# Large $|k|$ behavior for the reflection coefficient for Davey-Stewartson II
equations

Christian Klein  Institut de Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France
Institut Universitaire de France
E-mail<EMAIL_ADDRESS>,  Johannes Sjöstrand  Institut de
Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France
E-mail<EMAIL_ADDRESS>and  Nikola Stoilov  Institut de
Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France
and Laboratoire Jaques-Louis Lions, UMR 7598
Faculte des Sciences er Ingeniére, Sorbonne Université, 4 Place Jussieu 75005
Paris, France
E-mail<EMAIL_ADDRESS>
###### Abstract.

The study of complex geometric optics solutions to a system of d-bar equations
appearing in the context of electrical impedance tomography and the scattering
theory of the integrable Davey-Stewartson II equations for large values of the
spectral parameter $k$ in [18] is extended to the reflection coefficient. For
the case of potentials $q$ with compact support on some domain $\Omega$ with
smooth strictly convex boundary, improved asymptotic relations are provided.

This work is partially supported by the ANR-FWF project ANuI -
ANR-17-CE40-0035, the isite BFC project NAANoD, the EIPHI Graduate School
(contract ANR-17-EURE-0002) and by the European Union Horizon 2020 research
and innovation program under the Marie Sklodowska-Curie RISE 2017 grant
agreement no. 778010 IPaDEGAN

###### Contents

  1. 1 Introduction
    1. 1.1 State of the art
    2. 1.2 Main results
  2. 2 Estimates for the operator $AB$
  3. 3 Back to the reflection coefficient
  4. 4 Stationary phase approximation
    1. 4.1 Two term stationary phase expansion
    2. 4.2 Reduction to the case of an exact quadratic
    3. 4.3 Stationary phase approximation for the reflection coefficient
    4. 4.4 Example: Characteristic function of the unit disk
  5. 5 Conclusion

##  1\. Introduction

This paper addresses the scattering problem for the integrable Davey-
Stewartson (DS) II equation given by the Dirac system

(1.1)
$\begin{cases}\overline{\partial}\phi_{1}=\frac{1}{2}q\mathrm{e}^{\overline{kz}-kz}\phi_{2},\\\
\partial\phi_{2}=\sigma\frac{1}{2}\overline{q}\mathrm{e}^{kz-\overline{kz}}\phi_{1},\quad\sigma=\pm
1,\end{cases}$

subject to the asymptotic conditions

(1.2) $\lim_{|z|\to\infty}\phi_{1}=1,\quad\lim_{|z|\to\infty}\phi_{2}=0,$

where $q=q(x,y)$ is a complex-valued field, where the _spectral parameter_
$k\in\mathbb{C}$ is independent of $z=x+\mathrm{i}y$,
$(x,y)\in\mathbb{R}^{2}$, and where

$\partial:=\frac{1}{2}\left(\frac{\partial}{\partial
x}-\mathrm{i}\frac{\partial}{\partial
y}\right)\quad\text{and}\quad\bar{\partial}:=\frac{1}{2}\left(\frac{\partial}{\partial
x}+\mathrm{i}\frac{\partial}{\partial y}\right).$

The functions $\phi_{i}(z;k)$, $i=1,2$ depend on $z$ and $k$, and are called
_complex geometric optics_ (CGO) solutions. Note that they need not be
holomorphic in either variable.

In addition to the DS II system, the CGO solutions appear in the scattering
theory of two-dimensional integrable equations as the Kadomtsev-Petviashvili
and the Novikov-Veselov equation, see [17] for references, in _electrical
impedance tomography_ (EIT), see [30, 23], and Normal Matrix Models in Random
Matrix Theory, see e.g. [14]. Our main interest in this paper is in the
scattering data, the so-called _reflection coefficient_ $R$ defined by

(1.3)
$\overline{R}=\frac{2\sigma}{\pi}\int_{\mathbb{C}}\mathrm{e}^{kz-\overline{kz}}\overline{q}(z)\phi_{1}(z;k)L(\mathrm{d}z),$

where $L(dz)$ is the Lebesgue measure on the complex plane.

We are in particular interested in the case that the potential $q$ has compact
support on some simply connected domain $\Omega\subset\mathbb{C}$ with a
smooth boundary. This is a typical situation in EIT since the body of a
patient has obviously compact support. As an example for such a situation, we
study in this paper the case that $q$ is the characteristic function of the
domain $\Omega$. In the context of DS II such a setting would correspond to a
situation as in the seminal work by Gurevich and Pitaevski [12] for the
Korteweg-de Vries (KdV) equation. For dispersive PDEs as DS and KdV, rapid
modulated oscillations are expected in the vicinity of a discontinuity of the
initial data $q$ called _dispersive shock waves_ (DSW). A detailed study of
this case for DS would allow more insight into the formation of DSWs for the
DS II system.

Since it is analytically difficult to solve d-bar systems explicitly, a large
number of numerical approaches has been developed. The most popular ones are
based on discrete Fourier transforms applied to the solution of d-bar
equations in terms of the solid Cauchy transform, see [22, 23]. The first
approach along these lines with an exponential decrease of the numerical error
with the number of Fourier modes has been given in [14, 15] for Schwartz class
potentials. A similar dependence of the error on the numerical resolutions
could be achieved for potentials with compact support on a disk in [21].
However, it is only possible to reach _machine precision_ (here $10^{-16}$)
for values of $|k|\simeq 1000$. Therefore the numerical approach [21] was
complemented in [18] with explicit asymptotic formulae for large values of
$|k|$ for potentials being the characteristic function of a compact domain.
These results will be extended in the present paper to allow for sharper
asymptotic results and explicit expressions for the reflection coefficient.

###  1.1. State of the art

We briefly summarize the state of the art of the theory of the Dirac system
(1.1) and the results of [18, 19] to be generalized in this paper. The
question of existence and uniqueness of CGO solutions to system (1.1) with
$\sigma=1$ was studied in [3] for Schwartz class potentials and in [27, 28,
29] for potentials $q\in L^{\infty}(\mathbb{C})\cap L^{1}(\mathbb{C})$ such
that also $\hat{q}\in L^{\infty}(\mathbb{C})\cap L^{1}(\mathbb{C})$ where
$\hat{q}$ is the Fourier transform of $q$ (the potentials have to satisfy a
smallness condition in the focusing case $\sigma=-1$). In [6] this was
generalized respectively to real-valued, compactly supported potentials in
$L^{p}(\mathbb{C})$ and in [26] to potentials in $H^{1,1}(\mathbb{C})$ , and
in [25] to potentials in $L^{2}(\mathbb{C})$.

One application of the system (1.1) as shown in [9, 10] is that it gives both
the scattering and inverse scattering map for the Davey-Stewartson II equation

(1.4) $\begin{split}iq_{t}+(q_{xx}-q_{yy})+2\sigma(\Phi+|q|^{2})q&=0,\\\
\Phi_{xx}+\Phi_{yy}+2(|q|^{2})_{xx}&=0,\end{split}$

a two-dimensional nonlinear Schrödinger equation; DS is _defocusing_ for
$\sigma=1$, and _focusing_ for $\sigma=-1$. Note that DS systems appear in the
modulational regime of many dispersive equations as for instance the water
wave systems, see e.g., [17] for a review on DS equations and a comprehensive
list of references, and are only integrable for the choice of parameters in
(1.4).

The scattering data are given in terms of the reflection coefficient $R=R(k)$
in (1.3). The DS II equations (1.4) are completely integrable in the sense
that a Lax pair exists, the first part of the Lax pair being (1.1). Here
$\phi_{1}$, $\phi_{2}$, $q$ can be seen as having a dependence on the physical
time $t$ which is suppressed since it will not be studied in this paper.
However, it will play a role in the second equation of the Lax pair. We put
$\Theta=\begin{pmatrix}\phi_{1}e^{kz}\\\
\phi_{2}e^{\overline{kz}}\end{pmatrix}$ and get for the Lax pair

(1.7) $\displaystyle\Theta_{x}+i\sigma_{3}\Theta_{y}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&q\\\
\overline{q}&0\end{array}\right)\Theta,$ (1.13) $\displaystyle\Theta_{t}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}i\overline{\partial}^{-1}\partial\left(|q|^{2}\right)/2&-i\partial
q\\\
i\overline{\partial}\overline{q}&-i\overline{\partial}^{-1}\partial\left(|q|^{2}\right)/2\end{array}\right)\Theta$
$\displaystyle-\left(\begin{array}[]{cc}0&q\\\
\overline{q}&0\end{array}\right)\Theta_{y}+i\sigma_{3}\Theta_{yy}\ ,$

where $\sigma_{3}=\mbox{diag}(1,-1)$ is a Pauli matrix.

As $q$ in (1.4) evolves in time $t$, the reflection coefficient evolves
because of (1.13) by a trivial phase factor:

(1.14) $R(k;t)=R(k,0)e^{4it\Re(k^{2})}.$

The inverse scattering transform for DS II is then given by (1.1) and (1.2)
after replacing $q$ by $R$ and vice versa, the derivatives with respect to $z$
by the corresponding derivatives with respect to $k$, and asymptotic
conditions for $k\to\infty$ instead of $z\to\infty$, see [1].

The main interest in [18, 19] and the present paper is in the case when $|k|$
is large, i.e., $h:=1/|k|$ small. We introduce the following notation:

$kz-\overline{kz}=i|k|\Re(z\overline{\omega})=i|k|\langle
z,\omega\rangle_{\mathbb{R}^{2}},\ \ \omega=\frac{2i\overline{k}}{|k|},$

and for $u\in L^{2}(\mathbb{R}^{2})$

$\widehat{\tau}_{\omega}u=e^{kz-\overline{kz}}u,\quad\widehat{\tau}_{-\omega}u=e^{\overline{kz}-kz}u,\quad
E=(h\overline{\partial})^{-1},F=(h{\partial})^{-1}$

which leads for (1.1) to

$\begin{cases}h\overline{\partial}\phi_{1}=\widehat{\tau}_{-\omega}h(q/2)\phi_{2},\\\
h\partial\phi_{2}=\sigma\widehat{\tau}_{\omega}h(\overline{q}/2)\phi_{1}.\end{cases}$

See [18], Section 2, for the precise choice of $E$, $F$. Looking for solutions
of the form (1.17) - (1.18) below, leads to the inhomogeneous system

$\begin{cases}\phi^{1}_{1}-E\widehat{\tau}_{-\omega}\frac{hq}{2}\phi^{1}_{2}=Ef_{1},\\\
\phi^{1}_{2}-\sigma
F\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}\phi^{1}_{1}=Ff_{2},\end{cases}$

with $f_{1}=0$ and $f_{2}=\sigma\widehat{\tau}_{\omega}h\bar{q}/2$, or

$(1-\mathcal{K})\begin{pmatrix}\phi^{1}_{1}\\\
\phi^{1}_{2}\end{pmatrix}=\begin{pmatrix}Ef_{1}\\\ Ff_{2}\end{pmatrix},$

where

(1.15) $\mathcal{K}=\begin{pmatrix}0&A\\\ B&0\end{pmatrix},\ \
\begin{cases}A=E\widehat{\tau}_{-\omega}\frac{hq}{2},\\\ B=\sigma
F\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}\
(=\sigma\overline{A})\end{cases}.$

In ([18]) we showed,

######  Proposition 1.1.

Let $q\in\langle\cdot\rangle^{-2}H^{s}$ for some $s\in]1,2]$ and fix
$\epsilon\in]0,1]$. Define $\mathcal{K}$ as in (1.15). Then
$\mathcal{K}=\mathcal{O}(1):(\langle\cdot\rangle^{\epsilon}L^{2})^{2}\to(\langle\cdot\rangle^{\epsilon}L^{2})^{2}$,

$\mathcal{K}^{2}=\mathcal{O}(h^{s-1}):\,(\langle\cdot\rangle^{\epsilon}L^{2})^{2}\to(\langle\cdot\rangle^{\epsilon}L^{2})^{2}.$

For $h_{0}>0$ small enough and $0<h\leq h_{0}$,
$1-\mathcal{K}:(\langle\cdot\rangle^{\epsilon}L^{2})^{2}\to(\langle\cdot\rangle^{\epsilon}L^{2})^{2}$
has a uniformly bounded inverse,

(1.16)
$(1-\mathcal{K})^{-1}=(1-\mathcal{K}^{2})^{-1}(1+\mathcal{K})=\begin{pmatrix}(1-AB)^{-1}&0\\\
0&(1-BA)^{-1}\end{pmatrix}\begin{pmatrix}1&A\\\ B&1\end{pmatrix}.$

When $q$ is the characteristic function of a bounded strictly convex domain
with smooth boundary, the conclusions hold with $s=\frac{3}{2}$.

###  1.2. Main results

The main goal of this paper is to obtain improved asymptotics for the
reflection coefficient, in particular for the case of potentials with compact
support. To this end we solve the system (1.1) for
$q\in\langle\cdot\rangle^{-2}H^{s}$ for some $s\in]1,2]$ for small $h$ in the
form

(1.17)
$\phi_{1}=\phi_{1}^{0}+\phi_{1}^{1},\quad\phi_{2}=\phi_{2}^{0}+\phi_{2}^{1}.$

We start with

(1.18) $\phi_{1}^{0}=1,\ \phi_{2}^{0}=0.$

The functions $\phi_{1}^{1}$ and $\phi_{2}^{1}$ should satisfy

$\begin{cases}h\overline{\partial}\phi_{1}^{1}-\widehat{\tau}_{-\omega}\frac{hq}{2}\phi_{2}^{1}=f_{1},\\\
h\partial\phi_{2}^{1}-\sigma\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}\phi_{1}^{1}=f_{2},\end{cases}$

with $f_{1}=0$ and $f_{2}=\sigma\widehat{\tau}_{\omega}h\overline{q}/2$ which
is $\mathcal{O}(h)$ in $\langle\cdot\rangle^{-2}L^{2}$. We look for
$\phi_{j}^{1}\in\langle\cdot\rangle^{\epsilon}L^{2}$ for $\epsilon\in]0,1]$.
This is equivalent to

$\begin{cases}\phi_{1}^{1}-E\widehat{\tau}_{-\omega}\frac{hq}{2}\phi_{2}^{1}=Ef_{1},\\\
\phi_{2}^{1}-\sigma
F\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}\phi_{1}^{1}=Ff_{2},\end{cases}$

or

$(1-\mathcal{K})\begin{pmatrix}\phi_{1}^{1}\\\
\phi_{2}^{1}\end{pmatrix}=\begin{pmatrix}Ef_{1}\\\ Ff_{2}\end{pmatrix}.$

Here $Ff_{2}=\sigma F\widehat{\tau}_{\omega}h\overline{q}/2=\mathcal{O}(1)$ in
$\langle\cdot\rangle^{\epsilon}L^{2}$ and Proposition 1.1 gives us a unique
solution in $(\langle\cdot\rangle^{\epsilon}L^{2})^{2}$ which is
$\mathcal{O}(1)$ in that space. More precisely by (1.16), we get

######  Theorem 1.2.

The system (1.1) has the solution (1.17), (1.18) with

(1.19) $\phi_{1}^{1}=(1-AB)^{-1}A\sigma
F\widehat{\tau}_{w}\frac{h\overline{q}}{2}=(1-AB)^{-1}AB(1)$
$\begin{split}\phi_{2}^{1}&=(1-BA)^{-1}\sigma
F\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}\end{split}$

where $\sigma F\widehat{\tau}_{\omega}h\overline{q}/2=B(1)$ and $A\sigma
F\widehat{\tau}_{\omega}h\overline{q}/2=AB(1)$ are $\mathcal{O}(1)$ in
$\langle\cdot\rangle^{\epsilon}L^{2}$ by (1.15). Note that formally
$\phi_{1}=(1-AB)^{-1}(1)$, $\phi_{2}=\phi_{2}^{1}$.

On the way we improve the results of Prop. 1.1 for potentials $q$ being the
characteristic function of a compact domain,

######  Proposition 1.3.

If $q$ is the characteristic function of a bounded strictly convex domain with
smooth boundary, the conclusions of Prop. 1.1 hold with $s=2$.

In the following we will put $\sigma=1$ for the ease of presentation. When
$\Omega\in\mathbb{C}$ is a bounded domain, let

(1.20) $D_{\Omega}(z)=\frac{1}{\pi}\int_{\Omega}\frac{1}{z-w}L(dw),\quad
z\in\mathbb{C}$

be the solution of the d-bar problem

(1.21) $\begin{cases}\partial_{\bar{z}}D_{\Omega}=1_{\Omega},&\\\
D_{\Omega}(z)\to 0,&z\to\infty.\end{cases}$

The main theorem of this paper reads

######  Theorem 1.4.

Let $\Omega\Subset\mathbb{C}$ be open with a strictly convex smooth boundary,
and let $iu(w,k)$ be a holomorphic extension of $kw-\overline{kw}$ from
$\partial\Omega$ to $\mbox{neigh}(\partial\Omega,\mathbb{C})$. $D_{\Omega}$ is
continuous, $D_{\Omega}|_{\Omega}\in C^{\infty}(\overline{\Omega})$,
$D_{\Omega}|_{\mathbb{C}\backslash\Omega}\in
C^{\infty}(\mathbb{C}\backslash\Omega)$. Moreover

(1.22)
$\begin{split}\overline{R}&=\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}L(dz)\\\
&+\frac{1}{4i\pi|k|^{2}}\left(-\int_{\tilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw+\overline{\int_{\Gamma}D_{\Omega}(w)e^{-iu(w,k)}dw}\right)+\mathcal{O}(|k|^{-3}\ln|k|),\end{split}$

for $k\in\mathbb{C}$, $|k|\gg 1$. Here $\Gamma$ is the contour from Fig. 1,
and $\tilde{\Gamma}$ is defined as $\Gamma$ after replacing $k$ with $-k$.

We parametrize the positively oriented boundary $\partial\Omega$ by
$\gamma(t)$, $t\in[0,L]$ and put
$\varphi(t)=-(k\gamma(t)-\overline{k\gamma(t)})/|k|$ as well as
$a(t)=\dot{\gamma}(t)$. We denote the critical points of $\varphi$ by
$t_{\pm}$. Applying a stationary phase approximation to reflection coefficient
(1.22), we obtain

######  Corollary 1.5.

The leading order of the reflection coefficient for $|k|\gg 1$ is given by

(1.23)
$\begin{split}\overline{R}&=\frac{i\sqrt{2\pi}}{\pi\bar{k}}\sum_{t=t_{-},t_{+}}e^{-|k|\varphi(t)}\left[|k|^{-\frac{1}{2}}a(t)\varphi^{\prime\prime}(t)^{-\frac{1}{2}}+|k|^{-\frac{3}{2}}\left(\frac{a^{\prime\prime}(t)}{2}\varphi^{\prime\prime}(t)^{-\frac{3}{2}}\right.\right.\\\
&\left.\left.-\left(a(t)\frac{\varphi^{(4)}(t)}{6}+a^{\prime}(t)\frac{\varphi^{(3)}(t)}{2}\right)\varphi^{\prime\prime}(t)^{-\frac{5}{2}}+\frac{5}{24}a(t)\varphi^{(3)}(t)^{2}\varphi^{\prime\prime}(t)^{-\frac{7}{2}}\right)\right]\\\
&+\frac{\sqrt{2\pi}}{4\pi
i|k|^{2}}\sum_{t=t_{-},t_{+}}e^{-|k|\varphi(t)}|k|^{-\frac{1}{2}}\varphi^{\prime\prime}(t)^{-\frac{1}{2}}\left(-D_{\Omega}(\gamma(t))a(t)+\overline{D_{\Omega}(\gamma(t))\bar{a}(t)}\right)\\\
&+\mathcal{O}(|k|^{-3}\ln|k|).\end{split}$

The branches of the square roots are chosen as in Remark 4.1.

The paper is organized as follows: in section 2, we give estimates for the
operator $AB$. In section 3 these estimates are applied to the reflection
coefficient. In section 4 we provide explicit formulae via a stationary phase
approximation. We consider the example of the characteristic function of the
unit disk and give a partial proof of a conjecture in [18] for the reflection
coefficient in this case. We add some concluding remarks in section 5.

##  2\. Estimates for the operator $AB$

Let $\Omega\Subset\mathbb{C}$ be strictly convex with smooth boundary. The
central problem is to study $AB$ where $A$, $B$ are given in (1.15),

$A=E\widehat{\tau}_{-\omega}\frac{hq}{2},\quad B=\sigma
F\widehat{\tau}_{\omega}\frac{h\overline{q}}{2}(=\sigma\overline{A}),\quad
q=1_{\Omega},$
$Au(z)=\frac{1}{2\pi}\int_{\Omega}\frac{1}{z-w}e^{-kw+\overline{kw}}u(w)L(\mathrm{d}w),$
$Bu(z)=\frac{\sigma}{2\pi}\int_{\Omega}\frac{1}{\overline{z}-\overline{w}}e^{kw-\overline{kw}}u(w)L(\mathrm{d}w),$
(2.1)
$\begin{split}ABu(z)&=\frac{\sigma}{4\pi^{2}}\iint_{\Omega\times\Omega}\frac{1}{z-\zeta}e^{-k\zeta+\overline{k\zeta}}\frac{1}{\overline{\zeta}-\overline{w}}e^{kw-\overline{kw}}u(w)L(\mathrm{d}\zeta)L(\mathrm{d}w)\\\
&=\int_{\Omega}K(z,w)u(w)L(\mathrm{d}w),\end{split}$ (2.2)
$K(z,w)=\frac{\sigma}{4\pi^{2}}\iint_{\Omega}\frac{e^{\overline{k\zeta}}}{(z-\zeta)}\frac{e^{-k\zeta}}{(\overline{\zeta}-\overline{w})}\frac{\mathrm{d}\overline{\zeta}\wedge\mathrm{d}\zeta}{2\mathrm{i}}e^{kw-\overline{kw}},$

for the case of the characteristic function of $\Omega$.

We look for functions $\widetilde{f}$, $\widetilde{g}$ such that with $d$
denoting exterior differentiation with respect to $\zeta$,

$\frac{e^{\overline{k\zeta}}}{(z-\zeta)}\frac{e^{-k\zeta}}{(\overline{\zeta}-\overline{w})}\mathrm{d}\overline{\zeta}\wedge\mathrm{d}\zeta=\mathrm{d}\left(e^{\overline{k\zeta}-k\zeta}(\widetilde{f}\mathrm{d}\zeta+\widetilde{g}\overline{\mathrm{d}\zeta})\right)+\dots,$

i.e.

$\frac{e^{\overline{k\zeta}}}{(z-\zeta)}\frac{e^{-k\zeta}}{(\overline{\zeta}-\overline{w})}=\partial_{\overline{\zeta}}(e^{\overline{k\zeta}-k\zeta}\widetilde{f})-\partial_{{\zeta}}(e^{\overline{k\zeta}-k\zeta}\widetilde{g})+\dots\
.$

With

$\widetilde{f}=\frac{f}{\overline{k}(z-\zeta)(\overline{\zeta}-\overline{w})},\quad\widetilde{g}=\frac{g}{{k}(z-\zeta)(\overline{\zeta}-\overline{w})},$

we get

$\partial_{\overline{\zeta}}(e^{\overline{k\zeta}-k\zeta}\widetilde{f})=\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\left(\frac{e^{\overline{k\zeta}-k\zeta}f}{(z-\zeta)(\overline{\zeta}-\overline{w})}\right)=\\\
\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)(\overline{\zeta}-\overline{w})}\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f-\frac{1}{\overline{k}}\frac{e^{\overline{k\zeta}-k\zeta}}{(\overline{\zeta}-\overline{w})}\pi
f(z)\delta_{z}(\zeta),$
$-\partial_{{\zeta}}(e^{\overline{k\zeta}-k\zeta}\widetilde{g})=-\frac{1}{k}\partial_{\zeta}\left(\frac{e^{\overline{k\zeta}-k\zeta}g}{(z-\zeta)(\overline{\zeta}-\overline{w})}\right)=\\\
\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)(\overline{\zeta}-\overline{w})}\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g-\frac{1}{k}\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)}\pi
g(w)\delta_{w}(\zeta).$

Hence,

(2.3)
$\partial_{\overline{\zeta}}(e^{\overline{k\zeta}-k\zeta}\widetilde{f})-\partial_{\zeta}(e^{\overline{k\zeta}-k\zeta}\widetilde{g})=\\\
\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)(\overline{\zeta}-\overline{w})}\left[\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f+\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g\right]\\\
-\frac{\pi
e^{\overline{k\zeta}-k\zeta}}{\overline{k}(\overline{\zeta}-\overline{w})}f\delta_{z}(\zeta)-\frac{\pi
e^{\overline{k\zeta}-k\zeta}}{k(z-\zeta)}g\delta_{w}(\zeta)$

We would like to have

$\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f(\zeta)+\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g(\zeta)=1.$

We start by constructing a partition $1=\chi_{w}+\chi_{z}+r_{w,z}$, where
$r_{w,z}$ is supported in a region $|\zeta-z|$,
$|\zeta-w|<\mathcal{O}(\frac{1}{|k|})$ and then solve, up to asymptotic
errors,

(2.4)
$\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f=\chi_{w},$
(2.5)
$\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g=\chi_{z}.$

Put

(2.6) $d(z,w,k)=|z-w|+\frac{1}{|k|},$

######  Proposition 2.1.

Let

(2.7) $\widehat{d}(z,w,\zeta,k)=d(z,w,k)+\left|\zeta-\frac{z+w}{2}\right|$

and notice that $\widehat{d}(z,w,\zeta,k)$ is uniformly of the same order of
magnitude as $d(z,w,k)+|\zeta-z|$ and $d(z,w,k)+|\zeta-w|$, since $|z-w|\leq
d(z,w,k)$.

For all $(w,z,k)\in\mathbb{C}^{3}$ with $|k|\geq 1$, there exist
$\chi_{w},\chi_{z}\in C^{\infty}(\mathbb{C})$, $r_{w,z}\in
C_{0}^{\infty}(\mathbb{C})$ such that

(2.8) $\chi_{w},\chi_{z},r_{w,z}\geq 0,$ (2.9) $\chi_{w}+\chi_{z}+r_{w,z}=1,$
(2.10)
$\begin{split}|\zeta-z|&\geq\frac{1}{\mathcal{O}(1)}\widehat{d}(z,w,\zeta,k)\text{
on }\mathrm{supp\,}\chi_{z},\\\
|\zeta-w|&\geq\frac{1}{\mathcal{O}(1)}\widehat{d}(z,w,\zeta,k)\text{ on
}\mathrm{supp\,}\chi_{w},\end{split}$ (2.11) $r_{w,z}\hbox{ has its support in
}\left\\{\zeta\in\mathbb{C};\,|\zeta-z|,\,|\zeta-w|\leq\frac{3}{2|k|}\right\\},$
(2.12)
$\nabla^{\alpha}_{\zeta}\chi_{z}=\mathcal{O}\left(\widehat{d}^{-|\alpha|}\right),\quad\nabla^{\alpha}_{\zeta}\chi_{w}=\mathcal{O}\left(\widehat{d}^{-|\alpha|}\right),\quad\forall\alpha\in\mathbb{N}^{2},\quad|\alpha|=|\alpha|_{l^{1}}=\alpha_{1}+\alpha_{2},$
(2.13)
$\nabla^{\alpha}_{\zeta}r_{z,w}=\mathcal{O}_{N,\alpha}(1)\widehat{d}^{-|\alpha|}(|k|\widehat{d})^{-N},\forall\alpha\in\mathbb{N}^{2},\forall
N\in\mathbb{N}.$

These estimates are uniform with respect to $w,z,k$.

###### Proof.

Put

$\widetilde{\chi}_{z}(\zeta)=(1-\psi^{0})\left(\frac{\zeta-z}{d}\right),\ \
\widetilde{\chi}_{w}(\zeta)=(1-\psi^{0})\left(\frac{\zeta-w}{d}\right),$

where $\psi^{0}\in C^{\infty}_{0}(D(0,1/3))$ is real valued with
$1_{D(0,1/4)}\leq\psi^{0}\leq 1_{D(0,1/3)}$. Clearly
$\widetilde{\chi}_{z}(\zeta)+\widetilde{\chi}_{z}(\zeta)\geq 0$ and

(2.14)
$\widetilde{\chi}_{z}(\zeta)+\widetilde{\chi}_{z}(\zeta)=0\Rightarrow\begin{cases}|\frac{\zeta-z}{d}|\leq\frac{1}{3}\\\
|\frac{\zeta-w}{d}|\leq\frac{1}{3}\end{cases}\Leftrightarrow\begin{cases}|\zeta-z|\leq\frac{1}{3}|z-w|+\frac{1}{3}\frac{1}{|k|},\\\
|\zeta-w|\leq\frac{1}{3}|z-w|+\frac{1}{3}\frac{1}{|k|}.\end{cases}$

The last inequalities imply that $|z-w|\leq\frac{2}{3}|z-w|+\frac{2}{3|k|}$,
so

$|z-w|\leq\frac{2}{|k|},\ \ |\zeta-z|,\,|\zeta-w|\leq\frac{1}{|k|}.$

We have $\nabla^{\alpha}_{\zeta}\widetilde{\chi}_{z}$,
$\nabla^{\alpha}_{\zeta}\widetilde{\chi}_{w}=\mathcal{O}(d^{-|\alpha|})$ and

(2.15) $\begin{split}|\zeta-z|&\geq\frac{1}{\mathcal{O}(1)}d(z,w,k)\text{ on
}\mathrm{supp\,}\widetilde{\chi}_{z},\\\
|\zeta-w|&\geq\frac{1}{\mathcal{O}(1)}d(z,w,k)\text{ on
}\mathrm{supp\,}\widetilde{\chi}_{w},\end{split}$

where we can replace $d(z,w,k)$ with $\widehat{d}(z,w,\zeta,k)$. Moreover,
since $\widetilde{\chi}_{\cdot}(\zeta)=1$ when
$|\cdot-\zeta|\geq\frac{1}{3}d$, we have

$\nabla^{\alpha}_{\zeta}\widetilde{\chi}_{z}=\mathcal{O}\left((d+|z-\zeta|)^{-|\alpha|}\right),\quad\nabla^{\alpha}_{\zeta}\widetilde{\chi}_{w}=\mathcal{O}\left((d+|w-\zeta|)^{-|\alpha|}\right).$

Hence as noted in the statement of the proposition,

(2.16) $\nabla_{\zeta}^{\alpha}\widetilde{\chi}_{z},\
\nabla_{\zeta}^{\alpha}\widetilde{\chi}_{w}=\mathcal{O}(\widehat{d}^{-|\alpha|}).$

Let $\Psi^{1}\in C^{\infty}_{0}(D(0,\frac{3}{2}))$ satisfy
$1_{D(0,1)}\leq\Psi^{1}\leq 1$ and put
$\widetilde{r}_{z,w}=\Psi^{1}(|k|(\zeta-z))\Psi^{1}(|k|(\zeta-w))$. Then, by
(2.14) and the subsequent observation,

(2.17)
$f:=\widetilde{\chi}_{z}(\zeta)+\widetilde{\chi}_{w}(\zeta)+\widetilde{r}_{z,w}\asymp
1,$

and by construction the last term has its support in

$\left\\{\zeta\in\mathbb{C};|\zeta-z|,~{}|\zeta-w|\leq\frac{3}{2|k|}\right\\},$

so that $|z-w|\leq\frac{\mathcal{O}(1)}{|k|}$,
$\widehat{d}(z,w,\zeta,k)\asymp\frac{1}{|k|}$ on
$\mathrm{supp}~{}\widetilde{r}_{z,w}$. It follows that
$\widetilde{r}_{z,w}=\mathcal{O}_{N}(1)(|k|\widehat{d})^{-N},~{}\forall N\geq
0$ and more generally

(2.18)
$\nabla^{\alpha}_{\zeta}\widetilde{r}_{z,w}=\mathcal{O}_{N,\alpha}(1)\widehat{d}^{-|\alpha|}(|k|\widehat{d})^{-N}.$

From (2.16) we get

(2.19)
$\nabla^{\alpha}_{\zeta}f,~{}\nabla^{\alpha}_{\zeta}\frac{1}{f}=\mathcal{O}(1)\widehat{d}^{-|\alpha|}.$

Put

(2.20)
$\chi_{z}=\widetilde{\chi}_{z}/f,\quad\chi_{w}=\widetilde{\chi}_{w}/f,\quad
r_{z,w}=\widetilde{r}_{z,w}/f.$

Then we have

(2.21) $\chi_{z}+\chi_{w}+r_{z,w}=1,$

and (2.8)–(2.13) follow. ∎

We now return to the problem (2.4) – (2.5). As a first approximate solution,
we take $f^{0}=\chi_{w}$, $g^{0}=\chi_{z}$ and treat the other terms in the
LHSs as perturbations. We then get $f,g$ as formal Neumann series sums

(2.22)
$f=\sum^{\infty}_{0}\left(\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}-\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)^{\nu}\chi_{w},\quad
g=\sum^{\infty}_{0}\left(\frac{1}{k(z-\zeta)}+\frac{1}{k}\partial_{\zeta}\right)^{\nu}\chi_{z}.$

Recall (2.10) and (2.11). Then

$\begin{split}\nabla^{\nu_{0}}_{\zeta}\frac{1}{(\overline{\zeta}-\overline{w})^{\nu_{1}}}\partial_{\overline{\zeta}}^{\nu_{2}}\chi_{w}&=\mathcal{O}(\widehat{d}^{-(\nu_{0}+\nu_{1}+\nu_{2})}),\\\
~{}\\\
\nabla^{\nu_{0}}_{\zeta}\frac{1}{(z-\zeta)^{\nu_{1}}}\partial_{{\zeta}}^{\nu_{2}}\chi_{z}&=\mathcal{O}(\widehat{d}^{-(\nu_{0}+\nu_{1}+\nu_{2})}).\end{split}$

Thus

(2.23)
$\left.\begin{array}[]{ll}&\displaystyle\nabla_{\zeta}^{\nu_{0}}\left(\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}-\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)^{\nu}\chi_{w}\\\
&\displaystyle\nabla_{\zeta}^{\nu_{0}}\left(\frac{1}{k(z-\zeta)}+\frac{1}{k}\partial_{\zeta}\right)^{\nu}\chi_{z}\end{array}\right\\}=\mathcal{O}(\widehat{d}^{-\nu_{0}}(k\widehat{d})^{-\nu}).$

For $N\in\mathbb{N}$ define $f_{N},~{}g_{N}$ as in (2.22) but with finite sums

(2.24)
$f_{N}=\sum^{N}_{0}\left(\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}-\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)^{\nu}\chi_{w},\quad
g_{N}=\sum^{N}_{0}\left(\frac{1}{k(z-\zeta)}+\frac{1}{k}\partial_{\zeta}\right)^{\nu}\chi_{z}.$

Notice that by (2.23)

(2.25) $f_{N},\,g_{N}=\mathcal{O}(1).$

Then c.f. (2.4), (2.5)

(2.26)
$\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f_{N}=\chi_{w}-\left(\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}-\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)^{N+1}\chi_{w}=:\chi_{w}-S^{N+1}_{w}$
(2.27)
$\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g_{N}=\chi_{z}-\left(\frac{1}{k(z-\zeta)}+\frac{1}{k}\partial_{\zeta}\right)^{N+1}\chi_{z}=:\chi_{z}-T^{N+1}_{z}$

where

(2.28)
$\nabla^{\nu_{0}}_{\zeta}S^{N+1}_{w},~{}\nabla^{\nu_{0}}_{\zeta}T^{N+1}_{z}=\mathcal{O}(\widehat{d}^{-\nu_{0}}(k\widehat{d})^{-N-1}).$

Now, combine (2.26) - (2.28) with (2.21) and (2.18) (valid also for $r_{z,w}$)
to get

(2.29)
$\left(1-\frac{1}{\overline{k}(\overline{\zeta}-\overline{w})}+\frac{1}{\overline{k}}\partial_{\overline{\zeta}}\right)f_{N}+\left(1-\frac{1}{k(z-\zeta)}-\frac{1}{k}\partial_{\zeta}\right)g_{N}=\\\
1-S_{w}^{N+1}-T_{z}^{N+1}-r_{z,w}=:1-r^{N+1}$

where

(2.30)
$\nabla^{\nu_{0}}_{\zeta}r^{N+1}=\mathcal{O}(\widehat{d}^{-\nu_{0}}(k\widehat{d})^{-N-1}),\quad\nu_{0}\in\mathbb{N}.$

We use this in the discussion after (2.2). With

$\widetilde{f}_{N}=\frac{f_{N}}{\overline{k}(z-\zeta)(\overline{\zeta}-\overline{w})},\quad\widetilde{g}_{N}=\frac{g_{N}}{k(z-\zeta)(\overline{\zeta}-\overline{w})}$

we get (cf. (2.3))

$d\left(e^{\overline{k\zeta}-k\zeta}(\widetilde{f}_{N}d\zeta+\widetilde{g}_{N}\overline{d\zeta})\right)=\\\
\left[\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)(\overline{\zeta}-\overline{w})}-\frac{e^{\overline{k\zeta}-k\zeta}r^{N+1}}{(z-\zeta)(\overline{\zeta}-\overline{w})}-\frac{\pi
e^{\overline{k\zeta}-k\zeta}f_{N}\delta_{z}(\zeta)}{\overline{k}(\overline{\zeta}-\overline{w})}-\frac{\pi
e^{\overline{k\zeta}-k\zeta}g_{N}\delta_{w}(\zeta)}{k(z-\zeta)}\right]\overline{d\zeta}\wedge
d\zeta.$

We use this in (2.2) and apply Stokes’ formula:

(2.31) $\begin{split}K(z,w)=&\frac{\sigma
e^{kw-\overline{kw}}}{(2\pi)^{2}2\mathrm{i}}\left[\underbrace{\iint_{\Omega}\frac{e^{\overline{k\zeta}-k\zeta}}{(z-\zeta)(\overline{\zeta}-\overline{w})}r^{N+1}\overline{d\zeta}\wedge
d\zeta}_{=:K_{1}(z,w)}\right.\\\ &+\underbrace{\frac{2i\pi
e^{\overline{kz}-kz}}{\overline{k}(\overline{z}-\overline{w})}f_{N}(z,w,z)1_{\Omega}(z)}_{=:K_{2}(z,w)}+\underbrace{\frac{2i\pi
e^{\overline{kw}-kw}}{k(z-w)}g_{N}(z,w,w)1_{\Omega}(w)}_{=:K_{3}(z,w)}\\\
&+\left.\underbrace{\int_{\partial\Omega}\frac{e^{\overline{k\zeta}-k\zeta}f_{N}}{\overline{k}(\overline{\zeta}-\overline{w})(z-\zeta)}d\zeta}_{=:K_{4}(z,w)}+\underbrace{\int_{\partial\Omega}\frac{e^{\overline{k\zeta}-k\zeta}g_{N}}{k(\overline{\zeta}-\overline{w})(z-\zeta)}d\overline{\zeta}}_{=:K_{5}(z,w)}\right].\end{split}$

The factor in front of the big bracket is bounded, so it will suffice to
estimate the norms between various weighted $L^{p}$ spaces of the operators

(2.32) $A_{j}u(z)=\int_{\Omega}K_{j}(z,w)u(w)L(dw)$

with integral kernel $K_{j}$. Recalling (2.30), we get

$K_{1}(z,w)=\frac{\mathcal{O}(1)}{|k|^{N+1}}\int_{\Omega}\frac{1}{|z-\zeta||w-\zeta|}\frac{1}{\left(|z-w|+|\zeta-\frac{z+w}{2}|+\frac{1}{|k|}\right)^{N+1}}L(d\zeta).$

Consider separately the integrals over
$\Omega_{1}=\\{\zeta\in\Omega;~{}|\zeta-w|\geq|\zeta-z|\\}$ and
$\Omega_{2}=\\{\zeta\in\Omega;~{}|\zeta-w|<|\zeta-z|\\}$. The two integrals
can be handled similarly, and we only need to consider the first case
$|\zeta-w|\geq|\zeta-z|$. Here $|\zeta-w|\geq\frac{1}{2}|z-w|$ and the
corresponding integral is

$\leq\frac{\mathcal{O}(1)}{|z-w|}\frac{1}{|k|^{N+1}}\int_{\Omega_{1}}\frac{1}{|z-\zeta|}\frac{1}{\left(|z-w|+|\zeta-z|+\frac{1}{|k|}\right)^{N+1}}L(d\zeta)\\\
\leq\frac{\mathcal{O}(1)}{|z-w|}\frac{1}{|k|^{N+1}}\int_{\mathbb{C}}\frac{1}{|\zeta|}\frac{1}{\left(|\zeta|+\lambda\right)^{N+1}}L(d\zeta),$

where $\lambda:=|z-w|+1/|k|$. Putting

$\zeta=\lambda\widetilde{\zeta},\ \
L(d\zeta)=\lambda^{2}L(d\widetilde{\zeta}),$

gives the upper bound

$=\frac{\mathcal{O}(1)}{|z-w|}\frac{1}{|k|^{N+1}}\frac{\lambda^{2}}{\lambda^{N+2}}\int_{\mathbb{C}}\frac{1}{|\widetilde{\zeta}|}\frac{1}{\left(|\widetilde{\zeta}|+1\right)^{N+1}}L(d\zeta)\\\
=\mathcal{O}(1)\frac{1}{|k||z-w|\left(|k|\lambda\right)^{N}}=\frac{\mathcal{O}(1)}{|k||z-w|}\frac{1}{\left(|k||z-w|+1\right)^{N}}\\\
=\mathcal{O}(1)f_{k,N}(z-w).$

Here

$f_{k,N}(z)=\frac{1}{|k||z|(1+|k||z|)^{N}},$

and

$\int f_{k,N}(z)L(dz)=\int\frac{1}{|k||z|(1+|k||z|)^{N}}L(dz)=\\\
\frac{1}{|k|^{2}}\int\frac{1}{|\widetilde{z}|(1+|\widetilde{z}|)^{N}}L(d\widetilde{z})=\mathcal{O}(1)\frac{1}{|k|^{2}}.$

We deduce that $K_{1}$ is bounded by an $L^{1}$ convolution kernel, hence

$A_{1}=\mathcal{O}(1)\frac{1}{|k|^{2}}:L^{p}\to L^{p},\quad 1\leq
p\leq\infty.$

By (2.25) we have $K_{2}(z,w),~{}K_{3}(z,w)=\frac{\mathcal{O}(1)}{|k||z-w|}$
and it follows (here we integrate over $\Omega$ in (2.32)) that for every
bounded set $V\subset\mathbb{C}$,

(2.33) $1_{V}A_{2},\ 1_{V}A_{3}=\mathcal{O}(|k|^{-1}):L^{p}\to L^{p},\quad
1\leq p\leq\infty.$

In fact, $1/|z|$ is integrable on every bounded set.

We next estimate the contribution to $A_{2},~{}A_{3}$ from $|z|\gg 1$. For
$j=2,3$ let
$C_{j}=1_{\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})}A_{j}$. Then

$C_{j}u(z)=\int_{\Omega}\widetilde{K}_{j}(z,w)u(w)L(dw),\ \
|\widetilde{K}_{j}(z,w)|\leq\frac{1}{\langle z\rangle|k|},$
$|C_{j}u(z)|\leq\mathcal{O}(1)\langle
z\rangle^{-1}|k|^{-1}||u||_{L^{1}(\Omega)}\leq\mathcal{O}(1)\langle
z\rangle^{-1}|k|^{-1}||u||_{L^{p}(\Omega)},\ 1\leq p\leq\infty.$

Here

$\int\langle z\rangle^{-q}L(dz)<\infty\text{ if }q>2,$

so

$C_{j}=\mathcal{O}\left(1/|k|\right):L^{p}\to L^{q}\text{ if }1\leq
p\leq\infty,~{}q>2.$

If $q\leq 2$, $\epsilon>0$. Then,

$\|\langle\cdot\rangle^{-1-\epsilon}\|_{L^{q}}^{q}=\int\langle
z\rangle^{-(1+\epsilon)q}L(dz)<\infty$

iff $(1+\epsilon)q>2$, i.e. iff $\epsilon>2/q-1$. When $q=2$, this amounts to
$\epsilon>0$. We conclude that

$C_{j}=\mathcal{O}_{\epsilon}(1/|k|):L^{p}\to\langle\cdot\rangle^{\epsilon}L^{q}\text{
when }1\leq p\leq\infty,~{}1\leq q\leq 2,~{}\epsilon>\frac{2}{q}-1.$

Hence for $j=2,3$:

$A_{j}=\mathcal{O}(1/|k|):\begin{cases}L^{q}\to L^{q},\hbox{ when }q>2,\\\
L^{q}\to\langle\cdot\rangle^{\epsilon}L^{q},\hbox{ when }1\leq q\leq 2,\
\epsilon>2/q-1.\end{cases}$

We next estimate $A_{4},~{}A_{5}$ with kernels $K_{4},~{}K_{5}$ in (2.31). It
suffices to treat $A_{4},K_{4}$ since the expression for $K_{5}$ is very
similar. Let

$\Gamma_{z}=\\{\zeta\in\partial\Omega;|\zeta-z|\geq\frac{1}{2}|w-z|\\},\quad\Gamma_{w}=\\{\zeta\in\partial\Omega;|\zeta-w|\geq\frac{1}{2}|z-w|\\},$

so that $\partial\Omega\subset\Gamma_{z}\cup\Gamma_{w}$. Then

(2.34)
$\begin{split}|K_{4}(z,w)|&\leq\frac{\mathcal{O}(1)}{|k|}\left(\int_{\Gamma_{z}}\frac{|d\zeta|}{|z-\zeta||\zeta-w|}+\int_{\Gamma_{w}}\frac{|d\zeta|}{|z-\zeta||\zeta-w|}\right)\\\
&\leq\frac{\mathcal{O}(1)}{|k|}\left(\frac{1}{|w-z|}\int_{\partial\Omega}\frac{|d\zeta|}{|\zeta-w|}+\frac{1}{|w-z|}\int_{\partial\Omega}\frac{|d\zeta|}{|z-\zeta|}\right)\\\
&\leq\frac{\mathcal{O}(1)}{|k|}\left(\frac{1}{|w-z|}G(w)+\frac{1}{|w-z|}G(z)\right),\end{split}$

where

(2.35) $G(z)=\begin{cases}1+|\ln
d(\partial\Omega,z)|,&z\in\mathrm{neigh}(\overline{\Omega},\mathbb{C}),\\\
1/\langle
z\rangle,&z\in\mathbb{C}\setminus\mathrm{neigh}(\overline{\Omega},\mathbb{C}),\end{cases}$

and $d(\partial\Omega,w)$ denotes the distance between $\partial\Omega$ and
$w$.

$|\cdot-z|^{-\alpha}$ and $G^{\beta}$ are integrable on any bounded set when
$\beta\geq 0$, $0<\alpha<2$. Choose $\alpha=3/2$ and write
$\frac{|G(w)|}{|w-z|}$ as the geometric mean
$\left(\frac{1}{|w-z|}^{\frac{3}{2}}\right)^{\frac{2}{3}}\left(G(w)^{3}\right)^{\frac{1}{3}}$.
Using that geometric means are bounded by the arithmetic ones, we get

$\frac{1}{|w-z|}G(w)\leq\frac{2}{3}\frac{1}{|w-z|^{3/2}}+\frac{1}{3}G(w)^{3}.$

Using this and the corresponding estimate with $G(z)$ in (2.34) we get

(2.36)
$1_{\mathrm{neigh}(\overline{\Omega})}(z)|K_{j}(z,w)|1_{\Omega}(w)\leq\frac{\mathcal{O}(1)}{|k|}\frac{1}{|z-w|^{\frac{3}{2}}}+\frac{\mathcal{O}(1)}{|k|}G(w)^{3}+\frac{\mathcal{O}(1)}{|k|}G(z)^{3},$

when $j=4$. Clearly the same estimate holds when $j=5$. The first term is an
$L^{1}$-convolution kernel (neglecting a region $|z|\gg 1$ and recalling that
$w\in\Omega$), gives rise to an operator

$\mathcal{O}(1/|k|):L^{p}(\Omega)\to
L^{p}(\mathrm{neigh\,(\overline{\Omega})),}\quad p\in[1,\infty]$

for any bounded neighborhood of $\overline{\Omega}$. By the Hölder inequality
and the fact that $G^{\alpha}\in L^{1},\forall\alpha>0$, we see that the
second term gives rise to an operator

$\mathcal{O}(1/|k|):L^{p}\to L^{\infty},\ \forall p>1.$

Similarly the third term gives rise to an operator

$\mathcal{O}(1/|k|):L^{1}(\Omega)\to
L^{q}(\mathrm{neigh\,}(\overline{\Omega})),\quad q\in[1,\infty[$

for any bounded neighborhood of $\overline{\Omega}$. Recalling again that we
work on a bounded subset of $\mathbb{C}$ where $L^{p}\subset L^{q}$ for $1\leq
q\leq p\leq\infty$ we conclude that

(2.37)
$1_{\mathrm{neigh}(\overline{\Omega})}A_{4},~{}1_{\mathrm{neigh}(\overline{\Omega})}A_{5}=\mathcal{O}(1/|k|):L^{p}\to
L^{p},\quad 1<p<\infty.$

For $z\in\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})$,
$w\in\Omega$, (2.34) and (2.35) give

$|K_{4}(z,w)|\leq\frac{\mathcal{O}(1)}{|k|\langle z\rangle}(1+|\ln
d(w,\partial\Omega)|).$

This is the same estimate as for $K_{2},K_{3}$ except that the
$1_{\Omega}(w)$, belonging to all $L^{p^{\prime}}$ with $1\leq
p^{\prime}\leq+\infty$, is replaced by
$(1+|\ln(w,\partial\Omega)|)1_{\Omega}(w)$ belonging to all $L^{p^{\prime}}$
with $1\leq p^{\prime}<+\infty$. The estimates for
$1_{\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})}A_{j}$, $j=2,3$
extend to $1_{\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})}A_{4}$,
$1_{\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})}A_{5}:L^{p}\to
L^{q}$, for $1<p\leq\infty$: For $j=4,5,$

$1_{\mathbb{C}\setminus\mathrm{neigh\,}(\overline{\Omega})}A_{j}=\mathcal{O}(1/|k|):\begin{cases}L^{p}\to
L^{q},\quad 1<p\leq\infty,\ q>2,\\\
L^{p}\to\langle\cdot\rangle^{\epsilon}L^{q},\ 1<p\leq\infty,\ 1\leq q\leq 2,\
\epsilon>\frac{2}{q}-1.\end{cases}$

Hence with (2.37)

$A_{j}=\mathcal{O}(1/|k|):\begin{cases}L^{q}\to L^{q},&2<q<+\infty,\\\
L^{q}\to\langle\cdot\rangle^{\epsilon}L^{q},&1<q\leq
2,\epsilon>\frac{2}{q}-1.\end{cases}$

Combining the estimates for $A_{j}$, $1\leq j\leq 5$, we get

######  Theorem 2.2.

(2.38) $AB=\mathcal{O}(1/|k|):\begin{cases}L^{q}\to L^{q},&2<q<+\infty,\\\
L^{q}\to\langle\cdot\rangle^{\epsilon}L^{q},&1<q\leq 2,\
\epsilon>\frac{2}{q}-1.\end{cases}$

##  3\. Back to the reflection coefficient

Recall (1.3)

(3.1)
$\overline{R}=\frac{2\sigma}{\pi}\int_{\mathbb{C}}\mathrm{e}^{kz-\overline{kz}}\overline{q}(z)\phi_{1}(z;k)L(\mathrm{d}z),$

where $\phi_{1}=1+\phi^{1}_{1}$ and we assume $q=1_{\Omega}$ where
$\Omega\Subset\mathbb{C}$ is open with strictly convex smooth boundary.
$\phi_{1}^{1}$ is given by (1.19),

(3.2) $\phi_{1}^{1}=(1-AB)^{-1}AB(1)=AB(1)+(AB)^{2}(1)+...$

and $A$, $B$ are given in (1.15), now with $q=1_{\Omega}$

(3.3) $\begin{cases}A=E\widehat{\tau}_{-\omega}\frac{h}{2}1_{\Omega},\\\
B=\sigma F\widehat{\tau}_{\omega}\frac{h}{2}1_{\Omega}\
(=\sigma\overline{A})\end{cases}.$

By Theorem 2.2 we know that $1_{\Omega}AB=\mathcal{O}(1/|k|):L^{2}(\Omega)\to
L^{2}(\Omega)$ and we shall frequently use that $A=A\circ 1_{\Omega}$,
$B=B\circ 1_{\Omega}$. Combining (3.1), (3.2), we get (assuming $\sigma=1$ for
simplicity)

(3.4)
$\overline{R}=\frac{2}{\pi}\sum_{\nu=0}^{N-1}\int_{\Omega}e^{kz-\overline{kz}}(AB)^{\nu}(1)L(dz)+\mathcal{O}(|k|^{-N}),$

for every $N=1,2,...$. Here

$\left|\int_{\Omega}e^{kz-\overline{kz}}(AB)^{\nu}(1)L(dz)\right|\leq\mathrm{vol\,}(\Omega)^{1/2}\|(AB)^{\nu}(1)\|_{L^{2}(\Omega)}\leq
C(C/|k|)^{\nu}$

and we shall see that this estimate can be improved by using more information
about $A=A_{k}$ from Section 5 in [18] in the case when $\partial\Omega$ is
analytic. In remark 3.3 we explain how to extend the discussion to the case
when $\partial\Omega$ is merely smooth.

First, recall from (3.3) and the explicit formula for the fundamental solution
of $\overline{\partial}$ appearing in $E$, that

(3.5)
$Au(z)=\frac{1}{2\pi}\int_{\Omega}e^{-kw+\overline{kw}}\frac{1}{z-w}u(w)L(dw),$

or

(3.6) $A=A_{k}=A_{0}\circ e^{-k\cdot+\overline{k\cdot}},$

where

(3.7) $A_{0}u(z)=\frac{1}{2\pi}\int_{\Omega}\frac{1}{z-w}u(w)L(dw).$

$A_{0}$ is anti-symmetric for the standard bilinear scalar product on $L^{2}$;
$A_{0}^{\mathrm{t}}=-A_{0}$, so the transpose of $A_{k}$ is given by

(3.8)
$A_{k}^{\mathrm{t}}=-e^{-k\cdot+\overline{k\cdot}}A_{0}=-e^{-k\cdot+\overline{k\cdot}}A_{-k}e^{-k\cdot+\overline{k\cdot}},$

and $B_{k}$ is the complex conjugate of $A_{k}$ (here $\sigma=1$ for
simplicity):

$B_{k}=\overline{A_{k}}.$

In [18] we studied the function

(3.9) $f(z,k)=2\pi
A_{k}(1_{\Omega})(z)=\int_{\Omega}e^{-kw+\overline{kw}}\frac{1}{z-w}L(dw),$

when $\partial\Omega$ is analytic.

By (5.32), (5.31) in [18], we have

(3.10)
$f(z,k)=\frac{1}{2i\overline{k}}F(z)+(\pi/\overline{k})\left(e^{-iu(z,k)}(1_{\Omega_{-}}(z)-1_{\Omega_{+}}(z))+e^{-kz+\overline{kz}}1_{\Omega}(z)\right),$

where

(3.11) $F(z)=F_{\Gamma}(z)=\int_{\Gamma}\frac{1}{z-w}e^{-iu(w,k)}dw.$

Here the deformation $\Gamma$ of $\partial\Omega$ and the domains
$\Omega_{\pm}$ are defined in [18, Section 5], see Fig. 1. Further
$-iu(\cdot,k)$ is the holomorphic extension of $-k\cdot+\overline{k\cdot}$
from $\partial\Omega$ to a neighborhood.

$\Gamma_{-}$$\Gamma_{+}$$\Omega_{+}$$\Omega_{-}$$w_{+}$$w_{-}$ Figure 1. Real
analytic strictly convex boundary $\partial\Omega$ (solid) of some domain
$\Omega$ and the deformed contour $\Gamma$ (dashed) for this example.

In (5.69), (5.70) in [18] we have seen that

(3.12)
$F(z)=\mathcal{O}(1)(|z-w_{+}(k)||k|^{1/2}+1)^{-1}+\mathcal{O}(1)(|z-w_{-}(k)||k|^{1/2}+1)^{-1},$

where $w_{+}(k),\,w_{-}(k)\in\partial\Omega$ are the North and South poles,
determined by the fact that the interior unit normal of $\partial\Omega$ at
$w_{\pm}(k)$ is of the form $c_{\pm}\omega$ for some $c_{\pm}$ with $\pm
c_{\pm}>0$. Here $\omega\in\mathbb{R}^{2}\simeq\mathbb{C}$ is determined by
$kz-\overline{kz}=\mathrm{i}|k|\Re(z\overline{\omega})$ for all $z\in z$.

We deduced that

(3.13) $\|F\|_{L^{2}(\Omega)}=\mathcal{O}(1)\frac{(\ln|k|)^{1/2}}{|k|^{1/2}},$
(3.14)
$\|F/(2i\overline{k})\|_{L^{2}(\Omega)}=\mathcal{O}(1)\frac{(\ln|k|)^{1/2}}{|k|^{3/2}},$

and using (3.12) we now add the observation that

(3.15)
$\|F/(2i\overline{k})\|_{L^{1}(\Omega)}=\mathcal{O}(1)\frac{1}{|k|^{3/2}}.$

We next estimate $(\pi/\overline{k})e^{-iu(\cdot,k)}1_{\Omega_{+}}$, appearing
in (3.10). (Notice that $e^{-iu(\cdot,k)}1_{\Omega_{-}}$ is absent, since we
restrict the attention to $\Omega$ and $\Omega_{-}\cap\Omega=\emptyset$.) In
(5.75) in [18] we found that

(3.16)
$\left\|(\pi/\overline{k})e^{-iu(\cdot,k)}1_{\Omega_{+}}\right\|_{L^{2}(\Omega)}=\frac{\mathcal{O}(1)(\ln|k|)^{1/2}}{|k|^{3/2}},$

and we shall now apply the same procedure to the $L^{1}$-norm. We restrict the
attention to the contribution to the $L^{1}$-norm from a neighborhood of one
of the poles, say $w_{+}(k)$. (Away from such neighborhoods, the estimates are
simpler and lead to a stronger conclusion.) In suitable coordinates $\mu=t+is$
we have

$(\pi/\overline{k})e^{-iu(\cdot,k)}1_{\Omega_{+}}=\mathcal{O}(k^{-1})e^{-|k|ts/C},\
0\leq t\leq 1/\mathcal{O}(1),\ 0\leq s\leq t.$

(Away from a neighborhood of $\\{w_{+}(k),w_{-}(k)\\}$ we have the same
estimate, now for $1/\mathcal{O}(1)\leq t\leq\mathcal{O}(1)$, $\ 0\leq s\leq
1/\mathcal{O}(1)$.) The contribution to the $L^{1}$-norm of
$(\pi/\overline{k})e^{-iu(\cdot,k)}1_{\Omega_{+}}$ is

$\mathcal{O}(|k|^{-1})\int_{0}^{1}\int_{0}^{t}e^{-ts|k|/C}dsdt=\mathcal{O}(|k|^{-1})\int_{0}^{1}\frac{1}{t|k|}\left(1-e^{-t^{2}|k|/C}\right)dt\\\
=\frac{\mathcal{O}(1)}{|k|}\left(\int_{0}^{|k|^{-1/2}}tdt+\int_{|k|^{-1/2}}^{1}\frac{1}{t|k|}dt\right)=\frac{\mathcal{O}(1)\ln|k|}{|k|^{2}}.$

The contribution from
$\overline{\Omega}\setminus\mathrm{neigh\,}(\\{w_{+},w_{-}\\})$ to the
$L^{1}$-norm is $\mathcal{O}(1)|k|^{-2}$. Hence

(3.17)
$\left\|(\pi/\overline{k})e^{-iu(\cdot,k)}1_{\Omega_{+}}\right\|_{L^{1}(\Omega)}=\frac{\mathcal{O}(1)\ln|k|}{|k|^{2}}.$

Combining (3.10), (3.14), (3.16), we get as in [18] that

(3.18)
$A_{k}(1_{\Omega})=\frac{1}{2\overline{k}}e^{-kz+\overline{kz}}1_{\Omega}+r(z,k),$

i.e.,

(3.19) $r=\frac{F}{4\pi
i\overline{k}}-\frac{1}{2\overline{k}}e^{-iu}1_{\Omega_{+}}\hbox{ in }\Omega,$

where

(3.20) $r(\cdot,k)=\mathcal{O}(1)|k|^{-3/2}(\ln|k|)^{1/2}\hbox{ in
}L^{2}(\Omega).$

Using (3.15), (3.17) instead of (3.14), (3.16), we get

(3.21) $\|r(\cdot,k)\|_{L^{1}(\Omega)}=\mathcal{O}(1)|k|^{-3/2}.$

We now return to the expansion (3.4) for $\overline{R}$, and start with the
term for $\nu=1$. Let

$\langle u|v\rangle=\int_{\Omega}u(z)v(z)L(dz)$

denote the bilinear $L^{2}$ scalar product. We get with $A=A_{k}$, $B=B_{k}$
if nothing else is indicated,

$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)=\frac{2}{\pi}\langle
AB(1_{\Omega})|e^{k\cdot-\overline{k\cdot}}1_{\Omega}\rangle\\\
=\frac{2}{\pi}\langle
B(1_{\Omega})|A^{\mathrm{t}}e^{k\cdot-\overline{k\cdot}}(1_{\Omega})\rangle=-\frac{2}{\pi}\langle
B_{k}(1_{\Omega})|e^{-k\cdot+\overline{k\cdot}}A_{-k}(1_{\Omega})\rangle,$

where we used (3.8) in the last step. Applying (3.18) to $A_{-k}(1_{\Omega})$
and the fact that $B_{k}(1_{\Omega})=\overline{A_{k}(1_{\Omega})}$, we get

(3.22)
$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)\\\
=-\frac{2}{\pi}\langle\frac{1}{2k}e^{-\overline{k\cdot}+k\cdot}1_{\Omega}+\overline{r(\cdot,k)}|e^{-k\cdot+\overline{k\cdot}}(-\frac{1}{2\overline{k}}e^{k\cdot-\overline{k\cdot}}1_{\Omega}+r(\cdot,-k))\rangle\\\
=\frac{2}{\pi}\frac{1}{4|k|^{2}}\int_{\Omega}e^{kz-\overline{kz}}L(dz)-\frac{2}{\pi}\int_{\Omega}\frac{1}{2k}r(z,-k)L(dz)\\\
+\frac{2}{\pi}\int_{\Omega}\overline{r(z,k)}\frac{1}{2\overline{k}}L(dz)-\frac{2}{\pi}\int_{\Omega}e^{-kz+\overline{kz}}\overline{r(z,k)}r(z,-k)L(dz).$

As we have already seen, the integral in the first term in the last member is
$\mathcal{O}(|k|^{-3/2})$, so this term is $\mathcal{O}(|k|^{-7/2})$. By
(3.20) the last term in (3.22) is $\mathcal{O}(|k|^{-3}\ln|k|)$. Thus (3.22)
gives

(3.23)
$\begin{split}&\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)\\\
&=-\frac{2}{\pi}\int_{\Omega}\frac{1}{2k}r(z,-k)L(dz)+\frac{2}{\pi}\int_{\Omega}\overline{r(z,k)}\frac{1}{2\overline{k}}L(dz)+\mathcal{O}(|k|^{-3}\ln|k|).\end{split}$

(3.21) now yields

(3.24)
$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)=\mathcal{O}(|k|^{-5/2}).$

Before studying the leading asymptotics of the integrals in the left hand side
of(3.23), we shall gain a power of $k$ in the estimate of the general term in
(3.4) for $\nu\geq 2$:

(3.25)
$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}(AB)^{\nu}(1_{\Omega})(z)L(dz)=\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}A(BA)^{\nu-1}B(1_{\Omega})(z)L(dz)\\\
=-\frac{2}{\pi}\langle(BA)^{\nu-1}B(1_{\Omega})|e^{-k\cdot+\overline{k\cdot}}A_{-k}(1_{\Omega})\rangle=\mathcal{O}(1)|k|^{1-\nu}|k|^{-1}|k|^{-1}\\\
=\mathcal{O}(|k|^{-\nu-1})=\mathcal{O}(|k|^{-3}),$

since $(BA)^{\nu-1}=\mathcal{O}(|k|^{1-\nu}):\,L^{2}(\Omega)\to L^{2}(\Omega)$
and $B(1_{\Omega}),\,A_{-k}(1_{\Omega})=\mathcal{O}(1/|k|)$ in
$L^{2}(\Omega).$

Combining (3.4), (3.24), (3.25), we get

(3.26)
$\overline{R}=\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}L(dz)+\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})L(dz)+\mathcal{O}(|k|^{-3}),$

and in particular,

(3.27)
$\overline{R}=\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}L(dz)+\mathcal{O}(|k|^{-5/2}).$

We next study the second term in the right hand side of (3.26), starting from
(3.23). By (3.17) and (3.19) we have

(3.28) $r=\frac{F}{4\pi
i\overline{k}}+\frac{\mathcal{O}(1)\ln|k|}{|k|^{2}}\hbox{ in }L^{1}(\Omega),$

where $F=F(z,k)$. Using this in (3.23), we get

(3.29)
$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)=\\\
\frac{2}{\pi}\int_{\Omega}\frac{1}{2k}\frac{F(z,-k)}{4i\pi\overline{k}}L(dz)+\frac{2}{\pi}\int_{\Omega}\frac{\overline{F}(z,k)}{-4i\pi
k}\frac{1}{2\overline{k}}L(dz)+\mathcal{O}(|k|^{-3}\ln|k|)\\\
=\frac{1}{4i\pi^{2}|k|^{2}}\int_{\Omega}(F(z,-k)-\overline{F}(z,k))L(dz)+\mathcal{O}(|k|^{-3}\ln|k|).$

Here we recall (3.11) for $F(z,k)=F_{\Gamma}(z)$, where $\Gamma=\Gamma(k)$ is
a deformation of $\partial\Omega$ passing through the poles $w_{+}(k)$,
$w_{-}(k)$, situated outside $\overline{\Omega}$ near the boundary segment
$\Gamma_{-}$ from $w_{+}(k)$ to $w_{-}(k)$ and inside $\Omega$ near the
boundary segment $\Gamma_{+}$ from $w_{-}(k)$ to $w_{+}(k)$ (when following
the boundary with the positive orientation). This choice is given by the
method of steepest descent for $e^{-iu(\cdot,k)}$. When replacing $k$ with
$-k$, we have $e^{-iu(z,-k)}=e^{iu(z,k)}$ and $w_{\pm}(-k)=w_{\mp}(k)$.
Correspondingly, $\Gamma$ should be replaced by a contour $\widetilde{\Gamma}$
which is a deformation of $\partial\Omega$ inwards near the segment
$\Gamma_{-}$ from $w_{+}(k)$ to $w_{-}(k)$ and outwards near the segment
$\Gamma_{+}$ from $w_{-}(k)$ to $w_{+}(k)$.

We get

(3.30)
$\int_{\Omega}F(z,-k)L(dz)=\int_{\Omega}\int_{\widetilde{\Gamma}}\frac{1}{z-w}e^{iu(w,k)}dwL(dz)\\\
=\int_{\widetilde{\Gamma}}\int_{\Omega}\frac{1}{z-w}L(dz)e^{iu(w,k)}dw=-\pi\int_{\widetilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw,$

using that $\frac{1}{z-w}e^{iu(w,k)}$ is integrable on
$\Omega\times\widetilde{\Gamma}$ for the measure $L(dz)|dw|$. Here

(3.31) $D_{\Omega}(z)=\frac{1}{\pi}\int_{\Omega}\frac{1}{z-w}L(dw)$

is the solution to the $\overline{\partial}$-problem:

(3.32) $\begin{cases}\partial_{\overline{z}}D_{\Omega}=1_{\Omega},\\\
D_{\Omega}(z)\to 0,\ z\to\infty.\end{cases}$

Similarly,

(3.33)
$-\int_{\Omega}\overline{F(z,k)}L(dz)=\overline{\int_{\Omega}\int_{\Gamma}\frac{1}{w-z}e^{-iu(w,k)}dwL(dz)}\\\
=\overline{\int_{\Gamma}\int_{\Omega}\frac{1}{w-z}L(dz)e^{-iu(w,k)}dw}=\pi\overline{\int_{\Gamma}D_{\Omega}(w)e^{-iu(w,k)}dw}.$

Using (3.30), (3.33) in (3.29), we get

(3.34)
$\frac{2}{\pi}\int_{\Omega}e^{kz-\overline{kz}}AB(1_{\Omega})(z)L(dz)=\\\
\frac{1}{4i\pi|k|^{2}}\left(-\int_{\widetilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw+\overline{\int_{\Gamma}D_{\Omega}(w)e^{-iu(w,k)}dw}\right)+\mathcal{O}(|k|^{-3}\ln|k|).$

######  Remark 3.1.

It is not obvious whether the error term in (3.34) is optimal or a consequence
of the applied technique to prove the result.

######  Proposition 3.2.

$D_{\Omega}\in C(\mathbb{C})$ and the restrictions of this function to the
open sets $\Omega$ and $\mathbb{C}\setminus\overline{\Omega}$ extend to
functions in $C^{\infty}(\overline{\Omega})$ and $C^{\infty}({\bf
C}\setminus\Omega)$ respectively.

###### Proof.

The continuity of $D_{\Omega}$ is clear. We first look at $D_{\Omega}(z)$ in
$\mathbb{C}\setminus\overline{\Omega}$. Here $D_{\Omega}(z)$ is holomorphic
and

(3.35)
$\begin{split}\partial_{z}D_{\Omega}(z)=&\frac{1}{\pi}\int_{\Omega}\partial_{z}\left(\frac{1}{z-w}\right)L(dw)\\\
=&-\frac{1}{\pi}\int_{\Omega}\partial_{w}\left(\frac{1}{z-w}\right)\frac{d\overline{w}\wedge
dw}{2i}=\frac{1}{2\pi
i}\int_{\Omega}d_{w}\left(\frac{1}{z-w}d\overline{w}\right).\end{split}$

By Stokes’ formula,

(3.36) $\partial_{z}D_{\Omega}(z)=\frac{1}{2\pi
i}\int_{\partial\Omega}\frac{1}{z-w}d\overline{w}=\frac{1}{2\pi
i}\int_{0}^{L}\frac{1}{z-\gamma(t)}\overline{\dot{\gamma}(t)}dt$

where $\gamma:[0,L[\ni t\mapsto\gamma(t)\in\partial\Omega$ is a smooth
positively oriented parametrization of $\partial\Omega$.

It follows that

(3.37) $\partial_{z}D_{\Omega}(z)=\mathcal{O}(1)|\ln d(z,\partial\Omega)|,\
z\in(\mathbb{C}\setminus\overline{\Omega})\cap\mathrm{neigh\,}(\partial\Omega),$

where $d(z,\partial\Omega)$ denotes the distance from $z$ to $\partial\Omega$.

For $n=1,2,...$, we apply $\partial_{z}^{n}$ to (3.36):

(3.38) $\partial_{z}^{n+1}D_{\Omega}(z)=\frac{1}{2\pi
i}\int_{\partial\Omega}\partial_{z}^{n}\frac{1}{z-w}d\overline{w}\\\
=\frac{1}{2\pi
i}\int_{\partial\Omega}(-\partial_{w})^{n}\frac{1}{z-w}d\overline{w}=\frac{1}{2\pi
i}\int_{0}^{L}(-\dot{\gamma}^{-1}\partial_{t})^{n}\left(\frac{1}{z-\gamma(t)}\right)\overline{\dot{\gamma}}dt\\\
=\frac{1}{2\pi
i}\int_{0}^{L}\frac{1}{z-\gamma(t)}\underbrace{(\partial_{t}\circ\dot{\gamma}^{-1})^{n}\left(\overline{\dot{\gamma}}\right)}_{\mathcal{O}(1)}dt=\mathcal{O}(1)|\ln
d(z,\partial\Omega)|,$

still for $z\in\mathrm{neigh\,}(\partial\Omega)\setminus\overline{\Omega}$.
Integrating these estimates, we see that $\partial_{z}^{n}D_{\Omega}(z)$ is
bounded on $\mathbb{C}\setminus\overline{\Omega}$ for $n\in\Omega$ (which we
already knew to be valid away from a neighborhood of $\partial\Omega$) and
hence that $D_{\Omega}\in C^{\infty}(\mathbb{C}\setminus\Omega)$.

If instead of $D_{\Omega}$ we look at

(3.39)
$D_{\Omega,\psi}(z)=\frac{1}{\pi}\int_{\Omega}\frac{1}{z-w}\psi(w)L(dw)$

for some $\psi\in C^{\infty}(\overline{\Omega})$ with $\psi=1$ near
$\partial\Omega$, we still have $D_{\Omega,\psi}\in
C^{\infty}(\mathbb{C}\setminus\Omega)$. Indeed, we get very much as in (3.35),

$\partial_{z}D_{\Omega,\psi}(z)=\frac{1}{\pi}\int_{\Omega}\partial_{z}\left(\frac{1}{z-w}\right)\psi(w)L(dw)\\\
=-\frac{1}{\pi}\iint_{\Omega}\partial_{w}\left(\frac{1}{z-w}\right)\psi(w)\frac{d\overline{w}\wedge
dw}{2i}\\\ =\frac{1}{2\pi
i}\int_{\Omega}d_{w}\left(\frac{\psi(w)}{z-w}d\overline{w}\right)+\frac{1}{\pi}\iint_{\Omega}\frac{\partial_{w}\psi}{z-w}\frac{d\overline{w}\wedge
dw}{2i},$

where the last integral belongs to $C^{\infty}(\mathbb{C}\setminus\Omega)$ and
the second last integral is equal to $\frac{1}{2\pi
i}\int_{\partial\Omega}\frac{1}{z-w}d\bar{w}$ and is also in
$C^{\infty}(\mathbb{C}\setminus\Omega)$ as we have just seen.

We finally show that $D_{\Omega}(z)$, $z\in\Omega$, extends to a function in
$C^{\infty}(\overline{\Omega})$. Let $\chi\in C_{0}^{\infty}(\mathbb{C})$ be
equal to 1 near $\overline{\Omega}$. Then
$f(z)=\frac{1}{\pi}\int_{\mathbb{C}}(z-w)^{-1}\chi(w)L(dw)$ solves
$\partial_{\overline{z}}f=\chi$ and belongs to $C^{\infty}(\mathbb{C})$. For
$z\in\Omega$ we have

$D_{\Omega}(z)=f(z)-\frac{1}{\pi}\int_{\mathbb{C}\setminus\Omega}\frac{1}{z-w}\chi(w)L(dw).$

As in the remark above about $D_{\Omega,\psi}$ in
$\mathbb{C}\setminus\overline{\Omega}$ we see that
$\frac{1}{\pi}\int_{\mathbb{C}\setminus\Omega}\frac{1}{z-w}\chi(w)L(dw)$
belongs to $C^{\infty}(\overline{\Omega})$. Hence $D_{\Omega}(z)$ extends from
$\Omega$ to a smooth function on $\overline{\Omega}$. ∎

This completes the proof of Theorem 1.4 in the case when $\partial\Omega$ is
analytic.

The exponentials in the integrals in (3.34) behave like Gaussians peaked at
$w_{\pm}(k)$, and hence the integrals are $\mathcal{O}(|k|^{-1/2})$ and the
contribution from outside any fixed neighborhood of $\\{w_{+}(k),w_{-}(k)\\}$
is exponentially small. The proposition implies that $D_{\Omega}(w)$ is a
Lipschitz function and therefore we modify the integrals by
$\mathcal{O}(|k|^{-1})$ only, if we replace $D_{\Omega}(w)$ in a neighborhood
of $w_{\pm}(k)$ by $D_{\Omega}(w_{\pm}(k))$. Thus for instance

(3.40)
$\int_{\widetilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw=D_{\Omega}(w_{+}(k))\int_{\widetilde{\Gamma}\cap\mathrm{neigh\,}(w_{+}(k))}e^{iu(w,k)}dw\\\
+D_{\Omega}(w_{-}(k))\int_{\widetilde{\Gamma}\cap\mathrm{neigh\,}(w_{-}(k))}e^{iu(w,k)}dw+\mathcal{O}(k^{-1}).$

######  Remark 3.3.

We now drop the analyticity assumption and assume that
$\Omega\Subset\mathbb{C}$ is open with smooth boundary and strictly convex. We
have used the analyticity assumption in (3.10), (3.11), where $iu$ is the
holomorphic extension to a neighbourhood of $\partial\Omega$ of the function
$kw-\overline{kw}$, $w\in\partial\Omega$. In the merely smooth case we let
$iu(w,k)$ denote an almost holomorphic extension of $\partial\Omega\ni
w\mapsto kw-\overline{kw}$ and define $F$ as in (3.11), using the modified
function $u$. Stokes’ formula now produces a small error term to be added to
(3.11) and as in (2.18) in [19] and the subsequent discussion, we get

$f(z,k)=\frac{1}{2i\bar{k}}F(z)+\frac{\pi}{\bar{k}}\left(e^{-iu(z,k)}(1_{\Omega_{-}}(z)-1_{\Omega_{+}}(z))+e^{-kz+\overline{kz}}1_{\Omega}(z)\right)$
$+\mathcal{O}(\langle z\rangle^{-1}|k|^{-\infty}).$

The discussion after (3.11) goes through with only a minor change: In the
formula (3.19) for $r$ we have to add a remainder
$\mathcal{O}(|k|^{-\infty})$. But this does not affect the subsequent
estimates, and we get Theorem 1.4 also in the more general case of a smooth
boundary.

##  4\. Stationary phase approximation

To compute the leading orders in $1/|k|$ of the reflection coefficient, we
apply a standard stationary phase approximation. Since higher order terms in
this approximation are needed here, we briefly summarize some facts on the
approach.

###  4.1. Two term stationary phase expansion

We have in mind $\int_{\partial\Omega}e^{-kz+\overline{kz}}$ after a change of
contour from $\partial\Omega$ to $\Gamma$ and choosing a parametrisation. Let
$\varphi\in C^{\infty}(\mbox{neigh}(0,\mathbb{R}))$ satisfy $\varphi(0)=0$,
$\varphi^{\prime}(0)=0$, $\Re\varphi^{\prime\prime}(0)>0$, $a\in
C^{\infty}(\mbox{neigh}(0,\mathbb{R}))$. Consider

$I(\varphi,a;h)=h^{-\frac{1}{2}}\int_{V}e^{-\varphi(t)/h}a(t)dt,\quad
V=\mbox{neigh}(0,\mathbb{R}).$

We already know that $I\sim I_{0}+I_{1}h+\dots,$ where
$I_{0}=\frac{\sqrt{2\pi}}{\sqrt{\varphi^{\prime\prime}(0)}}$ with the natural
choice of the branch of the square root and our problem is to compute
$I_{1}h$. Write $\varphi(t)=\varphi_{2}t^{2}+\psi(t)$, where
$\psi(t)=\mathcal{O}(t^{3})$. Put

$\varphi(t;s)=\varphi_{2}t^{2}+s\psi,\quad 0\leq s\leq 1.$

Clearly $I(\varphi,a;h)$ is smooth in $s$ with
$\partial^{k}_{s}I(\varphi_{s},a;h)=\mathcal{O}(h^{k/2})$, $k=0,1,2,\dots$,
hence by a limited Taylor expansion

$\displaystyle I(\varphi,a;h)$
$\displaystyle=I(\varphi,a;h)|_{s=0}+(\partial_{s})I(\varphi,a;h)|_{s=0}+\frac{1}{2}(\partial^{2}_{s})I(\varphi,a;h)|_{s=0}+\mathcal{O}(h^{3/2})$
(4.1)
$\displaystyle=I(\varphi_{2}t^{2},a;h)-I(\varphi_{2}t^{2},a\psi/h;h)+\frac{1}{2}I(\varphi_{2}t^{2},a(\psi/h)^{2};h)+\mathcal{O}(h^{3/2}).$

Indeed

(4.2) $I(\varphi,\mathcal{O}(t^{n});h)=\mathcal{O}(h^{\frac{n}{2}}).$

Using again (4.2) we can replace $a$, $a\frac{\Psi}{h}$ and
$a(\frac{\Psi}{h})^{2}$ by their limited Taylor sums modulo
$\mathcal{O}(t^{3})$, $\mathcal{O}(\frac{t^{5}}{h})$ and
$\mathcal{O}(\frac{t^{7}}{h^{2}})$ respectively.
Writing $\psi\sim\varphi_{3}t^{3}+\varphi_{4}t^{4}+\dots$, $a\sim
a_{0}+a_{1}t+a_{2}t^{2}+\dots$ and using also that
$I(\varphi_{2}t^{2},t^{n};h)=0$ when $n$ is odd (up to an exponentially small
error if $V$ is not symmetric around $t=0$), we get

$\begin{split}I(\varphi_{2}t^{2},a;h)&=I(\varphi_{2}t^{2},a_{0},h)+I(\varphi_{2}t^{2},a_{2}t^{2},h)+\mathcal{O}(h^{2})\\\
&=a_{0}I(\varphi_{2}t^{2},1;1)+ha_{2}I(\varphi_{2}t^{2},t^{2};1)+\mathcal{O}(h^{2}),\end{split}$

and

$\begin{split}I(\varphi_{2}t^{2},\frac{a\psi}{h};h)&=I(\varphi_{2}t^{2},(a_{0}+a_{1}t)\frac{\varphi_{3}t^{3}+\varphi_{4}t^{4}}{h};h)+\mathcal{O}(h^{3/2})\\\
&=I(\varphi_{2}t^{2},\frac{(a_{0}\varphi_{4}+a_{1}\varphi_{3})t^{4}}{h};h)+\mathcal{O}(h^{3/2})\\\
&=h(a_{0}\varphi_{4}+a_{1}\varphi_{3})I(\varphi_{2}t^{2},t^{4};1))+\mathcal{O}(h^{3/2}),\end{split}$

as well as

$\begin{split}I(\varphi_{2}t^{2},a(\frac{\psi}{h})^{2};h)&=I(\varphi_{2}t^{2},\frac{a_{0}\varphi_{3}^{2}t^{6}}{h^{2}},h)+\mathcal{O}(h^{3/2})\\\
&=ha_{0}\varphi_{3}^{2}I(\varphi_{2}t^{2},t^{6};1)+\mathcal{O}(h^{3/2}).\end{split}$

We now have integrals with quadratic exponent and polynomial amplitudes and up
to exponentially small corrections from now on, we integrate over $\mathbb{R}$
instead of $V$.

###  4.2. Reduction to the case of an exact quadratic

Now we consider the reduction to the case of a quadratic exponential,

$I(\varphi_{0},t^{2n};1)=\int e^{-\varphi_{2}t^{2}}t^{2n}dt.$

Reparametrise $t^{2}=\frac{\tau^{2}}{2\varphi_{2}}$,
$t=\frac{1}{(2\varphi_{2})^{1/2}}\tau$, and
$dt=\frac{1}{\sqrt{2\varphi_{2}}}d\tau=\frac{1}{\sqrt{\varphi^{\prime\prime}(0)}}d\tau$

$I(\varphi_{2}t^{2},t^{2n};1)=(2\varphi_{2})^{-n-\frac{1}{2}}I(\frac{\tau^{2}}{2};\tau^{2n};1)=(\varphi^{\prime\prime}(0))^{-n-\frac{1}{2}}I(\frac{\tau^{2}}{2},\tau^{2n};1).$

Integrate by parts when $n\geq 1$:

$\begin{split}I(\tau^{2}/2,\tau^{2n};1)&=\int_{\mathbb{R}}e^{-\tau^{2}/2}\tau^{2n}d\tau=-\int_{\mathbb{R}}\partial_{\tau}(e^{-\tau^{2}/2})\tau^{2n-1}d\tau\\\
&=\int
e^{-\tau^{2}/2}(2n-1)\tau^{2(n-1)}d\tau=(2n-1)I(\tau^{2}/2,\tau^{2(n-1)};1).\end{split}$

In particular

$\begin{split}I(\tau^{2}/2,\tau^{2};1)&=I(\tau^{2}/2,1;1)=\sqrt{2\pi},\\\
I(\tau^{2}/2,\tau^{4};1)&=3I(\tau^{2}/2,1;1)=3\sqrt{2\pi},\\\
I(\tau^{2}/2,\tau^{6};1)&=15I(\tau^{2}/2,1;1)=15\sqrt{2\pi}.\end{split}$

We combine the different identities:

(4.3)
$\begin{split}I(\varphi,a;h)&=I(\varphi_{2}t^{2},a;h)-I(\varphi_{2}t^{2},a\frac{\psi}{h};h)+\frac{1}{2}I(\varphi_{2}t^{2},a(\frac{\psi}{h})^{2};h)+\mathcal{O}(h^{\frac{3}{2}})\\\
&=a_{0}I(\varphi_{2}t^{2},1;1)+ha_{2}I(\varphi_{2}t^{2},t^{2};1)-h(a_{0}\varphi_{4}+a_{1}\varphi_{3})I(\varphi_{2}t^{2},t^{4};1)\\\
&\quad\quad+\frac{1}{2}ha_{0}\varphi_{3}^{2}I(\varphi_{2}t^{2},t^{6};1)+\mathcal{O}(h^{\frac{3}{2}})\\\
&=\sqrt{2\pi}a(0)\varphi^{\prime\prime}(0)^{-\frac{1}{2}}+\sqrt{2\pi}h\left[\frac{1}{2}a^{\prime\prime}(0)\varphi^{\prime\prime}(0)^{-\frac{3}{2}}\right.\\\
&\quad\quad-\left(\frac{1}{8}a(0)\varphi^{(4)}(0)+\frac{1}{2}a^{\prime}(0)\varphi^{(3)}(0)\right)\varphi^{\prime\prime}(0)^{-\frac{5}{2}}\\\
&\quad\quad\quad\quad\left.+\frac{5}{24}a(0)\varphi^{(3)}(0)^{2}\varphi^{\prime\prime}(0)^{-\frac{7}{2}}\right]+\mathcal{O}(h^{\frac{3}{2}}).\end{split}$

Here we recall that $\varphi_{3}=\varphi^{(3)}(0)/6$ and
$\varphi_{4}=\varphi^{(4)}(0)/24$, $a_{0}=a(0)$, $a_{1}=a^{\prime}(0)$,
$a_{2}=\frac{1}{2}a^{\prime\prime}(0)$. We know that $I(\varphi,a;h)$ has an
asymptotic expansion in integer powers of $h$, so the remainder can be
improved to $\mathcal{O}(h^{2})$. In view of the application to
$\int_{\partial\Omega}e^{-kz+\overline{kz}}dz$, we try to express the result
in terms of $\varphi/h=:\Phi$, and it then seems convenient to replace
$I(a,\varphi;h)$ above by
$J(a,\Phi):=h^{\frac{1}{2}}I(a,\varphi;h)=\int_{V}e^{-\Phi(t)}a(t)dt$. From
(4.3) we get using $\varphi=h\Phi$,

$\displaystyle\frac{1}{\sqrt{2\pi}}J(\varphi,a)$
$\displaystyle=a(0)\Phi^{\prime\prime}(0)^{-\frac{1}{2}}+\frac{a^{\prime\prime}(0)}{2}\Phi^{\prime\prime}(0)^{-\frac{3}{2}}$
$\displaystyle-\left(a(0)\frac{\Phi^{(4)}(0)}{8}+a^{\prime}(0)\frac{\Phi^{(3)}(0)}{2}\right)\Phi^{\prime\prime}(0)^{-\frac{5}{2}}$
(4.4)
$\displaystyle+\frac{5}{24}a(0)\Phi^{(3)}(0)^{2}\Phi^{\prime\prime}(0)^{-\frac{7}{2}}+\mathcal{O}(h^{2}).$

Notice that the first term in the final expression is homogeneous of degree
$-\frac{1}{2}$ in $\Phi$, while the following one is homogeneous of degree
$-\frac{3}{2}$.

###  4.3. Stationary phase approximation for the reflection coefficient

We now apply the above results to the reflection coefficient (we only discuss
the analytic case here, see Remark 3.3 for a generalization to the smooth
case). We also assume in the following that $\Phi$ does not vanish at the
stationary points.

In application to $\int_{\partial\Omega}e^{kz-\overline{kz}}$, let $[0,L[\ni
t\mapsto\gamma(t)$ parametrize the boundary, so
$\int_{\partial\Omega}e^{kz-\overline{kz}}dz=\int_{0}^{L}e^{-\Phi(t)}a(t)dt$,
with

(4.5) $\Phi(t)=-(k\gamma(t)-\overline{k\gamma(t)}),\quad
a(t)=\dot{\gamma}(t).$

Apply (4.4) with $\mathcal{O}(h^{2})=\mathcal{O}(|k|^{-2})$.

######  Remark 4.1.

It remains to choose the correct branches of
$(\Phi^{\prime\prime}(t))^{\frac{1}{2}}$ at $t=t_{\pm}$. We adapt the notation
of [18], $iu(z,k)=kz-\overline{kz}$, $z\in\partial\Omega$. We have
$\Phi(t)=-iu(\gamma(t),k))=:-iU(t,k)$ which is purely imaginary with two non
degenerate critical points at $t=t_{+},t_{-}$ corresponding to the poles
$w_{+},w_{-}$ respectively. By contour deformation we see that the stationary
phase approximation is still valid with the branch of
$(\Phi^{\prime\prime}(t_{\pm}))^{1/2}$ obtained as the limit of
$(F_{\pm,\epsilon})^{1/2}$, where $F_{\pm,\epsilon}$ is a sequence converging
to $\Phi^{\prime\prime}(t_{\pm})$ when $\epsilon\searrow 0$ with the property
$\Re F_{\pm,\epsilon}>0$. We get $(\Phi^{\prime\prime}(t_{\pm}))^{1/2}=e^{\mp
i\pi/4}|U^{\prime\prime}(t_{\pm},k)|^{1/2}$.

We get for (3.40)

(4.6)
$\int_{\widetilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw=\sqrt{2\pi}\sum_{t=t_{+},t_{-}}e^{-\Phi(t)}D_{\Omega}(\gamma(t))a(t)(\Phi^{\prime\prime}(t))^{-\frac{1}{2}}+\mathcal{O}(|k|^{-1}),$

where $\Phi$ and $a$ are defined in (4.5), and where the signs of the roots
are chosen as detailed in Remark 4.1.

The leading term in the reflection coefficient (1.22) is due to the term
$\int_{\Omega}e^{kz-\overline{kz}}L(dz)$. We apply Stokes’ theorem as before
to write this in the form of an integral over $\partial\Omega$ and apply a
stationary phase approximation,

(4.7)
$\begin{split}\frac{i}{2\bar{k}}\int_{\partial\Omega}e^{kz-\overline{kz}}dz&=\frac{i\sqrt{2\pi}}{2\bar{k}}\sum_{t=t_{+},t_{-}}e^{-\Phi(t)}\left(a(t)(\Phi^{\prime\prime}(t))^{-\frac{1}{2}}+\frac{a^{\prime\prime}(t)}{2}(\Phi^{\prime\prime}(t))^{-\frac{3}{2}}\right.\\\
&-\left(\frac{a(t)}{8}\Phi^{(4)}(t)+\frac{1}{2}a^{\prime}(t){\Phi^{(3)}(t)}\right)(\Phi^{\prime\prime}(t))^{-\frac{5}{2}}\\\
&\quad\quad\quad\left.+\frac{5}{24}a(t)\Phi^{(3)}(t)^{2}(\Phi^{\prime\prime}(t))^{-\frac{7}{2}}\right)+\mathcal{O}(|k|^{-3}).\end{split}$

Again $\Phi$ and $a$ are defined in (4.5), and the sign of the roots are
chosen as explained in Remark 4.1. With (4.6) and (4.7) we get for the
reflection coefficient (1.22) relation (1.23).

###  4.4. Example: Characteristic function of the unit disk

In general, we cannot compute explicitly $D_{\Omega}(z)$, but in the special
case of the unit disc, we have

$D_{D(0,1)}(z)=\begin{cases}\overline{z},\ |z|\leq 1,\\\ 1/z,\ |z|\geq
1.\end{cases}$

For the stationary phase approximation, we parametrize $\partial\Omega$ via
$\gamma(t)=e^{it}$, $t\in[0,2\pi[$. Writing $k=|k|e^{i\vartheta}$,
$\vartheta\in\mathbb{R}$, we have $\Phi=-2i|k|\sin(t+\vartheta)$ and
$a=ie^{it}$ in (4.5). The critical points are
$t^{\pm}=\pm\frac{\pi}{2}-\vartheta$ (the relation to the previously
introduced $t_{\pm}$ is $t^{\pm}:=t_{\mp}$). We have
$\Phi(t^{\pm})=-\Phi^{\prime\prime}(t^{\pm})=\Phi^{(4)}(t^{\pm})=\mp 2i|k|$,
whereas $\Phi^{\prime}(t^{\pm})=\Phi^{(3)}(t^{\pm})=0$ and
$a(t^{\pm})=-a^{\prime\prime}(t^{\pm})=\mp e^{-i\vartheta}$. This implies for
the right hand side of (4.7

(4.8)
$\frac{i}{2\bar{k}}\int_{\partial\Omega}e^{kz-\overline{kz}}dz=\frac{\sqrt{\pi}}{|k|^{3/2}}\left(\sin(2|k|-\pi/4)+\frac{3}{16|k|}\cos(2|k|-\pi/4)\right)+\mathcal{O}(|k|^{-7/2}).$

Similarly we get for (4.6)

(4.9)
$\int_{\widetilde{\Gamma}}D_{\Omega}(w)e^{iu(w,k)}dw=\sqrt{\frac{\pi}{|k|}}2i\cos(2|k|-\pi/4)+\mathcal{O}(|k|^{-1}).$

Thus we get for the leading terms of the reflection coefficient (1.23) the
result conjectured in [18] (note that the formula for $R/2$ was given there),

(4.10)
$R\approx\frac{2}{\sqrt{\pi|k|^{3}}}\left(\sin(2|k|-\pi/4)-\frac{5}{16|k|}\cos(2|k|-\pi/4)\right).$

Note that the conjectured error term is smaller than what is proven in this
paper.

##  5\. Conclusion

In this paper, we have presented asymptotic relations for large $|k|$ for the
solutions to the Dirac system (1.1) subject to the asymptotic conditions
(1.2). Previous results for potentials being the characteristic function of a
compact domain with smooth convex boundary have been improved and extended to
the reflection coefficient, the scattering data in the context of an
integrable systems approach to the DS II equation. The results are now
extended to $\mathcal{O}(|k|^{-5/2})$ which makes it possible to apply these
formulae to complement numerical computations in order to get the reflection
coefficient for all $k\in\mathbb{C}$ with the same precision as discussed in
[18]. This allows to treat the reflection coefficient with a _hybrid approach_
combining numerical and analytical results.

An interesting question in the context of EIT would be to extend the results
of this paper to a compact domain with cavities. Since in applications to the
human body, the organs of a patient are of essentially constant conductivity,
this corresponds to a situations of a domain with compact support and cavities
all of which have smooth compact boundaries. The boundary data at the cavities
are a consequence of the conductivity in the interior. It will be the subject
of further work to adapt the present formulae to this case. An interesting
question to be addressed is also to find the optimal error term in Theorem
1.4.

## References

  * [1] M.J. Ablowitz, A.S. Fokas, On the inverse scattering transform of multidimensional nonlinear evolution equations related to first order systems in the plane, J. Math Phys. 25 no 8 (1984), 2494-2505.
  * [2] Assainova, O., Klein, C., McLaughlin, K. D. and Miller, P. D., A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit. Comm. Pure Appl. Math., 72: 1474-1547 (2019).
  * [3] R. Beals and R. Coifman, Multidimensional inverse scattering and nonlinear PDE Proc. Symp. Pure Math. (Providence: American Mathematical Society) 43, 45-70 (1985)
  * [4] R.M. Brown, Estimates for the scattering map associated with a two-dimensional first-order system. J. Nonlinear Sci. 11, no. 6, 459-471 (2001)
  * [5] R. Brown and P. Perry, Soliton solutions and their (in)stability for the focusing Davey-Stewartson II equation, Nonlinearity 31(9) 4290 doi.org/10.1088/1361-6544/aacc46 (2018)
  * [6] R.M. Brown, G.A. Uhlmann, Communications in partial differential equations 22 (5-6), 1009-1027 (1997)
  * [7] A.P. Calderón, On inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) pp 65-73 (Soc. Brasil. Mat.)
  * [8] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Series 269, Cambridge University Press 1999\.
  * [9] A.S. Fokas, On the Inverse Scattering of First Order Systems in the Plane Related to Nonlinear Multidimensional Equations, Phys. Rev. Lett. 51, 3-6 (1983)
  * [10] A.S. Fokas and M.J. Ablowitz, On a Method of Solution for a Class of Multi-Dimensional Nonlinear Evolution Equations, Phys. Rev. Lett. 51, 7-10 (1983)
  * [11] C. Kenig, J. Sjöstrand, G. Uhlmann. The Calderón problem with partial data. Annals of Mathematics 165 (2007), 567-591.
  * [12] A. G. Gurevich, L. P. Pitaevskii, Non stationary structure of a collisionless shock waves, JEPT Letters 17 (1973), 193-195.
  * [13] L. Hörmander, An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library 7 (North-Holland Publishing Co., Amsterdam, 1990)
  * [14] C. Klein and K. McLaughlin, Spectral approach to D-bar problems, Comm. Pure Appl. Math., DOI: 10.1002/cpa.21684 (2017)
  * [15] C. Klein, K. McLaughlin and N. Stoilov, Spectral approach to semi-classical d-bar problems with Schwartz class potentials, Physica D: Nonlinear Phenomena DOI: 10.1016/j.physd.2019.05.006 (2019)
  * [16] C. Klein and K. Roidot, Numerical Study of the semiclassical limit of the Davey-Stewartson II equations, Nonlinearity 27, 2177-2214 (2014).
  * [17] C. Klein and J.-C. Saut, Nonlinear dispersive equations — Inverse Scattering and PDE methods, Applied Mathematical Sciences 209 (Springer, 2002)
  * [18] C. Klein, J. Sjöstrand, N. Stoilov, Large $|k|$ behavior of complex geometric optics solutions to d-bar problems, accepted for publication in Comm. Pure Appl. Maths. https://arxiv.org/abs/2009.06909.
  * [19] C. Klein, J. Sjöstrand, N. Stoilov, Large $|k|$ behavior of d-bar problems for domains with a smooth boundary, ‘”Partial Differential Equations, Spectral Theory, and Mathematical Physics: The Ari Laptev Anniversary Volume’ (edited by Pavel Exner, Rupert L. Frank, Fritz Gesztesy, Helge Holden and Timo Weidl), EMS Press, https://arxiv.org/abs/2009.06909
  * [20] C. Klein and N. Stoilov, A numerical study of blow-up mechanisms for Davey-Stewartson II systems, Stud. Appl. Math., DOI : 10.1111/sapm.12214 (2018)
  * [21] C. Klein and N. Stoilov, Numerical scattering for the defocusing Davey-Stewartson II equation for initial data with compact support, Nonlinearity 32 (2019) 4258-4280
  * [22] K. Knudsen, J. L. Mueller, S. Siltanen. Numerical solution method for the d-bar equation in the plane. J. Comput. Phys. 198 no. 2, 500-517 (2004).
  * [23] J.L. Mueller and S. Siltanen. Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2012.
  * [24] P. Muller, D. Isaacson, J. Newell, and G. Saulnier. A Finite Difference Solver for the D-bar Equation. Proceedings of the 15th International Conference on Biomedical Applications of Electrical Impedance Tomography, Gananoque, Canada, 2014.
  * [25] A. I. Nachman, I. Regev, and D. I. Tataru, A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey-Stewartson equation and to the inverse boundary value problem of Calderon, Invent. Math. 220, 395–451 (2020).
  * [26] P. Perry. Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\mathbb{R}^{2})$. J. Spectr. Theory 6 (2016), no. 3, pp. 429–481.
  * [27] L.Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. I, J. Math. Anal. Appl. 183 (1) (1994), 121-154.
  * [28] L.Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. II, J. Math. Anal. Appl. 183 (2) (1994), 289-325.
  * [29] L.Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. III, J. Math. Anal. Appl. 183 , 477-494 (1994)
  * [30] G. Uhlmann. Electrical impedance tomography and Calderón’s problem. Inverse Problems, 25(12):123011, 2009.