On the bounds of the sum of eigenvalues for a Dirichlet problem involving
mixed fractional Laplacians

Huyuan<EMAIL_ADDRESS>
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang,
Jiangxi 330022, PR China

Mousomi Bhakta222<EMAIL_ADDRESS>
Department of Mathematics, Indian Institute of Science Education and Research
(IISER-Pune),
Pune 411008, India

Hichem Hajaiej333<EMAIL_ADDRESS>
California State University, Los Angeles, 5151, USA

###### Abstract

Our purpose in this paper is to study of the eigenvalues
$\\{\lambda_{i}(\mu)\\}_{i}$ of the Dirichlet problem

$(-\Delta)^{s_{1}}u=\lambda\big{(}(-\Delta)^{s_{2}}u+\mu u\big{)}\ \ {\rm in}\
\,\Omega,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^{N}\setminus\Omega,$

where $0<s_{2}<s_{1}<1$, $N>2s_{1}$ and $(-\Delta)^{s}$ is the fractional
Laplacian operator defined in the principle value sense.

We first show the existence of a sequence of eigenvalues, which approaches
infinity. Secondly we provide a Berezin–Li–Yau type lower bound for the sum of
the eigenvalues of the above problem. Furthermore, using a self-contained and
novel method, we establish an upper bound for the sum of eigenvalues of the
problem under study.

Keywords: Dirichlet eigenvalues; Fractional Laplacian, Berezin-Li-Yau method,
mixed nonlocal operator, mixed fractional Laplacian.

MSC2010: 35P15; 35R09.

##  1 Introduction and main results

Let $0<s_{2}<s_{1}<1$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$
with the integer $N\geq 1$, $N>2s_{1}$. The main goal of this paper is to
study the lower bounds of the eigenvalues for the Dirichlet problem

$\left\\{\begin{array}[]{lll}(-\Delta)^{s_{1}}u=\lambda\Big{(}(-\Delta)^{s_{2}}u+\mu
u\Big{)}&{\rm in}\quad\Omega,\\\\[5.69054pt]
\phantom{(-\Delta)^{s}}u=0&{\rm{in}}\
\quad\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (1.1)

where $(-\Delta)^{s}$ is the fractional laplacian defined in the following
sense (principle value)

$(-\Delta)^{s}u(x)=c_{N,s}\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}^{N}\setminus
B_{\epsilon}(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy$ (1.2)

with $s\in(0,1)$,
$c_{N,s}=2^{2s}\pi^{-\frac{N}{2}}s\frac{\Gamma(\frac{N+2s}{2})}{\Gamma(1-s)}$
and $\Gamma$ being the Gamma function, see e.g. [40]. Recall that, for
$s\in(0,1)$, the fractional Laplacian of a function $u\in
C^{\infty}_{c}(\mathbb{R}^{N})$ can also be defined by:

$(-\Delta)^{s}u(\xi)=\mathcal{F}^{-1}\bigg{(}|\xi|^{2s}\mathcal{F}(u)(\xi)\bigg{)}\qquad\text{for
all $\xi\in\mathbb{R}^{N}$},$

where $F(f)$ denotes the Fourier transform of $f$.

Problem (1.1) involves two fractional Laplacians with two different powers.
The terminology “mixed operators” refers to the differential or pseudo-
differential order of the operator, and to the type of the operator, which can
combine classical and fractional features. When $\lambda<0$, (1.1) involves a
sum of two fractional Laplacians of different orders. Indeed, such sum of
operators arises naturally from superposition of two stochastic processes with
a different random walk and a Lévy flight, this is the case when a particle
can follow either of these two processes according to a certain probability,
the associated limit diffusion equation is described by a sum of fractional
Laplacians, see for e.g [4] . While if $\lambda>0$, (1.1) models a difference
of two fractional Laplacians. The sum and the difference of two fractional
Laplacians appear in many circumstances. To mention few, problems of blood
circulation in the heart, responsible for causing heart problems and in many
circumstances coronary bypass surgeries, can be modelled by two to five mixed
fractional Laplacians, for e.g. see [16, 33, 34, 35]. It also appears in many
circumstances because of the anomalous blood circulation, but it is often not
the same anomaly in all the arteries, and the blood can follow either of the
five arteries. Other applications of mixed fractional operators with different
orders include plasma physics and population dynamics, ways to reduce
pandemics and so on. In view of these important applications, we strongly
believe that equation (1.1) and some of its variants described above will get
an increasing interest in the near future.

The most primitive model of (1.1) for the eigenvalues is

$\left\\{\begin{array}[]{lll}-\Delta u=\lambda u&{\rm
in}\quad\Omega,\\\\[4.2679pt]
\phantom{-\Delta}u=0&{\rm{on}}\quad\\!\partial\Omega,\end{array}\right.$ (1.3)

which has attracted the attention of mathematicians since 1912. Indeed, Weyl
in [43] showed that the $k$-th eigenvalue, $\lambda_{1,k}(\Omega)$ ($1$ stands
for $s=1$) of (1.3), has the asymptotic behavior

$\lambda_{1,k}(\Omega)\sim C_{N}(k|\Omega|)^{\frac{2}{N}}\quad{\rm as}\ \
k\to+\infty,$

where

$C_{N}=(2\pi)^{2}|B_{1}|^{-\frac{2}{N}}$

and $|B_{1}|$ is the volume of unit ball in $\mathbb{R}^{N}$. Later, Pólya
[38] (in 1960) provided a lower bound for the k-th eigenvalue,

$\lambda_{1,k}(\Omega)\geq C(k/|\Omega|)^{\frac{2}{N}}$ (1.4)

for any plane covering domain $D$ in $\mathbb{R}^{2}$ with $C=C_{N}$, (his
proof also works in dimension $N\geq 3$). $D$ is called a plane covering
domain in $\mathbb{R}^{2}$ if an infinity of domains congruent to $D$ cover
the whole plane without gaps and without overlapping except a set of measure
zero. In [38], Pólya also made a conjecture that (1.4) holds for any bounded
domain in $\mathbb{R}^{N}$ with $C=C_{N}$. To answer this conjecture, Lieb
[32] proved (1.4) with a positive constant $C$ for general bounded domain and
Li-Yau [31] improved the constant $C$ to $\frac{N}{N+2}C_{N}$. Until now, this
constant for lower bound is the best and (1.4) with $C=\frac{N}{N+2}C_{N}$ is
called Berezin-Lieb-Yau inequality. More related estimates on lower bounds for
the eigenvalues under various setting can be found in [14, 18, 32, 36]. On the
other hand, the upper bounds of Dirichlet eigenvalues were derived by Kröger
in [29] by calculating the Rayleigh quotient by using a sequence of functions
approaching the characterized function of $\Omega$. We also refer to Yang’s
upper bounds of the Dirichlet’s eigenvalues in [13, 8] in the following way:

$\lambda_{1,k}(\Omega)\leq
c(N,k)\lambda_{1,1}(\Omega)k^{\frac{2}{N}}\quad\text{ for some $c(N,k)>0$}.$

The unstopped interest in finding bounds for eigenvalues of the Dirichlet
problem is in part due to the following fact: The Hilbert-Polya conjecture is
to associate the zero of the Riemann Zeta function with the eigenvalue of a
Hermitian operator. This quest initiated the mathematical interest for
estimating the sum of Dirichlet eigenvalues of the Laplacian while in physics
the question is related to count the number of bound states of a one body
Schrödinger operator and to study their asymptotic distribution. The latter
constitutes in itself a branch in nonlinear analysis. During the last decade,
there has been a renewed and increasing interest in the study of linear and
nonlinear integral operators. The prototype is the fractional Laplacian. This
has been motivated by numerous applications in different fields motivated by
connections to real world life applications and by important advances in the
theory of linear and nonlinear partial differential equations, see basic
properties [24, 25, 26, 37], regularities [5, 39], Liouville property [6],
general nonlocal operator [11], fractional Pohozaev identity [40],
singularities [1, 7, 9], uniqueness [20], fractional variational setting [3,
15, 19, 27, 41] and the references therein.

When $\mu=0$, $s_{2}=0$ and $s=s_{1}\in(0,1)$, (1.1) reduces to the fractional
Laplacian problem

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=\lambda u&{\rm
in}\quad\Omega,\\\\[5.69054pt] \phantom{(-\Delta)^{s}}u=0&{\rm{in}}\
\quad\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (1.5)

for which the asymptotic behavior of eigenvalues $\lambda_{s,i}$ has been
studied, for Klein-Gordon operator i.e. $s=\frac{1}{2}$ in [23, Proposition
3.1] or for general order $s\in(0,1)$ in [21, Theorem 1],

$\lim_{k\to+\infty}k^{-\frac{2s}{N}}\lambda_{s,k}=a(N,s)|\Omega|^{-\frac{2s}{N}}$
(1.6)

where

$a(N,s)=(2\pi)^{2s}|B_{1}|^{{-\frac{2s}{N}}}.$ (1.7)

Moreover, a refinement of Berezin–Li–Yau-type lower bound for the sum of
eigenvalues was built by Yolcu and Yolcu in [45, Theorem 1.4] as follows

$\sum^{k}_{j=1}\lambda_{s,j}\geq\frac{N}{N+2s}a(N,s)|\Omega|^{-\frac{2s}{N}}k^{1+\frac{2s}{N}}+ck^{1-\frac{2-2s}{N}}$
(1.8)

for some $c>0$ depending on $|\Omega|$. In a recent work [44], Hajaiej and
Wang provided the asymptotic behavior of the sum of the eigenvalues of (1.5)

$\lim_{k\to+\infty}k^{-1-\frac{2s}{N}}\sum^{k}_{j=1}\lambda_{s,j}(\Omega)=\frac{N}{N+2s}a(N,s)|\Omega|^{-\frac{2s}{N}}.$
(1.9)

For more estimates on eigenvalues of the fractional Dirichlet problem, we
refer the readers to [12, 21, 23, 45, 46], a review [20] and the references
therein.

To analyze the fractional Dirichlet eigenvalues of (1.1), we denote
${\mathbb{H}}^{s}_{0}(\Omega)$ the space of all measurable functions
$u:\mathbb{R}^{N}\to\mathbb{R}$ with $u\equiv 0$ in
$\mathbb{R}^{N}\setminus\Omega$ and

${\iint_{\mathbb{R}^{2N}}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy<+\infty.$

It is well known that ${\mathbb{H}}^{s}_{0}(\Omega)$ is a Hilbert space
equipped with the inner product

$\mathcal{E}_{s}(u,w)=\frac{c_{N,s}}{2}{\iint_{\mathbb{R}^{2N}}}\frac{(u(x)-u(y)){(w(x)-w(y))}}{|x-y|^{N+2s}}dxdy$

and the induced norm

$\|u\|_{s}:=\sqrt{\mathcal{E}_{s}(u,u)}.$

We say a function $u\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)$ be an eigenfunction
of (1.1) corresponding to the eigenvalue $\lambda$ if

${\mathcal{E}}_{s_{1}}(u,w)=\lambda\Big{(}{\mathcal{E}}_{s_{2}}(u,w)+\mu\int_{\Omega}uw\,dx\Big{)}\quad\text{for
all $w\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)$.}$

For $\mu\geq-\lambda_{s_{2},1}$, we denote by
${\mathbb{H}}^{s_{2}}_{\mu,0}(\Omega)$ the space of all measurable functions
$u:\mathbb{R}^{N}\to\mathbb{R}$ with $u\equiv 0$ in
$\mathbb{R}^{N}\setminus\Omega$ and

$\|u\|_{s_{2},\mu}:=\bigg{(}{\mathcal{E}}_{s_{2}}(u,u)+\mu\int_{\Omega}u^{2}\,dx\bigg{)}^{\frac{1}{2}}<\infty.$

The corresponding inner product in ${\mathbb{H}}^{s_{2}}_{\mu,0}(\Omega)$ is
given by

${\langle
u,w\rangle_{s_{2},\mu}:=}\,{\mathcal{E}}_{s_{2}}(u,w)+\mu\int_{\Omega}uw\,dx,\quad\forall\,\,u,\,w\in{\mathbb{H}}^{s_{2}}_{0}(\Omega).$

Let $\lambda_{s_{2},1}$ be the first eigenvalue of (1.5) corresponding to
$s=s_{2}$. We note that $\|.\|_{s_{2},\mu}$ is equivalent to $\|.\|_{s_{2}}$
for $\mu>-\lambda_{s_{2},1}$.

Our first aim is to show the existence of a sequence of discrete eigenvalues
of (1.1) as follows.

######  Theorem 1.1.

Let $\mu>-\lambda_{s_{2},1}$, where $\lambda_{s_{2},1}>0$ be the first
eigenvalue of (1.5) with $s=s_{2}$. Then problem (1.1) admits a sequence of
real eigenvalues

$0<\lambda_{1}(\mu)\leq\lambda_{2}(\mu)\leq\cdots\leq\lambda_{j}(\mu)\leq\lambda_{j+1}(\mu)\leq\cdots$

and the corresponding eigenfunction $\phi_{i}$, $i\in\mathbb{N}$. Moreover, we
have the following properties:

  1. (i)

$\lambda_{j}(\mu)=\min\\{{\mathcal{E}}_{s_{1}}(u,u):\,u\in{\mathbb{H}}_{0,j}(\Omega),\,{\|u\|_{{\mathbb{H}}^{s_{2}}_{\mu,0}(\Omega)}=1}\\}$,
where

${{\mathbb{H}}_{0,1}(\Omega)={\mathbb{H}}^{s_{1}}_{0}(\Omega),\quad{\mathbb{H}}_{0,j}(\Omega):=\\{u\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)\>:\>\langle
u,\,\phi_{m}\rangle_{s_{2},\mu}=0\,\,\mbox{for}\,\,m=1,\dots,j-1\\}\,\,\text{for}\,\,j>1;}$

  2. (ii)

$\\{\phi_{j}\>:\>j\in\mathbb{N}\\}$ is an orthonormal basis of
${\mathbb{H}}^{s_{2}}_{\mu,0}(\Omega)$;

  3. (iii)

$\lim\limits_{j\to\infty}\lambda_{j}(\mu)=+\infty$;

  4. (iv)

For $\mu\in(-\lambda_{s_{2},1},+\infty)$, the map $\mu\mapsto\lambda_{1}(\mu)$
is decreasing and
$\displaystyle\limsup_{\mu\to{-\lambda_{s_{2},1}^{+}}}\lambda_{1}(\mu)<+\infty$.

We remark that

  1. (a)

from the appendix in [41], problem (1.5) has the property that the first
eigenvalue is simple and the corresponding eigenfunction $\phi_{s,1}$ is
positive; these properties are derived by the following:
${\mathcal{E}}_{s}(|\phi_{s,1}|)<{\mathcal{E}}_{s}(\phi_{s,1})$, if
$\phi_{s,1}$ is sign-changing. However, this argument fails for problem (1.1)
due to presence of multiple fractional Laplacians, and it is very interesting
but challenging to determine the one-fold of $\lambda_{1}(\mu)$ and the
positivity of the eigenfunction $\phi_{1}$ corresponding to the first
eigenvalue $\lambda_{1}$ for problem (1.1);

  2. (b)

thanks to the monotonicity and boundedness of $\lambda_{1}(\mu)$, assertion
$(iv)$ indicates that it is possible to obtain the existence of
$\\{\lambda_{j}(\mu)\\}_{j\in\mathbb{N}}$ for $\mu\leq-\lambda_{s_{2},1}$;

  3. (c)

it is known that eigenfunctions of (1.5) are $C^{\infty}(\Omega)$. To see
this, one uses bootstraps method to prove solutions of (1.5) are in
$L^{\infty}(\Omega)$ and then uses regularity results of [39]. While the
regularity for (1.1) seems to be much more complicated, because bootstraps
iteration has to work between different order fractional Laplacians.

We now provide a lower bound for the sum of eigenvalues of (1.1).

######  Theorem 1.2.

Let $\mu\geq 0$ and $\\{\lambda_{j}(\mu)\\}_{j\in\mathbb{N}}$ be the
increasing sequence of eigenvalues of problem (1.1) and $\omega_{{}_{N-1}}$
denote the surface area of the unit sphere in $\mathbb{R}^{N}$. Then for
$k\in\mathbb{N}$

$\displaystyle\sum^{k}_{j=1}\lambda_{j}(\mu)\geq
b_{1}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{1+\frac{2(s_{1}-s_{2})}{2s_{2}+N}}-\mu
b_{2}|\Omega|^{-\frac{2s_{1}}{N}}k^{1+\frac{2s_{1}-4s_{2}}{2s_{2}+N}},$ (1.10)

where

$b_{1}=\frac{(2s_{2}+N)^{\frac{2s_{1}+N}{2s_{2}+N}}}{2s_{1}+N}\Big{(}2^{-(N+1+2s_{2})}\pi^{-\frac{3N}{2}}\omega_{{}_{N-1}}^{2-\frac{2s_{2}}{N}}\frac{\Gamma(\frac{N-2s_{2}}{2})}{\Gamma(s_{2}+1)}N^{\frac{2s_{2}}{N}}\Big{)}^{-\frac{2(s_{1}-s_{2})}{2s_{2}+N}}$

and

$b_{2}=b_{1}\frac{2s_{1}+N}{N(2s_{2}+N)^{\frac{2s_{2}}{2s_{2}+N}}}\Big{(}2^{-(N+1+2s_{2})}\pi^{-\frac{3N}{2}}\omega_{{}_{N-1}}^{2-\frac{2s_{2}}{N}}\frac{\Gamma(\frac{N-2s_{2}}{2})}{\Gamma(s_{2}+1)}N^{\frac{2s_{2}}{N}}\Big{)}^{\frac{2s_{2}}{N+2s_{2}}}.$

We remark that when $s_{2}=0=\mu$, the constant $b_{1}(s_{1},0)$ in Theorem
1.2 reduces to $\frac{N}{N+2s_{1}}a(N,s_{1})$, where $a(N,s_{1})$ is defined
as in (1.7). Moreover, (1.10) coincides with Berezin-Li-Yau estimate for (1.3)
(with $s_{1}=1$).

The proof of Theorem 1.2 is inspired by the Berezin–Li–Yau method (see [31])
in which the authors treated mainly the function
$\Psi_{k}(x,y)=\sum^{k}_{j=1}\psi_{j}(x)\psi_{j}(y),$ where $\psi_{j}$’s are
the eigenfunctions corresponding to the eigenvalues $\lambda_{j}$ of (1.3).
Denote $F=\int_{\mathbb{R}^{N}}|({\mathcal{F}}_{x}\Psi_{k})(z,y)|^{2}dy$, the
key estimate is the following

$\int_{\mathbb{R}^{N}}Fdz\leq\Big{(}\frac{N+2}{N}\int_{\mathbb{R}^{N}}|z|^{2}Fdz\Big{)}^{\frac{N}{N+2}}\Big{(}|B_{1}|\|F\|_{L^{\infty}}\Big{)}^{\frac{2}{N+2}},$
(1.11)

where $|B_{1}|$ denotes the volume of unit ball in $\mathbb{R}^{N}$. For our
problem (1.1), the situation is much more complicated. More precisely, to
apply Berezin-Li-Yau method, we need to consider the function
$f=\int_{\mathbb{R}^{N}}|({\mathcal{F}}_{x}\Phi_{k})(z,y)|^{2}dy$, where

$\Phi_{k}(x,y)=\sum^{k}_{j=1}\phi_{j}(x)\tilde{\phi}_{j}(y),\quad\tilde{\phi}_{j}:=\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}^{\frac{1}{2}}\phi_{j}$

and $\phi_{j}$’s are eigenfunctions corresponding to the eigenvalues
$\lambda_{j}$ of (1.1). The most important estimate is
$\int_{\mathbb{R}^{N}}(|z|^{2s_{2}}+\mu)f(z)dz$, which is controlled by
$\int_{\mathbb{R}^{N}}|z|^{2s_{1}}fdz$ and $\|f\|_{L^{\infty}}$. These
difficulties arise from the non homogeneous property of lower oder term
$(|z|^{2s_{2}}+\mu)f$ and the estimate $\|f\|_{L^{\infty}}$. For
$\|f\|_{L^{\infty}}$, the original tool is the Bessel’s inequality, which
requires orthonormal property for $\\{\phi_{j}\\}$ in $L^{2}(\Omega)$, while
$\\{\phi_{j}\\}_{j}$ is not an orthonormal sequence in $L^{2}(\Omega)$ (it is
orthonormal in $H^{s_{2}}_{\mu,0}(\Omega)$). To overcome this difficulty, we
transform $\phi_{j}$ to $\tilde{\phi}_{j}$ which is orthonormal in
$L^{2}(\mathbb{R}^{N})$ but not supported in $\Omega$. Further using certain
delicate estimate, we have overcome the difficulty (see Section 3.2).

The following is a direct corollary from Theorem 1.2 using the monotonicity of
map $j\to\lambda_{j}(\mu)$.

######  Corollary 1.3.

Let $\mu\geq 0$ and $\\{\lambda_{j}(\mu)\\}_{j\in\mathbb{N}}$ be the
increasing sequence of eigenvalues of problem (1.1). Then for $k\in\mathbb{N}$

$\displaystyle\lambda_{k}(\mu)\geq
b_{1}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{\frac{2(s_{1}-s_{2})}{2s_{2}+N}}-\mu
b_{2}|\Omega|^{-\frac{2s_{1}}{N}}k^{\frac{2s_{1}-4s_{2}}{2s_{2}+N}},$ (1.12)

where $b_{1}$ and $b_{2}$ are same as in Theorem 1.2.

For the upper bounds, due to the numerous challenges mentioned earlier, we
only address the case that $\mu=0$ and a restriction on the upper range of
$s_{1}$. More specifically, we have the following estimates on upper bounds.

######  Theorem 1.4.

Let $0<s_{2}<s_{1}<\frac{1+s_{2}}{2}$, $\mu=0$ and
$\\{\lambda_{i}(\mu)\\}_{i\in\mathbb{N}}$ be the increasing sequence of
eigenvalues of problem (1.1) and $\Omega$ be a bounded $C^{2}$ domain. Then
there exists $c_{0}=c_{0}(N,s_{1},s_{2},\Omega)>0$ and
$\delta_{3}\in(0,1+\frac{3}{2}\frac{s_{1}-s_{2}}{N})$ such that for
$k\in\mathbb{N}$

$\displaystyle\sum^{k}_{j=1}\lambda_{j}(0)\leq
b_{3}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{1+\frac{2(s_{1}-s_{2})}{N}}+c_{0}k^{\delta_{3}},$
(1.13)

where

$b_{3}=(2\pi)^{2(s_{1}-s_{2})}\omega_{{}_{N-1}}^{-\frac{2(s_{1}-s_{2})}{N}}\frac{N^{1+\frac{2(s_{1}-s_{2})}{N}}}{N+2(s_{1}-s_{2})}.$

It is worth noting that

  1. (d)

the constant $b_{3}$ in the upper bound (1.13) coincides with (1.9) with
$s=s_{1}-s_{2}$;

  2. (e)

the upper bound and the lower bound for our problem (1.1) obtained in Theorem
1.2 and Theorem 1.4 are not enough to determine
$\sum^{k}_{j=1}\lambda_{j}(\mu)$, even for $\mu=0$.

  3. (f)

from our proofs, it isn’t too difficult to see that Theorem 1.1 and Theorem
1.2 can be extended to the case $s_{1}=1$. However, it fails for the upper
bound in Theorem 1.4 because of the restriction of
$s_{1}<\frac{1+s_{2}}{2}<1$, which is not essential since it appears due to
the technique difficulty.

The rest of the paper is organized as follows. In Section 2, we study
qualitative properties of the eigenvalues and the corresponding eigenfunctions
of problem (1.1), namely Theorem 1.1. Section 3 is devoted to show the lower
bound of the sum of eigenvalues, namely Theorem 1.2. Finally, in section 4, we
discuss the upper bounds aspects, namely Theorem 1.4.

Notations: Throughout this paper, $\omega_{N-1}$ denotes the surface area of
unit sphere in $\mathbb{R}^{N}$, $B_{r}(x)\subset\mathbb{R}^{N}$ is an open
ball of radius $r$ centered at $x\in\mathbb{R}^{N}$, and we set
$B_{r}:=B_{r}(0)$ for $r>0$. For any set $A$ of $\mathbb{R}^{N}$, $|A|$
denotes the Lebesgue measure of $A$ and $\mathcal{F}(f)$ denotes the Fourier
transform of a function $f$. By $C^{\infty}_{c}(\mathbb{R}^{N})$, we denote
$C^{\infty}$ functions in $\mathbb{R}^{N}$ with compact support.

##  2 Existence

We set $Q:=\mathbb{R}^{2N}\setminus(\Omega^{c}\times\Omega^{c})$, where
$\Omega^{c}=\mathbb{R}^{N}\setminus\Omega$ and for $1\leq p<\infty$ define

${W^{s,p}_{0}(\Omega)}:=\Big{\\{}u:\mathbb{R}^{N}\to\mathbb{R}\mbox{
measurable }\Big{|}u=0\ {\rm a.e.\ in\ }\Omega^{c}\mbox{ and
}\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy<\infty\Big{\\}}.$

Note that from the fractional Poincaré inequality, see [15], the space
${W^{s,p}_{0}(\Omega)}$ is endowed with the norm defined as

$\|u\|_{W^{s,p}_{0}(\Omega)}:=\left(\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p}.$

######  Theorem 2.1.

Let $0<s_{2}<s_{1}<1$, $p>1$ and $\Omega$ be a bounded domain in
$\mathbb{R}^{N}$, then ${W^{s_{1},p}_{0}(\Omega)\hookrightarrow
W^{s_{2},p}_{0}(\Omega)}$ is continuous and compact.

Before starting to prove the above theorem, we need to introduce the Besov
Space over $\Omega$. We follow the notations of [42, Section 2.3.1].

######  Definition 2.2.

Let $\mathcal{S}$ denote the Schwartz class functions on $\mathbb{R}^{N}$ and
${\mathcal{A}}$ be collection of all sequence
$\eta=\\{\eta_{i}\\}_{i=0}^{\infty}\in\mathcal{S}(\mathbb{R}^{N})$ such that

supp$(\eta_{0})\subset\\{x:|x|\leq 2\\}$,
supp$(\eta_{j})\subset\\{x:2^{j-1}\leq|x|\leq 2^{j+1}\\}$ for $j=1,2,\cdots$,

$\sum_{i=0}^{\infty}\eta_{j}(x)=1,$

and for every multi-index $\alpha$ there exists a positive number $c_{\alpha}$
such that

$2^{j\alpha}|D^{\alpha}\eta_{j}(x)|\leq
c_{\alpha}\quad\forall\,j=0,1,2,\dots\quad\mbox{and}\quad\forall\,x\in\mathbb{R}^{N}.$

######  Definition 2.3.

Let $s\in(-\infty,\infty)$ and $p,q\in(0,\infty]$ and
$\\{\eta_{i}\\}_{i=0}^{\infty}\in{\mathcal{A}}$. Then

$B^{s}_{p,q}(\mathbb{R}^{N}):=\Big{\\{}f\in\mathcal{S^{\prime}}(\mathbb{R}^{N}):\|f\|_{B^{s}_{p,q}(\mathbb{R}^{N})}^{\eta}:=\|2^{sj}{\mathcal{F}}^{-1}\big{(}\eta_{j}{\mathcal{F}}(f)\big{)}\|_{l_{q}\big{(}L^{p}(\mathbb{R}^{N})\big{)}}<\infty\Big{\\}}.$

It can be shown that the quasi-norm
$\|f\|_{B^{s}_{p,q}(\mathbb{R}^{N})}^{\eta}$ does not depend on the choice of
$\eta\in{\mathcal{A}}$ (see [42, Proposition 1 in 2.3.2]).

######  Definition 2.4.

Let $\mathcal{D^{\prime}}(\Omega)$ denote the set of all distributions over
$\Omega$. For $1\leq p,\,q\leq\infty$, and $0<s<1$, we set

$B^{s}_{p,q}(\Omega)=\Big{\\{}u\in\mathcal{D^{\prime}}(\Omega):\,\exists\,\,g\in
B^{s}_{p,q}(\mathbb{R}^{N})\quad\mbox{with}\quad g|_{\Omega}=u\Big{\\}}$

and

$\|u\|_{B^{s}_{p,q}(\Omega)}=\inf_{g\in
B^{s}_{p,q}(\mathbb{R}^{N}),\,g|_{\Omega}=u}\|g\|_{B^{s}_{p,q}(\mathbb{R}^{N})}.$

Here $B^{s}_{p,q}(\Omega)$ is called the Besov Space over $\Omega$.

######  Lemma 2.5.

[42, Theorem 3.1.1(i)] Let $p_{0},q_{0},p_{1},q_{1}\in(0,\infty)$,
$s_{0},s_{1}\in(-\infty,\infty)$ and $\Omega$ be a smooth bounded domain in
$\mathbb{R}^{N}$. Then the following embedding

$B^{s_{0}}_{p_{0},q_{0}}(\Omega)\hookrightarrow
B^{s_{1}}_{p_{1},q_{1}}(\Omega)$

is continuous if $s_{0}-\frac{N}{p_{0}}>s_{1}-\frac{N}{p_{1}}$.

######  Lemma 2.6.

[42, pg.233] Let $0<s_{2}<s_{1}$ and $p,q\in(0,\infty)$ and $\Omega$ be a
smooth bounded domain in $\mathbb{R}^{N}$. Then the embedding
$B^{s_{1}}_{p,q}(\Omega)\hookrightarrow B^{s_{2}}_{p,q}(\Omega)$ is compact.

Proof of Theorem 2.1.  From [2, Lemma 2.2], we know that the embedding
${W^{s_{1},p}_{0}(\Omega)\subset W^{s_{2},p}_{0}(\Omega)}$ is continuous.
Moreover, from [42, pg. 209]), it follows that

$\|u\|_{W^{s,p}(\Omega)}:=\|u\|_{L^{p}(\Omega)}+\displaystyle\bigg{(}\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\bigg{)}^{\frac{1}{p}}$

is an equivalent norm for $\|u\|_{B^{s}_{p,p}(\Omega)}$ for
$p\in(1,\infty),\,s\in(0,1)$. Therefore, by Lemma 2.6 we have that

$W^{s_{1},p}(\Omega)\hookrightarrow W^{s_{2},p}(\Omega)\quad\text{is
compact.}$ (2.1)

Now let $\\{u_{n}\\}$ be a bounded sequence in ${W^{s_{1},p}_{0}(\Omega)}$, to
prove the theorem we need to extract a convergent subsequence in
${W^{s_{2},p}_{0}(\Omega)}$. By Rellich compactness, up to a subsequence
$u_{n}\to u$ in $L^{p}(\Omega)$ for some $u\in{W^{s_{1},p}_{0}(\Omega)}$. For
that subsequence we define $v_{n}:=u_{n}-u$. Therefore, we need to show
$v_{n}\to 0$ in ${W^{s_{2},p}_{0}(\Omega)}$. As $v_{n}=0$ in $\Omega^{c}$,

$\displaystyle\|v_{n}\|_{{W^{s_{2},p}_{0}(\Omega)}}^{p}$ $\displaystyle=$
$\displaystyle\int_{\Omega\times\Omega}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy+2\int_{x\in\Omega}\int_{y\in\Omega^{c}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy$
(2.2) $\displaystyle\leq$
$\displaystyle\|v_{n}\|_{W^{s_{2},p}(\Omega)}^{p}+2\int_{x\in\Omega}\int_{y\in\Omega^{c}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy.$

By Póincare inequality in [15, Theorem 6.7] there exists a positive constant
$c>0$ such that

$\|v\|_{L^{p}(\Omega)}^{p}\leq
c\int_{\Omega\times\Omega}\frac{|v(x)-v(y)|^{p}}{|x-y|^{N+sp}}dxdy\quad{\rm
for\ any}\ v\in{W^{s,p}_{0}(\Omega)}.$

Thus, from the definition of ${W^{s_{1},p}_{0}(\Omega)}$ and
$W^{s_{1},p}(\Omega)$, there exists $c>0$ such that

$\|v_{n}\|_{W^{s_{1},p}(\Omega)}\leq c\|v_{n}\|_{{W^{s_{1},p}_{0}(\Omega)}}.$

Therefore, $\\{v_{n}\\}$ is a bounded sequence in $W^{s_{1},p}(\Omega)$.
Consequently, by (2.1), up to a subsequence $v_{n}\to 0$ in
$W^{s_{2},p}(\Omega)$. Therefore, to complete the proof, we only have to show
that

$\int_{x\in\Omega}\int_{y\in\Omega^{c}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy\to
0.$

To that end, let $\epsilon>0$ be arbitrary and we define
$M:=\sup_{n}\|v_{n}\|_{{W^{s_{1},p}_{0}(\Omega)}}$. Then

$\displaystyle\int_{x\in\Omega}\int_{y\in\Omega^{c}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy$
$\displaystyle=$
$\displaystyle\int_{x\in\Omega}\int_{y\in\Omega^{c}\cap|x-y|<{\varepsilon}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy$
$\displaystyle\quad+\int_{x\in\Omega}\int_{y\in\Omega^{c}\cap|x-y|\geq\epsilon}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{2}p}}dxdy$
$\displaystyle=$
$\displaystyle\int_{x\in\Omega}\int_{y\in\Omega^{c}\cap|x-y|<\epsilon}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{1}p}}|x-y|^{(s_{1}-s_{2})p}dxdy$
$\displaystyle\quad+\int_{x\in\Omega}\bigg{(}\int_{y\in\Omega^{c}\cap|x-y|\geq\epsilon}\frac{dy}{|x-y|^{N+s_{2}p}}\bigg{)}|v_{n}(x)|^{p}dx$
$\displaystyle\leq$
$\displaystyle\epsilon^{(s_{1}-s_{2})p}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+s_{1}p}}dxdy$
$\displaystyle\quad+C(\epsilon)\|v_{n}\|_{L^{p}(\Omega)}^{p}$
$\displaystyle\leq$ $\displaystyle\epsilon^{(s_{1}-s_{2})p}M+o(1),$

where $o(1)\to 0$ as $n\to\infty$. Therefore, the above integral can be made
arbitrary small. Hence from (2.2), we conclude that
$\|v_{n}\|_{{W^{s_{2},p}_{0}(\Omega)}}\to 0$. This completes the proof. $\Box$

In the particular case that $p=2$, for $s\in(0,1)$ we set

${\mathbb{H}}^{s}_{0}(\Omega)={W^{s,2}_{0}(\Omega)}$

which is a Hilbert space with the inner product ${\mathcal{E}}_{s}(u,v)$ for
$u,v\in{\mathbb{H}}^{s}_{0}(\Omega)$.

Proof of Theorem 1.1.  The functional

${\mathbb{H}}^{s_{1}}_{0}(\Omega)\to\mathbb{R},\qquad
u\mapsto\Phi(u):={\mathcal{E}}_{s_{1}}(u,u)$

is weakly lower semicontinuous. Moreover, let

${\mathcal{M}}_{1}:=\\{u\in{\mathbb{H}}^{s_{1}}_{0}(\Omega),\,\|u\|_{s_{2}}+\mu\|u\|_{L^{2}(\Omega)}=1\\},$

where $\|\cdot\|_{s_{2}}+\mu\|\cdot\|_{L^{2}(\Omega)}$ is equivalent to
$\|\cdot\|_{s_{2}}$ in the space ${\mathbb{H}}^{s_{2}}_{0}(\Omega)$ for
$\mu>-\lambda_{s_{2},1}$. Then we have that

$\Phi(u)<+\infty\qquad\text{for $u\in{\mathcal{M}}_{1}$.}$

Put $\lambda_{1}(\mu):=\inf_{{\mathcal{M}}_{1}}\Phi$. By Theorem 2.1, the
embedding $W^{s_{1},p}_{0}(\Omega)\hookrightarrow W^{s_{2},p}_{0}(\Omega)$ is
compact and therefore, it follows that $\lambda_{1}(\mu)$ is attained by a
function $\phi_{1}\in{\mathcal{M}}_{1}$. Consequently, there exists a Lagrange
multiplier $\lambda\in\mathbb{R}$ such that

${\mathcal{E}}_{s_{1}}(\phi_{1},w)=\frac{1}{2}\Phi^{\prime}(\phi_{1})w=\lambda\Big{(}{\mathcal{E}}_{s_{2}}(\phi_{1},w)+\mu\int_{\Omega}\phi_{1}w\,dx\Big{)}\qquad\text{for
all $w\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)$.}$

Choosing $w=\phi_{1}$ yields to $\lambda=\lambda_{1}(\mu)$. Hence $\phi_{1}$
is an eigenfunction of (1.1) corresponding to the eigenvalue
$\lambda_{1}(\mu)$. Moreover $\lambda_{1}(\mu)>0$. Next we proceed inductively
and assume that $\phi_{2},\dots,\phi_{k}\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)$
and $\lambda_{2}(\mu)\leq\dots\leq\lambda_{k}(\mu)$ are already given for some
$k\in\mathbb{N}$ with the properties that for $j=2,\dots,k$, the function
$\phi_{j}$ is a minimizer of $\Phi$ within the set

$\displaystyle{\mathcal{M}}_{j}:=\\{u\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)\>:\>\|u\|_{s_{2}}+\|u\|_{L^{2}(\Omega)}=1,\>\text{${\mathcal{E}}_{s_{2}}(u,\phi_{m})+\mu\int_{\Omega}u\phi_{m}\,dx=0$
for $m=1,\dots j-1$}\\},$

$\lambda_{j}(\mu)=\inf_{{\mathcal{M}}_{j}}\Phi=\Phi(\phi_{j})$, and

${\mathcal{E}}_{s_{1}}(\phi_{j},\varphi)=\lambda_{j}(\mu)\Big{(}{\mathcal{E}}_{s_{2}}(\phi_{j},\varphi)+\mu\int_{\Omega}\phi_{j}\varphi\,dx\Big{)}\qquad\text{for
all $\varphi\in\mathbb{H}^{s_{1}}_{0}(\Omega)$.}$ (2.3)

Again by the compact embedding in Theorem 2.1, the value $\lambda_{k+1}$ is
attained by a function $\phi_{k+1}\in{\mathcal{M}}_{k+1}$. Consequently, there
exists a Lagrange multiplier $\lambda\in\mathbb{R}$ such that

${\mathcal{E}}_{s_{1}}(\phi_{k+1},\varphi)=\lambda\Big{(}{\mathcal{E}}_{s_{2}}(\phi_{k+1},\varphi)+\mu\int_{\Omega}\phi_{k+1}\varphi\,dx\Big{)}\qquad\text{for
all $\varphi\in{\mathcal{M}}_{k+1}(\Omega)$.}$ (2.4)

Choosing $\varphi=\phi_{k+1}$, we have that $\lambda=\lambda_{k+1}(\mu)$.
Moreover, for $j=1,\dots,k$, we have, by (2.3) and the definition of
${\mathcal{M}}_{k+1}(\Omega)$, that

$\displaystyle{\mathcal{E}}_{s_{1}}(\phi_{k+1},\phi_{j})={\mathcal{E}}_{s_{1}}(\phi_{j},\phi_{k+1})$
$\displaystyle=\lambda_{j}(\mu)\Big{(}{\mathcal{E}}_{s_{2}}(\phi_{j},\phi_{k+1})+\mu\int_{\Omega}\phi_{j}\phi_{k+1}\,dx\Big{)}$
$\displaystyle=0$
$\displaystyle=\lambda_{k+1}(\mu)\Big{(}{\mathcal{E}}_{s_{2}}(\phi_{k+1},\phi_{j})+\mu\int_{\Omega}\phi_{j}\phi_{k+1}\,dx\Big{)}.$

Hence (2.4) holds with $\lambda=\lambda_{k+1}(\mu)$ for all
$\varphi\in{\mathbb{H}}^{s_{1}}_{0}(\Omega)$. Inductively, we have now
constructed a normalized sequence $(\phi_{k})_{k}$ in
${\mathbb{H}}^{s_{1}}_{0}(\Omega)$ and a nondecreasing sequence
$\\{\lambda_{k}\\}_{k}$ in $\mathbb{R}$ such that property (i) holds and such
that $\phi_{k}$ is an eigenfunction of (1.1) corresponding to
$\lambda=\lambda_{k}(\mu)$ for every $k\in\mathbb{N}$. Moreover, by
construction, the sequence $\\{\phi_{k}\\}_{k}$ forms an orthonormal system in
${\mathbb{H}}^{s_{2}}_{\mu,0}(\Omega)$.

Next we show property (iii), i.e.,
$\lim\limits_{k\to+\infty}\lambda_{k}(\mu)=+\infty.$ Supposing by
contradiction that $c:=\lim\limits_{k\to\infty}\lambda_{k}(\mu)<+\infty$, we
deduce that ${\mathcal{E}}_{s_{1}}(\phi_{k},\phi_{k})\leq c$ for every
$k\in\mathbb{N}$. Hence the sequence $(\phi_{k})$ is bounded in
${\mathbb{H}}^{s_{1}}_{0}(\Omega)$, and therefore by Rellich compactness
theorem, $(\phi_{k})$ contains a convergent subsequence $(\phi_{k_{j}})_{j}$
in $L^{2}(\Omega)$. This however is impossible since the functions
$\\{\phi_{k_{j}}\\}_{j\in\mathbb{N}}$ are orthonormal in
$H^{s_{2}}_{\mu,0}(\Omega)$. Hence (iii) is proved.

Next, to prove that $\\{\phi_{k}\>:\>k\in\mathbb{N}\\}$ is an orthonormal
basis of $H^{s_{2}}_{\mu,0}(\Omega)$, we first suppose by contradiction that
there exists $v\in\mathbb{H}^{s_{1}}_{0}(\Omega)$ with
$\|v\|_{s_{2}}+{\mu}\|v\|_{L^{2}(\Omega)}=1$ and
${\mathcal{E}}_{s_{2}}(v,\phi_{k})+\mu\int_{\Omega}v\phi_{k}\,dx=0$ for any
$k\in\mathbb{N}$. Since $\lim\limits_{k\to\infty}\lambda_{k}(\mu)=+\infty$,
there exists an integer $k_{0}>0$ such that

$\Phi(v)<\lambda_{k_{0}}(\mu)=\inf_{{\mathcal{M}}_{k_{0}}}\Phi(u),$

which by definition of ${\mathcal{M}}_{k_{0}}$ implies that
${\mathcal{E}}_{s_{2}}(v,\phi_{k})+\mu\int_{\Omega}v\phi_{k}\,dx\not=0$ for
some $k\in\\{1,\dots,k_{0}-1\\}$. This is a a contradiction to
${\mathcal{E}}_{s_{2}}(v,\phi_{k})+\mu\int_{\Omega}v\phi_{k}\,dx=0$ for any
$k\in\mathbb{N}$. Thus, we conclude that ${\mathbb{H}}^{s_{1}}_{0}(\Omega)$ is
contained in the $H^{s_{2}}_{\mu,0}$-closure of the span of
$\\{\phi_{k}\>:\>k\in\mathbb{N}\\}$. Since ${\mathbb{H}}^{s_{1}}_{0}(\Omega)$
is dense in $H^{s_{2}}_{\mu,0}(\Omega)$, we conclude that the span of
$\\{\phi_{k}\>:\>k\in\mathbb{N}\\}$ is dense in $H^{s_{2}}_{\mu,0}(\Omega)$,
and hence $\\{\phi_{k}\>:\>k\in\mathbb{N}\\}$ is an orthonormal basis of
$H^{s_{2}}_{\mu,0}(\Omega)$. This proves (ii).

Finally, we show $(iv)$. It follows by the definition of the first eigenvalue
$\lambda_{1}(\mu)$ that for $\mu\in(-\lambda_{s_{2},1},+\infty)$, the map
$\mu\mapsto\lambda_{1}(\mu)$ is decreasing.

Let $(\lambda_{s_{2},1},\varphi_{s_{2},1})$ be the first eigenvalue and the
corresponding eigenfunction of (1.5) with $s=s_{2}$. We may assume that
$\varphi_{s_{2},1}>0$ in $\Omega$, $\int_{\Omega}\varphi_{s_{2},1}^{2}dx=1$.
Let $\eta_{0}$ be a smooth function with compact support in $B_{r_{0}}(x_{0})$
such that $B_{2r_{0}}(x_{0})\subset\Omega$ with
$\int_{\Omega}\eta_{0}^{2}dx=1$. Note that
$\eta_{0}\not\equiv\varphi_{s_{2},1}$ in $\Omega$. Then
${\mathcal{E}}_{s_{2}}(\eta_{0},\eta_{0})>\lambda_{s_{2},1}$, otherwise
$\eta_{0}\equiv\varphi_{s_{2},1}$ since the first eigenfunction of (1.5) with
$s=s_{2}$ is simple.

Thus, we have that

$\displaystyle\lambda_{1}(\mu)$
$\displaystyle\leq\frac{{\mathcal{E}}_{s_{1}}(\eta_{0},\eta_{0})}{{\mathcal{E}}_{s_{2}}(\eta_{0},\eta_{0})+\mu\int_{\Omega}\eta_{0}^{2}dx}$
$\displaystyle\
\leq\frac{{\mathcal{E}}_{s_{1}}(\eta_{0},\eta_{0})}{{\mathcal{E}}_{s_{2}}(\eta_{0},\eta_{0})-\lambda_{s_{2},1}}\
<+\infty\quad{\rm as}\ \ \mu\to-\lambda_{s_{2},1}^{+}.$

This completes the proof. $\Box$

##  3 Lower bounds

###  3.1 Important estimate

To prove the Li-Yau’s type lower bound for (1.1), we need the following
results.

######  Proposition 3.1.

Let $\mu\geq 0$, $0<s_{2}<s_{1}<1$ and $f$ be a real-valued function defined
on $\mathbb{R}^{N}$ with $0\leq f\leq M_{1}$,

$\int_{\mathbb{R}^{N}}f(z)|z|^{2s_{1}}dz\leq M_{2},$

then

$\displaystyle\int_{\mathbb{R}^{N}}(|z|^{2s_{2}}+\mu)f(z)dz$
$\displaystyle\leq\frac{\omega_{{}_{N-1}}M_{1}}{2s_{2}+N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{\frac{2s_{2}+N}{2s_{1}+N}}\Big{[}1+\mu\frac{2s_{2}+N}{N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{-\frac{2s_{2}}{2s_{1}+N}}\Big{]}.$

Proof of Proposition 3.1.  Let

$h(z)=\left\\{\begin{array}[]{lll}(|z|^{2s_{2}}+\mu)M_{1}&{\rm
for}\quad|z|<R,\\\\[4.2679pt] \phantom{}0&{\rm{for}}\quad|z|\geq
R,\end{array}\right.$

where $R>0$ such that

$\int_{B_{R}}M_{1}|z|^{2s_{1}}dz=M_{2}.$

Direct computation shows that

$R^{2s_{1}+N}=\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}.$

Since

$\displaystyle|z|^{2s_{2}}\Big{(}|z|^{2(s_{1}-s_{2})}-R^{2(s_{1}-s_{2})}\Big{)}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}\geq
0$

and

$\displaystyle\mu\Big{(}|z|^{2s_{1}}-R^{2s_{1}}\Big{)}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}\geq
0,$

it follows that

$\displaystyle\int_{\mathbb{R}^{N}}|z|^{2s_{2}}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}dz$
$\displaystyle\leq$
$\displaystyle\frac{1}{R^{2(s_{1}-s_{2})}}\int_{\mathbb{R}^{N}}|z|^{2s_{2}}|z|^{2(s_{1}-s_{2})}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}dz$
$\displaystyle\leq$
$\displaystyle\frac{1}{R^{2(s_{1}-s_{2})}}\int_{\mathbb{R}^{N}}|z|^{2s_{1}}(f(z)-M_{1}\chi_{{}_{B_{R}}})dz\leq
0,$ (3.1)

and

$\displaystyle\mu\int_{\mathbb{R}^{N}}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}dz$
$\displaystyle\leq$
$\displaystyle\frac{\mu}{R^{2s_{1}}}\int_{\mathbb{R}^{N}}|z|^{2s_{1}}\Big{(}f(z)-(|z|^{2s_{2}}+\mu)^{-1}h(z)\Big{)}dz$
$\displaystyle\leq$
$\displaystyle\frac{\mu}{R^{2s_{1}}}\int_{\mathbb{R}^{N}}|z|^{2s_{1}}(f(z)-M_{1}\chi_{{}_{B_{R}}})dz\leq
0.$ (3.2)

Combining (3.1) and (3.1), we have

$\displaystyle\int_{\mathbb{R}^{N}}(|z|^{2s_{2}}+\mu)f(z)dz$
$\displaystyle\leq\int_{\mathbb{R}^{N}}h(z)dz$
$\displaystyle=\frac{\omega_{{}_{N-1}}M_{1}}{2s_{2}+N}R^{2s_{2}+N}\Big{(}1+\mu\frac{2s_{2}+N}{N}R^{-2s_{2}}\Big{)}$
$\displaystyle=\frac{\omega_{{}_{N-1}}M_{1}}{2s_{2}+N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{\frac{2s_{2}+N}{2s_{1}+N}}\Big{[}1+\mu\frac{2s_{2}+N}{N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{-\frac{2s_{2}}{2s_{1}+N}}\Big{]}.$

We complete the proof. $\Box$

We also need the following Lemma.

######  Lemma 3.2.

Assume that $1>\tau_{1}>\tau_{2}>0$ and $d_{1}>0.$

Let $r_{1}$ be the solution of

$r^{\tau_{1}}\Big{(}1+r^{-\tau_{2}}\Big{)}=d_{1},$

then

$d_{1}^{\frac{1}{\tau_{1}}}\Big{(}1-\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\Big{)}<r_{1}<d_{1}^{\frac{1}{\tau_{1}}}.$

Proof.  Let

$f(r)=r^{\tau_{1}}\Big{(}1+r^{-\tau_{2}}\Big{)},$

then $f(0)=0$, $\lim_{r\to+\infty}f(r)=+\infty$ and
$f^{\prime}(r)=\tau_{1}r^{\tau_{1}-1}+(\tau_{1}-\tau_{2})r^{\tau_{1}-\tau_{2}-1}>0$,
$f^{\prime\prime}(r)=\tau_{1}(\tau_{1}-1)r^{\tau_{1}-2}+(\tau_{1}-\tau_{2})(\tau_{1}-\tau_{2}-1)r^{\tau_{1}-\tau_{2}-2}<0$,
As a consequence, $f$ is strictly increasing, concave in $(0,+\infty)$ and for
any $d_{1}>0$, there exists a unique solution $r_{1}$ such that
$f(r_{1})=d_{1}$. Moreover, since $f(r)>0$ for $r>0$, we conclude $r_{1}>0$.

Let

$R_{1}=d_{1}^{\frac{1}{\tau_{1}}}\Big{(}1-\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\Big{)}_{+}\quad{\rm
and}\quad R_{2}=d_{1}^{\frac{1}{\tau_{1}}},$

where $a_{+}=\max\\{0,a\\}$. Note that

$\displaystyle\frac{f(R_{2})}{d_{1}}$
$\displaystyle=1+d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}>1$

and if $\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}<1$, then

$\displaystyle\frac{f(R_{1})}{d_{1}}$
$\displaystyle=\Big{(}1-\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\Big{)}^{\tau_{1}}+d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\Big{(}1-\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\Big{)}^{\tau_{1}-\tau_{2}}$
$\displaystyle\leq
1-d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}+d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}-{\frac{\tau_{1}-\tau_{2}}{\tau_{1}}d_{1}^{-\frac{2\tau_{2}}{\tau_{1}}}}$
$\displaystyle<1,$

where we used the fact that for $\tau\in(0,1)$,

$(1-t)^{\tau}\leq 1-\tau t\quad{\rm for\ any}\ t\in(0,1).$

On the other hand, if
$\frac{1}{\tau_{1}}d_{1}^{-\frac{\tau_{2}}{\tau_{1}}}\geq 1$, then $R_{1}=0$.
Therefore, $\frac{f(R_{1})}{d_{1}}=0<1$. Thus, in both the cases
$f(R_{1})<d_{1}<f(R_{2})$. Now using the fact that $f$ is strictly increasing,
continuous and $f(r_{1})=d_{1}$, we conclude $R_{1}<r_{1}<R_{2}$. This
completes the proof. $\Box$

###  3.2 Lower bound

The Bessel inequality plays an important role in the Li-Yau method for the
lower bound of eigenvalues: Let ${\mathbb{H}}$ be an inner product space or a
Hermitian product space together with its product function
$\langle\cdot,\cdot\rangle$. Let $e_{1},e_{2},\cdots$ be any (finite or
infinite) orthonormal sequence. Then for any $x\in{\mathbb{H}}$,

$\sum_{j\geq 1}|\langle x,e_{j}\rangle|^{2}\leq\langle
x,x\rangle=\|x\|_{{}_{\mathbb{H}}}^{2}.$

Proof of Theorem 1.2. Let

$\Phi_{k}(x,y)=\sum^{k}_{j=1}\phi_{j}(x)\tilde{\phi}_{j}(y),$

where

$\tilde{\phi}_{j}:=\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}^{\frac{1}{2}}\phi_{j}={\mathcal{F}}^{-1}\bigg{[}\big{(}|z|^{2s_{2}}+\mu\big{)}^{\frac{1}{2}}{\mathcal{F}}\big{(}\phi_{j}\big{)}\bigg{]}.$

Using the Fourier transform, we have that

${\mathcal{F}}_{x}(\Phi_{k})(z,y)=(2\pi)^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\Phi_{k}(x,y)e^{ix\cdot
z}dx$

and

${\mathcal{F}}\left(\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{\frac{1}{2}}{\phi_{j}}\right)(z)=\big{(}|z|^{2s_{2}}+\mu\big{)}^{\frac{1}{2}}{{\mathcal{F}}\big{(}\phi_{j}(z)\big{)}}\
\ \text{ for}\ \,\mu\geq 0.$

Therefore, doing a straightforward computation we have

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\Big{|}\big{(}|z|^{s_{2}}+\mu\big{)}^{\frac{1}{2}}{\mathcal{F}}_{x}(\Phi_{k})(z,y)\Big{|}^{2}dydz=\sum_{j=1}^{k}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\tilde{\phi}_{j}^{2}(x)\tilde{\phi}_{j}^{2}(y)dydx=k,$
(3.3)

by the orthonormality of $\\{\tilde{\phi}_{j}\\}_{j\in\mathbb{N}}$ in
$L^{2}(\mathbb{R}^{N})$. Next, we estimate
$\big{|}\int_{\mathbb{R}^{N}}\Big{|}{\mathcal{F}}_{x}(\Phi_{k})(z,y)\Big{|}^{2}dy\big{|}_{L^{\infty}(\mathbb{R}^{N})}$.
Here the main difficulty comes from the fact that $\\{\phi_{j}\\}_{j}$ is not
an orthonormal sequence in $L^{2}(\Omega)$.

Let $G_{\mu}$ be the fundamental solution of $(-\Delta)^{s_{2}}+\mu$ in
$\mathbb{R}^{N}$. Then

$0<G_{\mu}(x)\leq G_{0}(x)=a_{N,s_{2}}|x|^{2s_{2}-N}\quad{\rm for}\ \,\mu\geq
0,$

$a_{N,s_{2}}=2^{-2s_{2}}\pi^{-\frac{N}{2}}\frac{\Gamma(\frac{N-2s_{2}}{2})}{\Gamma(s_{2})}=2^{-2s_{2}}\pi^{-\frac{N}{2}}\frac{\Gamma(\frac{N-2s_{2}}{2})}{\Gamma(s_{2}+1)}s_{2}:=\tilde{a}_{N,s_{2}}s_{2}.$
(3.4)

Note that

$\displaystyle\big{(}(-\Delta)^{s_{2}}+\mu\big{)}^{-1}g=G_{\mu}\ast g\quad{\rm
in}\quad\mathbb{R}^{N}$

for $g\in L^{2}(\mathbb{R}^{N})$. Using Fourier transform, we can see that

${\mathcal{F}}\bigg{(}\big{(}(-\Delta)^{s_{2}}+\mu\big{)}^{-1}g\bigg{)}(z)=\big{(}|z|^{2s_{2}}+\mu\big{)}^{-1}{\mathcal{F}}(g(z)).$

Consequently, $\big{(}(-\Delta)^{s_{2}}+\mu\big{)}^{-1/2}g$ can be defined as
follows

$\big{(}(-\Delta)^{s_{2}}+\mu\big{)}^{-1/2}g:=\bigg{[}\big{(}(-\Delta)^{s_{2}}+\mu\big{)}^{-1}\bigg{]}^{1/2}g={\mathcal{F}}^{-1}\bigg{[}\big{(}(|z|^{2s_{2}}+\mu)^{-1}\big{)}^{1/2}{\mathcal{F}}(g)\bigg{]}.$

Now, in view of Parseval’s relation and Bessel’s inequality, we obtain

$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}{\mathcal{F}}_{x}(\Phi_{k})(z,y)\Big{|}^{2}dy$
$\displaystyle=(2\pi)^{-N}\sum^{k}_{j=1}\Big{|}\int_{\mathbb{R}^{N}}\phi_{j}(x)\big{(}e^{{\rm
i}x\cdot
z}\chi_{\Omega}(x)\big{)}dx\Big{|}^{2}\int_{\mathbb{R}^{N}}|\tilde{\phi}_{j}(y)|^{2}dy$
$\displaystyle=(2\pi)^{-N}\sum^{k}_{j=1}\Big{|}\int_{\mathbb{R}^{N}}\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{\frac{1}{2}}\phi_{j}(x)\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{-\frac{1}{2}}\big{(}e^{{\rm
i}x\cdot z}\chi_{\Omega}(x)\big{)}dx\Big{|}^{2}$
$\displaystyle=(2\pi)^{-N}\sum^{k}_{j=1}\Big{|}\int_{\mathbb{R}^{N}}\tilde{\phi}_{j}(x)\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{-\frac{1}{2}}\big{(}e^{{\rm
i}x\cdot z}\chi_{\Omega}(x)\big{)}dx\Big{|}^{2}$
$\displaystyle\leq(2\pi)^{-N}\int_{\mathbb{R}^{N}}\Big{|}\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{-\frac{1}{2}}\big{(}e^{{\rm
i}x\cdot z}\chi_{\Omega}(x)\big{)}\Big{|}^{2}dx$
$\displaystyle=(2\pi)^{-N}\int_{\mathbb{R}^{N}}\bigg{[}\Big{(}(-\Delta)^{s_{2}}+\mu\Big{)}_{x}^{-1}\big{(}e^{{\rm
i}x\cdot z}\chi_{\Omega}(x)\big{)}\bigg{]}\overline{\big{(}e^{{\rm i}x\cdot
z}\chi_{\Omega}(x)\big{)}}dx$
$\displaystyle\leq(2\pi)^{-N}\int_{\Omega}\Big{|}G_{\mu}\ast\big{(}e^{{\rm
i}x\cdot z}\chi_{\Omega}(x)\big{)}\Big{|}dx$
$\displaystyle\leq(2\pi)^{-N}\int_{\Omega}G_{0}\ast\chi_{\Omega}dx$
$\displaystyle=(2\pi)^{-N}a_{N,s_{2}}\int_{\Omega}\int_{\Omega}|x-y|^{2s_{2}-N}dydx$
$\displaystyle\leq(2\pi)^{-N}a_{N,s_{2}}|\Omega|\sup_{x\in\Omega}\int_{\Omega}|x-y|^{2s_{2}-N}dy.$

Now we choose $r>0$ such that $|\Omega|=\frac{\omega_{N-1}}{N}r^{N}$ and using
rearrangement inequality, we have

$\displaystyle\sup_{x\in\Omega}\int_{\Omega}|x-y|^{2s_{2}-N}dy$
$\displaystyle\leq\int_{B_{r}(x)}|x-y|^{2s_{2}-N}dy$
$\displaystyle=\frac{\omega_{{}_{N-1}}}{2s_{2}}r^{2s_{2}}=\frac{\omega_{{}_{N-1}}}{2s_{2}}\bigg{(}\frac{N|\Omega|}{\omega_{N-1}}\bigg{)}^{\frac{2s_{2}}{N}}=a_{0}|\Omega|^{\frac{2s_{2}}{N}},$

where

$a_{0}=\frac{1}{2s_{2}}N^{\frac{2s_{2}}{N}}\omega_{{}_{N-1}}^{1-\frac{2s_{2}}{N}}.$
(3.5)

As a consequence, we obtain that

$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}{\mathcal{F}}_{x}(\Phi_{k})(z,y)\Big{|}^{2}dy\leq(2\pi)^{-N}a_{N,s_{2}}a_{0}|\Omega|^{1+\frac{2s_{2}}{N}}=\tilde{a}_{N,s_{2}}\frac{N^{\frac{2s_{2}}{N}}\omega_{{}_{N-1}}^{1-\frac{2s_{2}}{N}}}{2^{N+1}\pi^{N}}|\Omega|^{1+\frac{2s_{2}}{N}}.$
(3.6)

Meanwhile, using the definition of $\Phi_{k}$, it also follows that

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|z|^{2s_{1}}\big{|}{\mathcal{F}}_{x}(\Phi_{k})(z,y)\big{|}^{2}dydz$
$\displaystyle=\int_{\Omega}\int_{\Omega}\Phi_{k}(x,y)(-\Delta)^{s_{1}}_{x}\Phi_{k}(x,y)dydx$
$\displaystyle=\sum^{k}_{j=1}\int_{\Omega}\phi_{j}(-\Delta)^{s_{1}}\phi_{j}dx\int_{\Omega}|\tilde{\phi}_{j}(y)|^{2}dy$
$\displaystyle=\sum^{k}_{j=1}\lambda_{j}(\mu).$

Now we apply Proposition 3.1 to the function

$f(z)=\int_{\mathbb{R}^{N}}|({\mathcal{F}}_{x}\Phi_{k})(z,y)|^{2}dy$

with

$M_{1}=(2\pi)^{-N}a_{N,s_{2}}a_{0}|\Omega|^{\frac{N+2s_{2}}{N}}\quad{\rm
and}\quad M_{2}=\sum^{k}_{j=1}\lambda_{j}(\mu),$

then we conclude that

$\displaystyle k$
$\displaystyle\leq\frac{\omega_{{}_{N-1}}M_{1}}{2s_{2}+N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{\frac{2s_{2}+N}{2s_{1}+N}}\cdot\left(1+\mu\frac{2s_{2}+N}{N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{-\frac{2s_{2}}{2s_{1}+N}}\right).$
(3.7)

Case: $\mu=0$.  In this case, (3.7) reduces to

$k\leq\frac{\omega_{{}_{N-1}}M_{1}}{2s_{2}+N}\Big{(}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}\Big{)}^{\frac{2s_{2}+N}{2s_{1}+N}},$

that is

$M_{2}\geq\frac{(2s_{2}+N)^{\frac{2s_{1}+N}{2s_{2}+N}}}{2s_{1}+N}(M_{1}\omega_{{}_{N-1}})^{-\frac{2(s_{1}-s_{2})}{2s_{2}+N}}k^{\frac{2s_{1}+N}{2s_{2}+N}},$

which implies that

$\displaystyle\sum^{k}_{j=1}\lambda_{j}(0)$
$\displaystyle\geq\frac{(2s_{2}+N)^{\frac{2s_{1}+N}{2s_{2}+N}}}{2s_{1}+N}\big{(}(2\pi)^{-N}a_{N,s_{2}}a_{0}\omega_{{}_{N-1}}\big{)}^{-\frac{2(s_{1}-s_{2})}{2s_{2}+N}}|\Omega|^{-\frac{N+2s_{2}}{N}\frac{2(s_{1}-s_{2})}{N+2s_{2}}}k^{1+\frac{2(s_{1}-s_{2})}{2s_{2}+N}}$
$\displaystyle=b_{1}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{1+\frac{2(s_{1}-s_{2})}{2s_{1}+N}}.$

Case: $\mu>0$.  By Lemma 3.2 with

$\tau_{1}=\frac{2s_{2}+N}{2s_{1}+N},\qquad\tau_{2}=\frac{2s_{2}}{2s_{1}+N}$

and

$r=\Big{(}\mu\frac{2s_{2}+N}{N}\Big{)}^{-\frac{2s_{1}+N}{2s_{2}}}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}},\qquad
d_{1}=\frac{2s_{2}+N}{\omega_{{}_{N-1}}M_{1}}\Big{(}\mu\frac{2s_{2}+N}{N}\Big{)}^{-\frac{2s_{2}+N}{2s_{2}}}k>0.$

Then

$\displaystyle\Big{(}\mu\frac{2s_{2}+N}{N}\Big{)}^{-\frac{2s_{1}+N}{2s_{2}}}\frac{(2s_{1}+N)M_{2}}{M_{1}\omega_{{}_{N-1}}}$
$\displaystyle\geq\Big{(}\frac{2s_{2}+N}{\omega_{{}_{N-1}}M_{1}}\Big{(}\mu\frac{2s_{2}+N}{N}\Big{)}^{-\frac{2s_{2}+N}{2s_{2}}}k\Big{)}^{\frac{2s_{1}+N}{2s_{2}+N}}\Big{[}1-\frac{2s_{1}+N}{2s_{2}+N}\Big{(}\frac{2s_{2}+N}{\omega_{{}_{N-1}}M_{1}}\big{(}\mu\frac{2s_{2}+N}{N}\big{)}^{-\frac{2s_{2}+N}{2s_{2}}}k\Big{)}^{-\frac{2s_{2}}{2s_{2}+N}}\Big{]},$

which is equivalent to

$\displaystyle
M_{2}\geq\frac{(2s_{2}+N)^{\frac{2s_{1}+N}{2s_{2}+N}}}{2s_{1}+N}(M_{1}\omega_{{}_{N-1}})^{-\frac{2(s_{1}-s_{2})}{2s_{2}+N}}k^{\frac{2s_{1}+N}{2s_{2}+N}}\Big{[}1-\frac{\mu(2s_{1}+N)}{N(2s_{2}+N)^{\frac{2s_{2}}{2s_{2}+N}}}(\omega_{N-1}M_{1})^{\frac{2s_{2}}{2s_{2}+N}}k^{-\frac{2s_{2}}{2s_{2}+N}}\Big{]},$

that is,

$\displaystyle\sum^{k}_{j=1}\lambda_{j}(\mu)$ $\displaystyle\geq
b_{1}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{1+\frac{2(s_{1}-s_{2})}{2s_{2}+N}}$
$\displaystyle\quad-\mu
b_{1}\frac{2s_{1}+N}{N(2s_{2}+N)^{\frac{2s_{2}}{2s_{2}+N}}}\Big{(}(2\pi)^{-N}\omega_{{}_{N-1}}c_{s_{2}}a_{0}\Big{)}^{\frac{2s_{2}}{N+2s_{2}}}|\Omega|^{-\frac{2s_{1}}{N}}k^{1+\frac{2s_{1}-4s_{2}}{2s_{2}+N}}.$

Substituting the value of $a_{0}$ and $a_{N,s}$ from (3.5) and (3.4)
respectively, we complete the proof. $\Box$

Proof of Corollary 1.3.  It follows by the nondecreasing monotonicity of
$k\mapsto\lambda_{k}(\mu)$ that

$\lambda_{k}(\mu)\geq\frac{1}{k}\sum^{k}_{j=1}\lambda_{j}(\mu).$

Applying the above inequality in (1.10) yields (1.12). $\Box$

##  4 Upper bounds

In order to prove the upper bounds, we need following lemmas.

######  Lemma 4.1.

Let $s\in(0,1)$ and for fixed $z\in\mathbb{R}^{N}\setminus\\{0\\}$

$v_{z}(x)=e^{{\rm i}x\cdot z},\quad\forall\,x\in\mathbb{R}^{N},$ (4.1)

then

$(-\Delta)^{s}v_{z}(x)=|z|^{2s}v_{z}(x),\quad\forall\,x\in\mathbb{R}^{N}.$
(4.2)

Proof.  Without loss of generality, it is enough to prove (4.2) with
$z=te_{1}$, where $t>0$ and $e_{1}=(1,0,\cdots,0)\in\mathbb{R}^{N}$. For this,
we write

$v_{t}(x)=\mu_{z}(x_{1})=e^{{\rm i}tx_{1}},\quad
x=(x_{1},x^{\prime})\in\mathbb{R}\times\mathbb{R}^{N-1}.$

Using [10, Lemma 3.1] we obtian that

$\displaystyle(-\Delta)^{s}v_{t}(x)=(-\Delta)^{s}_{\mathbb{R}}v_{t}(x_{1}).$

Now we claim that

$(-\Delta)^{s}_{\mathbb{R}}v_{t}(x_{1})=t^{2s}v_{t}(x_{1}),\quad\forall\,x_{1}\in\mathbb{R}.$
(4.3)

Indeed, observe that
$-\Delta_{\mathbb{R}}v_{t}:=-(v_{t})_{x_{1}x_{1}}=t^{2}v_{t}$ in $\mathbb{R}$
and then

$(|\xi_{1}|^{2}-t^{2})\widehat{v_{t}}(\xi_{1})={\mathcal{F}}\left(-\Delta_{\mathbb{R}}v_{t}-t^{2}v_{t}\right)(\xi_{1})=0,$

which implies that

${\rm supp}(\widehat{v_{t}})\subset\\{\pm t\\},$

which in turn implies

$(|\xi_{1}|^{2s}-t^{2s})\widehat{v_{t}}(\xi_{1})=0={\mathcal{F}}\left((-\Delta)^{s}_{\mathbb{R}}v_{t}-t^{2s}v_{t}\right)(\xi_{1}).$

and finally

$\left((-\Delta)^{s}_{\mathbb{R}}v_{t}-t^{2s}v_{t}\right)(\xi_{1})=0\quad{\rm
in}\ \ \mathbb{R},$

which yields

$\displaystyle(-\Delta)^{s}v_{t}(x)$
$\displaystyle=(-\Delta)^{s}_{\mathbb{R}}v_{t}=t^{2s}v_{t}(x),\quad\forall\,x\in\mathbb{R}^{N},$

which completes the proof. $\Box$

Let $\eta_{0}$ be a $C^{2}$ increasing function such that
${\|\eta_{0}\|_{C^{1}}},\,\|\eta_{0}\|_{C^{2}}\leq 2$,

$\eta_{0}(t)=1\ \ {\rm if}\ \,t\geq 1,\qquad\eta_{0}(t)=0\ \ {\rm if}\ \,t\leq
0.$

For $\sigma>0$, denote

$w_{\sigma}(x)=\eta_{0}(\sigma^{-1}\rho(x)),\quad\forall\,x\in\mathbb{R}^{N},$
(4.4)

where

$\rho(x)=\text{dist}(x,\mathbb{R}^{N}\setminus\Omega)\quad\text{for}\,\,x\in\mathbb{R}^{N}.$

Since $\Omega$ is a $C^{2}$ domain, then $\rho$ is $C^{2}$ in
$\\{x\in\mathbb{R}^{N}:\rho(x)<\delta_{0}\\}$ for some $\delta_{0}>0$.
Therefore, there is $\sigma_{0}\in(0,1]$ such that for
$\sigma\in(0,\sigma_{0}]$

$w_{\sigma}\in C^{2}(\mathbb{R}^{N}).$

Notice that

$w_{\sigma}\to 1\quad{\rm in}\ \,\Omega\ \ {\rm as}\ \ \sigma\to 0^{+}.$

Moreover, we have that

$|\Omega|\geq\int_{\Omega}w_{\sigma}dx\geq\int_{\Omega}w_{\sigma}^{2}dx\geq|\Omega_{\sigma}|,$
(4.5)

thanks to $w_{\sigma}=1$ in $\Omega_{\sigma}$.

######  Lemma 4.2.

Let $s\in(0,1)$ and $\Omega$ be a $C^{2}$ domain, then for
$\sigma\in(0,\sigma_{0}]$

$|(-\Delta)^{s}w_{\sigma}(x)|\leq
2c_{N,s}\omega_{{}_{N-1}}\sigma^{-2s}\quad{\rm for}\ \,x\in\Omega.$

Proof.  For $x\in\Omega$, we have that

$\displaystyle|2w_{\sigma}(x)-w_{\sigma}(x+\zeta)-w_{\sigma}(x-\zeta)|$
$\displaystyle\leq\min\\{4,\|w_{\sigma}\|_{C^{2}}|\zeta|^{2}\\}$
$\displaystyle\leq\min\\{4,\sigma^{-2}\|\eta_{0}\|_{C^{2}}|\zeta|^{2}\\}.$

We use an equivalent definition

$\displaystyle\frac{2}{c_{N,s}}|(-\Delta)^{s}w_{\sigma}(x)|$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{N}}\frac{2w_{\sigma}(x)-w_{\sigma}(x+\zeta)-w_{\sigma}(x-\zeta)}{|\zeta|^{N+2s}}d\zeta\Big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\frac{\min\\{4,\sigma^{-2}\|\eta_{0}\|_{C^{2}}|\zeta|^{2}\\}}{|\zeta|^{N+2s}}d\zeta$
$\displaystyle\leq
2\sigma^{-2}\int_{B_{\sigma}}\frac{|\zeta|^{2}}{|\zeta|^{N+2s}}d\zeta+\int_{\mathbb{R}^{N}\setminus
B_{\sigma}}\frac{4}{|\zeta|^{N+2s}}d\zeta$ $\displaystyle\leq
4\omega_{{}_{N-1}}\sigma^{-2s},$

where $\|\eta_{0}\|_{C^{2}}\leq 2$. This completes the proof. $\Box$

Note that if $w_{\sigma}$ and $v_{z}$ are defined by (4.1) and (4.4)
respectively then

$(-\Delta)^{s}(w_{\sigma}v_{z})(z)=v_{z}(x)(-\Delta)^{s}w_{\sigma}(x)+w_{\sigma}(x)(-\Delta)^{s}v_{z}(x)+{\mathcal{L}}^{s}_{z}w_{\sigma}(x),$

where

${\mathcal{L}}^{s}_{z}w_{\sigma}(x)=c_{N,s}\int_{\mathbb{R}^{N}}\frac{(w_{\sigma}(x)-w_{\sigma}(\tilde{x}))(e^{i\tilde{x}\cdot
z}-e^{ix\cdot z})}{|x-\tilde{x}|^{N+2s}}d\tilde{x}.$ (4.6)

######  Lemma 4.3.

Let $s\in(0,1)$, $\Omega$ be a $C^{2}$ domain and $R\geq 1$ be such that
$\Omega\subset B_{R}(0)$, then $\sigma\in(0,\sigma_{0}]$, $x\in\Omega$ and
$|z|>1$
$(i)$ for $s\in(\frac{1}{2},1)$,

$\displaystyle\frac{1}{c_{N,s}}|{\mathcal{L}}^{s}_{z}w_{\sigma}(x)|\leq\frac{\omega_{{}_{N-1}}}{1-s}\sigma^{-1}|z|^{2s-1}+\frac{4\sigma^{-1}\omega_{{}_{N-1}}}{2s-1}|z|^{2s-1}+\frac{\omega_{{}_{N-1}}}{2s}R^{-2s};$

$(ii)$ for $s=\frac{1}{2}$,

$\displaystyle\frac{1}{c_{N,s}}|{\mathcal{L}}^{s}_{z}w_{\sigma}(x)|\leq\frac{\omega_{{}_{N-1}}}{1-s}\sigma^{-1}|z|^{2s-1}+4\sigma^{-1}\omega_{{}_{N-1}}(\log|z|+\log(4R))+\frac{\omega_{{}_{N-1}}}{2s}R^{-1};$

$(iii)$ for $s\in(0,\frac{1}{2})$,

$\displaystyle\frac{1}{c_{N,s}}|{\mathcal{L}}^{s}_{z}w_{\sigma}(x)|\leq\frac{\omega_{{}_{N-1}}}{1-s}\sigma^{-1}|z|^{2s-1}+4\sigma^{-1}\frac{\omega_{{}_{N-1}}}{1-2s}(4R)^{1-2s}+\frac{\omega_{{}_{N-1}}}{2s}R^{-2s}.$

Proof.  Note that

$|e^{i\tilde{x}\cdot z}-e^{ix\cdot z}|\leq\min\\{2,|z||\tilde{x}-x|\\}$

and

$|w_{\sigma}(x)-w_{\sigma}(\tilde{x})|\leq\frac{2}{\sigma}|x-\tilde{x}|,\quad|\tilde{x}|<3R.$

For $x\in\Omega$ and $|z|>1$, we have that

$\displaystyle\frac{1}{c_{N,s}}|{\mathcal{L}}^{s}_{z}w_{\sigma}(x)|$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\frac{|w_{\sigma}(x)-w_{\sigma}(\tilde{x})|\,|e^{i\tilde{x}\cdot
z}-e^{ix\cdot z}|}{|x-\tilde{x}|^{N+2s}}d\tilde{x}$
$\displaystyle\leq\int_{B_{4R}}\frac{2\sigma^{-1}|x-\tilde{x}|\,\min\\{2,|z||\tilde{x}-x|\\}}{|x-\tilde{x}|^{N+2s}}d\tilde{x}+\int_{\mathbb{R}^{N}\setminus
B_{4R}}\frac{2}{|x-\tilde{x}|^{N+2s}}d\tilde{x}$ $\displaystyle\leq
2\sigma^{-1}|z|\int_{B_{\frac{1}{|z|}}(x)}|x-\tilde{x}|^{2-N-2s}d\tilde{x}+4\sigma^{-1}\int_{B_{4R}\setminus
B_{\frac{1}{|z|}}(x)}|x-\tilde{x}|^{1-N-2s}d\tilde{x}$
$\displaystyle\quad+\int_{\mathbb{R}^{N}\setminus
B_{R}}\frac{2}{|\tilde{x}|^{N+2s}}d\tilde{x},$

where

$2\sigma^{-1}|z|\int_{B_{\frac{1}{|z|}}(x)}|x-\tilde{x}|^{2-N-2s}d\tilde{x}\leq\frac{\sigma^{-1}\omega_{{}_{N-1}}}{1-s}|z|^{2s-1},$
$\int_{\mathbb{R}^{N}\setminus
B_{R}}\frac{2}{|\tilde{x}|^{N+2s}}d\tilde{x}\leq\frac{\omega_{{}_{N-1}}}{2s}R^{-2s}$

and

$4\sigma^{-1}\int_{B_{4R}\setminus
B_{\frac{1}{|z|}}(x)}|x-\tilde{x}|^{1-N-2s}d\tilde{x}\leq\left\\{\begin{array}[]{lll}4\frac{\sigma^{-1}\omega_{{}_{N-1}}}{2s-1}|z|^{2s-1}&{\rm
if}\ \,s\in(\frac{1}{2},1),\\\\[5.69054pt]
\phantom{}4\sigma^{-1}\omega_{{}_{N-1}}(\log|z|+\log(4R))&{\rm{if}}\
\,s=\frac{1}{2},\\\\[5.69054pt]
\phantom{}4\frac{\sigma^{-1}\omega_{{}_{N-1}}}{1-2s}(4R)^{1-2s}&{\rm{if}}\
\,s\in(0,\frac{1}{2}).\end{array}\right.$

This completes the proof. $\Box$

Proof of Theorem 1.4.  Denote

$\Psi_{k}(x,y)=\sum^{k}_{j=1}\tilde{\phi}_{j}(x)\phi_{j}(y)$

and

${\mathcal{F}}_{x}(\Psi_{k})(z,y)=(2\pi)^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\Psi_{k}(x,y)e^{ix\cdot
z}dx.$

Let $v_{\sigma}(z,y)$ be the solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{\frac{s_{2}}{2}}u=w_{\sigma}e^{{\rm
i}x\cdot z}&{\rm in}\quad\Omega,\\\\[5.69054pt]
\phantom{(-\Delta)^{\frac{s_{2}}{2}}}u=0&{\rm{in}}\
\quad\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (4.7)

We claim that $v_{\sigma}\in H^{s}_{0}(\Omega)$ for any
$s\in(0,\frac{1+s_{2}}{2})$.

In fact, from [39, Corollary 1.6], for
$\beta\in[\frac{s_{2}}{2},\frac{1+s_{2}}{2})$there exists $C_{\beta}>0$ such
that

$[v_{\sigma}]_{C^{\beta}(\Omega_{t})}\leq
C_{\beta}t^{s_{2}/2-\beta},\quad\forall\,t\in(0,r_{0})$

for some $r_{0}>0$, where

$\Omega_{t}=\\{x\in\Omega:\,\rho(x)>t\\}.$

From [39, Proposition 1.1], we also have $v_{\sigma}\in
C^{s_{2}/2}(\mathbb{R}^{N})$. First we note that for
$\beta\in[\frac{s_{2}}{2},\frac{1+s_{2}}{2})$, it holds

$\big{|}v_{\sigma}(x)-v_{\sigma}(y)\big{|}\leq
C_{\beta}\max\big{\\{}\rho^{s_{2}/2-\beta}(x),\rho^{s_{2}/2-\beta}(y)\big{\\}}\big{|}x-y\big{|}^{\beta}\quad{\rm
for\ any}\ \ x,\,y\in\Omega.$ (4.8)

Indeed, to see the above estimate, note that if $|x-y|\leq 2\rho(x)$ then
$\rho(y)\leq 3\rho(x)$, i.e., $x,y\in\Omega_{\frac{\rho(y)}{3}}$ and thus

$\big{|}v_{\sigma}(x)-v_{\sigma}(y)\big{|}\leq
C\rho^{s_{2}/2-\beta}(y)\big{|}x-y\big{|}^{\beta}.$

Similarly if $|x-y|\leq 2\rho(y)$ then it follows

$\big{|}v_{\sigma}(x)-v_{\sigma}(y)\big{|}\leq
C\rho^{s_{2}/2-\beta}(x)\big{|}x-y\big{|}^{\beta}.$

Finally when $|x-y|>2\max\\{\rho(x),\rho(y)\\}$, then from [39] we have

$\big{|}v_{\sigma}(x)-v_{\sigma}(y)\big{|}\leq|v_{\sigma}(x)|+|v_{\sigma}(y)|\leq
C(\rho^{\frac{s_{2}}{2}}(x)+\rho^{\frac{s_{2}}{2}}(y))\leq
C\max\big{\\{}\rho^{s_{2}/2-\beta}(x),\rho^{s_{2}/2-\beta}(y)\big{\\}}\big{|}x-y\big{|}^{\beta}.$

Hence (4.8) follows.

Next, for any given $s\in(\frac{s_{2}}{2},\frac{1+s_{2}}{2})$. We fix some
$\beta>0$ such that $\beta\in(s,\frac{1+s_{2}}{2})$. Then using (4.8), we have
that

$\displaystyle\int_{\Omega}\int_{\Omega}\frac{|v_{\sigma}(x)-v_{\sigma}(y)|^{2}}{|x-y|^{N+2s}}dydx$
$\displaystyle\leq
C\int_{\Omega}\int_{\Omega}\bigg{(}\rho^{s_{2}-2\beta}(x)+\rho^{s_{2}-2\beta}(y)\bigg{)}|x-y|^{2\beta-N-2s}dydx$
$\displaystyle\leq
2\int_{\Omega}\int_{\Omega}\rho^{s_{2}-2\beta}(y)|x-y|^{2\beta-N-2s}dydx$
$\displaystyle\leq
2\int_{\Omega}\bigg{(}\int_{|x-y|\leq\rho(y)}|x-y|^{2\beta-N-2s}dx\bigg{)}\rho^{s_{2}-2\beta}(y)dy$
$\displaystyle\quad\ \
+2\int_{\Omega}\bigg{(}\int_{|x-y|>\rho(y)}|x-y|^{2\beta-N-2s}dx\bigg{)}\rho^{s_{2}-2\beta}(y)dy$
$\displaystyle\leq
C\int_{\Omega}\rho^{s_{2}-2\beta}(y)dy+C\int_{\Omega}\rho^{2\beta-2s+s_{2}-2\beta}(y)dy$
$\displaystyle\leq
C\int_{t=0}^{r_{0}}\int_{\rho(y)=t}t^{s_{2}-2\beta}dSdt+C\int_{t=0}^{r_{0}}\int_{\rho(y)=t}t^{s_{2}-2s}dSdt$
$\displaystyle<\infty,$

for our choice that $\frac{s_{2}}{2}<s<\beta<\frac{1+s_{2}}{2}$. Similarly, we
also show that

$\displaystyle\int_{\Omega^{c}}\int_{\Omega}\frac{|v_{\sigma}(x)-v_{\sigma}(y)|^{2}}{|x-y|^{N+2s}}dydx$
$\displaystyle=\int_{\Omega^{c}}\int_{\Omega}\frac{|v_{\sigma}(y)|^{2}}{|x-y|^{N+2s}}dydx\leq\int_{\Omega^{c}}\int_{\Omega}\rho(y)^{2s_{2}}|x-y|^{-(N+2s)}dydx<\infty,$

as $s<\frac{1+s_{2}}{2}$ and for any $y\in\Omega$, it follows
$\\{x:\in\Omega^{c}:|x-y|<\rho(y)\\}=\emptyset$. Hence, $v_{\sigma}\in
H^{s}_{0}(\Omega)$ for $s<\frac{1+s_{2}}{2}$.

Denote

$\tilde{v}_{\sigma}(x,z)=(-\Delta)^{\frac{s_{2}}{2}}v_{\sigma}(x,z)$

and

$v_{\sigma,k}(z,y):=v_{\sigma}(y,z)-\sum^{k}_{j=1}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\phi_{j}(y).$

Note that

$(-\Delta)^{\frac{s_{2}}{2}}_{y}v_{\sigma,k}(z,y)={\tilde{v}_{\sigma}(y,z)}-\sum^{k}_{j=1}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}({\mathbb{R}^{N}})}\tilde{\phi}_{j}(y),$

thus we get $v_{\sigma,k}(z,\cdot)\in{\mathbb{H}}_{0,k+1}(\Omega)$ and the
Rayleigh-Ritz formula shows that

$\displaystyle\lambda_{k+1}\Big{|}\int_{\mathbb{R}^{N}}(-\Delta)^{\frac{s_{2}}{2}}_{y}v_{\sigma,k}(z,y)dy\Big{|}^{2}\leq\Big{|}\int_{\mathbb{R}^{N}}(-\Delta)^{\frac{s_{1}}{2}}_{y}v_{\sigma,k}(z,y)dy\Big{|}^{2}$

for any $z\in\mathbb{R}^{N}$ and $\sigma>0$. Thus we can conclude that

$\lambda_{k+1}(\mu)\leq\inf_{\sigma>0}\frac{\int_{B_{r}}\int_{\mathbb{R}^{N}}(-\Delta)^{s_{1}}_{y}v_{\sigma,k}(z,y)\,\overline{v_{\sigma,k}(z,y)}dydz}{\int_{B_{r}}\int_{\mathbb{R}^{N}}(-\Delta)^{s_{2}}_{y}v_{\sigma,k}(z,y)\,\overline{v_{\sigma,k}(z,y)}dydz}.$

An elementary calulation yields that

$\displaystyle\int_{B_{r}}\int_{\mathbb{R}^{N}}(-\Delta)^{s_{2}}_{y}v_{\sigma,k}(z,y)\,\overline{v_{\sigma,k}(z,y)}dydz$
$\displaystyle=\int_{B_{r}}\int_{\mathbb{R}^{N}}\big{|}(-\Delta)^{\frac{s_{2}}{2}}_{y}v_{\sigma,k}(z,y)\big{|}^{2}dydz$
$\displaystyle=\int_{B_{r}}\int_{\mathbb{R}^{N}}\big{|}(-\Delta)^{\frac{s_{2}}{2}}_{y}v_{\sigma}(z,y)\big{|}^{2}dydz-\int_{B_{r}}\sum^{k}_{j=1}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}\int_{\Omega}\tilde{\phi}_{j}^{2}dxdz$
$\displaystyle=\int_{B_{r}}\int_{{\mathbb{R}^{N}}}w_{\sigma}^{2}dxdz-\sum^{k}_{j=1}\int_{B_{r}}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}dz$

and

$\displaystyle\int_{B_{r}}\Big{(}\int_{{\mathbb{R}^{N}}}(-\Delta)^{s_{1}}_{y}v_{\sigma,k}(z,y)\,\overline{v_{\sigma,k}(z,y)}dydz$
$\displaystyle=\int_{B_{r}}\int_{\mathbb{R}^{N}}\Big{(}(-\Delta)^{s_{1}}_{y}v_{\sigma}(y,z)\,\overline{v_{\sigma}(y,z)}\Big{)}dydz-\int_{B_{r}}\sum^{k}_{j=1}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}\int_{\Omega}\phi_{j}(-\Delta)^{s_{1}}\phi_{j}dxdz$
$\displaystyle=\int_{B_{r}}\int_{\mathbb{R}^{N}}\Big{(}(-\Delta)^{s_{1}-s_{2}}_{y}(w_{\sigma}e^{{\rm
i}x\cdot z})\Big{)}\,\overline{(w_{\sigma}e^{{\rm i}x\cdot
z})}dydz-\sum^{k}_{j=1}\lambda_{j}(\mu)\int_{B_{r}}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}dz$
$\displaystyle=\int_{B_{r}}\int_{\mathbb{R}^{N}}\Big{(}{\mathcal{L}}_{z}^{s_{1}-s_{2}}(w_{\sigma})\,\overline{w_{\sigma}e^{{\rm
i}x\cdot
z}}+w_{\sigma}(-\Delta)^{s_{1}-s_{2}}w_{\sigma}+|z|^{2(s_{1}-s_{2})}w_{\sigma}^{2}(x)\Big{)}dxdz$
$\displaystyle\qquad-\sum^{k}_{j=1}\lambda_{j}(\mu)\int_{B_{r}}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}dz.$

Since $\Omega$ is $C^{2}$, there exists $t_{0}\in(0,1)$ such that

$|\partial\Omega_{t}|\leq 2|\partial\Omega|\quad{\rm for}\ t\in(0,t_{0}].$

From (4.5),

$\displaystyle|\Omega|\geq\int_{\Omega}w_{\sigma}^{2}(y)dy>|\Omega_{\sigma}|\geq|\Omega|-2\sigma|\partial\Omega|>\frac{|\Omega|}{2},$

when $\sigma$ is small enough. We choose $r>r_{0}$ for some $r_{0}>1$ such
that $r^{-\frac{s_{1}-s_{2}}{2}}=\sigma$ satisfies the above relation and
$\sigma<\frac{1}{4|\partial\Omega|}|\Omega|$. Therefore,

$|\Omega|\geq\int_{\Omega}w_{\sigma}^{2}(y)dy\geq|\Omega|-2r^{-\frac{s_{1}-s_{2}}{2}}|\partial\Omega|>\frac{|\Omega|}{2}.$
(4.9)

Note that for $s=s_{1}-s_{2}$ and $r>r_{0}$

$\displaystyle\frac{1}{c_{N,s}}\int_{B_{r}}\int_{\mathbb{R}^{N}}|{\mathcal{L}}^{s}_{z}w_{\sigma}\,\overline{w_{\sigma}e^{{\rm
i}x\cdot z}}|dydz$
$\displaystyle\leq\frac{\omega_{{}_{N-1}}r^{N+2s-1}}{\sigma(N+2s-1)(1-s)}|\Omega|+\frac{1}{\sigma}\varphi_{s}(r,R)|\Omega|+\frac{\omega_{{}_{N-1}}^{2}r^{N}}{2sN}R^{-2s}|\Omega|$
$\displaystyle\leq\frac{\omega_{{}_{N-1}}|\Omega|}{(N+2s-1)(1-s)}r^{N+2s+\frac{s}{2}-1}+r^{\frac{s}{2}}\varphi_{s}(r,R)|\Omega|+\frac{\omega_{{}_{N-1}}^{2}r^{N}}{2sN}R^{-2s}|\Omega|$
$\displaystyle\leq
c_{1}(N,s_{1},s_{2})|\Omega|\,r^{N+\max\\{2s+\frac{s}{2}-1,\,\frac{s}{2}\\}},$

where

$\varphi_{s}(r,R)=\left\\{\begin{array}[]{lll}\frac{4\omega_{{}_{N-1}}r^{N+2s-1}}{(2s-1)(N+2s-1)}&{\rm
if}\quad s\in(\frac{1}{2},1),\\\\[5.69054pt]
\phantom{}\frac{4\omega_{{}_{N-1}}r^{N}(\log r+\log(4R))}{N}&{\rm{if}}\
\,s=\frac{1}{2},\\\\[5.69054pt]
\phantom{}\frac{4\omega_{{}_{N-1}}^{2}r^{N}}{N(1-2s)}(4R)^{1-2s}&{\rm{if}}\
\,s\in(0,\frac{1}{2}).\end{array}\right.$ (4.10)

Moreover, we have that

$\displaystyle\int_{B_{r}}\int_{\mathbb{R}^{N}}|w_{\sigma}(-\Delta)^{s}w_{\sigma}|dydz$
$\displaystyle\leq
2c_{N,s}\frac{\omega_{{}_{N-1}}^{2}r^{N}}{N}\sigma^{-2s}|\Omega|=c_{2}(N,s_{1},s_{2})|\Omega|r^{N+s^{2}}.$

Furthermore,

$\displaystyle\int_{B_{r}}\int_{\mathbb{R}^{N}}|z|^{2s}w_{\sigma}^{2}(x)dydz=\frac{\omega_{{}_{N-1}}r^{N+2s}}{N+2s}\int_{\Omega}w_{\sigma}^{2}(y)dy.$

Let

$\delta_{0}=\max\big{\\{}2s+\frac{s}{2}-1,\,\frac{s}{2},s^{2}\big{\\}}=\frac{s}{2},$

where we have used the hypothesis that $s_{1}<\frac{1+s_{2}}{2}$.

Let

$\displaystyle P_{j}:$
$\displaystyle=\int_{B_{r}}\Big{|}\big{\langle}\tilde{v}_{\sigma}(.,z),\tilde{\phi}_{j}\big{\rangle}_{L^{2}(\mathbb{R}^{N})}\Big{|}^{2}dz$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\Big{|}\int_{\mathbb{R}^{N}}w_{\sigma}e^{{\rm
i}x\cdot z}\tilde{\phi}_{j}\Big{|}^{2}dz\
\leq(2\pi)^{N}\\!\int_{\mathbb{R}^{N}}\\!\Big{|}{\mathcal{F}}(w_{\sigma}\tilde{\phi}_{j})\Big{|}^{2}dz$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=\
\,(2\pi)^{N}\int_{\mathbb{R}^{N}}(w_{\sigma}\tilde{\phi}_{j})^{2}dx\
\,\leq\,(2\pi)^{N},$

using the Parseval’s inequality. Note that

$\displaystyle\lambda_{k+1}(0)\leq\frac{\frac{\omega_{{}_{N-1}}}{N+2(s_{1}-s_{2})}r^{N+2(s_{1}-s_{2})}\displaystyle\int_{\Omega}w_{\sigma}^{2}(y)dy+O(1)|\Omega|r^{N+\frac{s}{2}}-\displaystyle\sum^{k}_{j=1}\lambda_{j}(0)P_{j}}{\frac{\omega_{{}_{N-1}}}{N}r^{N}\displaystyle\int_{\Omega}w_{\sigma}^{2}(y)dy-\displaystyle\sum^{k}_{j=1}P_{j}},$

and now taking

$Q_{1}=\frac{\omega_{{}_{N-1}}}{N+2(s_{1}-s_{2})}r^{N+2(s_{1}-s_{2})}\int_{\Omega}w_{\sigma}^{2}(y)dy+O(1)|\Omega|r^{N+\frac{s}{2}}\quad{\rm
and}\quad
Q_{2}=\frac{\omega_{{}_{N-1}}}{N}r^{N}\int_{\Omega}w_{\sigma}^{2}(y)dy,$

we have that

$\displaystyle 0$
$\displaystyle\leq\frac{Q_{1}-\sum^{k}_{j=1}\lambda_{j}(0)P_{j}}{Q_{2}-\sum^{k}_{j=1}P_{j}}-\lambda_{k+1}(0)$
$\displaystyle=\frac{\big{(}Q_{1}-Q_{2}\lambda_{k+1}(0)\big{)}+\sum^{k}_{j=1}\Big{(}\lambda_{k+1}(0)-\lambda_{j}(0)\Big{)}P_{j}}{Q_{2}-\sum^{k}_{j=1}P_{j}}$
$\displaystyle\leq\frac{\big{(}Q_{1}-Q_{2}\lambda_{k+1}(0)\big{)}+(2\pi)^{N}\sum^{k}_{j=1}\Big{(}\lambda_{k+1}(0)-\lambda_{j}(0)\Big{)}}{Q_{2}-(2\pi)^{N}k},$

since $\lambda_{k+1}(0)\geq\lambda_{j}(\mu)$ for $j<k+1$ and
$P_{j}\in\big{(}0,(2\pi)^{N}]$. Therefore

$0<\lambda_{k+1}(0)\leq\frac{\frac{\omega_{{}_{N-1}}}{N+2(s_{1}-s_{2})}r^{N+2(s_{1}-s_{2})}\displaystyle\int_{\Omega}w_{\sigma}^{2}(y)dy+O(1)|\Omega|r^{N+\frac{s_{1}-s_{2}}{2}}-(2\pi)^{N}\displaystyle\sum^{k}_{j=1}\lambda_{j}(0)}{\frac{\omega_{{}_{N-1}}}{N}r^{N}\displaystyle\int_{\Omega}w_{\sigma}^{2}(y)dy-(2\pi)^{N}k}.$

There exists $k_{0}\geq 1$ such that for $k\geq k_{0}$, we can choose
$r>r_{0}$ satisfying

$\frac{\omega_{{}_{N-1}}}{N}r^{N}\int_{\Omega}w_{\sigma}^{2}(y)dy=(2\pi)^{N}(k+1).$
(4.11)

As a consequence, using (4.9) and the above relation, we obtain for $k\geq
k_{0}$

$\displaystyle\sum^{k}_{j=1}\lambda_{j}(0)$
$\displaystyle\leq(2\pi)^{-N}\frac{\omega_{{}_{N-1}}}{N+2(s_{1}-s_{2})}r^{N+2(s_{1}-s_{2})}\int_{\Omega}w_{\sigma}^{2}(y)dy+c|\Omega|r^{N+\frac{s_{1}-s_{2}}{2}}$
$\displaystyle=(2\pi)^{2(s_{1}-s_{2})}\frac{N}{N+2(s_{1}-s_{2})}\Big{(}N^{-1}\omega_{{}_{N-1}}\int_{\Omega}w_{\sigma}^{2}(y)dy\Big{)}^{-\frac{2(s_{1}-s_{2})}{N}}(k+1)^{1+2\frac{s_{1}-s_{2}}{N}}$
$\displaystyle\qquad+c_{3}k^{1+\frac{1}{2}\frac{s_{1}-s_{2}}{N}}$
$\displaystyle\leq
b_{3}|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}k^{1+2\frac{s_{1}-s_{2}}{N}}+c_{4}k^{1+\frac{3}{2}\frac{s_{1}-s_{2}}{N}}+c_{3}k^{1+\frac{1}{2}\frac{s_{1}-s_{2}}{N}},$

where $c_{3},c_{4}>0$ depends on $N,s_{1},s_{2}$, $\Omega$,

$(k+1)^{1+2\frac{s_{1}-s_{2}}{N}}\leq
k^{1+2\frac{s_{1}-s_{2}}{N}}+c_{5}k^{2\frac{s_{1}-s_{2}}{N}}\quad{\rm for}\
\,k\geq 1$

and we use the fact that for $c_{7},c_{8}>0$

$\displaystyle\Big{(}\int_{\Omega}w_{\sigma}^{2}(y)dy\Big{)}^{-\frac{2(s_{1}-s_{2})}{N}}$
$\displaystyle\leq\Big{(}|\Omega|-2r^{-\frac{s_{1}-s_{2}}{2}}|\partial\Omega|\Big{)}^{-\frac{2(s_{1}-s_{2})}{N}}$
$\displaystyle\leq|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}+c_{6}r^{-\frac{s_{1}-s_{2}}{2}}$
$\displaystyle\leq|\Omega|^{-\frac{2(s_{1}-s_{2})}{N}}+c_{7}k^{-\frac{s_{1}-s_{2}}{2N}}.$

Here for $k\leq k_{0}$, we only have to adjust the constant $c_{3}$ or
$c_{4}$. $\Box$

Acknowledgements: This work of H. Chen is supported by NNSF of China, No:
12071189 and 12001252 and the Jiangxi Provincial Natural Science Foundation,
No: 20202BAB201005, 20202ACBL201001.

The research of M. Bhakta is partially supported by the SERB MATRICS grant
MTR/2017/000168 and SERB WEA grant WEA/2020/000005.

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