2021/2023 UNC Greensboro PDE Conference,

Electronic Journal of Differential Equations, Conference 26 (2022/2025), pp. 201–217.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NORMAL SOLVABILITY AND FREDHOLM PROPERTIES FOR

SPECIAL CLASSES OF HYPOELLIPTIC OPERATORS

ANI TUMANYAN

Abstract. In this work, we establish normal solvability and a priori estimates
for hypoelliptic operators with special variable coefficients, associated with

multi-quasi-elliptic symbols, acting in weighted Sobolev spaces in Rn. We
obtain Fredholm criteria for the special classes of regular hypoelliptic operators

in various scales of multianisotropic spaces. We also provide applications to the

smoothness of solutions, index invariance on the scale, and spectral properties
of such operators.

1. Introduction, basic notions and definitions

We study the Fredholm properties of a class of regular hypoelliptic operators,
which is a special subclass of Hörmander’s hypoelliptic operators and has many
important applications (see [16]). The characteristic polynomials of these operators
are multi-quasi-elliptic, so they are a natural generalization of elliptic, parabolic,
2b-parabolic, and quasielliptic operators. These operators were introduced in the
late 60s-70s and studied by many authors: Nikolsky [22], Mikhailov [21], Friberg
[13], Volevich, Gindikin [31], Ghazaryan [14], and others.

The analysis of regular hypoelliptic operators has certain challenges, as corre-
sponding characteristic polynomials are not homogeneous like in the elliptic case.
Solvability conditions, a priori estimates, and Fredholm properties have been stud-
ied for special classes of hypoelliptic operators in various functional spaces, but
most of the results are related to elliptic and quasielliptic operators.

The Fredholm property for elliptic operators has been studied for different scales
of weighted spaces in Rn in the works of Bagirov [3], Lockhart, McOwen [20, 19],
Schrohe [24], and numerous others.

A priori estimates and the Fredholm solvability of quasielliptic operators have
been studied in the works of Bagirov [4], Karapetyan, Darbinyan [17], Darbinyan
and Tumanyan [8, 29], and others. Isomorphic characteristics for quasielliptic op-
erators with constant coefficients on a special scale of weighted spaces have been
derived in the works of Demidenko (see [10, 11]), and such operators have been

2020 Mathematics Subject Classification. 35H10, 35H30, 47A53.
Key words and phrases. Regular hypoelliptic operator; a priori estimate; Fredholm operator;

multianisotropic weighted space.
©2025 This work is licensed under a CC BY 4.0 license.

Published May 13, 2025.

201



202 A. TUMANYAN EJDE-2022/25/CONF/26

further studied in Hile’s work (see [15]). In this paper, we obtain a priori esti-
mates, normal solvability, and Fredholm criteria for a different class of quasielliptic
operators with variable coefficients on the scale of weighted anisotropic spaces.

Rodino, Boggiatto, and Buzano studied the Fredholm properties and the spec-
trum of special classes of pseudo-differential operators in multianisotropic spaces
with polynomial weights (see [5]). The spectral properties of Schrödinger type
hypoelliptic operators, as well as hypoelliptic pseudo-differential operators, which
are relatively bounded perturbations of constant-coefficients operators, have been
studied in the works of Buzano, Ziggioto (see [6, 7]). Fredholm criteria have been
established for specific subclasses of regular hypoelliptic operators in the works
[26, 28].

In this article, we obtain normal solvability and a priori estimates for two classes
of regular hypoelliptic operators with variable coefficients, acting on special scales
of weighted Sobolev spaces in Rn. Fredholm criteria are established for the consid-
ered classes of operators on multianisotropicHk,R,p

q (Rn) and anisotropicHk,ν,p
q (Rn)

scales of spaces with appropriate weight functions q. We study regularity of solu-
tions, index invariance, and spectral properties of these operators. The scales of
multianisotropic spaces and conditions on the coefficients considered are more gen-
eral than those in previous works (see [29, 26]).

Definition 1.1. A bounded linear operator A, acting from a Banach space X to
a Banach space Y , is called a normally solvable operator if the image of operator
A is closed (Im(A) = Im(A)).

An operator A is called an n-normally solvable (or n-normal) operator if it is
normally solvable, and the kernel of operator A is finite-dimensional (dimker(A) <
∞).

An operator A is called a Fredholm operator if it is n-normal, and the cokernel
of operator A is finite-dimensional (dim coker(A) = dimY/ Im(A) <∞).

Definition 1.2. For a closed operator A with a dense domain in a Banach space
X, essential spectrum of A is the set σes(A) of complex numbers λ such that A−λI
is not a Fredholm operator.

The difference between the dimension of the kernel and the cokernel of operator
A is called the index of the operator

ind(A) = dimker(A)− dim coker(A).

Definition 1.3. For a bounded linear operator A, acting from a Banach space X
to a Banach space Y , the bounded linear operators R1 : Y → X and R2 : Y → X
are called, respectively, left and right regularizers if the following holds: R1A =
IX + T1, AR2 = IY + T2, where IX , IY are the identity operators, T1 : X → X and
T2 : Y → Y are compact operators.

A bounded linear operator R : Y → X is called a regularizer for operator A if it
is a left and right regularizer.

Let n ∈ N and Rn be the Euclidean n-dimensional space, Zn
+, Nn be the sets of n-

dimensional multi-indices and multi-indices with natural components respectively.
Let N ⊂ Zn

+ be a finite set of multi-indices, R = R(N ) be a minimum convex
polyhedron containing all the points N .

Definition 1.4. A polyhedron R is called completely regular if the following holds:
(a) R is a complete polyhedron: R has a vertex at the origin and further vertices



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 203

on each coordinate axes in Rn; (b) all components of the outer normals of (n− 1)-
dimensional non-coordinate faces of R are positive.

Let R be a completely regular polyhedron. Denote by Rn−1
j (j = 1, . . . , In−1)

(n−1)-dimensional non-coordinate faces of R with the corresponding outer normals
µj such that all multi-indices α ∈ Rn−1

j satisfy (α : µj) = α1

µj
1

+ · · · + αn

µj
n

= 1,

∂R =
⋃In−1

j=1 Rn−1
j . For k > 0 denote by kR := {kα = (kα1, kα2 . . . , kαn) : α ∈ R}.

Consider the differential operator

P (x,D) =
∑
α∈R

aα(x)D
α, (1.1)

where Dα = Dα1
1 . . . Dαn

n , Dj = i−1 ∂
∂xj

, x = (x1, . . . , xn) ∈ Rn, aα(x) ∈ C(Rn).

Denote

P (x, ξ) =
∑
α∈R

aα(x)ξ
α. (1.2)

For ξ ∈ Rn denote

|ξ|R =
∑
α∈R

|ξα|, |ξ|∂R =
∑

α∈∂R

|ξα|.

Definition 1.5. A differential operator P (x,D) is called regular at a point x0 ∈ Rn,
if there exists a constant δ > 0 such that

1 + |P (x0, ξ)| ≥ δ|ξ|R,∀ξ ∈ Rn.

P (x,D) is called regular in Rn, if P (x,D) is regular at each point x ∈ Rn.
P (x,D) is called uniformly regular in Rn, if there exists a constant δ > 0 such

that:

1 + |P (x, ξ)| ≥ δ|ξ|R, ∀ξ ∈ Rn,∀x ∈ Rn.

A polyhedron R is called the characteristic or Newton polyhedron of P (x,D).

In the anisotropic case, a completely regular polyhedron R has only one (n−1)-
dimensional non-coordinate face with an outer normal ν ∈ Nn. The differential
operator P (x,D) with such characteristic polyhedron R can be written as

P (x, ξ) =
∑

(α:ν)≤1

aα(x)ξ
α. (1.3)

For ξ ∈ Rn and ν ∈ Nn, denote |ξ|ν :=
∑n

i=1 |ξ
νi
i |.

Example 1.6. We ahve the followfing examples of regular differential operators:

(1) Let m ∈ N and R be a Newton polyhedron for the set of points (0, 0, . . . , 0),
(m, 0, . . . , 0), . . . , (0, 0, . . . ,m). In this case conditions from definition 1.5
coincide with ellipticity conditions with |ξ|∂R = |ξ|m =

∑n
i=1 |ξmi |.

(2) Let ν ∈ Nn and R be a Newton polyhedron for the set of points (0, 0, . . . , 0),
(ν1, 0, . . . , 0), . . . , (0, 0, . . . , νn). In this case conditions from definition 1.5
coincide with quasiellipticity conditions with |ξ|∂R = |ξ|ν =

∑n
i=1 |ξ

νi
i |.

(3) Let n = 2 and R be a Newton polyhedron for the points (0, 0), (8, 0), (0, 8)
and (6, 4). Then

P (x,D) = a1D
8
1 + a2D

6
1D

4
2 + a3D

8
2 + q(x)

is a regular differential operator in R2 with some a1, a2, a3 > 0 and q ∈
C(R2).



204 A. TUMANYAN EJDE-2022/25/CONF/26

(4) Let n = 3 and R be a Newton polyhedron for the points (0, 0, 0), (8, 0, 0),
(0, 8, 0), (6, 4, 0), (6, 0, 6), (0, 6, 6) and (0, 0, 12). Then

P (x,D) = D8
1 +D6

1D
4
2 +D8

2 +D6
1D

6
3 +D6

2D
6
3 +D12

3 + q(x)

is a regular differential operator in R3 with q ∈ C(R3).

Let the sequence {ai}∞i=0 ⊂ R+ be such that the series
∑∞

i=0 ai diverges, and the
inequality ai+1 < γai with γ > 0 holds for i = 0, 1, . . ..

Using the sequence {ai}∞i=0, we define the special covering of Rn as {Wp}∞p=1 and
the sets of functions {φp}∞p=1 and {ψp}∞p=1, following the definitions in the works
[3] and [26].

These systems of functions have the following properties:

(1) suppφp ⊂ suppψp ⊂Wp;
(2) ψp(x)φp(x) = φp(x) for all x ∈ Rn;
(3) for each α ∈ Z+, there exists a constant Cα > 0 such that

|Dαψp(x)| ≤ Cα(a[ p−1
l ])

−|α|, |Dαφp(x)| ≤ Cα(a[ p−1
l ])

−|α|,

for all x ∈ Rn, p = 1, 2, . . . ;
(4)

∑∞
p=1 φp(x) ≡ 1.

We denote

Q := {g ∈ C(Rn) : g(x) > 0,∀x ∈ Rn}.
Further, we define two sets of special weight functions. For m ∈ Z+ and a

completely regular polyhedron R, denote as Qm,R a set of weight functions g ∈ Q,
which satisfy the following conditions:

(1) 1
g(x) ⇒ 0 when |x| → ∞;

(2) for β ∈ mR, β ̸= 0 Dβg(x) ∈ C(Rn) and there exists Cβ > 0 such that
|Dβg(x)|

g(x)1+(β:µj)
≤ Cβ for all x ∈ Rn, j = 1, . . . , In−1;

(3) for each ε > 0 there exist δ = δ(ε) > 0 and p0 = p0(ε) > 0 such that for all
p > p0 when maxj=1,...,l diamUj < δ the following holds:

max
x,y∈Wp

|g(x)− g(y)|
g(y)

< ε, max
x,y∈Wp

1

g(x)
1

µmax a[ p−1
l ]

< ε,

where µmax = max1≤i≤In−1
max1≤s≤n{µi

s}.
The considered class Qm,R includes polynomial functions and special exponential
functions such as: (1 + |x|R)l, exp (1 + |x|R)

r
when l, r > 0.

For m ∈ Z+ and ν ∈ Nn, denote as Q̃k,ν a set of weight functions g ∈ Q, which
satisfy the following conditions:

(1) there exists a constant C > 0 such that 0 < g(x) ≤ C for all x ∈ Rn;
(2) for β ∈ Zn

+, (β : ν) ≤ m,β ̸= 0 Dβg ∈ C(Rn) and there exists Cβ > 0 such

that |Dβg(x)|
g(x)1+(β:ν) ≤ Cβ for all x ∈ Rn;

(3) for each ε > 0 there exist δ = δ(ε) > 0 and p0 = p0(ε) > 0 such that for all
p > p0 when maxj=1,...,l diamUj < δ the following holds:

max
x,y∈Wp

|g(x)− g(y)|
g(y)

< ε, max
x,y∈Wp

1

g(x)
1

νmin a[ p−1
l ]

< ε,

where νmin = min1≤i≤n{νi}.



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 205

The class Q̃m,ν includes functions (1 + |x|ν)l, where − νmin

νmax
< l ≤ 0. For k ∈ R, a

completely regular polyhedron R and 1 < p <∞, we denote

Hk,R,p(Rn) := {u ∈ S′ : ∥u∥k,R,p := ∥F−1(1 + |ξ|∂R)kFu∥Lp(Rn) <∞},

where S′ is the set of tempered distributions.
For Ω ⊂ Rn, we denote by Ḣk,R,p(Ω) the completeness of C∞

0 (Ω) with the norm
∥ · ∥k,R,p.

For k ∈ Z+, a completely regular polyhedron R, 1 < p < ∞ and q ∈ Q, we
denote

Hk,R,p
q (Rn) :=

{
u : ∥u∥Hk,R,p

q (Rn) := ∥u∥k,R,p,q

:=
∑

α∈kR

∥Dαu · qk−maxi(α:µ
i)∥Lp(Rn) <∞}.

For Ω ⊂ Rn and x0 ∈ Rn, we denote

Hk,R,p
q (Ω) :=

{
u : ∥u∥Hk,R,p

q (Ω) :=
∑

α∈kR

∥Dαu · qk−maxi(α:µ
i)∥Lp(Ω) <∞

}
,

Hk,R,p
q(x0)

(Rn) :=
{
u : ∥u∥Hk,R,p

q(x0)
(Rn) := ∥u∥k,R,p,q(x0)

:=
∑

α∈kR

∥Dαu · q(x0)k−maxi(α:µ
i)∥Lp(Rn) <∞}.

For p = 2, we denote Hk,R(Rn) := Hk,R,2(Rn) and Hk,R
q (Rn) := Hk,R,2

q (Rn).
In the anisotropic case, when R has only one (n−1)-dimensional non-coordinate

face with an outer normal ν ∈ Nn, we analogously define the space Hk,ν,p(Rn)

for k ∈ R and the spaces Hk,ν,p
q (Rn), Hk,ν,p

q (Ω), Hk,ν,p
q(x0)

(Rn) for k ∈ Z+. The

introduced spaces generalize multianisotropic Sobolev-type spaces (see [14]).

2. A priori estimates and normal solvability

Let k ∈ Z+ and q ∈ Q. Consider the differential operator P (x,D) (see (1.1))
with the coefficients that satisfy the following conditions:

P (x,D) =
∑
α∈R

aα(x)D
α =

∑
α∈R

(
a0α(x)q(x)

1−maxi(α:µ
i) + a1α(x)

)
Dα, (2.1)

whereDβ(a0α(x)) = O
(
q(x)mini(β:µ

i)
)
andDβ(a1α(x)) = o

(
q(x)1−maxi(α−β:µi)

)
when

|x| → ∞ for all α ∈ R, β ∈ kR.
It is easy to check that P (x,D) generates a bounded linear operator, acting from

Hk+1,R,p
q (Rn) to Hk,R,p

q (Rn).
We consider also the special case, when R has only one (n−1)-dimensional non-

coordinate face with an outer normal ν ∈ Nn. In this case, the differential operator
P (x,D) can be expressed as

P (x,D) =
∑

(α:ν)≤1

aα(x)D
α =

∑
(α:ν)≤1

(
a0α(x)q(x)

1−(α:ν) + a1α(x)
)
Dα, (2.2)

where Dβ(a0α(x)) = O(q(x)(β:ν)), Dβ(a1α(x)) = o(q(x)1−(α−β:ν)) when |x| → ∞ for
all (α : ν) ≤ 1, (β : ν) ≤ k. For N > 0 and x0 ∈ Rn denote

KN (x0) := {x ∈ Rn : |x− x0| ≤ N},KN := KN (0).

Further, we use the following theorem, a consequence of [18, Theorem 7.1].



206 A. TUMANYAN EJDE-2022/25/CONF/26

Theorem 2.1. Let k ∈ Z+, q ∈ Q, and P (x,D) be the differential operator (2.1).
Then the operator P (x,D) : Hk+1,R,p

q (Rn) → Hk,R,p
q (Rn) is an n−normal operator

if and only if there exist constants κ > 0 and N > 0 such that

∥u∥k+1,R,p,q ≤ κ(∥Pu∥k,R,p,q + ∥u∥Lp(KN )), ∀u ∈ Hk+1,R,p
q (Rn). (2.3)

Further, we consider two classes of regular hypoelliptic operators defined using

the weight functions from Qk,R and Q̃k,ν , respectively. We derive special conditions
on the symbol of operators for the fulfillment of a priori estimates (2.3) for these
special classes of regular hypoelliptic operators.

Theorem 2.2. Let k ∈ Z+, q ∈ Qk,R, and P (x,D) be the differential operator given
by (2.1) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)− a0α(y)| = 0 for
all α ∈ R. Suppose there exists a constant κ > 0 such that

∥u∥k+1,R,p,q ≤ κ(∥Pu∥k,R,p,q + ∥u∥Lp(Rn)), ∀u ∈ Hk+1,R,p
q (Rn). (2.4)

Then, P (x,D) is regular in Rn, and there exist constants δ > 0 and M > 0 such
that ∣∣ ∑

α∈R
a0α(x)ξ

α
∣∣ ≥ δ(1 + |ξ|∂R), ∀ξ ∈ Rn, |x| > M. (2.5)

Proof. The proof of [26, Theorem 2.2] for p = 2 can be generalized for the spaces
Hk+1,R,p

q (Rn) in a similar way. □

Theorem 2.3. Let k ∈ Z+, q ∈ Q̃k,ν , and P (x,D) be the differential operator given
by (2.2) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)− a0α(y)| = 0 for

all α ∈ Zn
+, (α : ν) ≤ 1. Suppose there exist constants κ > 0 and N > 0 such that:

∥u∥k+1,ν,p,q ≤ κ(∥Pu∥k,ν,p,q + ∥u∥Lp(KN )),∀u ∈ Hk+1,ν,p
q (Rn). (2.6)

Then, P (x,D) is regular in Rn, and there exist constants δ > 0 and M > 0 such
that ∣∣ ∑

(α:ν)≤1

a0α(x)ξ
α
∣∣ ≥ δ(1 + |ξ|ν),∀ξ ∈ Rn, |x| > M. (2.7)

Proof. From [29, Theorem 2.1] follows that P (x,D) is regular in Rn. Thus, we need
to prove (2.7).

Let {xm}∞m=1 ⊂ Rn is such a sequence that |xm| → ∞ when m → ∞. Without
loss of generality assume xm ∈ Wm. Let ξ ∈ Rn. Consider the function ũm(x) =

exp(i(q(xm)
1
ν ξ, x))ψm(x). Since q ∈ Q̃k,ν , then for any ε > 0 there exist δ(ε) > 0

and m0(ε) > 0 such that for all m > m0 and maxj=1,...,l diamUj < δ

|q(x)− q(y)| ≤ εq(y),∀x, y ∈Wm.

Then, for any r > 0 it holds

|q(x)r − q(xm)r| ≤ τr(ε)q(xm)r,∀x ∈Wm, (2.8)

where τr(ε) → 0 when ε→ 0.
From inequality (2.8) and supp ũm ⊂Wm follows that there exists τ(ε) such that

τ(ε) → 0 when ε→ 0 and the following inequalities hold:

∥ũm∥k+1,ν,p,q ≥ (1− τ(ε))∥ũm∥k+1,ν,p,q(xm), (2.9)

∥Pũm∥k,ν,p,q ≤ (1 + τ(ε))∥Pũm∥k,ν,p,q(xm). (2.10)



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 207

For sufficiently largem0 ∈ N and a small enough maxj=0,...,l diamUj form > m0,
it holds

∥ũm∥k+1,ν,p,q ≥ 1

2
∥ũm∥k+1,ν,p,q(xm), (2.11)

∥Pũm∥k,ν,p,q ≤ 1

2
∥Pũm∥k,ν,p,q(xm). (2.12)

Considering that q ∈ Q̃k,ν and the properties of {ψm}∞m=1, we obtain that for all
(γ : ν) ≤ k+1 and ε > 0 there exist δ(ε) > 0 and m0(ε) > 0 such that for m > m0

and maxj=1,...,l diamUj < δ, the following inequality holds

|Dγψm(x)|
q(x)(γ:ν)

=
|Dγψm(x)|a|γ|

[m−1
l ]

q(x)
(γ:ν)− |γ|

νmin q(x)
|γ|

νmin a
|γ|
[m−1

l ]

≤ ωγ(ε), (2.13)

where ωγ(ε) → 0 when ε→ 0.
For β ∈ Zn

+ with some constants C1 > 0 and σ = σ(ν) > 0 the following holds:

∥Dβ ũm∥Lp(Rn)q(xm)k+1−(β:ν)

≥ |ξβ |q(xm)k+1∥ψm∥Lp(Rn) − C1

∑
0≤γ<β

|ξγ |q(xm)k+1−(β−γ:ν)∥Dβ−γψm∥Lp(Rn).

Using estimate (2.13) and the properties of {ψm}∞m=1, the following estimate holds

∥Dβ ũm∥Lp(Rn)q(xm)k+1−(β:ν)

≥ |ξβ |q(xm)k+1µ(Wm)− ω1(ε)
∑

0≤γ<β

|ξγ |q(xm)k+1µ(Wm),

where µ(Wm) is the measure of Wm, ω1(ε) → 0 when ε→ 0.
Then, it is easy to check that the following holds:

∥ũm∥k+1,ν,p,q(xm)

≥
∑

(β:ν)≤k+1

|ξβ |q(xm)k+1µ(Wm)− ω2(ε)
∑

(γ:ν)≤k+1

|ξγ |q(xm)k+1µ(Wm), (2.14)

where ω2(ε) → 0 when ε→ 0.
For β ∈ Zn

+(β : ν) ≤ k, we have

∥Dβ(P (x,D)ũm)∥Lp(Rn)q(xm)k−(β:ν)

≤
∥∥Dβ

( ∑
(α:ν)≤1

a0α(x)q(x)
1−(α:ν)Dαũm

)∥∥
Lp(Rn)

q(xm)k−(β:ν)

+
∥∥Dβ(

∑
α∈R

a1α(x)D
αũm)

∥∥
Lp(Rn)

q(xm)k−(β:ν).

(2.15)

Taking into account that Dβ(a1α(x)) = o(q(x)1−(α−β:ν)) when |x| → ∞ for all
α, β ∈ Zn

+, (α : ν) ≤ 1, (β : ν) ≤ k and (2.13), it is easy to check that for a
sufficiently large m0 and m > m0, it holds∥∥Dβ(

∑
α∈R

a1α(x)D
αũm)

∥∥
Lp(Rn)

q(xm)k−(β:ν)

≤ ω3(ε)
∑

(γ:ν)≤k+1

|ξγ |q(xm)k+1µ(Wm),
(2.16)



208 A. TUMANYAN EJDE-2022/25/CONF/26

where ω3(ε) → 0 when ε→ 0. From conditions (2.1), limm→∞ maxx,y∈Wm
|a0α(x)−

a0α(y)| = 0 for all (α : ν) ≤ 1, q ∈ Q̃k,ν , and inequality (2.8), we conclude that for
(α : ν) ≤ 1 and (β : ν) ≤ k when m0 is large enough and maxj=1,...,l diamUj is
small enough, for m > m0, the following holds:

|Dβ(a0α(x)q(x)
1−(α:ν) − a0α(xm)q(xm)1−(α:ν))| ≤ τα,β(ε)q(xm)1−(α:ν)+(β:ν), (2.17)

where τα,β(ε) → 0 when ε → 0. Utilizing (2.17) and following a similar approach
to the proof in [27, Theorem 2.4], we obtain that for a sufficiently large m0 and
m > m0, the following estimate holds:∥∥Dβ

( ∑
(α:ν)≤1

a0α(x)q(x)
1−(α:ν)Dαũm

)∥∥
Lp(Rn)

q(xm)k−(β:ν)

≤
∣∣ ∑
(α:ν)≤1

a0α(xm)ξα
∣∣|ξβ |q(xm)k+1∥ψm∥Lp(Rn)

+ C2

∑
0≤γ<β

|ξγ |q(xm)k+1−(β−γ:ν)∥Dβ−γψm∥Lp(Rn)

≤
∣∣ ∑
(α:ν)≤1

a0α(xm)ξα
∣∣|ξβ |q(xm)k+1µ(Wm) + ω4(ε)

∑
0≤γ<β

|ξγ |q(xm)k+1µ(Wm)

(2.18)
where ω4(ε) → 0 when ε→ 0.

From estimates (2.16)–(2.18) for a sufficiently large m0, for all m > m0, we
obtain

∥Pũm∥k,ν,p,q(xm) ≤
∣∣ ∑
(α:ν)≤1

a0α(xm)ξα
∣∣ ∑
(β:ν)≤k

|ξβ |q(xm)k+1µ(Wm)

+ ω5(ε)
∑

(γ:ν)≤k+1

|ξγ |q(xm)k+1µ(Wm),
(2.19)

where ω5(ε) → 0 when ε→ 0. Then, from (2.4), using (2.14) and (2.19), we obtain∑
(β:ν)≤k+1

|ξβ |q(xm)k+1µ(Wm)− ω2(ε)
∑

(γ:ν)<k+1

|ξγ |q(xm)k+1µ(Wm)

≤ κ
(
|

∑
(α:ν)≤1

a0α(xm)ξα|
∑

(β:ν)≤k

|ξβ |q(xm)k+1µ(Wm)

+ ω5(ε)
∑

(γ:ν)≤k+1

|ξγ |q(xm)k+1µ(Wm)
)
.

Since {a0α(x) : (α : ν) ≤ 1} are bounded functions and xm → ∞ when m → ∞,
there exist convergent subsequences of sequences {a0α(xm) : (α : ν) ≤ 1}. Without
loss of generality, assume that the sequences {a0α(xm) : (α : ν) ≤ 1} are convergent.
So, for each α ∈ Zn

+, (α : ν) ≤ 1 there exists a constant ã0α such that a0α(xm) ⇒ ã0α
when m → ∞. Then, dividing by q(xm)k+1µ(Wm) and choosing m0 sufficiently
large, for m > m0 we obtain the following inequality:∑

(β:ν)≤k+1

|ξβ | − ω6(ε)
∑

(γ:ν)≤k+1

|ξγ | ≤ κ
∣∣ ∑
(α:ν)≤1

ã0αξ
α
∣∣ ∑
(β:ν)≤k

|ξβ |,

where ω6(ε) → 0 when ε→ 0.



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 209

By appropriately choosing ε, we obtain that with some constant C3 > 0 it holds

C3

∑
(α:ν)≤k+1

|ξα| ≤
∣∣ ∑
(α:ν)≤1

ã0αξ
α
∣∣ ∑
(β:ν)≤k

|ξβ |.

From this inequality, using the [29, estimates (2.12)], we obtain that there exists a
constant δ > 0 such that ∣∣ ∑

(α:ν)≤1

ã0αξ
α
∣∣ ≥ δ(1 + |ξ|ν).

Since the last inequality holds for all the partial limits of sequences {a0α(xm) : (α :
ν) ≤ 1}, where |xm| → ∞ when m → ∞, we conclude the existence of constants
δ > 0 and M > 0 such that

|
∑

(α:ν)≤1

a0α(x)ξ
α| ≥ δ(1 + |ξ|ν),∀ξ ∈ Rn, |x| > M.

□

It turns out that the necessary conditions obtained on the symbol of operators
are also sufficient for fulfilling the a priori estimates (2.4) in the considered spaces.

Theorem 2.4. Let k ∈ Z+, q ∈ Qk,R, and P (x,D) be the differential form given
by (2.1) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)− a0α(y)| = 0 for

all α ∈ R. Assume P (x,D) is regular in Rn, and there exist constants δ > 0 and
M > 0 such that∣∣ ∑

α∈R
a0α(x)ξ

α
∣∣ ≥ δ(1 + |ξ|∂R), ∀ξ ∈ Rn, |x| > M. (2.20)

Then there exist constants κ > 0 and N > 0 such that

∥u∥k+1,R,p,q ≤ κ(∥Pu∥k,R,p,q + ∥u∥Lp(KN )),∀u ∈ Hk+1,R,p
q (Rn). (2.21)

Theorem 2.5. Let k ∈ Z+, q ∈ Q̃k,ν , and P (x,D) be the differential form given by
(2.2) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)−a0α(y)| = 0 for all

α ∈ Zn
+, (α : ν) ≤ 1. Assume P (x,D) is regular in Rn, and there exist constants

δ > 0 and M > 0 such that∣∣ ∑
(α:ν)≤1

a0α(x)ξ
α
∣∣ ≥ δ(1 + |ξ|ν), ∀ξ ∈ Rn, |x| > M. (2.22)

Then there exist constants κ > 0 and N > 0 such that

∥u∥k+1,ν,p,q ≤ κ(∥Pu∥k,ν,p,q + ∥u∥Lp(KN )), ∀u ∈ Hk+1,ν,p
q (Rn). (2.23)

Proof. We combine the proof of these two theorems for completely regular polyhe-

dron R. When necessary, a distinction between the weight functions from Q̃k,ν and
Qk,R and the corresponding spaces Hk+1,ν,p

q (Rn) and Hk+1,R,p
q (Rn) are provided.

Let m0 ∈ N. Using the a priori estimates for bounded domains from the work [23]
with some constants C1 > 0 and N1 > 0 we have

m0∑
m=1

∥φmu∥k+1,R,p,q ≤ C1

(
∥Pu∥k,R,p,q + ∥u∥Lp(KN1

)

)
, (2.24)

for all u ∈ Hk+1,R,p
q (Rn), where N1 is such that

⋃m0

i=1Wi ⊂ KN1
. We denote

Pm(x,D) :=
∑
α∈R

[
ψm(x)

(
a0α(x)q(x)

1−maxi(α:µ
i) − a0α(xm)q(xm)1−maxi(α:µ

i)
)



210 A. TUMANYAN EJDE-2022/25/CONF/26

+ a0α(xm)q(xm)1−maxi(α:µ
i)
]
Dα, m = 1, 2, . . . .

Taking into consideration that q ∈ Qk,R and the properties of {φm}∞m=1, we obtain
that for all γ ∈ kR and ε > 0 there exist δ(ε) > 0 and m0(ε) > 0 such that for all
m > m0 and maxj=1,...,l diamUj < δ the following inequality holds

|Dγφm(x)|
q(x)(γ:µi)

=
|Dγφm(x)|a|γ|

[m−1
l ]

q(x)(γ:µi)− |γ|
µmax q(x)

|γ|
µmax a

|γ|
[m−1

l ]

≤ τ1,γ(ε), (2.25)

where τ1,γ(ε) → 0 when ε→ 0. For q ∈ Q̃k,ν and (γ : ν) ≤ k, we have

|Dγφm(x)|
q(x)(γ:ν)

=
|Dγφm(x)|a|γ|

[m−1
l ]

q(x)
−(

|γ|
νmin

−(γ:ν))
q(x)

|γ|
νmin a

|γ|
[m−1

l ]

≤ τ2,γ(ε), (2.26)

where τ2,γ(ε) → 0 as ε→ 0.
The corresponding inequalities apply to the functions {ψm}∞m=1 in a similar

manner. Using these estimates, the conditions on the coefficients

lim
m→∞

max
x,y∈Wm

, |a0α(x)− a0α(y)| = 0

and [28, Lemma 3.1], analogously to the proof [17, Theorem 2.2], one can verify that
for a sufficiently large m0 and m > m0, the operators Pm(x,D) : Hk+1,R,p

q (Rn) →
Hk,R,p

q (Rn) have bounded inverse operators. Since (2.5) holds, they have uniformly
bounded norms, and with some C2 > 0 it holds

∥φmu∥k+1,R,p,q ≤ C2∥Pm(φmu)∥k,R,p,q, ∀u ∈ Hk+1,R,p
q (Rn).

Since Pm(φmu) = P0(φmu), for all u ∈ Hk+1,R,p
q (Rn) and m > m0, we obtain

∥φmu∥k+1,R,p,q ≤ C2∥Pm(φmu)∥k,R,p,q

≤ C2∥P0(φmu)∥k,R,p,q, ∀u ∈ Hk+1,R,p
q (Rn).

Using the properties of the functions {φm}∞m=1 and estimate (2.25) for q ∈ Qk,R

and (2.26) for q ∈ Q̃k,ν , it can be shown that for a sufficiently large m0 and a
small enough maxj=1,...,l diamUj for m > m0 with some constants C3, C4 > 0 the
following estimate holds:

∥φmP0u− P0(φmu)∥pk,R,p,q

≤ C3

∥∥ ∑
α∈R

∑
β+γ=α,|γ|>0

a0α(x)D
βuDγφmq(x)

1−maxi(α:µ
i)
∥∥p
k,R,p,q

≤ C4

∥∥ ∑
α∈R

∑
β+γ=α,|γ|>0

a0α(x)D
βuDγφm

1

q(x)mini(γ:µi)
q(x)1−maxi(β:µ

i)∥pk,R,p,q

≤ ω1(ε)∥u∥p
Hk+1,R,p

q (Wm)
,

where ω1(ε) → 0 when ε→ 0.
Summing up for allm > m0 and taking into account the property (ii) of {φm}∞m=1

and {Wm}∞m=1, with some constant C5 > 0 we obtain
∞∑

m=m0+1

∥φmu∥pk+1,R,p,q ≤ C5(∥P0u∥pk,R,p,q + ω1(ε)∥u∥pk+1,R,p,q), (2.27)



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 211

for all u ∈ Hk+1,R,p
q (Rn). Using the properties of the functions {φm}∞m=1, along

with (2.24) and (2.27), we establish that with some constant C6 > 0 the following
holds:

∥u∥k+1,R,p,q ≤
m0∑
m=1

∥φmu∥k+1,R,p,q +

∞∑
m=m0+1

∥φmu∥k+1,R,p,q

≤ C6

(
∥Pu∥k,R,p,q + ∥u∥Lp(KN1

) + ∥P0u∥k,R,p,q

+ ω2(ε)∥u∥k+1,R,p,q), ∀u ∈ Hk+1,R,p
q (Rn),

(2.28)

where ω2(ε) → 0 as ε→ 0.
We have P0(x,D) = P (x,D)− L(x,D). Then

∥P0u∥k,R,p,q ≤ ∥Pu∥k,R,p,q + ∥Lu∥k,R,p,q, ∀u ∈ Hk+1,R,p
q (Rn).

Considering the conditions Dβ(a1α(x)) = o(q(x)1−maxi (α−β:µi)) when |x| → ∞,
α ∈ R, β ∈ kR it can be verified that for a sufficiently large m0

∥Lu∥k,R,p,q ≤ ω3(ε)∥u∥k+1,R,p,q + C7∥u∥Hk+1,R,p(KN1
),∀u ∈ Hk+1,R,p

q (Rn),

where ω3(ε) → 0 when ε→ 0 and N1 is such that
⋃m0

i=1Wi ⊂ KN1 .
Similarly to (2.24), applying the a priori estimate from [23], with some constant

C8 > 0, we obtain

∥Lu∥k,R,p,q ≤ ω3(ε)∥u∥k+1,R,p,q + C8(∥Pu∥k,R,p,q + ∥u∥Lp(KN1
)).

Combining the last estimate with (2.28), we obtain

∥u∥k+1,R,p,q

≤ C6(∥Pu∥k,R,p,q + ∥u∥Lp(KN1
) + ∥P0u∥k,R,p,q + ω2(ε)∥u∥k+1,R,p,q)

≤ C6

(
∥Pu∥k,R,p,q + ∥u∥Lp(KN1

) + ω3(ε)∥u∥k+1,R,p,q

+ C8(2∥Pu∥k,R,p,q + ∥u∥Lp(KN1
)) + ω2(ε)∥u∥k+1,R,p,q), ∀u ∈ Hk+1,R,p

q (Rn).

Choosing m0 sufficiently large and maxj=1,...,l diamUj sufficiently small such that

C6(ω3(m0) + ω2(m0)) < 1/2,

then, with some constant C9 > 0, we obtain

∥u∥k+1,R,p,q ≤ C9

(
∥Pu∥k,R,p,q + ∥u∥Lp(KN1

)

)
, ∀u ∈ Hk+1,R,p

q (Rn).

□

Corollary 2.6. Let k ∈ N, q ∈ Q̃k,ν , and P (x,D) be the differential form given by
(2.2) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)−a0α(y)| = 0 for all

α ∈ Zn
+, (α : ν) ≤ 1. Assume P (x,D) is regular in Rn, and there exist constants

δ > 0 and M > 0 such that∣∣ ∑
(α:ν)≤1

a0α(x)ξ
α
∣∣ ≥ δ(1 + |ξ|ν),∀ξ ∈ Rn, |x| > M. (2.29)

Then, if u ∈ Hk,ν,p
q (Rn), P (x,D)u ∈ Hk,ν,p

q (Rn), then u ∈ Hk+1,ν,p
q (Rn).



212 A. TUMANYAN EJDE-2022/25/CONF/26

Proof. Applying Theorem 2.5, we obtain that with some constant C1 > 0 for u ∈
Hk,ν,p

q (Rn) the following estimate holds

∥u∥k,ν,p,q ≤ C1(∥Pu∥k−1,ν,p,q + ∥u∥Lp(KN )). (2.30)

Using arguments similar to those in [1, Theorem 15.1], one can check that there
exists a constant C2 > 0 such that∑

(α:ν)=k+1

∥Dαu∥Lp(Rn) ≤ C2

(
∥Pu∥k,ν,p,q + ∥u∥H1,ν,p(KN )

)
. (2.31)

Using the property 1 of q ∈ Q̃k,ν and estimate (2.30) with some constant C3 > 0,
we obtain∑

(α:ν)≤k

∥Dαu · qk+1−(α:ν)∥Lp(Rn) ≤ C3

∑
(α:ν)≤k

∥Dαu · qk−(α:ν)∥Lp(Rn)

≤ C4(∥Pu∥k−1,ν,p,q + ∥u∥Lp(KN )).

(2.32)

Taking into consideration that u ∈ Hk,ν,p
q (Rn) and the fact that Hk,ν,p

q (Rn) is

embedded in H1,ν,p(KN ), from (2.31) and (2.32), we obtain that u ∈ Hk+1,ν,p
q (Rn).

□

3. Fredholm criteria

Lemma 3.1. For k ∈ Z+ and q ∈ Qk,R, the space Hk+1,R,p
q (Rn) is compactly

embedded in Hk,R,p
q (Rn).

Proof. Since 1
q(x) ⇒ 0 as |x| → ∞, for each ε > 0 there exist N = N(ε) > 0

and ϕε ∈ C∞
0 (Rn) such that suppϕε ⊂ KN , 0 ≤ ϕε(x) ≤ 1 for all x ∈ Rn,

ϕε(x) = 1 for |x| ≤ N/2 and ϕε(x) = 0 for |x| ≥ N that with some constants
C1 = C1(ε) > 0, C2 = C2(ε) > 0 the following estimate holds

∥u∥k,R,p,q

=
∑

α∈kR

∥Dαu · qk−maxi(α:µ
i)∥Lp(Rn)

=
∑

α∈kR

∥Dαu((1− ϕε(x))
1

q(x)
q(x)k+1−maxi(α:µ

i) + ϕε(x)q(x)
k−maxi(α:µ

i))∥Lp(Rn)

≤ ε
∑

α∈(k+1)R

∥(1− ϕε)D
αu · qk+1−maxi(α:µ

i)∥Lp(Rn) + C1∥ϕεu∥Hk,R,p
q (KN )

≤ ε∥u∥k+1,R,p,q + C2∥ϕεu∥Ḣk,R,p(KN ).

Since Ḣk+1,R,p(KN ) is compactly embedded in Ḣk,R,p(KN ), applying the previous
estimate and [2, Proposition 10.8], we obtain that there exists a constant C3 =
C3(ε) > 0 such that

∥u∥k,R,p,q ≤ τ(ε)∥u∥k+1,R,p,q + C3∥u∥Lp(KN ), ∀u ∈ Hk+1,R,p
q (Rn), (3.1)

where τ(ε) → 0 as ε→ 0.
From (3.1) and [2, Proposition 10.8], we obtain that Hk+1,R,p

q (Rn) is compactly

embedded in Hk,R,p
q (Rn). □

Further, we use the following theorem (see [12, Theorem 3.14]).



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 213

Theorem 3.2. Let A be a bounded linear operator acting from a Banach space X
to a Banach space Y . Then the following holds

(1) if the operator A has a left regularizer, then kernel of operator A in X is
finite dimensional;

(2) if the operator A has a right regularizer, then the image of operator A is
closed in Y and the cokernel is finite dimensional;

(3) the operator A has left and right regularizers if and only if A is a Fredholm
operator.

Theorem 3.3. Let k ∈ Z+, q ∈ Qk,R, and P (x,D) be the differential operator given
by (2.1) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)− a0α(y)| = 0 for

all α ∈ R. Then, the operator P (x,D) : Hk+1,R,p
q (Rn) → Hk,R,p

q (Rn) is a Fredholm
operator if and only if P (x,D) is regular in Rn and there exist constants δ > 0 and
M > 0 such that

|
∑
α∈R

a0α(x)ξ
α| ≥ δ(1 + |ξ|∂R), ∀ξ ∈ Rn, |x| > M. (3.2)

Theorem 3.4. Let k ∈ Z+, q ∈ Q̃k,ν , and P (x,D) be the differential operator given
by (2.2) with the coefficients satisfying limm→∞ maxx,y∈Wm

|a0α(x)− a0α(y)| = 0 for

all α ∈ Zn
+, (α : ν) ≤ 1. Then, the operator P (x,D) : Hk+1,ν,p

q (Rn) → Hk,ν,p
q (Rn) is

a Fredholm operator if and only if P (x,D) is regular in Rn and there exist constants
δ > 0 and M > 0 such that∣∣ ∑

(α:ν)≤1

a0α(x)ξ
α
∣∣ ≥ δ(1 + |ξ|ν), ∀ξ ∈ Rn, |x| > M. (3.3)

Proof. Since a Fredholm operator is n-normal, the necessary part is a consequence
of Theorem 2.1 and Theorem 2.2 for Theorem 3.3, and Theorem 2.3 for Theorem
3.4.

Now, let us prove the sufficient part. We combine the proofs of these two
theorems for completely regular polyhedron R. When necessary, a distinction

between the weight functions from Q̃k,ν and Qk,R and the corresponding spaces
Hk+1,ν,p

q (Rn) and Hk+1,R,p
q (Rn) will be provided. Applying Theorem 2.4, we con-

clude that the operator P (x,D) : Hk+1,R,p
q (Rn) → Hk,R,p

q (Rn) is n-normal. It

remains to prove that the cokernel of the operator P (x,D) : Hk+1,R,p
q (Rn) →

Hk,R,p
q (Rn) is finite dimensional.
Let m0 ∈ N and xm ∈Wm,m = 1, 2 . . .. For m ≤ m0, we denote

Pm(x,D) :=
∑
α∈R

(ψm(x)(aα(x)− aα(xm)) + aα(xm))Dα,

Pm,0(x,D) :=
∑

α∈∂R

(ψm(x)(aα(x)− aα(xm)) + aα(xm))Dα,

Rm,0 := F−1 |ξ|∂R
(1 + |ξ|∂R)Pm,0(xm, ξ)

F.

Since P (x,D) is regular in Rn, for sufficiently small diameters of {Wm}m0
m=1, from

[27, Lemma 3.1], it follows that for m ≤ m0 the following representation holds:

Pm(x,D)Rm,0 = I + Tm
1 + Tm

2 , (3.4)



214 A. TUMANYAN EJDE-2022/25/CONF/26

where Tm
1 : Hk,R,p(Rn) → Hk+σ,R,p(Rn) with σ = σ(R) > 0 and the operator

Tm
2 : Hk,R,p(Rn) → Hk,R,p(Rn) satisfies ∥Tm

2 ∥ < 1. We denote

Rm := Rm,0(I + Tm
2 )−1.

Then

PmRm = I + Tm, (3.5)

where Tm : Hk,R(Rn) → Hk+σ,R(Rn) with some σ = σ(R) > 0.
For m > m0, we denote

Pm(x,D) :=
∑
α∈R

[
ψm(x)

(
a0α(x)q(x)

1−maxi(α:µ
i) − a0α(xm)q(xm)1−maxi(α:µ

i)
)

+ a0α(xm)q(xm)1−maxi(α:µ
i)
]
Dα.

Similarly as in the proof of Theorem 2.4, we can take m0 large enough such that
for m > m0 operators Pm : Hk+1,R,p

q (Rn) → Hk,R,p
q (Rn) have uniformly bounded

inverse operators Rm : Hk,R,p
q (Rn) → Hk+1,R,p

q (Rn). Consider

Rf :=

∞∑
l=0

ψlR
l(φlf), f ∈ Hk,R,p

q (Rn). (3.6)

Similarly to the proof of [27, Theorem 2.6], it can be checked that the following
representation holds

P (x,D)Ru = u+ ϕT1u+ T2u,

where ϕ ∈ C∞
0 (Rn), T1 : Hk,R,p(Rn) → Hk+σ,R,p(Rn) with σ = σ(R) > 0 and

operator T2 : Hk,R,p
q (Rn) → Hk,R,p

q (Rn) satisfies ∥T2∥ < 1.

For q ∈ Qk,R, by using Lemma 3.1 and [27, Theorem 2.6], it can be checked
that R : Hk,R,p

q (Rn) → Hk+1,R,p
q (Rn) is a right regularizer. Let us prove this

for the case q ∈ Q̃k,ν . Since ϕ ∈ C∞
0 (Rn), suppϕ ⊂ KN1

with some N1 > 0
and T1 : Hk,ν,p(Rn) → Hk+σ,ν,p(Rn) with σ = σ(ν) > 0, there exist constants
C1, C2, C3, C4 > 0 such that

∥ϕT1u∥k,ν,p,q ≤ C1∥ϕT1u∥Ḣk,ν,p(KN1
) ≤ C2∥ϕT1u∥Ḣk+σ,ν,p(KN1

)

≤ C3∥u∥Ḣk,ν,p(KN1
) ≤ C4∥u∥k,ν,p,q,∀u ∈ Hk,ν,p

q (Rn).

Consider the bounded sequence {un}∞n=1 ⊂ Hk,ν,p
q (Rn). Using the previous estimate

and the fact that Ḣk+σ,ν,p(KN1
) is compactly embedded in Ḣk,ν,p(KN1

), we have
a convergent subsequence of {ϕT1un}∞n=1. Thus, the operator ϕT1 : Hk,ν,p

q (Rn) →
Hk,ν,p

q (Rn) is compact.

Since the operator T2 : Hk,R,p
q (Rn) → Hk,R,p

q (Rn) satisfies ∥T2∥ < 1, there exists

(I + T2)
−1. Applying this operator to both sides, we obtain

P (x,D)R̃u = u+ T̃ u,

where R̃ := R(I + T2)
−1 and T̃ := ϕT1(I + T2)

−1 : Hk,R,p
q (Rn) → Hk,R,p

q (Rn)
is a compact operator since ϕT1 is a compact. Then, applying Theorem 3.2, we
conclude that the cokernel of the operator P (x,D) : Hk+1,R,p

q (Rn) → Hk,R,p
q (Rn) is

finite dimensional. Therefore, the operator P (x,D) : Hk+1,R,p
q (Rn) → Hk,R,p

q (Rn)
is a Fredholm operator. □



EJDE-2022/2025/CONF/26 NORMAL SOLVABILITY AND FREDHOLM PROPERTIES 215

4. Properties on the scales of weighted spaces

We denote

ker(P ;Hk,R,p
q ) := {u ∈ Hk+1,R,p

q (Rn) : P (x,D)u = 0},

Im(P ;Hk,R,p
q ) := {f ∈ Hk,R,p

q (Rn) : ∃u ∈ Hk+1,R,p
q (Rn) s.t. P (x,D)u = f},

coker(P ;Hk,R,p
q ) := Hk,R,p

q (Rn)/Im(P ;Hk,R,p
q ),

ind(P ;Hk,R,p
q ) := dimker(P ;Hk,R,p

q )− dim coker(P ;Hk,R,p
q ).

Corollary 4.1. Let k ∈ Z+, q ∈ Qk,R, and P (x,D) be the differential operator
(2.1). Assume (3.2) holds and the coefficients satisfy limm→∞ maxx,y∈Wm

|a0α(x)−
a0α(y)| = 0 for all α ∈ R. Then ker(P ;Hk,R,p

q ), coker(P ;Hk,R,p
q ), and ind(P ;Hk,R,p

q )
are independent of k and p.

Proof. The analogous construction (3.6) can be done for the left regularizer. Since
the left regularizer exists, using [28, Corollary 3.2], we obtain that for k1, k2 ∈ Z+

the following equalities hold: ker(P ;Hk1,R,p
q ) = ker(P ;Hk2,R,p

q ), coker(P ;Hk1,R,p
q ) =

coker(P ;Hk2,R,p
q ), ind(P ;Hk1,R,p

q ) = ind(P ;Hk2,R,p
q ). So, we establish the indepen-

dence from k. So, for k ∈ Z+ we have Ker(P ;Hk,R,p
q ) ⊂

⋂
s≥0H

s,R,p
q (Rn). Since

1
q(x) ⇒ 0 when |x| → ∞, it is easy to show that

⋂
s≥0H

s,R,p
q (Rn) ⊂ S. Thus, the

kernel is independent of k and p. Analogously, this is true for the kernel of adjoint
operator. Then, using [18, Theorem 3.1], we obtain the independence from k and p
for the cokernel. Therefore, the index of the operator is also independent of k and
p. □

Corollary 4.2. Let P (x,D) : Hk+1,R
q (Rn) → Hk,R

q (Rn) be an operator from The-
orem 3.3, considered as an unbounded operator in L2(Rn), and assume that (3.2)
holds. Then, one of the following statements holds:

• σ(P ) = C;
• σ(P ) is discrete and ind(P ;Hk,R

q ) = 0.

Proof. From Lemma 3.1 and Theorem 3.3, it follows that, for every λ ∈ C, the
operator P (x,D) − λI : Hk+1,R

q (Rn) → Hk,R
q (Rn) is a Fredholm operator. Then,

utilizing arguments similar to those in [25, Theorem 8.4] and taking into account
Lemma 3.1, we establish that one of the statements from the corollary is true. □

For the case q ∈ Q̃k,ν , the properties on the scale of spaces Hk,ν,p
q (Rn) can differ

from the previous class. Further, we consider the case q ≡ 1 and p = 2.

Corollary 4.3. Let q ≡ 1 and P (x,D) be the differential operator (2.2). Assume
that (3.3) holds and the coefficients satisfy limm→∞ maxx,y∈Wm

|a0α(x)−a0α(y)| = 0

for all (α : ν) ≤ 1. Then ker(P ;Hk,ν), coker(P ;Hk,ν), and ind(P ;Hk,ν) are
independent of k.

Proof. Using Corollary 2.6, it is easy to check that for k1, k2 ∈ Z+ the following
equality holds: ker(P ;Hk1,ν) = ker(P ;Hk2,ν). Then, from Corollary 2.6 and [30,
Lemma 2.1], we obtain a similar equality for the cokernels. Therefore, the kernels,
cokernels, and, consequently, the index of the operator are independent of k. □



216 A. TUMANYAN EJDE-2022/25/CONF/26

Corollary 4.4. Let q ≡ 1 and P (x,D) : Hk+1,ν(Rn) → Hk,ν(Rn) be an operator
from Theorem 3.4, considered as an unbounded operator in L2(Rn). Let there exist
constants ãα such that aα(x) ⇒ ãα when |x| → ∞, (α : ν) ≤ 1. Then

σes(P ) =
{ ∑

(α:ν)≤1

ãαξ
α : ξ ∈ Rn

}
.

Proof. From Theorem 3.4 and conditions on the coefficients, we obtain that oper-
ator P (x,D) − λI : Hk+1,ν(Rn) → Hk,ν(Rn) is a Fredholm operator if and only if
there exists a constant δ > 0 such that∣∣ ∑

(α:ν)≤1

ãαξ
α − λ

∣∣ ≥ δ(1 + |ξ|ν). (4.1)

It follows from (4.1) that σes(P ) =
{∑

(α:ν)≤1 ãαξ
α : ξ ∈ Rn

}
. □

Remark 4.5. Using condition (4.1) and [25, Proposition 8.1], it is easy to verify
that, for λ ̸∈ σes(P ), the index of the operator P (x,D) − λI : Hk+1,ν(Rn) →
Hk,ν(Rn) from Corollary 4.4 is 0, but the dimensions of the kernel and the cokernel
for such operators may differ from 0 (see examples from [9]).

Acknowledgements. This work was partially supported by the Thematic Fund-
ing of the Russian - Armenian University and the Ministry of Education and Science
of the Russian Federation, and by the Science Committee of the Republic of Arme-
nia, scientific project N25RG-1A205.

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Ani Tumanyan

Institute of Mathematics, Russian-Armenian University, Armenia
Email address: ani.tumanyan@rau.am


	1. Introduction, basic notions and definitions
	2. A priori estimates and normal solvability
	3. Fredholm criteria
	4. Properties on the scales of weighted spaces
	Acknowledgements

	References