^{2}\|\Phi_{m,\varepsilon}\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}^{2}+\frac{1}{m^{2}}\|f\|_{L^{2}_{y}(J_{2}^{c})}^{2}.$

###### Proof.

For $y_{n}=y_{0}+\frac{n\beta}{m}$, the lemma follows from the energy
inequality

$\frac{1}{2}\int_{y_{3}}^{1}\left[|\partial_{y}\Phi_{m,\varepsilon}|^{2}+m^{2}|\Phi_{m,\varepsilon}|^{2}\right]\mathrm{d}y\leq\frac{m^{2}}{\beta^{2}}\int_{y_{2}}^{y_{3}}|\Phi_{m,\varepsilon}|^{2}\mathrm{d}y+\int_{y_{2}}^{1}|f||\Phi_{m,\varepsilon}|\mathrm{d}y,$

and Young’s inequality to absorb the potential term. We omit the details. ∎

With the above lemma we are in position to provide bounds on the solution to
(TGf).

######  Proposition 7.2.

Let $\Phi_{m,\varepsilon}$ be the solution to (TGf). Then

  * •

If $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}\neq 1/4$, then

$|y-y_{0}+i\varepsilon|^{-\frac{1}{2}+\mu}|\Phi_{m,\varepsilon}(y,y_{0})|+|y-y_{0}+i\varepsilon|^{\frac{1}{2}+\mu}|\partial_{y}\Phi_{m,\varepsilon}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}\|f\|_{L^{2}_{y}}.$

  * •

If $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}=1/4$, then

$|y-y_{0}+i\varepsilon|^{-\frac{1}{2}}|\Phi_{m,\varepsilon}(y,y_{0})|+|y-y_{0}+i\varepsilon|^{\frac{1}{2}}|\partial_{y}\Phi_{m,\varepsilon}(y,y_{0})|\lesssim\frac{1}{m}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)\|f\|_{L^{2}_{y}}.$

  * •

If $m|y-y_{0}|\geq 3\beta$ then

$m\|\Phi_{m,\varepsilon}(y,y_{0})\|_{L^{2}_{y}(J_{3}^{c})}+\|\partial_{y}\Phi_{m,\varepsilon}(y,y_{0})\|_{L^{2}_{y}(J_{3}^{c})}\lesssim\frac{1}{m}\|f\|_{L^{2}_{y}}$

and

$|\partial_{y}\Phi_{m,\varepsilon}(y,y_{0})|\lesssim\|f\|_{L^{2}_{y}}.$

###### Proof.

The first part is a straightforward application of the bounds on the Green’s
function from Theorem 2 and the Cauchy-Schwartz inequality, once we write
$\Phi_{m,\varepsilon}(y,y_{0})=\int_{0}^{1}{\mathcal{G}}_{m,\varepsilon}^{+}(y,y_{0},z)f(z,y_{0})\mathrm{d}z$.
The second part of the proposition follows from the first part, which gives
$m\|\Phi_{m,\varepsilon}\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim\frac{1}{m}\|f\|_{L^{2}_{y}}$ and Lemma 7.1. For the pointwise
bound, assume without loss of generality that $y_{0}+\frac{3\beta}{m}<y\leq
1$. Then, let $y_{3}=y_{0}+\frac{3\beta}{m}$ and write

$\partial_{y}\Phi_{m,\varepsilon}(y,y_{0})=\partial_{y}\Phi_{m,\varepsilon}(y_{3},y_{0})+\int_{y_{3}}^{y}\left[\left(m^{2}-\beta^{2}\frac{1}{(y^{\prime}-y_{0}+i\varepsilon)^{2}}\right)\Phi_{m,\varepsilon}(y^{\prime},y_{0})+f(y^{\prime})\right]\mathrm{d}y^{\prime}.$

Now, $|y_{3}-y_{0}|=\frac{3\beta}{m}$ so that we estimate
$|\partial_{y}\Phi_{m,\varepsilon}(y_{3},y_{0})|\lesssim\frac{1}{m^{1+\mu}}\left|\frac{\beta}{m}\right|^{-\frac{1}{2}-\mu}\lesssim
m^{-\frac{1}{2}}$. Similarly, we use the second part of the proposition to
estimate the remaining terms in $L^{2}_{y}(J_{3}^{c})$ and obtain the desired
conclusion. ∎

##  8\. Boundary terms estimates

The purpose of this section is to obtain estimates on the boundary terms that
appear in the expressions for
$\partial_{y_{0}}\psi_{m,\varepsilon}^{\pm}(y,y_{0})$ and other related
derivatives. We begin by recording the following results, which will be used
throughout the entire section.

######  Proposition 8.1.

Let $\beta^{2}\neq 1/4$. There exists $\varepsilon_{0}>0$ such that for all
$y,y_{0}\in[0,1]$ with $m|y-y_{0}|\leq 3\beta$ there holds

$|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}-\mu}|\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|+|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}+\mu}|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{\frac{1}{2}-\mu}\frac{1}{|M_{-}(y_{0}\mp i\varepsilon))|},$

and

$|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}-\mu}|\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},1)|+|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}+\mu}|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},1)|\lesssim
m^{\frac{1}{2}-\mu}\frac{1}{|M_{-}(1-y_{0}\pm i\varepsilon))|},$

for all $0\leq\varepsilon\leq\varepsilon_{0}$.

###### Proof.

For $z=0$, note that we have the explicit expression

$\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)=\frac{M_{+}(1-y_{0}\pm
i\varepsilon)M_{-}(y-y_{0}\pm i\varepsilon)-M_{-}(1-y_{0}\pm
i\varepsilon)M_{+}(y-y_{0}\pm i\varepsilon)}{M_{+}(1-y_{0}\pm
i\varepsilon)M_{-}(-y_{0}\pm i\varepsilon)-M_{-}(1-y_{0}\pm
i\varepsilon)M_{+}(-y_{0}\pm i\varepsilon)},$

so that

$\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)=2m\frac{M_{+}(1-y_{0}\pm
i\varepsilon)M_{-}^{\prime}(y-y_{0}\pm i\varepsilon)-M_{-}(1-y_{0}\pm
i\varepsilon)M_{+}^{\prime}(y-y_{0}\pm i\varepsilon)}{M_{+}(1-y_{0}\pm
i\varepsilon)M_{-}(-y_{0}\pm i\varepsilon)-M_{-}(1-y_{0}\pm
i\varepsilon)M_{+}(-y_{0}\pm i\varepsilon)}.$

If $m|y-y_{0}|\leq 3\beta$, we use the bounds on the Wronskian from
Proposition 4.1. For $\beta^{2}>1/4$, the conclusion is straightforward. For
$\beta^{2}<1/4$, we take a closer look to the Wronskian estimates obtained on
the proof of Proposition 4.3. The bounds are a consequence of Lemma A.5,
A.15-A.17. The argument for $z=1$ is similar, we omit the details. ∎

######  Proposition 8.2.

Let $\beta^{2}=1/4$. There exists $\varepsilon_{0}>0$ such that for all
$y,y_{0}\in[0,1]$ with $m|y-y_{0}|\leq 3\beta$ there holds

$|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}}|\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|+|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}}|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{\frac{1}{2}}\frac{1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}}{|M_{0}(y_{0}\mp i\varepsilon))|},$

and

$|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}}|\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},1)|+|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}}|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},1)|\lesssim
m^{\frac{1}{2}}\frac{1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}}{|M_{0}(1-y_{0}\mp i\varepsilon))|},$

for all $0\leq\varepsilon\leq\varepsilon_{0}$.

###### Proof.

Since $m|y-y_{0}|\leq 3\beta$, the proof follows the same ideas to show
Proposition 5.1, with the help of Lemma A.5, A.10-A.12, we omit the details. ∎

###  8.1. Estimates for first order boundary terms

This subsection is devoted to obtain estimates on

${\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},z)=\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\partial_{z}\varphi_{m,\varepsilon}^{\pm}(z,y_{0})$

for $z=0$ and $z=1$ under the assumption that $m|y-y_{0}|\leq 3\beta$. In what
follows, we shall argue for $z=0$, the statements and proofs for $z=1$ are
similar and we thus omit them. We begin by providing bounds for
$\partial_{z}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})$.

######  Proposition 8.3.

Let $y_{0}\in[0,1]$, we have the following.

  * •

If $my_{0}\leq 3\beta$, then
$|\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})|\lesssim
m^{-\frac{1}{2}}Q_{0,m}.$

  * •

If $my_{0}\geq 3\beta$, then
$|\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})|\lesssim Q_{0,m}$.

For the proof, we assume that $y_{0}<1/2$. Otherwise, the proposition follows
from Proposition 7.2. Note that from (2.8) and (2.12), there holds

$\displaystyle\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})$
$\displaystyle=\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)F_{m,\varepsilon}^{\pm}(z,y_{0})\mathrm{d}z$
$\displaystyle=\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)\left(F_{m}(z)+\frac{y_{0}\mp
i\varepsilon}{\beta^{2}}{\Delta}_{m}{\omega}_{m}^{0}(z)\right)\mathrm{d}z,$

Further observe that, due to (H) we have

$\displaystyle\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)F_{m}^{\pm}(z,0)\mathrm{d}z$
$\displaystyle=-4(\mu+i\nu)m\int_{0}^{1}\frac{\phi_{u,m,\varepsilon}^{\pm}(z,y_{0})}{{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})}F_{m}(z,0)\mathrm{d}z$
$\displaystyle=-4(\mu+i\nu)m\int_{0}^{1}\frac{\phi_{u,m,\varepsilon}^{\pm}(z,y_{0})-\phi_{u,m}(z)}{{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})}F_{m}(z,0)\mathrm{d}z$

and we define

$f_{m,\varepsilon}^{\pm}(z,y_{0}):=\phi_{u,m,\varepsilon}^{\pm}(z,y_{0})-\phi_{u,m}(z).$

####  8.1.1. Estimates on $f_{m,\varepsilon}^{\pm}$ for $\beta^{2}\neq 1/4$

From the explicit formulas (3.4) and (3.8), we have

$\displaystyle f_{m,\varepsilon}^{\pm}(z,y_{0})$
$\displaystyle=M_{+}(1-y_{0}\pm i\varepsilon)M_{-}(z-y_{0}\pm
i\varepsilon)-M_{+}(1)M_{-}(z)$ $\displaystyle\quad-M_{-}(1-y_{0}\pm
i\varepsilon)M_{+}(z-y_{0}\pm i\varepsilon)+M_{-}(1)M_{+}(z)$

and we can obtain the next result.

######  Proposition 8.4.

Let $z,y_{0}\in[0,1]$ such that $my_{0}\leq 3\beta$ and $mz\leq 6\beta$. Let
$0\leq\varepsilon\leq\min\left(\frac{\beta}{m},\frac{1}{2m}\right)$. Then,

$|f_{m,\varepsilon}(z,y_{0})|\lesssim m^{\frac{1}{2}-\mu}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}|M_{+}(1-y_{0}\pm i\varepsilon)|.$

In particular, $\|f_{m,\varepsilon}\|_{L^{2}_{y}(J)}\lesssim m^{-\mu}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}|M_{+}(1-y_{0}\pm i\varepsilon)|$.

###### Proof.

We shall assume $\beta^{2}<1/4$, the case $\beta^{2}>1/4$ is analogous and
easier. We write

$\displaystyle M_{+}(1-y_{0}\pm i\varepsilon)M_{-}(z-y_{0}\pm
i\varepsilon)-M_{+}(1)M_{-}(z)$ $\displaystyle=M_{+}(1-y_{0}\pm
i\varepsilon)\Big{(}M_{-}(z-y_{0}\pm i\varepsilon)-M_{-}(z)\Big{)}$
$\displaystyle\quad+M_{-}(z)\Big{(}M_{+}(1-y_{0}\pm
i\varepsilon)-M_{+}(1)\Big{)}$

and

$\displaystyle M_{-}(1-y_{0}\pm i\varepsilon)M_{+}(z-y_{0}\pm
i\varepsilon)-M_{-}(1)M_{+}(z)$ $\displaystyle=M_{-}(1-y_{0}\pm
i\varepsilon)\Big{(}M_{+}(z-y_{0}\pm i\varepsilon)-M_{+}(z)\Big{)}$
$\displaystyle\quad+M_{+}(z)\Big{(}M_{-}(1-y_{0}\pm
i\varepsilon)-M_{-}(1)\Big{)}.$

Firstly, we estimate

$M_{+}(1-y_{0}\pm
i\varepsilon)-M_{+}(1)=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}M_{+}(1+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s=(-y_{0}\pm
i\varepsilon)\int_{0}^{1}M_{+}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s$

and we divide our argument as follows. Let $N_{\mu,0}$ be given as in Lemma
A.15.

For $m\leq N_{\mu,0}$, we use Lemma A.3 and the fact that $y_{0}\leq 1/2$ to
bound

$\displaystyle|M_{+}(1-y_{0}\pm i\varepsilon)-M_{+}(1)|$
$\displaystyle\lesssim m^{\frac{1}{2}+\mu}|y_{0}\pm
i\varepsilon|\int_{0}^{1}\frac{\mathrm{d}s}{|1+s(-y_{0}\pm
i\varepsilon)|^{\frac{1}{2}-\mu}}$ $\displaystyle\lesssim
m^{\frac{1}{2}+\mu}|y_{0}\pm i\varepsilon|$ $\displaystyle\lesssim|y_{0}\pm
i\varepsilon||M_{+}(1-y_{0}\pm i\varepsilon)|.$

In the last inequality, we have used Lemma A.5, A.17 and A.15. Similarly,

$|M_{-}(1-y_{0}\pm i\varepsilon)-M_{-}(1)|\lesssim|y_{0}\pm
i\varepsilon||M_{-}(1-y_{0}\pm i\varepsilon)|\lesssim|y_{0}\pm
i\varepsilon||M_{+}(1-y_{0}\pm i\varepsilon)|,$

where we have used Lemma A.5 and Lemma A.17 to deduce $|M_{-}(1-y_{0}\pm
i\varepsilon)|\lesssim|M_{+}(1-y_{0}\pm i\varepsilon)|$.

For $m\geq N_{\mu,0}$, we claim that

$\left|\frac{M_{+}^{\prime}(1+s(-y_{0}\pm i\varepsilon))}{M_{+}(1-y_{0}\pm
i\varepsilon)}\right|\lesssim 1.$

Indeed, this follows from

$\left|\frac{M_{+}^{\prime}(1+s(-y_{0}\pm i\varepsilon))}{M_{+}(1-y_{0}\pm
i\varepsilon)}\right|=\left|\frac{M_{+}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))}{M_{+}(1+s(-y_{0}\pm
i\varepsilon))}\right|\left|\frac{M_{+}(1+s(-y_{0}\pm
i\varepsilon))}{M_{+}(1-y_{0}\pm i\varepsilon)}\right|$

and the corresponding bounds from Lemma A.6 since $2m(1-y_{0})\geq m\geq
N_{\mu,0}$. Hence, we have

$\displaystyle|M_{+}(1-y_{0}\pm i\varepsilon)-M_{+}(1)|$ $\displaystyle\leq
2m|y_{0}\pm i\varepsilon|\int_{0}^{1}\left|M_{+}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))\right|\mathrm{d}s$ $\displaystyle\lesssim m|y_{0}\pm
i\varepsilon|\left|M_{+}(1-y_{0}\pm i\varepsilon)\right|.$

Similarly, we also have

$|M_{-}(1-y_{0}\pm i\varepsilon)-M_{-}(1)|\lesssim m|y_{0}\pm
i\varepsilon|\left|M_{-}(1-y_{0}\pm i\varepsilon)\right|\lesssim m|y_{0}\pm
i\varepsilon|\left|M_{+}(1-y_{0}\pm i\varepsilon)\right|,$

where we have used Lemma A.15 to deduce $\left|M_{-}(1-y_{0}\pm
i\varepsilon)\right|\lesssim\left|M_{+}(1-y_{0}\pm i\varepsilon)\right|$. We
next turn our attention to the bounds for $M_{-}(z-y_{0}\pm
i\varepsilon)-M_{-}(z)$. As before, we consider two cases.

$\bullet$ Case 1. For $2y_{0}\leq z$ we estimate

$M_{-}(z-y_{0}\pm
i\varepsilon)-M_{-}(z)=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}M_{-}(z+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s=(-y_{0}\pm
i\varepsilon)\int_{0}^{1}M_{-}^{\prime}(z+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s.$

From Lemma A.3,
$M_{-}^{\prime}(\zeta)\lesssim\zeta^{-\frac{1}{2}-\mu}m^{\frac{1}{2}-\mu}$,
and since $2y_{0}\leq z$, we have that $s|y_{0}\pm
i\varepsilon|\leq|z+s(-y_{0}\pm i\varepsilon)|$, for all $s\in(0,1)$. Thus,

$\displaystyle|M_{-}(z-y_{0}\pm i\varepsilon)-M_{-}(z)|$
$\displaystyle\lesssim m^{\frac{1}{2}-\mu}|y_{0}\pm
i\varepsilon|\int_{0}^{1}\frac{\mathrm{d}s}{|z+s(-y_{0}\pm
i\varepsilon)|^{\frac{1}{2}+\mu}}$ $\displaystyle\lesssim
m^{\frac{1}{2}-\mu}|y_{0}\pm
i\varepsilon|\int_{0}^{1}\frac{\mathrm{d}s}{|s(y_{0}\pm
i\varepsilon)|^{\frac{1}{2}+\mu}}$ $\displaystyle\lesssim
m^{\frac{1}{2}-\mu}|y_{0}\pm i\varepsilon|^{\frac{1}{2}-\mu}.$

$\bullet$ Case 2. For $z\leq 2y_{0}$, we directly estimate using Lemma A.3,
that is,

$\displaystyle|M_{-}(z-y_{0}\pm i\varepsilon)-M_{-}(z)|\leq|M_{-}(z-y_{0}\pm
i\varepsilon)|+|M_{-}(z)|$ $\displaystyle\lesssim
m^{\frac{1}{2}-\mu}\left(|z-y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}+|z|^{\frac{1}{2}-\mu}\right)$
$\displaystyle\lesssim m^{\frac{1}{2}-\mu}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}.$

∎

From this localised estimates, we are able to obtain bounds on
$f_{m,\varepsilon}(z,y_{0})$ for $mz\geq 6\beta$. For this, we first deduce
useful estimates on $\phi_{u,m}^{\pm}(z)$.

######  Lemma 8.5.

The function $\phi_{u,m}(z)=M_{+}(1)M_{-}(z)-M_{-}(1)M_{+}(z)$ satisfies

${\Delta}_{m}\phi_{u,m}^{\pm}(z)+\beta^{2}\frac{\phi_{u,m}^{\pm}(z)}{z^{2}}=0,\qquad\phi_{u,m}^{\pm}(1)=0.$

For $J_{6}=\\{z\in[0,1]:mz\leq 6\beta\\}$ and $J_{6}^{c}=[0,1]\setminus
J_{6}$, it is such that

$\|\phi_{u,m}\|_{L^{\infty}(J_{6})}\lesssim
m^{\frac{1}{2}-\mu}|z|^{\frac{1}{2}-\mu}|M_{+}(1)|,\quad\|\phi_{u,m}\|_{L^{2}_{y}(J_{6})}\lesssim
m^{-\frac{1}{2}}|M_{+}(1)|$

and

$\|\partial_{z}\phi_{u,m}\|_{L^{2}_{y}(J_{6}^{c})}+m\|\phi_{u,m}\|_{L^{2}_{y}(J_{6}^{c})}\lesssim
m^{\frac{1}{2}}|M_{+}(1)|.$

###### Proof.

The statements for $\|\phi_{u,m}\|_{L^{\infty}(J_{6})}$ and
$\|\phi_{u,m}\|_{L^{2}_{y}(J_{6})}$ follow from the asymptotic expansions for
small argument given by Lemma A.3. The integral estimates follow from the
$\|\phi_{u,m}\|_{L^{2}_{y}(J_{6})}$ bounds using Lemma 7.1. ∎

The following proposition obtains $L^{2}$ bounds on
$f_{m,\varepsilon}^{\pm}(\cdot,y_{0})$ from the localized bounds of
Proposition 8.4 and the above lemma.

######  Proposition 8.6.

We have that

$\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}}\lesssim
m^{-\frac{1}{2}}(m|y_{0}-i\varepsilon|)^{\frac{1}{2}-\mu}|M_{+}(1-y_{0}+i\varepsilon)|.$

###### Proof.

It is straightforward to see that $f_{m,\varepsilon}^{\pm}(z,y_{0})$ solves

${\Delta}_{m}f_{m,\varepsilon}^{\pm}+\beta^{2}\frac{f_{m,\varepsilon}^{\pm}}{(z-y_{0}\pm
i\varepsilon)^{2}}=\beta^{2}(-y_{0}\pm
i\varepsilon)\left(\frac{2}{z(z-y_{0}\pm i\varepsilon)^{2}}+\frac{-y_{0}\pm
i\varepsilon}{z^{2}(z-y_{0}\pm i\varepsilon)^{2}}\right)\phi_{u,m}$

and $f_{m,\varepsilon}^{\pm}(1,y_{0})=0$. Hence, using the same strategy from
Lemma 4.4, we have that

$\displaystyle\frac{1}{2}\int_{\frac{6\beta}{m}}^{1}|\partial_{z}f_{m,\varepsilon}^{\pm}|^{2}+m^{2}|f_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}z$
$\displaystyle\leq\frac{m^{2}}{\beta^{2}}\int_{\frac{5\beta}{m}}^{\frac{6\beta}{m}}|f^{\pm}_{m,\varepsilon}|^{2}\mathrm{d}z$
(8.1) $\displaystyle+\beta^{2}|-y_{0}\pm
i\varepsilon|\int_{\frac{5\beta}{m}}^{1}\left(\frac{2}{z}+\frac{|-y_{0}\pm
i\varepsilon|}{z^{2}}\right)\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z.$

Now, from Proposition 8.4, we have

$\frac{m^{2}}{\beta^{2}}\int_{\frac{5\beta}{m}}^{\frac{6\beta}{m}}|f^{\pm}_{m,\varepsilon}|^{2}\mathrm{d}z\lesssim\frac{m}{\beta}\left(m|y_{0}-i\varepsilon|\right)^{1-2\mu}|M_{+}(1-y_{0}+i\varepsilon)|^{2},$

while we write

$\beta^{2}|-y_{0}\pm
i\varepsilon|^{2}\int_{\frac{5\beta}{m}}^{1}\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z^{2}|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z=\beta^{2}|-y_{0}\pm
i\varepsilon|^{2}\left(\int_{\frac{5\beta}{m}}^{\frac{6\beta}{m}}+\int_{\frac{6\beta}{m}}^{1}\right)\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z^{2}|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z.$

For example, with the bounds of Proposition 8.4 and Lemma 8.5, and the fact
that $z\geq\frac{5\beta}{m}$ and $y_{0}\leq\frac{3\beta}{m}$, we have
$|z-y_{0}\pm i\varepsilon|^{-2}\lesssim m^{2}$ and

$\displaystyle\beta^{2}|-y_{0}\pm
i\varepsilon|^{2}\int_{\frac{5\beta}{m}}^{\frac{6\beta}{m}}\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z^{2}|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z$
$\displaystyle\lesssim{m^{2}}|y_{0}-i\varepsilon|^{2}\|f_{m,\varepsilon}^{\pm}\|_{L^{\infty}(J)}|M_{+}(1)|^{2}\int_{y_{2}}^{y_{2}+\frac{\beta}{m}}m^{\frac{1}{2}-\mu}z^{-\frac{3}{2}-\mu}\mathrm{d}z$
$\displaystyle\lesssim m^{\frac{7}{2}-\mu}|y_{0}\pm
i\varepsilon|^{\frac{5}{2}-\mu}|M_{+}(1)||M_{+}(1-y_{0}\pm i\varepsilon)|.$

On the other hand, Young’s inequality and Lemma 8.5 gives

$\displaystyle\beta^{2}|-y_{0}\pm
i\varepsilon|^{2}\int_{\frac{6\beta}{m}}^{1}\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z^{2}|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z$
$\displaystyle\leq\frac{m^{2}}{8}\int_{y_{2}+\frac{\beta}{m}}^{1}|f_{m,\varepsilon}^{\pm}(z)|^{2}\mathrm{d}z+Cm^{5}|y_{0}-i\varepsilon|^{4}{|M_{+}(1)|^{2}},$

for some $C>0$ large enough. Similarly, we bound

$\displaystyle\beta^{2}|-y_{0}\pm i\varepsilon|$
$\displaystyle\int_{\frac{5\beta}{m}}^{\frac{6\beta}{m}}\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z\lesssim m^{\frac{5}{2}-\mu}|y_{0}\pm
i\varepsilon|^{\frac{3}{2}-\mu}|M_{+}(1)||M_{+}(1-y_{0}\pm i\varepsilon)|$

and

$\displaystyle\beta^{2}|-y_{0}\pm
i\varepsilon|\int_{\frac{6\beta}{m}}^{1}\frac{|\phi_{u,m}(z)|f_{m,\varepsilon}^{\pm}(z)|}{z|z-y_{0}\pm
i\varepsilon|^{2}}\mathrm{d}z$
$\displaystyle\leq\frac{m^{2}}{8}\int_{\frac{6\beta}{m}}^{1}|f_{m,\varepsilon}^{\pm}(z)|^{2}\mathrm{d}z+Cm^{3}|y_{0}-i\varepsilon|^{2}{|M_{+}(1)|^{2}},$

for some $C>0$ large enough. Hence, we absorb the potential term on the left
hand side of (8.1) and conclude that

$\frac{1}{4}\int_{y_{2}+\frac{\beta}{m}}^{1}|\partial_{z}f_{m,\varepsilon}^{\pm}|^{2}+m^{2}|f_{m,\varepsilon}^{\pm}|^{2}\mathrm{d}z\lesssim
m(m|y_{0}-i\varepsilon|)^{1-2\mu}|M_{+}(1-y_{0}+i\varepsilon)|^{2}.$

and the lemma follows. ∎

####  8.1.2. Estimates on $f_{m,\varepsilon}^{\pm}$ for $\beta^{2}=1/4$

From the explicit formulas (3.12) and (3.15), we now have

$\displaystyle f_{m,\varepsilon}^{\pm}(z,y_{0})$
$\displaystyle=W_{0}(1-y_{0}\pm i\varepsilon)M_{0}(z-y_{0}\pm
i\varepsilon)-W_{0}(1)M_{0}(z)$ $\displaystyle\quad-M_{0}(1-y_{0}\pm
i\varepsilon)W_{0}(z-y_{0}\pm i\varepsilon)+M_{0}(1)W_{0}(z)$

from which we obtain the following result.

######  Proposition 8.7.

Let $z,y_{0}\in[0,1]$ such that $my_{0}\leq 3\beta$ and $mz\leq 6\beta$. Let
$0\leq\varepsilon\leq\min\left(\frac{\beta}{m},\frac{1}{2m}\right)$. Then,

$|f_{m,\varepsilon}(z,y_{0})|\lesssim(m|y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(my_{0}\right)\big{|}\right)|M_{0}(1-y_{0}\pm
i\varepsilon)|.$

In particular, $\|f_{m,\varepsilon}\|_{L^{2}_{y}(J)}\lesssim|y_{0}\pm
i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right)|M_{0}(1-y_{0}\pm i\varepsilon)|$.

###### Proof.

We write

$\displaystyle W_{0}(1-y_{0}\pm i\varepsilon)M_{0}(z-y_{0}\pm
i\varepsilon)-W_{0}(1)M_{0}(z)$ $\displaystyle=W_{0}(1-y_{0}\pm
i\varepsilon)\Big{(}M_{0}(z-y_{0}\pm i\varepsilon)-M_{0}(z)\Big{)}$
$\displaystyle\quad+M_{0}(z)\Big{(}W_{0}(1-y_{0}\pm
i\varepsilon)-W_{0}(1)\Big{)}$

We shall now estimate the differences involving the Whittaker function
$W_{0}$, the estimates for the differences involving $M_{0}$ follow similarly
as for the case $\beta^{2}\neq 1/4$ and they are

$\left|M_{0}(z-y_{0}\pm i\varepsilon)-M_{0}(z)\right|\lesssim
m^{\frac{1}{2}}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}},\quad\left|M_{0}(1-y_{0}\pm
i\varepsilon)-M_{0}(1)\right|\lesssim m|y_{0}\pm
i\varepsilon||M_{0}(1-y_{0}\pm i\varepsilon)|.$

Firstly, we estimate

$W_{0}(1-y_{0}\pm
i\varepsilon)-W_{0}(1)=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}W_{0}(1+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s=2m(-y_{0}\pm
i\varepsilon)\int_{0}^{1}W_{0}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s$

and we divide our argument as follows. Let $N_{\mu,0}$ be given as in Lemma
A.15.

For $m\leq N_{\mu,0}$, we use Lemma A.4 and the fact that
$y_{0}\leq\frac{1}{2}$ to bound

$\displaystyle|W_{0}(1-y_{0}\pm i\varepsilon)-W_{0}(1)|$
$\displaystyle\lesssim m^{\frac{1}{2}}|y_{0}\pm
i\varepsilon|\int_{0}^{1}\frac{1+\big{|}\log m|1+s(-y_{0}\pm
i\varepsilon)|\big{|}}{|1+s(-y_{0}\pm
i\varepsilon)|^{\frac{1}{2}}}\mathrm{d}s$ $\displaystyle\lesssim
m^{\frac{1}{2}}|y_{0}\pm i\varepsilon|$ $\displaystyle\lesssim|y_{0}\pm
i\varepsilon||M_{0}(1-y_{0}\pm i\varepsilon)|.$

In the last inequality, we have used Lemma A.5, A.17 and A.15.

For $m\geq N_{\mu,0}$, we claim that

$\left|\frac{W_{0}^{\prime}(1+s(-y_{0}\pm i\varepsilon))}{M_{0}(1-y_{0}\pm
i\varepsilon)}\right|\lesssim 1.$

Indeed, this follows from

$\left|\frac{W_{0}^{\prime}(1+s(-y_{0}\pm i\varepsilon))}{M_{0}(1-y_{0}\pm
i\varepsilon)}\right|=\left|\frac{W_{0}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))}{W_{0}(1+s(-y_{0}\pm
i\varepsilon))}\right|\left|\frac{W_{0}(1+s(-y_{0}\pm
i\varepsilon))}{W_{0}(1-y_{0}\pm
i\varepsilon)}\right|\left|\frac{W_{0}(1-y_{0}\pm
i\varepsilon)}{M_{0}(1-y_{0}\pm i\varepsilon)}\right|$

and the corresponding bounds from Lemma A.6 since $2m(1-y_{0})\geq m\geq
N_{\mu,0}$. Hence, we have

$\displaystyle|W_{0}(1-y_{0}\pm i\varepsilon)-W_{0}(1)|$ $\displaystyle\leq
2m|y_{0}\pm i\varepsilon|\int_{0}^{1}\left|W_{0}^{\prime}(1+s(-y_{0}\pm
i\varepsilon))\right|\mathrm{d}s$ $\displaystyle\lesssim m|y_{0}\pm
i\varepsilon|\left|M_{0}(1-y_{0}\pm i\varepsilon)\right|.$

We next turn our attention to the bounds for $W_{0}(z-y_{0}\pm
i\varepsilon)-W_{0}(z)$. As before, we consider two cases.

$\bullet$ Case 1. For $2y_{0}\leq z$ we estimate

$W_{0}(z-y_{0}\pm
i\varepsilon)-W_{0}(z)=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}W_{0}(z+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s=(-y_{0}\pm
i\varepsilon)\int_{0}^{1}W_{0}^{\prime}(z+s(-y_{0}\pm
i\varepsilon))\mathrm{d}s.$

From Lemma A.4, $W_{0}^{\prime}(\zeta)\lesssim
m^{\frac{1}{2}}\zeta^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|\zeta|\right)\big{|}\right)$,
and since $2y_{0}\leq z$, we have that $s|y_{0}\pm
i\varepsilon|\leq|z+s(-y_{0}\pm i\varepsilon)|$, for all $s\in(0,1)$. Thus,

$\displaystyle|W_{0}(z-y_{0}\pm i\varepsilon)-W_{0}(z)|$
$\displaystyle\lesssim m^{\frac{1}{2}}|y_{0}\pm
i\varepsilon|\int_{0}^{1}\frac{1+\big{|}\log\left(m|z+s(-y_{0}\pm
i\varepsilon)|\right)\big{|}}{|z+s(-y_{0}\pm
i\varepsilon)|^{\frac{1}{2}}}\mathrm{d}s$ $\displaystyle\lesssim
m^{\frac{1}{2}}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}}\int_{0}^{1}\frac{1+\big{|}\log\left(ms|y_{0}\pm
i\varepsilon|\right)\big{|}}{s^{\frac{1}{2}}}\mathrm{d}s$
$\displaystyle\lesssim(m|y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right).$

$\bullet$ Case 2. For $z\leq 2y_{0}$, we directly estimate using Lemma A.4,
that is,

$\displaystyle\begin{aligned} |M_{-}(z-y_{0}\pm
i\varepsilon)-M_{-}(z)|&\leq|M_{-}(z-y_{0}\pm i\varepsilon)|+|M_{-}(z)|\\\
&\lesssim m^{\frac{1}{2}}|z-y_{0}\pm
i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log\left(m|z-y_{0}\pm
i\varepsilon|\right)\big{|}\right)\\\
&\quad+m^{\frac{1}{2}}z^{\frac{1}{2}}\left(1+\big{|}\log\left(mz\right)\big{|}\right)\\\
&\lesssim(m|y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right).\end{aligned}$

∎

From this localised estimates, we are able to obtain bounds on
$f_{m,\varepsilon}(z,y_{0})$ for $mz\geq 6\beta$. For this, we first deduce
useful estimates on $\phi_{u,m}^{\pm}(z)$.

######  Lemma 8.8.

The function $\phi_{u,m}(z)=W_{0}(1)M_{0}(z)-M_{0}(1)W_{0}(z)$ satisfies

${\Delta}_{m}\phi_{u,m}^{\pm}(z)+\beta^{2}\frac{\phi_{u,m}^{\pm}(z)}{z^{2}}=0,\qquad\phi_{u,m}^{\pm}(1)=0.$

For $J_{6}=\\{z\in[0,1]:mz\leq 6\beta\\}$ and $J_{6}^{c}=[0,1]\setminus
J_{6}$, it is such that

$\|\phi_{u,m}\|_{L^{\infty}(J)}\lesssim(mz)^{\frac{1}{2}}\left(1+\big{|}\log\left(mz\right)\big{|}\right)|M_{0}(1)|,\quad\|\phi_{u,m}\|_{L^{2}_{y}(J)}\lesssim
m^{-\frac{1}{2}}|M_{0}(1)|$

and

$\|\partial_{z}\phi_{u,m}\|_{L^{2}_{y}(J^{c})}+m\|\phi_{u,m}\|_{L^{2}_{y}(J^{c})}\lesssim
m^{\frac{1}{2}}|M_{+}(1)|$

###### Proof.

The statement for $\|\phi_{u,m}\|_{L^{\infty}(J_{6})}$ follows from the
asymptotic expansions for small argument given by Lemma A.3. For the integral
estimates estimate, note that the change of variables $u=mz$ provides

$\displaystyle\|\phi_{u,m}\|_{L^{2}_{y}(J_{6})}^{2}\lesssim\int_{0}^{\frac{6\beta}{m}}(mz)\left(1+\big{|}\log\left(mz\right)\big{|}\right)^{2}|M_{0}(1)|^{2}\mathrm{d}z$
$\displaystyle=\frac{|M_{0}(1)|^{2}}{m}\int_{0}^{6\beta}\eta\left(1+|\log\left(\eta\right)|\right)^{2}\mathrm{d}\eta$
$\displaystyle\lesssim\frac{|M_{0}(1)|^{2}}{m}.$

The result follows using Lemma 7.1. ∎

The following proposition obtains $L^{2}(0,1)$ bounds on
$f_{m,\varepsilon}^{\pm}(\cdot,y_{0})$ from the localized bounds of
Proposition 8.7 and the above Lemma. We omit its proof due to its similarity
to the one for Proposition 8.6.

######  Proposition 8.9.

We have that

$\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}(0,1)}\lesssim
m^{-\frac{1}{2}}(m|y_{0}-i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right)|M_{0}(1-y_{0}\pm i\varepsilon)|$

We are now able to compare
$\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}(0,1)}$ and the Wronskian
$|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|$.

######  Lemma 8.10.

Let $y_{0}\in[0,1]$ such that $my_{0}\leq 3\beta$. There exists
$\varepsilon_{0}>0$ such that

$\frac{\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}(0,1)}}{|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|}\lesssim
m^{-\frac{3}{2}}.$

###### Proof.

Let $N_{0}>0$ be given by Lemma A.10, $\delta_{1}>0$ be given by Lemma A.11
and $\delta_{2}>0$ be given by Lemma A.13. From Lemma 5.2, there holds the
following,

$\bullet$ Case 1. For $m\leq N_{0}$ and $2m|y_{0}\pm
i\varepsilon|\leq\delta_{1}$, we have

$|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|\gtrsim m|M_{0}(1-y_{0}\pm
i\varepsilon)||W_{0}(y_{0}\mp i\varepsilon)|.$

where further

$\left|\frac{M_{0}(y_{0}\pm i\varepsilon)}{W_{0}(y_{0}\pm
i\varepsilon)}\right|\leq\frac{1}{2\sqrt{\pi}}.$

Now, from Lemma A.13, if $\delta_{1}\leq\delta_{2}$, then,

$|2m(y_{0}\pm i\varepsilon)|^{\frac{1}{2}}\left(1+\big{|}\log\left(2m|y_{0}\pm
i\varepsilon|\right)\big{|}\right)\lesssim|W_{0}(y_{0}\pm i\varepsilon)|,$

and the conclusion follows. On the other hand, for $\delta_{2}\leq 2m|y_{0}\pm
i\varepsilon|\leq\delta_{1}$, we have that

$m\frac{\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}}}{|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|}\lesssim
m^{-\frac{1}{2}}\frac{(m|y_{0}\pm i\varepsilon|)^{\frac{1}{2}}}{M_{0}(y_{0}\pm
i\varepsilon)}\lesssim m^{-\frac{1}{2}},$

due to Lemma A.14.

$\bullet$ Case 2. For $m\leq N_{0}$ and $\delta_{1}\leq 2my_{0}\leq N_{0}$, we
have now

$|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|\gtrsim m|M_{0}(1-y_{0}\pm
i\varepsilon)||M_{0}(y_{0}\mp i\varepsilon)|,$

the conclusion follows using Lemma A.14 and the fact that
$\left(1+\big{|}\log|2m(y_{0}\pm i\varepsilon)|\big{|}\right)\lesssim 1$.

$\bullet$ Case 3. For $m\geq N_{0}$, and $2m|y_{0}\pm
i\varepsilon|\leq\delta_{1}$, we have

$|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|\gtrsim m|M_{0}(1-y_{0}\pm
i\varepsilon)||W_{0}(y_{0}\mp i\varepsilon)|.$

and also

$\left|\frac{M_{0}(y_{0}\pm i\varepsilon)}{W_{0}(y_{0}\pm
i\varepsilon)}\right|\leq\frac{1}{2\sqrt{\pi}},$

we proceed as in Case 1, we omit the details.

$\bullet$ Case 4. For $m\geq N_{0}$ and $\delta_{1}\leq 2my_{0}\leq N_{0}$, we
have

$|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|\gtrsim m|M_{0}(1-y_{0}\pm
i\varepsilon)||M_{0}(y_{0}\mp i\varepsilon)|,$

we proceed as in Case 2, we omit the details. ∎

We are now in position to prove Proposition 8.3.

###### Proof of Proposition 8.3.

For $my_{0}\geq 3\beta$ we appeal to Proposition 7.2 to obtain the desired
bound. On the other hand, for $my_{0}\leq 3\beta$ let us recall that we can
write

$\displaystyle\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})$
$\displaystyle=\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)\left(F_{m}(z)+\frac{y_{0}\mp
i\varepsilon}{\beta^{2}}{\Delta}_{m}{\omega}_{m}^{0}(z)\right)\mathrm{d}z$

For $\beta^{2}\neq 1/4$, it is straightforward to see from Proposition 7.2
that

$\left|\frac{y_{0}\mp
i\varepsilon}{\beta^{2}}\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z){\Delta}_{m}{\omega}_{m}^{0}(z)\mathrm{d}z\right|\lesssim\frac{1}{m^{1+\mu}}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}\|{\omega}_{m}^{0}\|_{H^{2}_{y}},$

while, thanks to Proposition 8.6, the lower bounds on the Wronskian from
Proposition 4.3 and Lemma A.16 we bound

$\displaystyle\left|\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)F_{m}^{\pm}(z,0)\mathrm{d}z\right|$
$\displaystyle=\left|4(\mu+i\nu)m\int_{0}^{1}\frac{f_{m,\varepsilon}^{\pm}(z,y_{0})}{{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})}F_{m}^{\pm}(z,0)\mathrm{d}z\right|$
$\displaystyle\lesssim
m\frac{\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}_{z}}}{|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|}\|F_{m}^{\pm}(z,0)\|_{L^{2}_{z}}$
$\displaystyle\lesssim m^{-\frac{1}{2}}\|F_{m}\|_{L^{2}}.$

Similarly, for $\beta^{2}=1/4$, using again Proposition 7.2,

$\displaystyle\left|\frac{y_{0}\mp
i\varepsilon}{\beta^{2}}\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z){\Delta}_{m}{\omega}_{m}^{0}(z)\mathrm{d}z\right|$
$\displaystyle\lesssim\frac{1}{m}|y_{0}\pm
i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right)\|{\omega}_{m}^{0}\|_{H^{2}_{y}}$
$\displaystyle\lesssim m^{-\frac{3}{2}}\|{\omega}_{m}^{0}\|_{H^{2}_{y}},$

while, thanks to Lemma 8.10 we have

$\displaystyle\left|\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)F_{m}^{\pm}(z,0)\mathrm{d}z\right|$
$\displaystyle=\left|4(\mu+i\nu)m\int_{0}^{1}\frac{f_{m,\varepsilon}^{\pm}(z,y_{0})}{{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})}F_{m}^{\pm}(z,0)\mathrm{d}z\right|$
$\displaystyle\lesssim
m\frac{\|f_{m,\varepsilon}^{\pm}(\cdot,y_{0})\|_{L^{2}_{z}}}{|{\mathcal{W}}_{m,\varepsilon}^{\pm}(y_{0})|}\|F_{m}^{\pm}(z,0)\|_{L^{2}_{z}}$
$\displaystyle\lesssim m^{-\frac{1}{2}}\ \|F_{m}\|_{L^{2}}.$

With this, the proof is finished. ∎

We next provide pointwise localized bounds on
${\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)$.

######  Proposition 8.11.

Let $\beta^{2}\neq 1/4$ and $0\leq\varepsilon\leq\varepsilon_{0}$. Let
$y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq 3\beta$. Then,

  * •

If $my_{0}\leq 3\beta$, we have

$\left|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)\right|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{1}{2}+\mu}|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}-\mu}{Q_{0,m}}.$

  * •

If $my_{0}\geq 3\beta$, we have

$\left|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)\right|\lesssim(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}-\mu}{Q_{0,m}}.$

######  Proposition 8.12.

Let $\beta^{2}=1/4$ and $0\leq\varepsilon\leq\varepsilon_{0}$. Let
$y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq 3\beta$. Then,

  * •

If $my_{0}\leq 3\beta$, we have

$\left|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)\right|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{1}{2}}|y-y_{0}\pm
i\varepsilon|^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right){Q_{0,m}}.$

  * •

If $my_{0}\geq 3\beta$, we have

$\left|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)\right|\lesssim(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right){Q_{0,m}}.$

With the above pointwise bounds, one deduces the following integral estimates
for all $\beta^{2}>0$.

######  Corollary 8.13.

Let $y_{0}\in[0,1]$. Then,

  * •

If $my_{0}\leq 3\beta$, we have

$\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{-\frac{3}{2}}y_{0}^{-\frac{1}{2}}{Q_{0,m}}.$

  * •

If $my_{0}\geq 3\beta$, we have

$\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{-\frac{1}{2}}{Q_{0,m}}.$

The two propositions are a consequence of Propositions 8.1, 8.2, the lower
bounds from Lemma A.9, A.14, A.18 and the pointwise estimates on
$\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})$ from Proposition 8.3.

###  8.2. Boundary pointwise estimates on Green’s Function’s derivatives

This subsection estimates derivatives of the Green’s function
${\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)$ evaluated at the boundary
values $y,z\in\\{0,1\\}$.

######  Lemma 8.14.

We have that for $my_{0}\geq 3\beta$,

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},0)|\lesssim
m$

while for $my_{0}\leq 3\beta$,

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},0)|\lesssim\frac{1}{y_{0}}.$

###### Proof.

For $\beta^{2}>1/4$, it follows from the proof of Proposition 4.1 and Lemma
A.9. For $\beta^{2}=1/4$, it follows from Lemma 5.2, Lemma A.13 and Lemma
A.14. For $b^{2}<1/4$, it follows from the proof of Proposition 4.3 and Lemma
A.18. ∎

######  Lemma 8.15.

For $m\geq 6\beta$, we have

  * •

If $my_{0}\leq 3\beta$, then $m(1-y_{0})\geq 3\beta$ and

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},0)|\lesssim{m^{\frac{1}{2}+\mu}}{y_{0}^{-\frac{1}{2}+\mu}}.$

  * •

If $m(1-y_{0})\leq 3\beta$, then $my_{0}\geq 3\beta$ and

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},0)|\lesssim{m^{\frac{1}{2}+\mu}}{(1-y_{0})^{-\frac{1}{2}+\mu}}.$

  * •

If $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$, then

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},0)|\lesssim
m.$

On the other hand, for $m\leq 6\beta$, we have that

  * •

If $y_{0}\leq\frac{1}{2}$, then

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},0)|\lesssim{y_{0}^{-\frac{1}{2}+\mu}}.$

  * •

If $1-y_{0}\leq\frac{1}{2}$, then

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},0)|\lesssim{(1-y_{0})^{-\frac{1}{2}+\mu}}.$

###### Proof.

It is straightforward the lower bounds on the Wronskian and Lemmas A.9, A.18,
A.13, for $\beta^{2}>1/4$, $\beta^{2}=1/4$ and $\beta^{2}<1/4$, respectively.
∎

######  Lemma 8.16.

The same bounds as in Lemma 8.15 hold for
$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},1)|$.

######  Lemma 8.17.

We have that for $m(1-y_{0})\geq 3\beta$,

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},1)|\lesssim
m$

while for $m(1-y_{0})\leq 3\beta$,

$|\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(1,y_{0},1)|\lesssim\frac{1}{1-y_{0}}.$

###  8.3. Estimates for second order boundary terms

In what follows, we shall consider only the case $m\geq 6\beta$, since the
setting $m\leq 6\beta$ is analogous and easier. With the pointwise derivatives
bounds obtained in the four previous lemmas, we are now in position to
estimate

$\displaystyle\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})$
$\displaystyle=-\frac{2}{\beta^{2}}\partial_{y}{\omega}_{m}^{0}(y)-F_{m,\varepsilon}^{\pm}(y,y_{0})+2\partial_{y}\Big{.}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\Big{]}_{z=0}^{z=1}$
$\displaystyle\quad+2\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\left(\partial_{z}F_{m,\varepsilon}^{\pm}(z,y_{0})+\partial_{y_{0}}F_{m,\varepsilon}^{\pm}(z,y_{0})\right)\mathrm{d}z$

for both $y=0$ and $y=1$. For simplicity we only discuss the case $y=0$; the
results and proofs are the same for the case $y=1$.

######  Proposition 8.18.

Let $m\geq 6\beta$ and $y_{0}\in[0,1]$. Then, we have that

  * •

For $my_{0}\leq 3\beta$ and $\beta^{2}\neq 1/4$,

$|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\left(1+\frac{1}{m^{1+\mu}}y_{0}^{-\frac{1}{2}-\mu}+m^{-\frac{1}{2}}y_{0}^{-1}+m^{\frac{1}{2}}y_{0}^{-\frac{1}{2}}\right)Q_{1,m}$

  * •

For $my_{0}\leq 3\beta$ and $\beta^{2}=1/4$,

$|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\left(1+\frac{1}{m}y_{0}^{-\frac{1}{2}}\left(1+\big{|}\log\left(my_{0}\right)\big{|}\right)+m^{-\frac{1}{2}}y_{0}^{-1}+m^{\frac{1}{2}}y_{0}^{-\frac{1}{2}}\right)Q_{1,m}$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\leq 3\beta$,

$|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\left(m+(1-y_{0})^{-\frac{1}{2}}\right)Q_{1,m},$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$,

$|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim
mQ_{1,m}.$

###### Proof.

For $y=0$, we can estimate

$|\partial_{y}{\omega}_{m}^{0}(0)|+|F_{m,\varepsilon}^{\pm}(0,y_{0})|\lesssim
Q_{1,m}$

thanks to the Sobolev embedding. On the other hand, from Proposition 7.2, for
$my_{0}\leq 3\beta$ and $\beta^{2}\neq 1/4$ we have

$\left|\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)\left(\partial_{z}F_{m,\varepsilon}^{\pm}(z,y_{0})+\partial_{y_{0}}F_{m,\varepsilon}^{\pm}(z,y_{0})\right)\mathrm{d}z\right|\lesssim\frac{1}{m^{1+\mu}}|y_{0}\pm
i\varepsilon|^{-\frac{1}{2}-\mu}Q_{1,m},$

while for $my_{0}\leq 3\beta$ and $\beta^{2}=1/4$ we have

$\displaystyle\left|\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)\left(\partial_{z}+\partial_{y_{0}}\right)F_{m,\varepsilon}^{\pm}(z,y_{0})\mathrm{d}z\right|\lesssim\frac{1}{m}|y_{0}\pm
i\varepsilon|^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m},$

whereas for $my_{0}\geq 3\beta$, we have

$\left|\int_{0}^{1}\partial_{y}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},z)\left(\partial_{z}+\partial_{y_{0}}\right)F_{m,\varepsilon}^{\pm}(z,y_{0})\mathrm{d}z\right|\lesssim
Q_{1,m}.$

Now, for the solid boundary terms
$\partial_{y}\Big{.}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\Big{]}_{z=0}^{z=1}$,
we shall use Proposition 8.3 as well as Lemmas 8.14-8.17. Indeed, for
$\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},0)=\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},0)\partial_{y}\varphi_{m,\varepsilon}^{\pm}(0,y_{0})$,
Lemma 8.14 provides

  * •

For $my_{0}\leq 3\beta$, we have that
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},0)|\lesssim
m^{-\frac{1}{2}}y_{0}^{-1}Q_{0,m}$.

  * •

For $my_{0}\geq 3\beta$, we have that
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},0)|\lesssim
mQ_{0,m}$.

Similarly, for
$\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},1)=\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(0,y_{0},1)\partial_{y}\varphi_{m,\varepsilon}^{\pm}(1,y_{0})$,
we have from Lemma 8.16 that

  * •

For $my_{0}\leq 3\beta$, then
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},1)|\lesssim
m^{\frac{1}{2}+\mu}y_{0}^{-\frac{1}{2}+\mu}Q_{0,m}$.

  * •

For $my_{0}\geq 3\beta$, we further distinguish

    * –

For $m(1-y_{0})\leq 3\beta$, then
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},1)|\lesssim
m^{\mu}(1-y_{0})^{-\frac{1}{2}+\mu}Q_{0,m}$.

    * –

For $m(1-y_{0})\geq 3\beta$, then
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(0,y_{0},1)|\lesssim
mQ_{0,m}$.

As a result, for $my_{0}\leq 3\beta$ and $\beta^{2}\neq 1/4$ we have that

$\displaystyle|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|$
$\displaystyle\lesssim
Q_{1,m}+\frac{1}{m^{1+\mu}}y_{0}^{-\frac{1}{2}-\mu}Q_{1,m}+m^{-\frac{1}{2}}y_{0}^{-1}Q_{0,m}+m^{\frac{1}{2}+\mu}y_{0}^{-\frac{1}{2}+\mu}Q_{0,m}$
$\displaystyle\lesssim\left(1+\frac{1}{m^{1+\mu}}y_{0}^{-\frac{1}{2}-\mu}+m^{-\frac{1}{2}}y_{0}^{-1}+m^{\frac{1}{2}}y_{0}^{-\frac{1}{2}}\right)Q_{1,m},$

while for $my_{0}\leq 3\beta$ and $\beta^{2}=1/4$ we have that

$\displaystyle|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|$
$\displaystyle\lesssim
Q_{1,m}+\frac{1}{m}y_{0}^{-\frac{1}{2}}\left(1+\big{|}\log\left(my_{0}\right)\big{|}\right)Q_{1,m}+m^{-\frac{1}{2}}y_{0}^{-1}Q_{0,m}+m^{\frac{1}{2}}y_{0}^{-\frac{1}{2}}Q_{0,m}$
$\displaystyle\lesssim\left(1+\frac{1}{m}y_{0}^{-\frac{1}{2}}\left(1+\big{|}\log\left(my_{0}\right)\big{|}\right)+m^{-\frac{1}{2}}y_{0}^{-1}+m^{\frac{1}{2}}y_{0}^{-\frac{1}{2}}\right)Q_{1,m}.$

Similarly, for $my_{0}\geq 3\beta$ and $m(1-y_{0})\leq 3\beta$ we conclude

$\displaystyle|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|$
$\displaystyle\lesssim
Q_{1,m}+mQ_{0,m}+(1-y_{0})^{-\frac{1}{2}}Q_{0,m}\lesssim\left(m+(1-y_{0})^{-\frac{1}{2}}\right)Q_{1,m},$

whereas for $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$ we obtain

$\displaystyle|\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|$
$\displaystyle\lesssim Q_{1,m}+mQ_{0,m}\lesssim mQ_{1,m}$

and the proof is finished. ∎

We next present estimates for

$\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},z)=\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\left(\partial_{y}+\partial_{y_{0}}\right)^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0}),$

at $z=0$. As before, we only obtain these bounds under the assumption that
$m|y-y_{0}|\leq 3\beta$. We state them for $z=0$; the result for $z=1$ is the
same and thus we omit the details. The next two Propositions are a direct
consequence of Propositions 8.1, 8.2, 8.18, as well as Lemma A.9 A.13, A.14,
A.18, depending on $\beta^{2}$.

######  Proposition 8.19.

Let $\beta^{2}\neq 1/4$ and $y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq
3\beta$. Then,

  * •

For $my_{0}\leq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{3}{2}}(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}-\mu}Q_{1,m}$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\leq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim\left(m+(1-y_{0})^{-\frac{1}{2}}\right)(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}-\mu}Q_{1,m},$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim
m(m|y-y_{0}\pm i\varepsilon|)^{\frac{1}{2}-\mu}Q_{1,m}.$

######  Proposition 8.20.

Let $\beta^{2}=1/4$ and $y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq 3\beta$.
Then,

  * •

For $my_{0}\leq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{3}{2}}(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m}$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\leq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim\left(m+(1-y_{0})^{-\frac{1}{2}}\right)(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m},$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$,

$|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(y,y_{0},0)|\lesssim
m(m|y-y_{0}\pm
i\varepsilon|)^{\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m}.$

We upgrade the above pointwise estimates to integral bounds for $y\in[0,1]$
such that $2\beta\leq m|y-y_{0}|\leq 3\beta$, which will be useful later on.

######  Corollary 8.21.

Let $0\leq\varepsilon\leq\varepsilon_{0}\leq\frac{\beta}{m}$. Then,

  * •

For $my_{0}\leq 3\beta$,

$\|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{-1}y_{0}^{-\frac{3}{2}}Q_{1,m}$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\leq 3\beta$,

$\|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim
m^{-\frac{1}{2}}\left(m+(1-y_{0})^{-\frac{1}{2}}\right)Q_{1,m},$

  * •

For $my_{0}\geq 3\beta$ and $m(1-y_{0})\geq 3\beta$,

$\|\widetilde{{{\mathcal{B}}_{m,\varepsilon}^{\pm}}}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{\frac{1}{2}}Q_{1,m}.$

We finish the section with estimates for

$\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},z)=\partial_{y}\partial_{z}{\mathcal{G}}_{m,\varepsilon}^{\pm}(y,y_{0},z)\partial_{z}\varphi_{m,\varepsilon}^{\pm}(z,y_{0})$

for $z=0$ and $z=1$ under the localizing assumption that $m|y-y_{0}|\leq
3\beta$. The next two results follows directly from Proposition 8.1,
Proposition 8.2 and Proposition 8.3.

######  Proposition 8.22.

Let $\beta^{2}\neq 1/4$. Let $y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq
3\beta$. Then,

  * •

For $my_{0}\leq 3\beta$, we have that

$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{1}{2}+\mu}|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}-\mu}Q_{0,m}.$

  * •

For $my_{0}\geq 3\beta$, we have that

$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{\frac{1}{2}-\mu}|y-y_{0}\pm i\varepsilon|^{-\frac{1}{2}-\mu}Q_{0,m}.$

######  Proposition 8.23.

Let $b^{2}=1/4$ and $y,y_{0}\in[0,1]$ such that $m|y-y_{0}|\leq 3\beta$. Then,

  * •

For $my_{0}\leq 3\beta$, we have that

$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{-\frac{1}{2}}y_{0}^{-\frac{1}{2}}|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{0,m}.$

  * •

For $my_{0}\geq 3\beta$, we have that

$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},0)|\lesssim
m^{\frac{1}{2}}|y-y_{0}\pm
i\varepsilon|^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{0,m}.$

Finally, we state the integral bounds that are deduced from the above
estimates.

######  Corollary 8.24.

Let $0\leq\varepsilon\leq\varepsilon_{0}$. Then,

  * •

For $my_{0}\leq 3\beta$, we have that

$\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim(my_{0})^{-\frac{1}{2}}Q_{0,m}.$

  * •

For $my_{0}\geq 3\beta$, we have that

$\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},0)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{\frac{1}{2}}Q_{0,m}.$

##  9\. Estimates for the Generalized Stream-functions

This section is devoted to obtaining estimates for the generalized stream-
functions $\psi_{m,\varepsilon}^{\pm}(y,y_{0})$ and densities
$\rho_{m,\varepsilon}^{\pm}(y,y_{0},z)$, as well as for some of their
derivatives. Moreover, we define

$\widetilde{\psi_{m}}(y,y_{0}):=\lim_{\varepsilon\rightarrow
0}\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0}),$

and similarly

$\widetilde{\rho_{m}}(y,y_{0}):=\lim_{\varepsilon\rightarrow
0}\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0}).$

We state the following proposition regarding estimates for
$\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})$ and
$\partial_{y,y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})$, from which one
obtains the corresponding estimates for
$\partial_{y_{0}}\widetilde{\psi_{m}}(y,y_{0})$ and
$\partial_{y,y_{0}}^{2}\widetilde{\psi_{m}}(y,y_{0})$, respectively.

######  Proposition 9.1.

The following holds true.

  * $(i)$

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}\neq 1/4$, we have that

$|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{1,m}+\sum_{\sigma=0,1}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|,$

and

$|\partial_{y,y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{3}{2}-\mu}Q_{1,m}+\sum_{\sigma=0,1}|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|,$

where the bounds for $|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$
and $|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$ for
$\sigma=0,1$ are given in Propositions 8.11 and 8.22, respectively.

  * $(ii)$

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}=1/4$, we have that

$|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\frac{1}{m}|y-y_{0}|^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m}+\sum_{\sigma=0,1}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|,$

and

$|\partial_{y,y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\frac{1}{m}|y-y_{0}|^{-\frac{3}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m}+\sum_{\sigma=0,1}|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|,$

where the bounds for $|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$
and $|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$ for
$\sigma=0,1$ are given in Propositions 8.12 and 8.23, respectively.

  * $(iii)$

For $m|y-y_{0}|\geq 3\beta$, we have that

$\|\partial_{y}\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}\|_{L^{2}_{y}(J_{3}^{c})}^{2}+m^{2}\|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}\|_{L^{2}_{y}(J_{3}^{c})}^{2}\lesssim
Q_{1,m}^{2}+m^{2}\sum_{\sigma=0,1}\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}^{2},$

where the bounds for
$\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$ are given in Corollary 8.13.

In particular, these bounds also apply to
$\partial_{y_{0}}\widetilde{\psi_{m}}(y,y_{0})$ and
$\partial_{y,y_{0}}^{2}\widetilde{\psi_{m}}(y,y_{0})$.

###### Proof.

Both $(i)$ and $(ii)$ follows from Proposition 3.5 and Proposition 7.2. As for
$(iii)$, we argue assuming that $\beta^{2}\neq 1/4$. Taking a
$\partial_{y_{0}}$ derivative in (2.11), we see that

$\textsc{TG}_{m,\varepsilon}^{\pm}\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}=\partial_{y_{0}}F_{m,\varepsilon}^{\pm}-2\beta^{2}\frac{1}{(y-y_{0}\pm
i\varepsilon)^{3}}\varphi_{m,\varepsilon}^{\pm}.$

In order to use Lemma 7.1, we need to control
$\left\|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}\right\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$ and $\left\|\frac{1}{(y-y_{0}\pm
i\varepsilon)^{3}}\varphi_{m,\varepsilon}^{\pm}\right\|_{L^{2}_{y}(J_{2}^{c})}$.
We begin by estimating

$\int_{y_{0}+\frac{2\beta}{m}}^{y_{0}+\frac{3\beta}{m}}|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|^{2}\mathrm{d}y\lesssim\sum_{\sigma=0,1}\int_{y_{0}+\frac{2\beta}{m}}^{y_{0}+\frac{3\beta}{m}}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|^{2}\mathrm{d}y+Q_{1,m}^{2}\int_{y_{0}+\frac{2\beta}{m}}^{y_{0}+\frac{3\beta}{m}}\frac{1}{m^{2+2\mu}}|y-y_{0}|^{-1-2\mu}\mathrm{d}y$

Now, for $\beta^{2}<1/4$ we have $\mu\neq 0$ and

$Q_{1,m}^{2}\int_{y_{0}+2\frac{\beta}{m}}^{y_{0}+3\frac{\beta}{m}}\frac{1}{m^{2+2\mu}}|y-y_{0}|^{-1-2\mu}\mathrm{d}y\lesssim\frac{1}{m^{2}}Q_{1,m}^{2},$

while for $\beta^{2}>1/4$, we have $\mu=0$ and therefore the bound still
becomes

$Q_{1,m}^{2}\int_{y_{0}+2\frac{\beta}{m}}^{y_{0}+3\frac{\beta}{m}}\frac{1}{m^{2}}|y-y_{0}|^{-1}\mathrm{d}y\lesssim\frac{1}{m^{2}}Q_{1,m}^{2}\left(\log\left(\frac{3\beta}{m}\right)-\log\left(\frac{2\beta}{m}\right)\right)\lesssim\frac{1}{m^{2}}Q_{1,m}^{2}.$

Therefore, we conclude that

$\left\|\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}\right\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim\frac{1}{m}Q_{1,m}+\sum_{\sigma=0,1}\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$

On the other hand, we use Proposition 7.2 applied to
$\varphi_{m,\varepsilon}^{\pm}(y,y_{0})$ to estimate

$\left\|\frac{1}{(y-y_{0}\pm
i\varepsilon)^{3}}\varphi_{m,\varepsilon}^{\pm}\right\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}^{2}\lesssim\int_{2\beta\leq m|y-y_{0}|\leq
3\beta}\frac{1}{m^{2+2\mu}}|y-y_{0}\pm
i\varepsilon|^{-5-2\mu}Q_{0,m}\mathrm{d}y\lesssim m^{2}Q_{0,m}$

and

$\left\|\frac{1}{(y-y_{0}\pm
i\varepsilon)^{3}}\varphi_{m,\varepsilon}^{\pm}\right\|_{L^{2}_{y}(J_{3}^{c})}^{2}\lesssim
m^{6}\|\varphi_{m,\varepsilon}^{\pm}\|_{L^{2}_{y}(J_{3}^{c})}\lesssim
m^{2}Q_{0,m}.$

The result follows from applying Lemma 7.1. ∎

The next proposition gives bounds on
$\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}$ and therefore also on
$\partial_{y_{0}}^{2}\widetilde{\psi_{m}}(y,y_{0})$.

######  Proposition 9.2.

The following holds true.

  * •

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}\neq 1/4$, we have that

$|\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{3}{2}-\mu}Q_{2,m}+\sum_{\sigma=0,1}\Big{(}|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|+|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(y,y_{0},\sigma)|\Big{)},$

where the bounds for
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$ and
$|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(y,y_{0},\sigma)|$ are given
in Propositions 8.22 and 8.19, respectively.

  * •

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}=1/4$, we have that

$\displaystyle|\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|$
$\displaystyle\lesssim\frac{1}{m}|y-y_{0}|^{-\frac{3}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{2,m}$
$\displaystyle\quad+\sum_{\sigma=0,1}\Big{(}|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|+|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(y,y_{0},\sigma)|\Big{)},$

where the bounds for
$|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$ and
$|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(y,y_{0},\sigma)|$ are given
in Propositions 8.23 and 8.20, respectively.

  * •

For $m|y-y_{0}|\geq 3\beta$, we have that

$\displaystyle\|\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}\|_{L^{2}_{y}(J_{3}^{c})}$
$\displaystyle\lesssim
Q_{2,m}+\sum_{\sigma=0,1}\left(\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}+\|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\right)$
$\displaystyle\quad+m\sum_{\sigma=0,1}\|B_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})},$

where the estimates for
$\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$,
$\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$ and
$\|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$ are given in Corollaries 8.13, 8.21 and 8.24, respectively.

In particular, these bounds also apply to
$\partial_{y_{0}}^{2}\widetilde{\psi_{m}}(y,y_{0})$.

###### Proof.

The first two statements of the proposition follow from Proposition 3.5 and
Proposition 7.2. For the third part of the proposition, we argue for
$\beta^{2}\neq 1/4$. Taking $\partial_{y_{0}}^{2}$ derivatives to (2.11), we
see that $\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})$ solves

$\textsc{TG}_{m,\varepsilon}^{\pm}\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}=\partial_{y_{0}}^{2}F_{m,\varepsilon}^{\pm}-4\beta^{2}\frac{\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{3}}-6\beta^{2}\frac{\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{4}}.$

In order to use Lemma 7.1, we need to bound
$\|\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$, as well as
$\|\frac{\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{3}}\|_{L^{2}_{y}(J_{2}^{c}\cap J_{3})}$ and
$\|\frac{\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{4}}\|_{L^{2}_{y}(J_{2}^{c}\cap J_{3})}$. We estimate

$\displaystyle\int_{2\beta\leq m|y-y_{0}|\leq 3\beta}$
$\displaystyle|\partial_{y_{0}}^{2}\varphi_{m,\varepsilon}^{\pm}(y,y_{0})|^{2}\mathrm{d}y$
$\displaystyle\lesssim Q_{2,m}\int_{2\beta\leq|y-y_{0}|\leq
3\beta}\frac{1}{m^{2+2\mu}}|y-y_{0}|^{-3-2\mu}\mathrm{d}y$
$\displaystyle\qquad+\sum_{\sigma=0,1}\Big{(}\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}+\|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\Big{)}$ $\displaystyle\lesssim
Q_{2,m}+\sum_{\sigma=0,1}\Big{(}\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}+\|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\pm}}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\Big{)}.$

Similarly, from Proposition 9.1 we have that

$\left\|\frac{\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{3}}\right\|_{L^{2}_{y}(J_{2}^{c})}\lesssim
m^{3}\|{\partial_{y_{0}}\varphi_{m,\varepsilon}^{\pm}}\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim
m^{2}Q_{1,m}+m^{3}\sum_{\sigma=0,1}\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(\cdot,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})},$

while using Proposition 7.2 we obtain

$\left\|\frac{\varphi_{m,\varepsilon}^{\pm}}{(y-y_{0}\pm
i\varepsilon)^{4}}\right\|_{L^{2}_{y}(J_{2}^{c})}\lesssim
m^{4}\|{\varphi_{m,\varepsilon}^{\pm}}\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{2}Q_{0,m}.$

With this, the proof is complete. ∎

We finish the subsection by providing the estimates for $\widetilde{\rho_{m}}$
and $\partial_{y_{0}}\widetilde{\rho_{m}}$.

######  Proposition 9.3.

The following holds true.

  * •

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}\neq 1/4$, we have that

$|\widetilde{\rho_{m}}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{0,m}.$

and

$|\partial_{y_{0}}\widetilde{\rho_{m}}(y,y_{0})|\lesssim\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{3}{2}-\mu}Q_{1,m}+\sup_{0\leq\varepsilon\leq\varepsilon_{0}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}|y-y_{0}+\kappa
i\varepsilon|^{-1}|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)|,$

where the bounds for $|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$
for $\sigma=0,1$ are given in Proposition 8.11.

  * •

For $m|y-y_{0}|\leq 3\beta$ and $\beta^{2}=1/4$, we have that

$|\widetilde{\rho_{m}}(y,y_{0})|\lesssim\frac{1}{m}|y-y_{0}|^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{0,m}.$

and

$\displaystyle|\partial_{y_{0}}\widetilde{\rho_{m}}(y,y_{0})|$
$\displaystyle\lesssim\frac{1}{m}|y-y_{0}|^{-\frac{3}{2}}\left(1+\big{|}\log\left(m|y-y_{0}\pm
i\varepsilon|\right)\big{|}\right)Q_{1,m}$
$\displaystyle\quad+\sup_{0\leq\varepsilon\leq\varepsilon_{0}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}|y-y_{0}+\kappa
i\varepsilon|^{-1}|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)|,$

where the bounds for $|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|$
for $\sigma=0,1$ are given in Proposition 8.12.

  * •

For $m|y-y_{0}|\geq 3\beta$, we have that

$\|\widetilde{\rho_{m}}\|_{L^{2}_{y}(J_{3}^{c})}\lesssim\frac{1}{m}Q_{0,m}$

and

$\|\partial_{y_{0}}\widetilde{\rho_{m}}\|_{L^{2}_{y}(J_{3}^{c})}\lesssim
Q_{1,m}+m\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}$

###### Proof.

The bounds follow directly from Proposition 3.6, Proposition 7.2 and
Proposition 9.1. ∎

##  10\. Time-decay estimates

This section is devoted to the proof of the time decay estimate rates of the
stream function $\psi_{m}(t,y)$, its $\partial_{y}\psi_{m}(t,y)$ derivative
and the density $\rho_{m}(t,y)$. Let us recall that we can write

$\psi_{m}(t,y)=\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow 0}\int_{0}^{1}{\rm
e}^{-imy_{0}t}\left(\psi^{-}_{m}(y,y_{0})-\psi_{m}^{+}(y,y_{0})\right)\mathrm{d}y_{0},$

and

$\rho_{m}(t,y)=\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow 0}\int_{0}^{1}{\rm
e}^{-imy_{0}t}\left(\rho^{-}_{m}(y,y_{0})-\rho_{m}^{+}(y,y_{0})\right)\mathrm{d}y_{0}.$

A simple integration by parts provides

$\displaystyle\psi_{m}(t,y)$ $\displaystyle=-\frac{1}{2\pi
i}\frac{1}{imt}\lim_{\varepsilon\rightarrow 0}\big{[}{\rm
e}^{-imy_{0}t}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\big{]}_{y_{0}=0}^{y_{0}=1}$
$\displaystyle\quad+\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\int_{0}^{1}{\rm
e}^{-imy_{0}t}\left(\partial_{y_{0}}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y_{0}}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle=\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\int_{0}^{1}{\rm
e}^{-imy_{0}t}\left(\partial_{y_{0}}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y_{0}}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$

where we use Theorem 4 to show that the solid terms associated to the spectral
boundary vanish. Throughout the entire section, let us consider $\beta^{2}\neq
1/4$, unless we state otherwise.

We begin proving the following result.

######  Proposition 10.1.

Let $t\geq 1$. Then,

$\displaystyle\|\psi_{m}(t)\|_{L^{2}_{y}}$ $\displaystyle\lesssim
m^{-\frac{3}{2}}t^{-\frac{3}{2}+\mu}Q_{2,m}$

###### Proof.

We write

$\displaystyle\psi_{m}(t,y)$ $\displaystyle=\frac{1}{2\pi
i}\frac{1}{imt}\lim_{\varepsilon\rightarrow 0}\int_{0}^{1}{\rm
e}^{-imy_{0}t}\left(\partial_{y_{0}}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y_{0}}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}.$

Let us denote $\delta_{0}:=\min\left(\frac{3\beta}{m},\frac{1}{2}\right)$ and
let $\delta\in\left(0,\frac{\delta_{0}}{2}\right)$. In particular, we note
that $m\delta\leq 3\beta$, it is bounded. We shall first show the decay rates
for $\|\psi_{m}(t)\|_{L^{2}_{y}(\delta,1-\delta)}$ and then for
$\|\psi_{m}(t)\|_{L^{2}_{y}(0,\delta)}$ and
$\|\psi_{m}(t)\|_{L^{2}_{y}(1-\delta,1)}$. $\bullet$ Step 1. For
$y\in(\delta,1-\delta)$, we write

$\displaystyle\psi_{m}(t,y)$ $\displaystyle=\frac{1}{2\pi
i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y-\frac{\delta}{2}}+\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right){\rm
e}^{-imy_{0}t}\left(\partial_{y_{0}}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y_{0}}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle=\mathcal{T}_{1}+\mathcal{T}_{2}+\mathcal{T}_{3}.$

and we begin with estimating $\mathcal{T}_{2}$. There, we have that
$|y-y_{0}|\leq\frac{\delta}{2}\leq\frac{\delta_{0}}{4}$ and we can use the
bounds from Proposition 9.1 to bound

$\displaystyle|\mathcal{T}_{2}|$
$\displaystyle\lesssim\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}+\mu}Q_{1,m}\mathrm{d}y_{0}+\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)|\mathrm{d}y_{0}$
$\displaystyle=\mathcal{T}_{2,1}+\mathcal{T}_{2,2}.$

We can integrate directly to obtain

$|\mathcal{T}_{2,1}|\lesssim\frac{1}{m^{2+\mu}t}\delta^{\frac{1}{2}-\mu}Q_{1,m}.$

For $\mathcal{T}_{2,2}$, for $\sigma=0$ we decompose

$\displaystyle\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|\mathrm{d}y_{0}=\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\left(\chi_{y_{0}\leq\frac{3\beta}{m}}+\chi_{y_{0}>\frac{3\beta}{m}}\right)|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|\mathrm{d}y_{0}=\mathcal{T}_{2,2,1}+\mathcal{T}_{2,2,2}.$

We use Proposition 8.11 to compute

$\mathcal{T}_{2,2,1}\lesssim\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}m^{-1+\mu}y_{0}^{-\frac{1}{2}+\mu}Q_{0,m}\mathrm{d}y_{0}\lesssim\frac{1}{m^{2-\mu}t}\delta^{\frac{1}{2}+\mu}Q_{0,m}$

and

$\mathcal{T}_{2,2,2}\lesssim\frac{1}{mt}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}Q_{0,m}\mathrm{d}y_{0}\lesssim\frac{1}{mt}\delta
Q_{0,m}.$

The bounds for the terms of $T_{2,2}$ for $\sigma=1$ are the same, we omit the
details. We summarize these estimates into

$\|{\mathcal{T}}_{2}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}\left(m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}+m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}+\mu}+\delta\right)Q_{1,m}$

We shall next estimate $\mathcal{T}_{1}$, the bounds of $\mathcal{T}_{3}$ are
the same and the arguments to prove them are identical. For $\mathcal{T}_{1}$,
note that we can further integrate by parts,

$\displaystyle\mathcal{T}_{1}$ $\displaystyle=-\frac{1}{2\pi
i}\frac{1}{m^{2}t^{2}}\lim_{\varepsilon\rightarrow 0}\big{[}{\rm
e}^{-imy_{0}t}\partial_{y_{0}}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\big{]}_{y_{0}=\frac{\delta}{2}}^{y_{0}=y-\frac{\delta}{2}}$
$\displaystyle\quad+\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\int_{0}^{\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\partial_{y_{0}}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{2\pi
i}\frac{1}{m^{2}t^{2}}\lim_{\varepsilon\rightarrow
0}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\partial_{y_{0}}^{2}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle=\mathcal{T}_{1,1}+\mathcal{T}_{1,2}+\mathcal{T}_{1,3}.$

We shall treat each $\mathcal{T}_{1,i}$, for $i=1,2,3$ separately.

$\diamond$ Estimates for $\mathcal{T}_{1,1}$. For the boundary terms of
$\mathcal{T}_{1,1}$, consider first $y_{0}=y-\frac{\delta}{2}$. Then,
$|y-y_{0}|=\frac{\delta}{2}\leq\frac{\delta_{0}}{4}$, so that from Proposition
9.1 we have

$\frac{1}{m^{2}t^{2}}\left|\partial_{y_{0}}\widetilde{\psi_{m}}\left(y,y-\tfrac{\delta}{2}\right)\right|\lesssim\frac{1}{m^{2}t^{2}}\left(\frac{1}{m^{1+\mu}}\delta^{-\frac{1}{2}-\mu}Q_{1,m}+\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\left|{\mathcal{B}}_{m,\varepsilon}^{\kappa}\left(y,y-\tfrac{\delta}{2},\sigma\right)\right|\right).$

Now, from Proposition 8.11 we have

$\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\left|{\mathcal{B}}_{m,\varepsilon}^{\kappa}\left(y,y-\tfrac{\delta}{2},\sigma\right)\right|\lesssim\left(1+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}+\mu}\right)Q_{0,m},$

since $y\in(\delta,1-\delta)$ ensures $y-\frac{\delta}{2}>\frac{\delta}{2}$.

For the boundary term $\mathcal{T}_{1,1}$ associated to
$y_{0}=\frac{\delta}{2}$, since $y\in(\delta,1-\delta)$, we have that
$1-\frac{\delta}{2}\geq y-y_{0}\geq\frac{\delta}{2}$. Hence, for those
$y\in(\delta,1-\delta)$ such that $m|y-y_{0}|\leq 3\beta$, we use Proposition
9.1 to pointwise estimate

$\frac{1}{m^{2}t^{2}}\left|\partial_{y_{0}}\widetilde{\psi_{m}}\left(y,\tfrac{\delta}{2}\right)\right|\lesssim\frac{1}{m^{2}t^{2}}\left(\frac{1}{m^{1+\mu}}\delta^{-\frac{1}{2}-\mu}Q_{1,m}+\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\left|{\mathcal{B}}_{m,\varepsilon}^{\kappa}\left(y,\tfrac{\delta}{2},\sigma\right)\right|\right),$

where we further have from Proposition 8.11 that

$\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\|{\mathcal{B}}_{m,\varepsilon}^{\pm}\left(y,\tfrac{\delta}{2},0\right)\|_{L^{2}_{y}(J_{3})}\lesssim\left(m^{-\frac{3}{2}}\delta^{-\frac{1}{2}}+m^{-\frac{1}{2}}\right)Q_{0,m}.$

Next, for those $y\in(\delta,1-\delta)$ such that $m|y-y_{0}|\geq 3\beta$ we
can directly estimate in $L^{2}_{y}$ using Proposition 9.1 to deduce that

$\frac{1}{m^{2}t^{2}}\left\|\partial_{y_{0}}\widetilde{\psi_{m}}\left(y,\tfrac{\delta}{2}\right)\right\|_{L^{2}_{y}(J_{3}^{c})}\lesssim\frac{1}{m^{2}t^{2}}\left(\frac{1}{m}Q_{1,m}+\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\|{\mathcal{B}}_{m,\varepsilon}^{\kappa}\left(y,\tfrac{\delta}{2},\sigma\right)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\right),$

while from Corollary 8.13 we are able to bound

$\|{\mathcal{B}}_{m,\varepsilon}^{\pm}\left(y,\tfrac{\delta}{2},0\right)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim
m^{-\frac{3}{2}}\delta^{-\frac{1}{2}}Q_{0,m},\quad\|{\mathcal{B}}_{m,\varepsilon}^{\pm}\left(y,\tfrac{\delta}{2},1\right)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\lesssim m^{-\frac{1}{2}}Q_{0,m}.$

Therefore, we have

$\|{\mathcal{T}}_{1,1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{m^{2}t^{2}}\left(1+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}+\mu}+m^{-\frac{3}{2}}\delta^{-\frac{1}{2}}+m^{-\frac{1}{2}}\right)Q_{0,m}$

This concludes the analysis of $\mathcal{T}_{1,1}$.

$\diamond$ Estimates for $\mathcal{T}_{1,2}$. We begin by splitting

$|\mathcal{T}_{1,2}|\lesssim\frac{1}{mt}\int_{0}^{\frac{\delta}{2}}\left(\chi_{m|y-y_{0}|\leq
3\beta}+\chi_{m|y-y_{0}|>3\beta}\right)|\partial_{y_{0}}\widetilde{\psi_{m}}(y,y_{0})|\mathrm{d}y_{0}=\mathcal{T}_{1,2,1}+\mathcal{T}_{1,2,2}$

We use Proposition 9.1 to estimate

$|{\mathcal{T}}_{1,2,1}|\lesssim\frac{1}{mt}\int_{0}^{\frac{\delta}{2}}\left(\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{1,m}+\chi_{m|y-y_{0}|\leq
3\beta}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|\right)\mathrm{d}y_{0}$

Now, since $y\in(\delta,1-\delta)$ and $y_{0}\leq\frac{\delta}{2}$, we have
$|y-y_{0}|\geq\frac{\delta}{2}$. Hence,

$\int_{0}^{\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{1,m}\mathrm{d}y_{0}\lesssim\frac{1}{m^{1+\mu}}\delta^{\frac{1}{2}-\mu}Q_{1,m}.$

Moreover, Proposition 8.11 provides

$\displaystyle\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}$
$\displaystyle\left\|\int_{0}^{\frac{\delta}{2}}\chi_{m|y-y_{0}|\leq
3\beta}|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)|\mathrm{d}y_{0}\right\|_{L^{2}_{y}(\delta,1-\delta)}$
$\displaystyle\lesssim\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{0}^{\frac{\delta}{2}}\left\|{\mathcal{B}}_{m,\varepsilon}^{\pm}(y,y_{0},\sigma)\right\|_{L^{2}_{y}(J_{3})}\mathrm{d}y_{0}$
$\displaystyle\lesssim
m^{-\frac{1}{2}}\delta^{\frac{1}{2}}\left(m^{-1}+\delta^{\frac{1}{2}}\right)Q_{0,m}.$

As a result, we are able to bound

$\|{\mathcal{T}}_{1,2,1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}\left(m^{-1-\mu}\delta^{\frac{1}{2}-\mu}+m^{-\frac{3}{2}}\delta^{\frac{1}{2}}+m^{-\frac{1}{2}}\delta\right)Q_{1,m}.$

We again use Proposition 9.1 and Corollary 8.13 to estimate

$\displaystyle\|{\mathcal{T}}_{1,2,2}\|_{L^{2}_{y}(\delta,1-\delta)}$
$\displaystyle\lesssim\frac{1}{mt}\int_{0}^{\frac{\delta}{2}}\|\partial_{y_{0}}\widetilde{\psi_{m}}(y,y_{0})\|_{L^{2}_{y}(J_{3}^{c})}\mathrm{d}y_{0}$
$\displaystyle\lesssim\frac{1}{mt}\int_{0}^{\frac{\delta}{2}}\left(\frac{1}{m}Q_{1,m}+\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\|{\mathcal{B}}_{m,\varepsilon}^{\kappa}\left(y,\tfrac{\delta}{2},\sigma\right)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\right)\mathrm{d}y_{0}$
$\displaystyle\lesssim\frac{1}{mt}\left(m^{-1}\delta+m^{-\frac{3}{2}}\delta^{\frac{1}{2}}+m^{-\frac{1}{2}}\delta\right)Q_{1,m}$

so that we can conclude

$\|{\mathcal{T}}_{1,2}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}\left(m^{-1-\mu}\delta^{\frac{1}{2}-\mu}+m^{-1}\delta+m^{-\frac{3}{2}}\delta^{\frac{1}{2}}+m^{-\frac{1}{2}}\delta\right)Q_{1,m}.$

$\diamond$ Estimates for $\mathcal{T}_{1,3}$. We shall split again

$|{\mathcal{T}}_{1,3}|\lesssim\frac{1}{m^{2}t^{2}}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\left(\chi_{m|y-y_{0}|\leq
3\beta}+\chi_{m|y-y_{0}|>3\beta}\right)|\partial_{y_{0}}^{2}\widetilde{\psi_{m}}(y,y_{0})|\mathrm{d}y_{0}=\mathcal{T}_{1,3,1}+\mathcal{T}_{1,3,2}.$

Now, we use Proposition 9.2 to estimate

$\displaystyle|{\mathcal{T}}_{1,3,1}|$
$\displaystyle\lesssim\frac{1}{m^{2}t^{2}}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{3}{2}-\mu}Q_{2,m}\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{m^{2}t^{2}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\chi_{m|y-y_{0}|\leq
3\beta}\Big{(}|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)|+|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\kappa}}(y,y_{0},\sigma)|\Big{)}\mathrm{d}y_{0}.$

Clearly, since $y\in(\delta,1-\delta)$, we have that

$\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{3}{2}-\mu}Q_{2,m}\mathrm{d}y_{0}\lesssim\frac{1}{m^{1+\mu}}\left[(y-y_{0})^{-\frac{1}{2}-\mu}\right]_{y_{0}=\frac{\delta}{2}}^{y_{0}=y-\frac{\delta}{2}}Q_{2,m}\lesssim\frac{1}{m^{1+\mu}}\delta^{-\frac{1}{2}-\mu}Q_{2,m}.$

Similarly, Proposition 8.19 provides

$\displaystyle\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}$
$\displaystyle\left\|\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\chi_{m|y-y_{0}|\leq
3\beta}|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\kappa}}(y,y_{0},\sigma)|\mathrm{d}y_{0}\right\|_{L^{2}_{y}(\delta,1-\delta)}$
$\displaystyle\lesssim\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{\frac{\delta}{2}}^{1-\frac{\delta}{2}}\left\|\widetilde{{\mathcal{B}}_{m,\varepsilon}^{\kappa}}(y,y_{0},\sigma)\right\|_{L^{2}_{y}(J_{3})}\mathrm{d}y_{0}$
$\displaystyle\lesssim\left(m^{-1}\delta^{-\frac{1}{2}}+m^{-\frac{1}{2}}\delta^{\frac{1}{2}}+m^{\frac{1}{2}}\right)Q_{1,m},$

while Proposition 8.22 gives

$\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}\left|\partial_{y}{{\mathcal{B}}_{m,\varepsilon}^{\kappa}}(y,y_{0},\sigma)\right|\mathrm{d}y_{0}\lesssim\left(m^{\frac{1}{2}-\mu}+(m\delta)^{\frac{1}{2}-\mu}+m^{-\frac{1}{2}}\delta^{-\frac{1}{2}+\mu}\right)Q_{0,m}.$

Therefore, we have

$\|{\mathcal{T}}_{1,3,1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}+m(m\delta)^{\frac{1}{2}-\mu}+(m\delta)^{-\frac{1}{2}}\right)Q_{2,m}.$

For ${\mathcal{T}}_{1,3,2}$, we use Minkowski inequality, Proposition 9.2 and
Corollaries 8.13, 8.21 and 8.24 to estimate

$\displaystyle\|{\mathcal{T}}_{1,3,2}\|_{L^{2}_{y}(\delta,1-\delta)}$
$\displaystyle\lesssim\frac{1}{m^{2}t^{2}}\int_{\frac{\delta}{2}}^{1-\frac{\delta}{2}}\left(Q_{2,m}+m\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\right)\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{m^{2}t^{2}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{\frac{\delta}{2}}^{1-\frac{\delta}{2}}\|\partial_{y}{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{m^{2}t^{2}}\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{\frac{\delta}{2}}^{1-\frac{\delta}{2}}\|\widetilde{B_{m,\varepsilon}^{\kappa}}(y,y_{0},\sigma)\|_{L^{2}_{y}(J_{2}^{c}\cap
J_{3})}\mathrm{d}y_{0}$
$\displaystyle\lesssim\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}\delta^{\frac{1}{2}}+m^{-1}\delta^{-\frac{1}{2}}\right)Q_{2,m}.$

Hence, we conclude that

$\|{\mathcal{T}}_{1,3}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}+m(m\delta)^{\frac{1}{2}-\mu}+m^{-\frac{1}{2}}\delta^{\frac{1}{2}}\right)Q_{2,m}$

and thus

$\|{\mathcal{T}}_{1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}}Q_{2,m}+\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{2,m}.$

In particular, gathering the estimates for ${\mathcal{T}}_{2}$ and
${\mathcal{T}}_{1,i}$, for $i=1,2,3$, we obtain

$\|\psi_{m}(t,y)\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{m^{2}t}(m\delta)^{\frac{1}{2}-\mu}Q_{2,m}+\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{2,m}$

$\bullet$ Step 2. For $y\in(0,\delta)$, we have that

$\psi_{m}(t,y)=\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right)e^{imy_{0}t}\left(\partial_{y_{0}}\psi_{m}^{-}(y,y_{0})-\psi_{m}^{+}(y,y_{0})\right)\mathrm{d}y_{0}=\widetilde{{\mathcal{T}}}_{1}+\widetilde{{\mathcal{T}}}_{2}.$

One can see that the bounds for $\widetilde{{\mathcal{T}}}_{2}$ here are the
same as the ones for ${\mathcal{T}}_{3}$, the procedure to obtain them is the
same, we thus omit the details. On the other hand, for
$\widetilde{{\mathcal{T}}}_{1}$ we argue as follows. Note that for $0\leq
y_{0}\leq y+\frac{\delta}{2}$, we have that
$|y-y_{0}|\leq\delta\leq\frac{3\beta}{m}$ and therefore we have from
Proposition 9.1,

$|\widetilde{{\mathcal{T}}}_{1}|\lesssim\frac{1}{mt}\int_{0}^{y+\frac{\delta}{2}}\left(\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{1,m}+\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\left|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)\right|\right)\mathrm{d}y_{0}.$

Since $y\in(0,\delta)$, we trivially have that

$\frac{1}{mt}\int_{0}^{y+\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}Q_{1,m}\mathrm{d}y_{0}\lesssim\frac{1}{mt}m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{1,m}.$

Similarly, using the bounds from Proposition 8.11,

$\displaystyle\sum_{\sigma=0,1}\sum_{\kappa\in\\{+,-\\}}\int_{0}^{y+\frac{\delta}{2}}\left|{\mathcal{B}}_{m,\varepsilon}^{\kappa}(y,y_{0},\sigma)\right|\mathrm{d}y_{0}$
$\displaystyle\lesssim\int_{0}^{y+\frac{\delta}{2}}\left(1+m^{-1+\mu}y_{0}^{-\frac{1}{2}+\mu}\right)\mathrm{d}y_{0}$
$\displaystyle\lesssim
y+\tfrac{\delta}{2}+m^{-1+\mu}\left(y+\tfrac{\delta}{2}\right)^{\frac{1}{2}+\mu}.$

As a result, we compute

$\|\widetilde{{\mathcal{T}}}_{1}\|_{L^{2}_{y}(0,\delta)}\lesssim\frac{1}{mt}\left(m^{-2}(m\delta)^{1-\mu}+\delta^{\frac{3}{2}}+m^{-2}(m\delta)^{1+\mu}\right)Q_{1,m}\lesssim\frac{1}{mt}m^{-\frac{3}{2}}(m\delta)^{1-\mu}Q_{1,m},$

and thus we obtain

$\|\psi_{m}(t)\|_{L^{2}_{y}(0,\delta)}\lesssim\frac{1}{mt}m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}}Q_{2,m}+\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{2,m}.$

The same bounds can be achieved for $\|\psi_{m}(t)\|_{L^{2}_{y}(1-\delta,1)}$,
the proof of which follows along the same ideas, we omit the details. With all
these bounds, we are now able to estimate

$\displaystyle\|\psi_{m}(t)\|_{L^{2}_{y}}$
$\displaystyle\lesssim\frac{1}{mt}m^{-1}(m\delta)^{\frac{1}{2}-\mu}Q_{2,m}+\frac{1}{m^{2}t^{2}}\left(m^{\frac{1}{2}}+m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{2,m}$
$\displaystyle\lesssim m^{-\frac{3}{2}}t^{-\frac{3}{2}+\mu}Q_{2,m}$

once we choose $\delta=\frac{c_{0}}{mt}$, for
$c_{0}=\frac{1}{1000}\min(\beta,1)$. The proof is complete. ∎

######  Proposition 10.2.

Let $t\geq 1$. Then,

$\|\partial_{y}\psi_{m}(t)\|_{L^{2}_{y}}\lesssim
m^{-\frac{1}{2}}t^{-\frac{1}{2}+\mu}Q_{1,m}.$

###### Proof.

The argument follows the same lines as the proof for
$\|\psi_{m}(t,y)\|_{L^{2}_{y}}$. Hence, we shall only give the bounds for the
terms involved in the computation. Their proof have already been carried out
in the proof of Proposition 10.1.

$\bullet$ Step 1. For $y\in(\delta,1-\delta)$ we shall write

$\displaystyle\partial_{y}\psi_{m}(t,y)$ $\displaystyle=\frac{1}{2\pi
i}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y-\frac{\delta}{2}}+\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right)e^{-imy_{0}t}\left(\partial_{y}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle={\mathcal{I}}_{1}+{\mathcal{I}}_{2}+{\mathcal{I}}_{3}$

We begin by using Proposition 7.2 to bound

$\left\|{\mathcal{I}}_{2}\right\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\left\|\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\frac{1}{m^{1+\mu}}|y-y_{0}|^{-\frac{1}{2}-\mu}\mathrm{d}y_{0}\right\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}.$

As before, for ${\mathcal{I}}_{1}$ we split it into

$\displaystyle{\mathcal{I}}_{1}$ $\displaystyle=-\frac{1}{2\pi
i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\left[e^{-imy_{0}t}\left(\partial_{y}\psi_{m,\varepsilon}^{-}(y,y_{0})-\partial_{y}\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\right]_{y_{0}=\frac{\delta}{2}}^{y_{0}=y-\frac{\delta}{2}}$
$\displaystyle\quad+\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow
0}\int_{0}^{\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\partial_{y}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\partial_{y,y_{0}}^{2}\left(\psi_{m,\varepsilon}^{-}(y,y_{0})-\psi_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle=\mathcal{I}_{1,1}+\mathcal{I}_{1,2}+\mathcal{I}_{1,3}.$

From Proposition 7.2 we see that

$\|{\mathcal{I}}_{1,1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}Q_{0,m},\quad\|{\mathcal{I}}_{1,2}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}.$

Similarly, from Proposition 9.1, we obtain

$\|{\mathcal{I}}_{1,3}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}.$

The bounds for ${\mathcal{I}}_{3}$ are the same as the ones for
${\mathcal{I}}_{1}$, we omit the details. Recovering all terms, we conclude
that

$\|\partial_{y}\psi_{m}(t)\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}+\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}.$

$\bullet$ Step 2. For $y\in(0,\delta)$, we shall split now

$\partial_{y}\psi_{m}(t,y)=\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right)e^{-imy_{0}t}\left(\partial_{y}\psi_{m}^{-}(y,y_{0})-\partial_{y}\psi_{m}^{+}(y,y_{0})\right)\mathrm{d}y_{0}=\widetilde{{\mathcal{I}}}_{1}+\widetilde{{\mathcal{I}}}_{2}.$

As before, the bound for $\widetilde{{\mathcal{I}}}_{2}$ is the same as the
bound for ${\mathcal{I}}_{3}$. For $\widetilde{{\mathcal{I}}}_{1}$, note that
$|y-y_{0}|\leq\delta\leq\frac{3\beta}{m}$ so that we shall use Proposition 7.2
to find that

$\|\widetilde{{\mathcal{I}}}_{1}\|_{L^{2}_{y}(0,\delta)}\lesssim
m^{-2}(m\delta)^{1-\mu}Q_{0,m}.$

Gathering the previous bound, we obtain

$\|\partial_{y}\psi_{m}(t)\|_{L^{2}_{y}}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}+\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}.$

As before, the conclusion follows for $\delta=\frac{c_{0}}{mt}$, with
$c_{0}=\frac{1}{1000}\min\left(\beta,1\right)$. ∎

We next obtain the decay rates for the perturbed density.

######  Proposition 10.3.

Let $t\geq 1$. Then,

$\|\rho_{m}(t)\|_{L^{2}_{y}}\lesssim
m^{-\frac{1}{2}}t^{-\frac{1}{2}+\mu}Q_{1,m}.$

###### Proof.

The demonstration also follows the same strategy as the proof for Proposition
10.1, we just present the main ideas and bounds.

$\bullet$ Step 1. For $y\in(\delta,1-\delta)$ we write

$\displaystyle\rho_{m}(t,y)$ $\displaystyle=\frac{1}{2\pi
i}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y-\frac{\delta}{2}}+\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right)e^{-imy_{0}t}\left(\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle={\mathcal{S}}_{1}+{\mathcal{S}}_{2}+{\mathcal{S}}_{3}$

We use Proposition 9.3 to bound

$\|{\mathcal{S}}_{2}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}.$

As before, both the bounds for ${\mathcal{S}}_{3}$ and ${\mathcal{S}}_{1}$ and
the manner to show them are the same, we just comment on ${\mathcal{S}}_{1}$,
which we split as follows.

$\displaystyle{\mathcal{S}}_{1}$ $\displaystyle=-\frac{1}{2\pi
i}\frac{1}{imt}\lim_{\varepsilon\rightarrow 0}\left[{\rm
e}^{-imy_{0}t}\left(\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})\right)\right]_{y_{0}=\frac{\delta}{2}}^{y_{0}=y-\frac{\delta}{2}}$
$\displaystyle\quad+\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow
0}\int_{0}^{\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\left(\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle\quad+\frac{1}{2\pi i}\frac{1}{imt}\lim_{\varepsilon\rightarrow
0}\int_{\frac{\delta}{2}}^{y-\frac{\delta}{2}}{\rm
e}^{-imy_{0}t}\partial_{y_{0}}\left(\rho_{m,\varepsilon}^{-}(y,y_{0})-\rho_{m,\varepsilon}^{+}(y,y_{0})\right)\mathrm{d}y_{0}$
$\displaystyle=\mathcal{S}_{1,1}+\mathcal{S}_{1,2}+\mathcal{S}_{1,3}.$

From Proposition 9.3 we easily deduce

$\|{\mathcal{S}}_{1,1}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}-\mu}Q_{0,m},\quad\|{\mathcal{S}}_{1,2}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}.$

On the other hand, Proposition 9.3 also yields

$\|{\mathcal{S}}_{1,3}\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}.$

Gathering the bounds we get

$\|\rho_{m}(t)\|_{L^{2}_{y}(\delta,1-\delta)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}+\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}.$

$\bullet$ Step 2. For $y\in(0,\delta)$ we shall now consider

$\rho_{m}(t,y)=\frac{1}{2\pi i}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{y+\frac{\delta}{2}}+\int_{y+\frac{\delta}{2}}^{1}\right)e^{-imy_{0}t}\left(\rho_{m}^{-}(y,y_{0})-\rho_{m}^{+}(y,y_{0})\right)\mathrm{d}y_{0}=\widetilde{{\mathcal{S}}}_{1}+\widetilde{{\mathcal{S}}}_{2}.$

The bounds for $\widetilde{{\mathcal{S}}_{2}}$ are the same as the ones for
${\mathcal{S}}_{3}$, we just focus on $\widetilde{{\mathcal{S}}_{1}}$. From
Proposition 9.3, we see that

$\|\widetilde{{\mathcal{S}}_{1}}\|_{L^{2}_{y}(0,\delta)}\lesssim
m^{-2}(m\delta)^{1-\mu}Q_{0,m}.$

With this, it follows that

$\|\rho_{m}(t)\|_{L^{2}_{y}(0,1)}\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}-\mu}Q_{0,m}+\frac{1}{mt}\left(m^{\frac{1}{2}}+(m\delta)^{-\frac{1}{2}-\mu}\right)Q_{1,m}$

and thus the Proposition is proved once we choose $\delta=\frac{c_{0}}{mt}$,
with $c_{0}=\frac{1}{1000}\min\left(\beta,1\right)$. ∎

We next prove the inviscid damping decay estimates for the case
$\beta^{2}=1/4$. The precise bounds are recorded in the following proposition.

######  Proposition 10.4.

Let $t\geq 1$. Then,

$\displaystyle\|\psi_{m}(t)\|_{L^{2}_{y}}$ $\displaystyle\lesssim
m^{-\frac{3}{2}}t^{-\frac{3}{2}}(1+\log(t))\left(\|\rho_{m}^{0}\|_{H^{4}_{y}}+\|{\omega}_{m}^{0}\|_{H^{4}_{y}}\right)$
$\displaystyle\|\partial_{y}\psi_{m}(t)\|_{L^{2}_{y}}$ $\displaystyle\lesssim
m^{-\frac{1}{2}}t^{-\frac{1}{2}}(1+\log(t))\left(\|\rho_{m}^{0}\|_{H^{3}_{y}}+\|{\omega}_{m}^{0}\|_{H^{3}_{y}}\right)$
$\displaystyle\|\rho_{m}(t)\|_{L^{2}_{y}}$ $\displaystyle\lesssim
m^{-\frac{1}{2}}t^{-\frac{1}{2}}(1+\log(t))\left(\|\rho_{m}^{0}\|_{H^{3}_{y}}+\|{\omega}_{m}^{0}\|_{H^{3}_{y}}\right).$

###### Proof.

The proof follows along the same lines for the case $\beta^{2}\neq 1/4$, the
only difference is the logarithmic singularity present in the bounds of
several quantities. For this, we note that for $m\delta<1$,

$\displaystyle\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\frac{1}{m}|y-y_{0}|^{-\frac{1}{2}}$
$\displaystyle\left(1+\big{|}\log\left(m|y-y_{0}|\right)\big{|}\right)\mathrm{d}y_{0}$
$\displaystyle\lesssim
m^{-\frac{1}{2}}\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}(m|y-y_{0}|)^{-\frac{1}{2}}\left(1+\big{|}\log\left(m|y-y_{0}|\right)\big{|}\right)\mathrm{d}y_{0}$
$\displaystyle\lesssim
m^{-\frac{3}{2}}\int_{-m\frac{\delta}{2}}^{m\frac{\delta}{2}}|\eta|^{-\frac{1}{2}}(1-\log|\eta|)\mathrm{d}\eta$
$\displaystyle\lesssim
m^{-\frac{3}{2}}(m\delta)^{\frac{1}{2}}\left(1+\big{|}\log\left(m\delta\right)\big{|}\right).$

Here, we have used that, for $0<m\delta\leq 1$,

$\displaystyle-\int_{0}^{m\delta}\eta^{-\frac{1}{2}}\log(\eta)\mathrm{d}\eta=-\int_{0}^{m\delta}2\partial_{\eta}(\eta^{\frac{1}{2}})\log(\eta)\mathrm{d}\eta$
$\displaystyle=\left[-2\eta^{\frac{1}{2}}\log(\eta)\right]_{\eta=0}^{\eta=m\delta}+2\int_{0}^{m\delta}\eta^{-\frac{1}{2}}\mathrm{d}\eta$
$\displaystyle\lesssim(m\delta)^{\frac{1}{2}}\left(1+\big{|}\log\left(m\delta\right)\big{|}\right).$

The same argument also yields

$\int_{y-\frac{\delta}{2}}^{y+\frac{\delta}{2}}\frac{1}{m}|y-y_{0}|^{-\frac{3}{2}}\left(1+\big{|}\log\left(m|y-y_{0}|\right)\big{|}\right)\mathrm{d}y_{0}\lesssim
m^{-\frac{1}{2}}(m\delta)^{-\frac{1}{2}}\left(1+\big{|}\log\left(m\delta\right)\big{|}\right).$

With this, the result follows thanks to the estimates obtained in Propositions
9.1-9.3, we omit the details. ∎

Finally, Theorem 1 is a direct consequence of Propositions 10.1-10.4 together
with Parseval identity.

##  Appendix A Properties of the Whittaker functions

Here we state and prove some properties of the Whittaker functions that are
used throughout the paper, we refer to [NIST] for a complete description of
the Whittaker functions.

###  A.1. Basic definitions and asymptotic expansions

For $\gamma,\zeta\in{\mathbb{C}}$, the Whittaker function
$M_{0,\gamma}(\zeta)$ is given by

$M_{0,\gamma}(\zeta)=e^{-\frac{1}{2}\zeta}\zeta^{\frac{1}{2}+\gamma}M\left(\tfrac{1}{2}+\gamma,1+2\gamma,\zeta\right),\quad
M(a,b,\zeta)=\sum_{s=0}^{\infty}\frac{(a)_{s}}{(b)_{s}s!}\zeta^{s},$

where $(a)_{s}=a(a+1)(a+2)\dots(a+s-1)$ denotes the Pochhammer symbol. For
$\gamma=0$, we also introduce

$W_{0,0}(\zeta)=e^{-\frac{1}{2}\zeta}\sqrt{\frac{\zeta}{\pi}}\sum_{s=0}^{\infty}\frac{\left(\tfrac{1}{2}\right)_{s}}{(s!)^{2}}\zeta^{s}\left(2\frac{\Gamma^{\prime}(1+s)}{\Gamma(1+s)}-\frac{\Gamma^{\prime}(\tfrac{1}{2}+s)}{\Gamma(\tfrac{1}{2}+s)}-\log(\zeta)\right),$

where $\Gamma(x)$ denotes the Gamma function.

We recall that $\mu={\rm Re}\left(\sqrt{1/4-\beta^{2}}\right)$ and $\nu={\rm
Im}\left(\sqrt{1/4-\beta^{2}}\right)$, and set $\gamma=\mu+i\nu$. We begin by
recording some basic properties regarding complex conjugation for
$M_{0,\gamma}(\zeta)$, which can be deduce from the series definition of
$M_{0,\gamma}$ and $W_{0,0}$.

######  Lemma A.1.

We have the following

  * •

For $\beta^{2}>1/4$, then
$M_{0,i\nu}(\zeta)=\overline{M_{0,-i\nu}\left(\overline{\zeta}\right)}$.

  * •

For $\beta^{2}\leq 1/4$, then
$M_{0,\mu}(\zeta)=\overline{M_{0,\mu}\left(\overline{\zeta}\right)}$.
Additionally, for $x\in{\mathbb{R}}$ then
$M_{0,\mu}(x),W_{0,0}(x)\in{\mathbb{R}}$.

We next state an analytic continuation property, which is key in studying the
Wronskian of the Green’s function and is directly determined by the analytic
continuation of the nonentire term of $M_{0,\gamma}(\zeta)$, which is
$\zeta^{\frac{1}{2}+\gamma}$, for $\zeta\in{\mathbb{C}}$.

######  Lemma A.2 ([NIST]).

Let $\beta^{2}>0$. Then

$M_{0,\gamma}(\zeta{\rm e}^{\pm\pi i})=\pm i{\rm e}^{\pm\gamma\pi
i}M_{0,\gamma}(\zeta),\quad W_{0,0}(\zeta e^{\pm
i\pi})=\sqrt{\pi}M_{0,0}(\zeta)\pm iW_{0,0}(\zeta)$

for all $\zeta\in{\mathbb{C}}$.

The next result gives a precise description of the asymptotic expansion of
$M_{\pm}(\zeta)$ and its derivatives, for $\zeta$ in a bounded domain.

######  Lemma A.3.

Let $\zeta\in{\mathbb{C}}$. Let $B_{R}\subset{\mathbb{C}}$ denote the closed
unit ball of radius $R>0$ centered in the origin. Then,

$\displaystyle M_{0,\pm\gamma}(\zeta)$
$\displaystyle=\zeta^{\frac{1}{2}\pm\gamma}\mathcal{E}_{0,\pm\gamma}(\zeta),\quad
M_{0,\pm\gamma}^{\prime}(\zeta)=\zeta^{-\frac{1}{2}\pm\gamma}\mathcal{E}_{1,\pm\gamma}(\zeta),$

where $\mathcal{E}_{j,\pm\gamma}\in L^{\infty}(B_{R})$ and
$\|\mathcal{E}_{j,\pm\gamma}\|_{L^{\infty}(B_{R})}\lesssim_{\gamma,R}1$, for
all $j\in\left\\{0,1,2\right\\}$.

Moreover, for $R_{m}:=\frac{R}{2m}$ and
$M_{\pm}(\zeta)=M_{0,\pm\gamma}(2m\zeta)$, let $B_{R_{m}}\subset{\mathbb{C}}$
denote the closed ball centered in the origin of radius $R_{m}$. We have that

$\displaystyle M_{\pm}(\zeta)$
$\displaystyle=\zeta^{\frac{1}{2}\pm\gamma}\mathcal{E}_{m,0,\pm\gamma}(\zeta),\quad
M_{\pm}^{\prime}(\zeta)=\zeta^{-\frac{1}{2}\pm\gamma}\mathcal{E}_{m,1,\pm\gamma}(\zeta),$

where $\mathcal{E}_{m,j,\pm\gamma}\in L^{\infty}(B_{R_{m}})$ and
$\|\mathcal{E}_{m,j,\pm\gamma}\|_{L^{\infty}(B_{R_{m}})}\lesssim_{\gamma}(2m)^{\frac{1}{2}\pm\mu}$,
for all $j\in\left\\{0,1,2\right\\}$.

###### Proof.

Firstly, from [NIST] we know that

$\displaystyle M_{0,\pm\gamma}(\zeta)$ $\displaystyle={\rm
e}^{-\frac{1}{2}\zeta}\zeta^{\frac{1}{2}\pm\gamma}M\left(\frac{1}{2}\pm\gamma,1\pm
2\gamma,\zeta\right)=\zeta^{\frac{1}{2}\pm\gamma}\mathcal{E}_{0,\pm\gamma}(\zeta),$

where $\mathcal{E}_{0,\pm\gamma}(\zeta)$ is entire and
$\|\mathcal{E}_{0,\pm\gamma}\|_{L^{\infty}(B_{R})}\lesssim_{\gamma,R}1$. On
the other hand, note that

$M_{0,\pm\gamma}^{\prime}(\zeta)=-\frac{1}{2}M_{0,\pm\gamma}(\zeta)+\left(\frac{1}{2}\pm\gamma\right)\frac{M_{0,\pm\gamma}(\zeta)}{\zeta}+\frac{1}{2}\zeta^{-\frac{1}{2}}M_{-\frac{1}{2},\frac{1}{2}\pm\gamma}(\zeta),$
(A.1)

where further

$M_{-\frac{1}{2},\frac{1}{2}\pm\gamma}(\zeta)={\rm
e}^{-\frac{1}{2}\zeta}\zeta^{1\pm\gamma}M\left(\frac{3}{2}\pm\gamma,2\pm
2\gamma,\zeta\right)=\zeta^{1\pm\gamma}\mathcal{H}_{\pm\gamma}(\zeta),$

with $\mathcal{H}_{\pm\gamma}(\zeta)$ entire and thus uniformly bounded in
$B_{R}$. Hence,

$M_{0,\pm\gamma}^{\prime}(\zeta)=\zeta^{-\frac{1}{2}-\gamma}\left(\left(\frac{1}{2}\pm\gamma\right)\mathcal{E}_{0,\pm\gamma}(\zeta)+\frac{1}{2}\zeta\left(\mathcal{H}_{\pm\gamma}(\zeta)-\mathcal{E}_{0,\pm\gamma}(\zeta)\right)\right)=\zeta^{-\frac{1}{2}-\gamma}\mathcal{E}_{1,\pm\gamma}(\zeta),$

with
$\|\mathcal{E}_{1,\pm\gamma}(\zeta)\|_{L^{\infty}(B_{R})}\lesssim_{\gamma,R}1$.
The formulas and bounds for $M_{\pm}(\zeta)=M_{0,\pm\gamma}(2m\zeta)$ and its
derivatives follow from those for $M_{0,\pm\gamma}$, the chain rule and the
observation that $2m\zeta\in B_{R}$ provided that $\zeta\in B_{R_{m}}$. ∎

######  Lemma A.4.

Let $\beta^{2}=1/4$ and $\zeta\in{\mathbb{C}}$. Let $B_{R}\subset{\mathbb{C}}$
denote the closed ball of radius $R>0$ centered at the origin. Then,

$W_{0,0}(\zeta)=\zeta^{\frac{1}{2}}\big{(}{\mathcal{E}}_{0,1}(\zeta)-\log(\zeta){\mathcal{E}}_{0,2}(\zeta)\big{)},\quad
W_{0,0}^{\prime}(\zeta)=\zeta^{-\frac{1}{2}}\big{(}{\mathcal{E}}_{1,1}(\zeta)-\log(\zeta){\mathcal{E}}_{1,2}(\zeta)\big{)},$

where ${\mathcal{E}}_{j,k}(\zeta)$ are entire functions in ${\mathbb{C}}$ and
$\|{\mathcal{E}}_{j,k}\|_{L^{\infty}(B_{R})}\lesssim 1$, for $j=0,1$ and
$k=1,2$.

###### Proof.

We begin with noting that
$W_{0,0}(2\zeta)=\sqrt{\frac{2\zeta}{\pi}}K_{0}(\zeta)$, where $K_{0}(\cdot)$
is the modified Bessel function of second kind of order 0. Moreover, we have
that

$K_{0}(\zeta)=-\left(\ln\left(\frac{\zeta}{2}\right)+\varsigma\right)I_{0}(\zeta)+2\sum_{k=1}^{\infty}\frac{I_{2k}(\zeta)}{k},$

where

$I_{2k}(\zeta)=\left(\frac{\zeta}{2}\right)^{2k}\sum_{j\geq
0}\frac{\left(\frac{\zeta^{2}}{4}\right)^{j}}{j!(2k+j)!}.$

Here, $I_{j}(\zeta)$ denotes the modified Bessel function of first kind of
order $j\in{\mathbb{N}}$. In particular, one observes that
$|I_{2k}(\zeta)|\leq I_{2k}(|\zeta|)$. Additionally, since
$\cosh(|\zeta|)=I_{0}(|\zeta|)+2\sum_{k=1}^{\infty}I_{2k}(|\zeta|)$, see
[NIST], we can bound

$\left|2\sum_{k=1}^{\infty}\frac{I_{2k}(z)}{k}\right|\leq
2\sum_{k=1}^{\infty}I_{2k}(|\zeta|)=\cosh(|\zeta|)-I_{0}(|\zeta|)<\cosh(|\zeta|).$

Therefore, since $I_{j}(\zeta)$ is analytic in ${\mathbb{C}}$ for all
$j\in{\mathbb{N}}$ and $\frac{1}{2}\zeta\in B_{R}$, we can write

$W_{0,0}(\zeta)=\zeta^{\frac{1}{2}}\big{(}{\mathcal{E}}_{0,1}(\zeta)-\log(\zeta){\mathcal{E}}_{0,2}(\zeta)\big{)},$

where

${\mathcal{E}}_{0,1}(\zeta)=\left(\log(2)-\varsigma\right)I_{0}(\zeta)+2\sum_{k=1}^{+\infty}\frac{I_{2k}(\zeta)}{k},\quad{\mathcal{E}}_{0,2}(\zeta)=I_{0}(\zeta)$

and they are such that
$\|{\mathcal{E}}_{0,j}(\zeta)\|_{L^{\infty}(B_{R})}\lesssim 1$, for $j=1,2$.

For $W_{0,0}^{\prime}(\zeta)$, note that
$W_{0,0}^{\prime}(\zeta)=\frac{1}{2\sqrt{\pi\zeta}}\left(K_{0}(\zeta/2)+\zeta
K_{0}^{\prime}(\zeta/2)\right)$. As before, we can write

$K_{0}^{\prime}(\zeta)=-K_{1}(\zeta)=-\left[\frac{1}{\zeta}I_{0}(\zeta)+\left(\log\left(\frac{1}{2}\zeta\right)+\varsigma-1\right)I_{1}(\zeta)-\sum_{k\geq
1}\frac{(1+2k)}{k(1+k)}I_{1+2k}(\zeta)\right].$

Since $\sinh(\zeta)=I_{1}(\zeta)+2\sum_{k\geq 1}I_{1+2k}(\zeta)$, confer
[NIST], we bound

$\left|\sum_{k\geq
1}\frac{(1+2k)}{k(1+k)}I_{1+2k}(\zeta)\right|\leq\sinh(|\zeta|)-I_{1}(|\zeta|)\leq\sinh(|\zeta|).$

and we conclude the existence of two entire functions
${\mathcal{E}}_{1,1}(\zeta)$ and ${\mathcal{E}}_{1,2}(\zeta)$ such that
$\|{\mathcal{E}}_{1,j}(\zeta)\|_{L^{\infty}(B_{R})}\lesssim 1$, for $j=1,2$