Solvability of some integro-differential equations

with the logarithmic Laplacian

Vitali Vougalter1, Vitaly Volpert2 3

1 Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4,
Canada

e-mail<EMAIL_ADDRESS>

2 Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1

Villeurbanne, 69622, France

3 Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow,
117198, Russia

e-mail<EMAIL_ADDRESS>

Abstract. We address the existence in the sense of sequences of solutions for
a certain integro-differential type problem involving the logarithmic
Laplacian. The argument is based on the fixed point technique when such
equation contains the operator without the Fredholm property. It is
established that, under the reasonable technical conditions, the convergence
in $L^{1}({\mathbb{R}}^{d})$ of the integral kernels yields the existence and
convergence in $L^{2}({\mathbb{R}}^{d})$ of the solutions.

Keywords: solvability conditions, non-Fredholm operators, logarithmic
Laplacian, integral kernel

AMS subject classification: 35P30, 45K05, 47G20

##  1 Introduction

We recall that a linear operator $L$ acting from a Banach space $E$ into
another Banach space $F$ satisfies the Fredholm property if its image is
closed, the dimension of its kernel and the codimension of its image are
finite. Consequently, the problem $Lu=f$ is solvable if and only if
$\phi_{i}(f)=0$ for a finite number of functionals $\phi_{i}$ from the dual
space $F^{*}$. Such properties of Fredholm operators are broadly used in many
methods of the linear and nonlinear analysis.

Elliptic equations in bounded domains with a sufficiently smooth boundary
satisfy the Fredholm property if the ellipticity condition, proper ellipticity
and Lopatinskii conditions are fulfilled (see e.g. [2], [11], [27], [30]).
This is the main result of the theory of linear elliptic problems. In the
situation of unbounded domains, such conditions may not be sufficient and the
Fredholm property may not be satisfied. For instance, the Laplace operator,
$Lu=\Delta u$, in $\mathbb{R}^{d}$ fails to satisfy the Fredholm property when
considered in Hölder spaces, $L:C^{2+\alpha}(\mathbb{R}^{d})\to
C^{\alpha}(\mathbb{R}^{d})$, or in Sobolev spaces, $L:H^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$.

Linear elliptic equations in unbounded domains satisfy the Fredholm property
if and only if, in addition to the conditions given above, the limiting
operators are invertible (see [31]). In some simple cases, the limiting
operators can be constructed explicitly. For example, if

$Lu=a(x)u^{\prime\prime}+b(x)u^{\prime}+c(x)u,\;\;\;x\in\mathbb{R},$

where the coefficients of the operator have limits at the infinities,

$a_{\pm}=\lim_{x\to\pm\infty}a(x),\;\;\;b_{\pm}=\lim_{x\to\pm\infty}b(x),\;\;\;c_{\pm}=\lim_{x\to\pm\infty}c(x),$

the limiting operators are equal to:

$L_{\pm}u=a_{\pm}u^{\prime\prime}+b_{\pm}u^{\prime}+c_{\pm}u.$

Since the coefficients here are constants, the essential spectrum of such
operator, that is the set of complex numbers $\lambda$ for which the operator
$L-\lambda$ fails to satisfy the Fredholm property, can be explicitly found by
means of the Fourier transform:

$\lambda_{\pm}(\xi)=-a_{\pm}\xi^{2}+b_{\pm}i\xi+c_{\pm},\;\;\;\xi\in\mathbb{R}.$

The invertibility of the limiting operators is equivalent to the condition
that the essential spectrum does not contain the origin.

In the case of general elliptic equations, the same assertions are valid. The
Fredholm property is satisfied if the origin does not belong to the the
essential spectrum or if the limiting operators are invertible. However, these
conditions may not be explicitly formulated.

In the situation of non-Fredholm operators the usual solvability relations may
not be applicable and solvability conditions are, in general, not known. There
are some classes of operators for which solvability relations are obtained.
Let us illustrate them with the following example. Consider the equation

$Lu\equiv\Delta u+au=f$ (1.1)

in $\mathbb{R}^{d},\ d\in{\mathbb{N}}$, where $a$ is a positive constant. The
operator $L$ here coincides with its limiting operators. The homogeneous
equation has a nontrivial bounded solution. Thus, the Fredholm property is not
satisfied. However, since the operator has constant coefficients, we can apply
the Fourier transform and find the solution explicitly. The solvability
conditions can be formulated as follows. If $f\in L^{2}(\mathbb{R}^{d})$ and
$xf\in L^{1}(\mathbb{R}^{d})$, then there exists a unique solution of such
problem in $H^{2}(\mathbb{R}^{d})$ if and only if

$\Bigg{(}f(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Bigg{)}_{L^{2}(\mathbb{R}^{d})}=0,\quad
p\in S_{\sqrt{a}}^{d}\quad a.e.$

(see [37]). Throughout the article $S_{r}^{d}$ denotes the sphere in
$\mathbb{R}^{d}$ of radius $r$ centered at the origin. Thus, though the
operator fails to satisfy the Fredholm property, the solvability relations are
stated similarly. However, such similarity is only formal because the range of
the operator is not closed.

In the case of the operator involving a scalar potential,

$Lu\equiv\Delta u+a(x)u=f,$

the Fourier transform is not directly applicable. Nevertheless, the
solvability conditions in ${\mathbb{R}}^{3}$ can be obtained by a rather
sophisticated application of the theory of self-adjoint operators (see [35]).
As before, the solvability relations are given in terms of the orthogonality
to the solutions of the homogeneous adjoint equation. There are several other
examples of linear elliptic non-Fredholm operators for which the solvability
conditions can be derived (see [31], [32], [34], [37]).

Solvability relations play a crucial role in the analysis of nonlinear
elliptic problems. In the case of the operators without the Fredholm property,
in spite of a certain progress in the understanding of the linear equations,
there exist only few examples where nonlinear non-Fredholm operators are
studied (see [9], [10], [15], [16], [36], [37], [38], [39]). The work [12]
addresses the finite and infinite dimensional attractors for the evolution
equations of mathematical physics. The large time behavior of solutions of a
class of fourth-order parabolic problems defined on unbounded domains using
the Kolmogorov $\varepsilon$-entropy as a measure was studied in [14]. In [20]
the authors discuss the attractor for a nonlinear reaction-diffusion system in
an unbounded domain in ${\mathbb{R}}^{3}$. The articles [22] and [29] are
important for the understanding of the Fredholm and properness properties of
quasilinear elliptic systems of second order and of the operators of this kind
on ${\mathbb{R}}^{N}$. The work [23] is dedicated to the exponential decay and
the Fredholm properties in second order quasilinear elliptic systems. The
article [13] is a systematic study of a dynamical systems approach to
investigating the symmetrization and stabilization properties of the
nonnegative solutions of nonlinear elliptic equations in asymptotically
symmetric unbounded domains.

In the present work we consider another class of stationary nonlinear
equations, for which the Fredholm property may not be satisfied:

$\Big{[}-\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}u+au+\int_{{\mathbb{R}}^{d}}G(x-y)F(u(y),y)dy=0,\quad
x\in{\mathbb{R}}^{d},\quad d\in{\mathbb{N}},$ (1.2)

where $a\in{\mathbb{R}}$ is a constant.

The logarithmic Laplacian $\hbox{ln}(-\Delta)$ is the operator with Fourier
symbol $2\hbox{ln}|p|$. It arises as the formal derivative
$\partial_{s}|_{s=0}(-\Delta)^{s}$ of fractional Laplacians at $s=0$. The
operator $(-\Delta)^{s}$ is extensively used, for instance in the studies of
the anomalous diffusion problems (see e.g. [28], [39] and the references
therein). Spectral properties of the logarithmic Laplacian in an open set of
finite measure with Dirichlet boundary conditions were considered in [26] (see
also [7]). The studies of $\hbox{ln}(-\Delta)$ are important for the
understanding of the asymptotic spectral properties of the family of
fractional Laplacians in the limit $s\to 0^{+}$. In [24] it was established
that such operator allows to characterize the $s$-dependence of solution to
fractional Poisson equations for the full range of exponents $s\in(0,1)$. The
solvability of the integro-differential equation similar to (1.2) with the
standard Laplace operator was discussed in [36].

The operator involved in problem (1.2) is given by

$l_{a}:=\frac{1}{2}\hbox{ln}(-\Delta)-a,\quad a\in{\mathbb{R}}$ (1.3)

and is considered on $L^{2}({\mathbb{R}}^{d}),\ d\in{\mathbb{N}}$. By virtue
of the standard Fourier transform, it can be trivially obtained that the
essential spectrum of (1.3) equals to

$\lambda_{a}(p)=\hbox{ln}|p|-a,\quad p\in{\mathbb{R}}^{d},\quad
a\in{\mathbb{R}}.$ (1.4)

As distinct from many previous articles devoted to the operators without the
Fredholm property, (1.4) fills not a semi-axis but the whole real line, which
can be easily observed from (1.4). Hence, the inverse of (1.3) is unbounded.

The article [17] deals with the solvability of certain linear nonhomogeneous
equations containing the logarithmic Laplacian and the logarithmic Schrödinger
operator with a shallow, short-range scalar potential. These results were
generalized to the case of the logarithmic Schrödinger operators in higher
dimensions in [18]. The work [19] is devoted to the solvability of linear and
nonlinear nonhomogeneous equations in one dimension involving the logarithmic
Laplacian and the transport term. The logarithmic Schrödinger operator and
associated Dirichlet problems were covered in [21]. Symmetry of positive
solutions for Lane-Emden systems involving the logarithmic Laplacian was
discussed in [40]. The logarithmic Laplacian is also used in the geometric
context of the $0$-fractional perimeter (see [8]).

In the population dynamics the integro-differential problems describe the
models with the intra-specific competition and the nonlocal consumption of
resources (see e.g. [4], [5]). Solving them is important for the understanding
of the emergence and propagation of patterns in the theory of speciation (see
[33]). Standing lattice solitons in the discrete NLS equation with saturation
were covered in [3].

Let us use the explicit form of the solvability conditions and explore the
existence of solutions of our nonlinear equation.

##  2 Formulation of the results

The nonlinear part of our problem (1.2) will satisfy the following regularity
conditions.

Assumption 1. Let $d\in{\mathbb{N}}$. Function
$F(u,x):{\mathbb{R}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ is satisfying the
Caratheodory condition (see [25]), such that

$|F(u,x)|\leq k|u|+h(x)\quad for\quad u\in{\mathbb{R}},\quad
x\in{\mathbb{R}}^{d}$ (2.1)

with a constant $k>0$ and $h(x):{\mathbb{R}}^{d}\to{\mathbb{R}}^{+},\quad
h(x)\in L^{2}({\mathbb{R}}^{d})$. Moreover, it is a Lipschitz continuous
function, so that

$|F(u_{1},x)-F(u_{2},x)|\leq l|u_{1}-u_{2}|\quad for\quad any\quad
u_{1,2}\in{\mathbb{R}},\quad x\in{\mathbb{R}}^{d}$ (2.2)

with a constant $l>0$.

The work [6] deals with the solvability of a local elliptic equation in a
bounded domain in ${\mathbb{R}}^{N}$. The nonlinear function involved there
was allowed to have a sublinear growth.

In order to establish the existence of the solution of problem (1.2), we will
use the auxiliary equation

$\Big{[}\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}u-au=\int_{{\mathbb{R}}^{d}}G(x-y)F(v(y),y)dy,\quad
x\in{\mathbb{R}}^{d},\quad d\in{\mathbb{N}},$ (2.3)

where $a\in{\mathbb{R}}$ is a constant.

Let us define

$(f_{1}(x),f_{2}(x))_{L^{2}({\mathbb{R}}^{d})}:=\int_{{\mathbb{R}}^{d}}f_{1}(x)\bar{f_{2}}(x)dx,$
(2.4)

with a slight abuse of notations when such functions do not belong to
$L^{2}({\mathbb{R}}^{d})$, like for instance those contained in orthogonality
relations (4.4) and (4.5) below. Clearly, if $f_{1}(x)\in
L^{1}({\mathbb{R}}^{d})$ and $f_{2}(x)\in L^{\infty}({\mathbb{R}}^{d})$, then
the integral in the right side of (2.4) is well defined.

The main issue for our equation (2.3) is that we deal with the self-adjoint,
non-Fredholm operator (1.3) involved in it, which is the obstacle to solve our
problem. The similar situations but in linear problems, both self-adjoint and
non self-adjoint containing the differential operators without the Fredholm
property were considered extensively in recent years (see [31], [32], [35],
[37]).

We establish that under the reasonable auxiliary conditions, problem (2.3)
defines a map $t_{a}:L^{2}({\mathbb{R}}^{d})\to L^{2}({\mathbb{R}}^{d})$ with
the constant $a\in{\mathbb{R}}$, which is a strict contraction.

Theorem 1. Let $G(x):{\mathbb{R}}^{d}\to{\mathbb{R}},\ G(x)\in
L^{1}({\mathbb{R}}^{d}),\ xG(x)\in L^{1}({\mathbb{R}}^{d})$ with
$d\in{\mathbb{N}},\ a\in{\mathbb{R}}$ and Assumption 1 is valid.

We also assume that orthogonality conditions (4.4) hold when $d=1$ and (4.5)
are valid for $d\geq 2$ and $(2\pi)^{\frac{d}{2}}N_{a}l<1$ with $N_{a}$
defined in (4.3). Then the map $v\mapsto t_{a}v=u$ on
$L^{2}({\mathbb{R}}^{d})$ defined by equation (2.3) has a unique fixed point
$v_{a}$, which is the only solution of problem (1.2) in
$L^{2}({\mathbb{R}}^{d})$.

The fixed point $v_{a},\ a\in{\mathbb{R}}$ is nontrivial in the whole space
provided the intersection of supports of the Fourier transforms of functions

$supp\widehat{F(0,x)}\cap supp\widehat{G}$

is a set of nonzero Lebesgue measure in ${\mathbb{R}}^{d}$.

Let us introduce the sequence of approximate equations related to problem
(1.2), such that

$\Big{[}-\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}u_{m}+au_{m}+\int_{{\mathbb{R}}^{d}}G_{m}(x-y)F(u_{m}(y),y)dy=0,\quad
x\in{\mathbb{R}}^{d},\quad d\in{\mathbb{N}}$ (2.5)

with a constant $a\in{\mathbb{R}}$ and $m\in{\mathbb{N}}$. The sequence of
kernels $\displaystyle{\\{G_{m}(x)\\}_{m=1}^{\infty}}$ tends to $G(x)$ as
$m\to\infty$ in the appropriate function spaces specified below. We establish
that, under the certain technical conditions, each of equations (2.5) admits a
unique solution $u_{m}(x)\in L^{2}({\mathbb{R}}^{d})$, the limiting problem
(1.2) possesses a unique solution $u(x)\in L^{2}({\mathbb{R}}^{d})$, and
$u_{m}(x)\to u(x)$ in $L^{2}({\mathbb{R}}^{d})$ as $m\to\infty$. This is the
so-called existence of solutions in the sense of sequences (see [38], also
[31]). In this case, the solvability conditions can be stated for the iterated
kernels $G_{m}$. They imply the convergence of the kernels in terms of the
Fourier transforms (see the Appendix) and, consequently, the convergence of
the solutions (Theorem 2). The similar ideas in the sense of the standard
Schrödinger type operators were used in [34]. Our second main statement is as
follows.

Theorem 2. Let $m\in{\mathbb{N}},\ G_{m}(x):{\mathbb{R}}^{d}\to{\mathbb{R}},\
G_{m}(x)\in L^{1}({\mathbb{R}}^{d}),\ xG_{m}(x)\in L^{1}({\mathbb{R}}^{d})$
with $d\in{\mathbb{N}},\ a\in{\mathbb{R}}$, such that $G_{m}(x)\to G(x)$ in
$L^{1}({\mathbb{R}}^{d})$ and $xG_{m}(x)\to xG(x)$ in
$L^{1}({\mathbb{R}}^{d})$ as $m\to\infty$ and Assumption 1 holds.

Let us also assume that orthogonality relations (4.22) are valid for $d=1$ and
orthogonality conditions (4.23) hold when $d\geq 2$ and

$(2\pi)^{\frac{d}{2}}N_{a,\ m}l\leq 1-\varepsilon$

for all $m\in{\mathbb{N}}$ with a certain fixed $0<\varepsilon<1$, where
$N_{a,\ m}$ is introduced in (4.21).

Then each equation (2.5) has a unique solution $u_{m}(x)\in
L^{2}({\mathbb{R}}^{d})$, and limiting problem (1.2) admits a unique solution
$u(x)\in L^{2}({\mathbb{R}}^{d})$.

Moreover, $u_{m}(x)\to u(x)$ in $L^{2}({\mathbb{R}}^{d})$ as $m\to\infty$.

The unique solution $u_{m}(x)$ of each equation (2.5) does not vanish
identically in the whole space provided that the intersection of supports of
the Fourier transforms of functions

$supp\widehat{F(0,x)}\cap supp\widehat{G_{m}}$

is a set of nonzero Lebesgue measure in ${\mathbb{R}}^{d}$. Analogously, the
unique solution $u(x)$ of limiting problem (1.2) is nontrivial if

$supp\widehat{F(0,x)}\cap supp\widehat{G}$

is a set of nonzero Lebesgue measure in ${\mathbb{R}}^{d}$.

Remark 1. The importance of Theorem 2 above is the continuous dependence of
the solutions with respect to the integral kernels.

##  3 Solvability of the integro-differential equation

Proof of Theorem 1. Let us first suppose that for some $v(x)\in
L^{2}({\mathbb{R}}^{d})$ there exist two solutions $u_{1,2}(x)\in
L^{2}({\mathbb{R}}^{d})$ of equation (2.3). Then the difference function
$w(x):=u_{1}(x)-u_{2}(x)\in L^{2}({\mathbb{R}}^{d})$ satisfies the homogeneous
equation

$\Big{[}\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}w-aw=0.$ (3.1)

Since the operator in the left side of (3.1) introduced in (1.3) does not have
any nontrivial square integrable zero modes, $w(x)$ vanishes in
${\mathbb{R}}^{d}$.

We choose arbitrarily $v(x)\in L^{2}({\mathbb{R}}^{d})$ and apply the standard
Fourier transform (4.1) to both sides of problem (2.3). This gives us

$\widehat{u}(p)=(2\pi)^{\frac{d}{2}}\frac{\widehat{G}(p)\widehat{\varphi}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}},$
(3.2)

where $\widehat{\varphi}(p)$ stands for the Fourier image of $F(v(x),x)$.

Evidently, the upper bound

$|\widehat{u}(p)|\leq(2\pi)^{\frac{d}{2}}N_{a}|\widehat{\varphi}(p)|$

holds. Clearly, $N_{a}<\infty$ by virtue of Lemma A1 of the Appendix under
orthogonality relations (4.4) when $d=1$ and under (4.5) for $d\geq 2$.

This allows us to derive the estimate from above on the norm as

$\|u\|_{L^{2}({\mathbb{R}}^{d})}\leq(2\pi)^{\frac{d}{2}}N_{a}\|F(v(x),x)\|_{L^{2}({\mathbb{R}}^{d})}.$
(3.3)

Let us recall inequality (2.1) of Assumption 1 above. Hence, under the given
conditions the right side of bound (3.3) is finite. This means that for an
arbitrary $v(x)\in L^{2}({\mathbb{R}}^{d})$ there exists a unique solution
$u(x)\in L^{2}({\mathbb{R}}^{d})$ of equation (2.3) with its Fourier transform
given by (3.2), such that the map $t_{a}:L^{2}({\mathbb{R}}^{d})\to
L^{2}({\mathbb{R}}^{d})$ is well defined.

This enables us to choose arbitrary $v_{1,2}(x)\in L^{2}({\mathbb{R}}^{d})$,
so that their images $u_{1,2}:=t_{a}v_{1,2}\in L^{2}({\mathbb{R}}^{d})$. From
(2.3) we easily deduce that

$\Big{[}\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}u_{1}-au_{1}=\int_{{\mathbb{R}}^{d}}G(x-y)F(v_{1}(y),y)dy,$
(3.4)
$\Big{[}\frac{1}{2}\hbox{ln}(-\Delta)\Big{]}u_{2}-au_{2}=\int_{{\mathbb{R}}^{d}}G(x-y)F(v_{2}(y),y)dy.$
(3.5)

Let us apply the standard Fourier transform (4.1) to both sides of equations
(3.4) and (3.5). This yields

$\widehat{u}_{1}(p)=(2\pi)^{\frac{d}{2}}\frac{\widehat{G}(p)\widehat{\varphi_{1}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}},\quad\widehat{u}_{2}(p)=(2\pi)^{\frac{d}{2}}\frac{\widehat{G}(p)\widehat{\varphi_{2}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}},$

where $\widehat{\varphi_{1}}(p)$ and $\widehat{\varphi_{2}}(p)$ denote the
Fourier transforms of $F(v_{1}(x),x)$ and $F(v_{2}(x),x)$ respectively.

It can be trivially checked that the estimates from above

$|\widehat{u}_{1}(p)-\widehat{u}_{2}(p)|\leq(2\pi)^{\frac{d}{2}}N_{a}|\widehat{\varphi_{1}}(p)-\widehat{\varphi_{2}}(p)|$

is valid, which allows us to obtain the upper bound for the norm as

$\|u_{1}-u_{2}\|_{L^{2}({\mathbb{R}}^{d})}\leq(2\pi)^{\frac{d}{2}}N_{a}\|F(v_{1}(x),x)-F(v_{2}(x),x)\|_{L^{2}({\mathbb{R}}^{d})}.$

We recall inequality (2.2) of Assumption 1. Thus,

$\|F(v_{1}(x),x)-F(v_{2}(x),x)\|_{L^{2}({\mathbb{R}}^{d})}\leq
l\|v_{1}(x)-v_{2}(x)\|_{L^{2}({\mathbb{R}}^{d})},$

so that

$\|t_{a}v_{1}-t_{a}v_{2}\|_{L^{2}({\mathbb{R}}^{d})}\leq(2\pi)^{\frac{d}{2}}N_{a}l\|v_{1}-v_{2}\|_{L^{2}({\mathbb{R}}^{d})}.$
(3.6)

The constant in the right side of (3.6) is less than one as we assume. By
virtue of the Fixed Point Theorem, there exists a unique function $v_{a}\in
L^{2}({\mathbb{R}}^{d})$ with the property $t_{a}v_{a}=v_{a}$. This is the
only solution of equation (1.2) in $L^{2}({\mathbb{R}}^{d})$.

Let us suppose $v_{a}(x)$ vanishes identically in ${\mathbb{R}}^{d}$. This
will contradict to our assumption that the Fourier transforms of $G(x)$ and
$F(0,x)$ are nontrivial on a set of nonzero Lebesgue measure in
${\mathbb{R}}^{d}$.

Then we turn our attention to establishing the existence in the sense of
sequences of the solutions for our integro-differential equation in
${\mathbb{R}}^{d}$.

Proof of Theorem 2. According to the result of Theorem 1 above, each problem
(2.5) admits a unique solution $u_{m}(x)\in L^{2}({\mathbb{R}}^{d}),\
m\in{\mathbb{N}}$. Limiting equation (1.2) has a unique solution $u(x)\in
L^{2}({\mathbb{R}}^{d})$ by virtue of Lemma A2 below along with Theorem 1.

We apply the standard Fourier transform (4.1) to both sides of (1.2) and (2.5)
and arrive at

$\widehat{u}(p)=(2\pi)^{\frac{d}{2}}\frac{\widehat{G}(p)\widehat{f}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}},\quad\widehat{u_{m}}(p)=(2\pi)^{\frac{d}{2}}\frac{\widehat{G_{m}}(p)\widehat{f_{m}}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}},\quad
m\in{\mathbb{N}},$ (3.7)

where $\widehat{f}(p)$ and $\widehat{f_{m}}(p)$ denote the Fourier images of
$F(u(x),x)$ and $F(u_{m}(x),x)$ respectively.

Evidently,

$|\widehat{u_{m}}(p)-\widehat{u}(p)|\leq(2\pi)^{\frac{d}{2}}\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}|\widehat{f}(p)|+$
$(2\pi)^{\frac{d}{2}}N_{a,\ m}|\widehat{f_{m}}(p)-\widehat{f}(p)|$

with $N_{a,\ m}$ defined in (4.21). Then

$\|u_{m}-u\|_{L^{2}({\mathbb{R}}^{d})}\leq(2\pi)^{\frac{d}{2}}\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Bigg{(}\frac{|p|}{e^{a}}\Bigg{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\|F(u(x),x)\|_{L^{2}({\mathbb{R}}^{d})}+$
$(2\pi)^{\frac{d}{2}}N_{a,\
m}\|F(u_{m}(x),x)-F(u(x),x)\|_{L^{2}({\mathbb{R}}^{d})}.$ (3.8)

Let us recall bound (2.2) of Assumption 1 above. Hence,

$\|F(u_{m}(x),x)-F(u(x),x)\|_{L^{2}({\mathbb{R}}^{d})}\leq
l\|u_{m}(x)-u(x)\|_{L^{2}({\mathbb{R}}^{d})}.$ (3.9)

By means of (3.8) and (3.9), we derive

$\|u_{m}(x)-u(x)\|_{L^{2}({\mathbb{R}}^{d})}\\{1-(2\pi)^{\frac{d}{2}}N_{a,\
m}l\\}\leq$
$(2\pi)^{\frac{d}{2}}\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\|F(u(x),x)\|_{L^{2}({\mathbb{R}}^{d})}.$
(3.10)

Using (3.10) and (4.24), we have

$\|u_{m}(x)-u(x)\|_{L^{2}({\mathbb{R}}^{d})}\leq\frac{(2\pi)^{\frac{d}{2}}}{\varepsilon}\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\|F(u(x),x)\|_{L^{2}({\mathbb{R}}^{d})}.$

By virtue of inequality (2.1) of our Assumption 1, if $u(x)\in
L^{2}({\mathbb{R}}^{d})$ then $F(u(x),x)\in L^{2}({\mathbb{R}}^{d})$ as well.
Therefore,

$u_{m}(x)\to u(x),\quad m\to\infty$ (3.11)

in $L^{2}({\mathbb{R}}^{d})$ via the result of Lemma A2 of the Appendix.

We suppose that the unique solution $u_{m}(x)$ of problem (2.5) considered
above is trivial in ${\mathbb{R}}^{d}$ for some $m\in{\mathbb{N}}$. This will
contradict to our condition that the Fourier transforms of $G_{m}(x)$ and
$F(0,x)$ do not vanish identically on a set of nonzero Lebesgue measure in
${\mathbb{R}}^{d}$. The similar reasoning holds for the unique solution $u(x)$
of limiting equation (1.2).

##  4 Appendix

Let $G(x)$ be a function, $G(x):{\mathbb{R}}^{d}\to{\mathbb{R}},\
d\in{\mathbb{N}}$. We denote its standard Fourier transform using the hat
symbol as

$\widehat{G}(p):={1\over(2\pi)^{\frac{d}{2}}}\int_{{\mathbb{R}}^{d}}G(x)e^{-ipx}dx,\quad
p\in{\mathbb{R}}^{d},$ (4.1)

such that

$\|\widehat{G}(p)\|_{L^{\infty}({\mathbb{R}}^{d})}\leq{1\over(2\pi)^{\frac{d}{2}}}\|G(x)\|_{L^{1}({\mathbb{R}}^{d})}$
(4.2)

and
$\displaystyle{G(x)={1\over(2\pi)^{\frac{d}{2}}}\int_{{\mathbb{R}}^{d}}\widehat{G}(q)e^{iqx}dq,\
x\in{\mathbb{R}}^{d}.}$

For the technical purposes we introduce the auxiliary quantity

$N_{a}:=\Bigg{\|}\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})},$
(4.3)

where $a\in{\mathbb{R}}$ is a constant.

Lemma A1. Let $G(x):{\mathbb{R}}^{d}\to{\mathbb{R}},\ G(x)\in
L^{1}({\mathbb{R}}^{d}),\ xG(x)\in L^{1}({\mathbb{R}}^{d})$ with
$d\in{\mathbb{N}}$ and $a\in{\mathbb{R}}$.

a) If $d=1$ then $N_{a}<\infty$ if and only if the orthogonality relations

$\Big{(}G(x),\frac{e^{\pm
ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}=0$ (4.4)

are valid.

b) If $d\geq 2$ then $N_{a}<\infty$ if and only if the orthogonality
conditions

$\Big{(}G(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Big{)}_{L^{2}({\mathbb{R}}^{d})}=0,\quad
p\in S_{e^{a}}^{d}$ (4.5)

hold.

Proof. First of all, we discuss the case b) of the lemma, such that the
dimension of the problem $d\geq 2$.

Let us introduce the spherical layer for the technical purposes, namely

$A_{\delta}:=\\{p\in{\mathbb{R}}^{d}\ |\ e^{a}(1-\delta)\leq|p|\leq
e^{a}(1+\delta)\\},\quad 0<\delta<1,\quad d\geq 2,$ (4.6)

such that

$\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}=\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}+\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c}}.$
(4.7)

Here and below $A^{c}$ will denote the complement of a set
$A\subseteq{\mathbb{R}}^{d}$. The characteristic function of a set $A$ is
being designated as $\chi_{A}$.

Evidently,

$A_{\delta}^{c}=A_{\delta}^{c+}\cup A_{\delta}^{c-},$

where

$A_{\delta}^{c+}:=\\{p\in{\mathbb{R}}^{d}\ |\ |p|>e^{a}(1+\delta)\\},$ (4.8)
$A_{\delta}^{c-}:=\\{p\in{\mathbb{R}}^{d}\ |\ |p|<e^{a}(1-\delta)\\}.$ (4.9)

Hence, the second term in the right side of (4.7) can be written as

$\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c+}}+\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c-}}.$
(4.10)

Let us estimate using (4.2)

$\frac{|\widehat{G}(p)|}{|\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}|}\chi_{A_{\delta}^{c+}}\leq\frac{\|G(x)\|_{L^{1}({\mathbb{R}}^{d})}}{(2\pi)^{\frac{d}{2}}\hbox{ln}(1+\delta)}<\infty$

as we assume.

Similarly,

$\frac{|\widehat{G}(p)|}{|\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}|}\chi_{A_{\delta}^{c-}}\leq\frac{\|G(x)\|_{L^{1}({\mathbb{R}}^{d})}}{-(2\pi)^{\frac{d}{2}}\hbox{ln}(1-\delta)}<\infty.$

We express

$\widehat{G}(p)=\widehat{G}(e^{a},\sigma)+\int_{e^{a}}^{|p|}\frac{\partial\widehat{G}(s,\sigma)}{\partial
s}ds.$

Here and further down $\sigma$ will denote the angle variables on the sphere.
This allows us to write the first term in the right side of (4.7) as

$\frac{\widehat{G}(e^{a},\sigma)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}+\frac{\int_{e^{a}}^{|p|}\frac{\partial\widehat{G}(s,\sigma)}{\partial
s}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}.$ (4.11)

By virtue of the definition of the standard Fourier transform, we easily
derive

$\Big{|}\frac{\partial\widehat{G}(p)}{\partial|p|}\Big{|}\leq\frac{1}{(2\pi)^{\frac{d}{2}}}\|xG(x)\|_{L^{1}({\mathbb{R}}^{d})},\quad
p\in{\mathbb{R}}^{d},\quad d\geq 2.$ (4.12)

Let us estimate the second term in (4.11) from above in the absolute value by

$\frac{1}{(2\pi)^{\frac{d}{2}}}\|xG(x)\|_{L^{1}({\mathbb{R}}^{d})}\Bigg{|}\frac{|p|-e^{a}}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{|}\chi_{A_{\delta}}\leq
C\|xG(x)\|_{L^{1}({\mathbb{R}}^{d})}<\infty$

as assumed. Here and below $C$ will stand for a finite, positive constant.

Thus, it remains to consider the term

$\frac{\widehat{G}(e^{a},\sigma)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}.$
(4.13)

Clearly, (4.13) is bounded if and only if $\widehat{G}(e^{a},\sigma)$
vanishes, which is equivalent to orthogonality relations (4.5).

We complete the proof of the lemma by considering the case a), such that the
dimension of the problem $d=1$. Let us introduce the intervals on the real
line

$I_{\delta}^{+}:=[e^{a}(1-\delta),e^{a}(1+\delta)],\quad
I_{\delta}^{-}:=[-e^{a}(1+\delta),-e^{a}(1-\delta)],$ (4.14)

with $0<\delta<1$, such that

$I_{\delta}:=I_{\delta}^{+}\cup I_{\delta}^{-}.$

Obviously,

$I_{\delta}^{c}:=I_{\delta}^{c+}\cup I_{\delta}^{c-},$

where

$I_{\delta}^{c+}:=(-\infty,-e^{a}(1+\delta))\cup(e^{a}(1+\delta),+\infty),$
(4.15) $I_{\delta}^{c-}:=(-e^{a}(1-\delta),e^{a}(1-\delta)).$ (4.16)

Evidently, we can write

$\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}=\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}+\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c+}}+\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}+\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c-}}.$
(4.17)

By means of (4.2), we obtain the upper bound on the second term in the right
side of (4.17) in the absolute value as

$\frac{|\widehat{G}(p)|}{\Big{|}\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}\Big{|}}\chi_{I_{\delta}^{c+}}\leq\frac{\|G(x)\|_{L^{1}({\mathbb{R}})}}{\sqrt{2\pi}\hbox{ln}(1+\delta)}<\infty$

due to our assumption.

Analogously, for the fourth term in the right side of (4.17) we have the
trivial inequality

$\frac{|\widehat{G}(p)|}{\Big{|}\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}\Big{|}}\chi_{I_{\delta}^{c-}}\leq\frac{\|G(x)\|_{L^{1}({\mathbb{R}})}}{-\sqrt{2\pi}\hbox{ln}(1-\delta)}<\infty.$

Note that

$\widehat{G}(p)=\widehat{G}(e^{a})+\int_{e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds.$

Hence, the first term in the right side of (4.17) equals to

$\frac{\widehat{G}(e^{a})}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}+\frac{\int_{e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}.$
(4.18)

Using the definition of the standard Fourier transform (4.1), we easily derive

$\Big{|}\frac{d\widehat{G}(p)}{dp}\Big{|}\leq\frac{1}{\sqrt{2\pi}}\|xG(x)\|_{L^{1}({\mathbb{R}})}.$
(4.19)

Clearly, formula (4.19) is the analog of bound (4.12) in one dimension.

Let us obtain the estimate from above in the absolute value on the second term
in (4.18) as

$\frac{1}{\sqrt{2\pi}}\|xG(x)\|_{L^{1}({\mathbb{R}})}\Bigg{|}\frac{p-e^{a}}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{|}\chi_{I_{\delta}^{+}}\leq
C\|xG(x)\|_{L^{1}({\mathbb{R}})}<\infty$

as we assume.

Obviously, the first term in (4.18) is bounded if and only if
$\widehat{G}(e^{a})=0$, which is equivalent to the orthogonality condition

$\Big{(}G(x),\frac{e^{ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}=0.$

Evidently,

$\widehat{G}(p)=\widehat{G}(-e^{a})+\int_{-e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds.$

Thus, the third term in the right side of (4.17) is equal to

$\frac{\widehat{G}(-e^{a})}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}+\frac{\int_{-e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}.$
(4.20)

The second term in (4.20) can be easily bounded from above in the absolute
value by

$\frac{1}{\sqrt{2\pi}}\|xG(x)\|_{L^{1}({\mathbb{R}})}\Bigg{|}\frac{p+e^{a}}{\hbox{ln}\Big{(}\frac{-p}{e^{a}}\Big{)}}\Bigg{|}\chi_{I_{\delta}^{-}}\leq
C\|xG(x)\|_{L^{1}({\mathbb{R}})}<\infty$

as assumed.

The first term in (4.20) is bounded if and only if $\widehat{G}(-e^{a})$
vanishes. This is equivalent to the orthogonality relation

$\Big{(}G(x),\frac{e^{-ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}=0,$

which completes the proof of the lemma.

We define the following technical expressions, which will help to treat
equations (2.5).

$N_{a,\
m}:=\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})},$
(4.21)

where $a\in{\mathbb{R}}$ is a constant, $d\in{\mathbb{N}}$ and
$m\in{\mathbb{N}}$ as well. Our final auxiliary proposition is as follows.

Lemma A2. Let $m\in{\mathbb{N}},\ G_{m}(x):{\mathbb{R}}^{d}\to{\mathbb{R}},\
G_{m}(x)\in L^{1}({\mathbb{R}}^{d}),\ xG_{m}(x)\in L^{1}({\mathbb{R}}^{d})$
with $d\in{\mathbb{N}}$ and $a\in{\mathbb{R}}$, such that $G_{m}(x)\to G(x)$
in $L^{1}({\mathbb{R}}^{d})$ and $xG_{m}(x)\to xG(x)$ in
$L^{1}({\mathbb{R}}^{d})$ as $m\to\infty$.

a) If $d=1$, let the orthogonality conditions

$\Big{(}G_{m}(x),\frac{e^{\pm
ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}=0,\quad m\in{\mathbb{N}}$
(4.22)

hold.

b) If $d\geq 2$, let the orthogonality relations

$\Big{(}G_{m}(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Big{)}_{L^{2}({\mathbb{R}}^{d})}=0,\quad
p\in S_{e^{a}}^{d},\quad m\in{\mathbb{N}}$ (4.23)

be valid.

Let in addition

$(2\pi)^{\frac{d}{2}}N_{a,\ m}l\leq 1-\varepsilon$ (4.24)

for all $m\in{\mathbb{N}}$ with some fixed $0<\varepsilon<1$.

Then

$\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\to\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}},\quad
m\to\infty,$ (4.25)

in $L^{\infty}({\mathbb{R}}^{d})$, so that

$\Bigg{\|}\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\to\Bigg{\|}\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})},\quad
m\to\infty.$ (4.26)

Furthermore,

$(2\pi)^{\frac{d}{2}}N_{a}l\leq 1-\varepsilon.$ (4.27)

Proof. By means of bound (4.2), we have

$\|\widehat{G_{m}}(p)-\widehat{G}(p)\|_{L^{\infty}({\mathbb{R}}^{d})}\leq\frac{1}{(2\pi)^{\frac{d}{2}}}\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}}^{d})}\to
0,\quad m\to\infty$ (4.28)

as assumed.

Evidently, (4.26) follows easily from (4.25) via the standard triangle
inequality.

Estimate (4.27) is an easy consequence of (4.24) and (4.26).

Let us first consider the situation b) of our lemma when the dimension of the
problem $d\geq 2$. In such case orthogonality conditions (4.23) hold as we
assume. We easily establish that the analogous statement will be valid in the
limit. Obviously, for $p\in S_{e^{a}}^{d}$

$\Big{|}\Big{(}G(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Big{)}_{L^{2}({\mathbb{R}}^{d})}\Big{|}=\Big{|}\Big{(}G(x)-G_{m}(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Big{)}_{L^{2}({\mathbb{R}}^{d})}\Big{|}\leq\frac{1}{(2\pi)^{\frac{d}{2}}}\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}}^{d})}\to
0$

as $m\to\infty$ due to the one of our assumptions, so that

$\Big{(}G(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Big{)}_{L^{2}({\mathbb{R}}^{d})}=0,\quad
p\in S_{e^{a}}^{d}$ (4.29)

holds. We have

$\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}=\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}+\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c}}.$
(4.30)

Clearly, the second term in the right side of (4.30) equals to

$\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c+}}+\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c-}}.$
(4.31)

Let us recall estimate (4.28). Hence

$\Bigg{\|}\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c+}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\leq\frac{\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}}^{d})}}{(2\pi)^{\frac{d}{2}}\hbox{ln}(1+\delta)}\to
0,\quad m\to\infty$

as assumed. Similarly,

$\Bigg{\|}\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}^{c-}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\leq\frac{\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}}^{d})}}{-(2\pi)^{\frac{d}{2}}\hbox{ln}(1-\delta)}\to
0,\quad m\to\infty.$

By virtue of orthogonality relations (4.29) and (4.23), we obtain

$\widehat{G}(e^{a},\sigma)=0,\quad\widehat{G_{m}}(e^{a},\sigma)=0,\quad
m\in{\mathbb{N}}.$ (4.32)

By means of (4.32),

$\widehat{G}(p)=\int_{e^{a}}^{|p|}\frac{\partial\widehat{G}(s,\sigma)}{\partial
s}ds,\quad\widehat{G_{m}}(p)=\int_{e^{a}}^{|p|}\frac{\partial\widehat{G_{m}}(s,\sigma)}{\partial
s}ds,\quad m\in{\mathbb{N}}.$

Then the first term in the right side of (4.30) is given by

$\frac{\int_{e^{a}}^{|p|}\Big{[}\frac{\partial\widehat{G_{m}}(s,\sigma)}{\partial
s}-\frac{\partial\widehat{G}(s,\sigma)}{\partial
s}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}},\quad
m\in{\mathbb{N}}.$ (4.33)

Let us use the definition of the standard Fourier transform (4.1) to derive

$\Big{|}\frac{\partial\widehat{G_{m}}(p)}{\partial|p|}-\frac{\partial\widehat{G}(p)}{\partial|p|}\Big{|}\leq\frac{1}{(2\pi)^{\frac{d}{2}}}\||x|G_{m}(x)-|x|G(x)\|_{L^{1}({\mathbb{R}}^{d})},\quad
p\in{\mathbb{R}}^{d},\quad d\geq 2,\quad m\in{\mathbb{N}},$ (4.34)

such that expression (4.33) can be easily bounded from above in the absolute
value by

$\frac{1}{(2\pi)^{\frac{d}{2}}}\||x|G_{m}(x)-|x|G(x)\|_{L^{1}({\mathbb{R}}^{d})}\Bigg{|}\frac{|p|-e^{a}}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{|}\chi_{A_{\delta}}\leq
C\||x|G_{m}(x)-|x|G(x)\|_{L^{1}({\mathbb{R}}^{d})},\quad m\in{\mathbb{N}}.$

Therefore,

$\Bigg{\|}\frac{\int_{e^{a}}^{|p|}\Big{[}\frac{\partial\widehat{G_{m}}(s,\sigma)}{\partial
s}-\frac{\partial\widehat{G}(s,\sigma)}{\partial
s}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{A_{\delta}}\Bigg{\|}_{L^{\infty}({\mathbb{R}}^{d})}\leq
C\||x|G_{m}(x)-|x|G(x)\|_{L^{1}({\mathbb{R}}^{d})}\to 0,\quad m\to\infty$

as we assume. This completes the proof of the case b) of the lemma.

Finally, we discuss the situation a) when the dimension of the problem $d=1$.
Hence, orthogonality relations (4.22) are valid as assumed. Let us demonstrate
that the analogous conditions will hold in the limit. Indeed,

$\Big{|}\Big{(}G(x),\frac{e^{\pm
ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}\Big{|}=\Big{|}\Big{(}G(x)-G_{m}(x),\frac{e^{\pm
ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}\Big{|}\leq\frac{1}{\sqrt{2\pi}}\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}})}\to
0$

as $m\to\infty$ due to the one of our conditions, such that

$\Big{(}G(x),\frac{e^{\pm
ie^{a}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}({\mathbb{R}})}=0.$ (4.35)

Obviously,

$\frac{\widehat{G_{m}}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}-\frac{\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}=\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}+$
$\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c+}}+\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}+\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c-}},\quad
m\in{\mathbb{N}}.$ (4.36)

By virtue of (4.28), the second term in the right side of (4.36) can be
estimated from above in the norm as

$\Bigg{\|}\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c+}}\Bigg{\|}_{L^{\infty}({\mathbb{R}})}\leq\frac{\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}})}}{\sqrt{2\pi}\hbox{ln}(1+\delta)}\to
0,\quad m\to\infty.$

Similarly, the fourth term in the right side of (4.36) can be bounded from
above in the norm as

$\Bigg{\|}\frac{\widehat{G_{m}}(p)-\widehat{G}(p)}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{c-}}\Bigg{\|}_{L^{\infty}({\mathbb{R}})}\leq\frac{\|G_{m}(x)-G(x)\|_{L^{1}({\mathbb{R}})}}{-\sqrt{2\pi}\hbox{ln}(1-\delta)}\to
0,\quad m\to\infty.$

Let us recall orthogonality relations (4.35) and (4.22). Hence,

$\widehat{G}(e^{a})=0,\quad\widehat{G_{m}}(e^{a})=0,\quad m\in{\mathbb{N}}.$
(4.37)

By means of (4.37), we have

$\widehat{G}(p)=\int_{e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds,\quad\widehat{G_{m}}(p)=\int_{e^{a}}^{p}\frac{d\widehat{G_{m}}(s)}{ds}ds,\quad
m\in{\mathbb{N}},$

so that the first term in the right side of (4.36) is equal to

$\frac{\int_{e^{a}}^{p}\Big{[}\frac{d\widehat{G_{m}}(s)}{ds}-\frac{d\widehat{G}(s)}{ds}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}.$
(4.38)

Using the definition of the standard Fourier transform (4.1), we easily arrive
at

$\Bigg{|}\frac{d\widehat{G_{m}}(p)}{dp}-\frac{d\widehat{G}(p)}{dp}\Bigg{|}\leq\frac{1}{\sqrt{2\pi}}\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})},\quad
p\in{\mathbb{R}},\quad m\in{\mathbb{N}}.$ (4.39)

Therefore, expression (4.38) can be estimated from above in the absolute value
by

$\frac{1}{\sqrt{2\pi}}\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})}\Bigg{|}\frac{p-e^{a}}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{|}\chi_{I_{\delta}^{+}}\leq
C\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})},\quad m\in{\mathbb{N}},$

such that

$\Bigg{\|}\frac{\int_{e^{a}}^{p}\Big{[}\frac{d\widehat{G_{m}}(s)}{ds}-\frac{d\widehat{G}(s)}{ds}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{+}}\Bigg{\|}_{L^{\infty}({\mathbb{R}})}\leq
C\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})}\to 0,\quad m\to\infty$

as assumed.

By virtue of orthogonality conditions (4.35) and (4.22), we have

$\widehat{G}(-e^{a})=0,\quad\widehat{G_{m}}(-e^{a})=0,\quad m\in{\mathbb{N}}.$
(4.40)

Using (4.40), we easily obtain

$\widehat{G}(p)=\int_{-e^{a}}^{p}\frac{d\widehat{G}(s)}{ds}ds,\quad\widehat{G_{m}}(p)=\int_{-e^{a}}^{p}\frac{d\widehat{G_{m}}(s)}{ds}ds,\quad
m\in{\mathbb{N}}.$

Then the third term in the right side of (4.36) is given by

$\frac{\int_{-e^{a}}^{p}\Big{[}\frac{d\widehat{G_{m}}(s)}{ds}-\frac{d\widehat{G}(s)}{ds}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}.$
(4.41)

Let us use inequality (4.39) to derive the upper bound on (4.41) in the
absolute value by

$\frac{1}{\sqrt{2\pi}}\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})}\Bigg{|}\frac{p+e^{a}}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\Bigg{|}\chi_{I_{\delta}^{-}}\leq
C\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})},\quad m\in{\mathbb{N}}.$

Then

$\Bigg{\|}\frac{\int_{-e^{a}}^{p}\Big{[}\frac{d\widehat{G_{m}}(s)}{ds}-\frac{d\widehat{G}(s)}{ds}\Big{]}ds}{\hbox{ln}\Big{(}\frac{|p|}{e^{a}}\Big{)}}\chi_{I_{\delta}^{-}}\Bigg{\|}_{L^{\infty}({\mathbb{R}})}\leq
C\|xG_{m}(x)-xG(x)\|_{L^{1}({\mathbb{R}})}\to 0,\quad m\to\infty$

due to the one of our assumptions, which completes the proof of the lemma in
the situation a).

Note that under the given conditions

$N_{a}<\infty,\quad N_{a,\ m}<\infty,\quad m\in{\mathbb{N}},$

where $a\in{\mathbb{R}}$ and $d\in{\mathbb{N}}$ according to the result of
Lemma A1 above.

## Acknowledgements

The first author is grateful to Israel Michael Sigal for the partial support
by the NSERC grant NA 7901. The second author has been supported by the RUDN
University Strategic Academic Leadership Program.

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