EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1716-1737
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Boundedness of non regular pseudo-differential
operators and their adjoints on variable exponent

Besov-Morrey spaces

Mohamed Congo1,∗, Marie Françoise Ouedraogo1

1 Département de Mathematiques. UFR Sciences Exactes et Appliquées/ Université Joseph
KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Abstract. This paper deal with the boundedness property of non regular pseudo-differential oper-
ators a(x,D) and their adjoints a(x,D)∗ on variable exponent BM spaces. For this purpose, given
such an operator, we use the technique of decomposition of its symbol into elementary symbols
already used in other spaces.

2020 Mathematics Subject Classifications: 42B37, 46E30, 35S05

Key Words and Phrases: Pseudo-differential operators, adjoints, Non regular symbols, Elemen-
tary symbol, Variable exponent Besov-Morrey spaces.

1. Introduction

Besov-Morrey spaces denoted N s
p,u,q were initially investigated by Kozono and Ya-

mazaki in [8] to study the solutions of the Navier-Stokes equations with critical regularity.
The theory of Besov-Morrey spaces and their applications to non-linear PDEs were fur-
ther studied by Mazzucato [13]. They are modified Besov spaces where the base norm is
of Morrey-type. A first generalisation of the Besov-Morrey spaces N s

p,u,q into N s
p(·),u(·),q

where only the exponents p and u varied was introduced by Fu and Xu in [6] and a full

generalisation to variable exponent Besov-Morrey spaces denoted N s(·)
p(·),u(·),q(·) with all ex-

ponents variable is due to Almeida and Caetano [1].
Now the boundedness of an operator is a fundamental property for it’s use. One can find
in several works the study of boundedness of pseudo-differential operators: on Lebesgue
spaces, Besov spaces, Triebel-Lizorkin spaces and Sobolev spaces (see [2], [3], [11] and
[12]). In particular, the boundedness of pseudo-differential operators on Besov-Morrey
(BM) spaces with constant exponents denoted N s

p,u,q was studied by Mazzucato in [13].
We are concerned in this paper with the boundedness of pseudo-differential operators on

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4553

Email addresses: mohamed.congo@yahoo.fr (M. Congo), omfrancoise@yahoo.fr (M. F. Ouedraogo)

https://www.ejpam.com 1716 © 2022 EJPAM All rights reserved.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1717

Besov-Morrey spaces with variable exponents denoted N s(·)
p(·),u(·),q(·)(see [1]). Since the

symbol class Sm
1,δ is too restrictive for applications to non-linear equations, we use symbols

in the class Cℓ
∗S

m
1,δ where the x regularity is measured in Hölder-Zygmund spaces. The

results of this paper generalize those of [13] and complement the studies done on Triebel-
Lizorkin-Morrey spaces (see [4]). We further extended the study of the boundedness of
such pseudo-differential operators to their adjoints.

Our approach is as follows: we consider pseudo-differential operators in N s(·)
p(·),u(·),q(·) whose

symbols belong to the class Cℓ
∗S

m
1,δ. We use the decomposition of these symbols into ele-

mentary symbols following the method of [2], [11] and [13]. We then set up intermediary
results useful to prove the main results in theorem [2] and theorem [3].
This paper is structured in 4 sections: the section 2 concerns preliminaries and set up
notations as well as definitions and properties of Morrey spaces and Besov-Morrey spaces
with variable smoothness and integrability. In section 3, we recall tools that are nec-
essary to establish lemmas and the main theorems of the next section. The section 4
contains the results and the proof of the main theorem of the boundedness of non regular

pseudo-differential operators in the space N s(·)
p(·),u(·),q(·) as well as the adjoint estimate.

2. Preliminaries

2.1. General Notation

We denote by Rn the n-dimensional real Euclidean space, N the collection of all natural
numbers and N0 = N ∪ {0}. We write B(x, r) for the open ball in Rn centered at x ∈ Rn

with radius r > 0. We use c as a generic positive constant, i.e. a constant whose value
may change with each appearance. If ξ belongs to Rn and r to R, the expression |ξ| ∼ r
means that there exists two constants c1, c2 > 0 such that c1r ≤ |ξ| ≤ c2r. The expression
f ≲ g means that f ≤ cg for some independent constant c, and f ≈ g means f ≲ g ≲ f .
Throughout the paper we denote by M(Rn) the family of all complex or extended real-
valued measurable functions on Rn.
By suppf we denote the support of the function f , i.e. the closure of its non-zero set. If
E ⊂ Rn is a measurable set, then χE denotes its characteristic function. We denote by
S = S(Rn) the set of all Schwartz functions on Rn. We denote by S ′ = S ′(Rn) the dual
space of all tempered distributions on Rn. The Fourier transform denoted Ff(ξ) or f̂ is
defined on S by

f̂(ξ) :=

∫
Rn

e−ix·ξf(x)dx

and extended to S ′ by duality. The inverse Fourier transform denoted F−1f(x) or f̌ is
defined by

f̌(x) :=
1

(2π)n

∫
Rn

eix·ξf(ξ)dξ.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1718

For two complex or extended real-valued measurable functions f, g on Rn the convolution
f ∗ g is given, in the usual way, by

(f ∗ g)(x) :=
∫
Rn

f(x− y)g(y)dy, x ∈ Rn and supp(f ∗ g) ⊂ suppf + suppg.

2.2. Variable exponents

In this sub-section we recall the definition and some properties of variable exponents.
For more details see [10] and [5].
• We denote by P(Rn) the set of all measurable functions p : Rn → (0,∞] (called variable
exponents) which are essentially bounded away from zero. We denote p+Rn := ess supRnp(x)
and p−Rn := ess infRnp(x); we abbreviate p+ = p+Rn and p− = p−Rn .
• The function ϕp is defined as follows:

ϕp(x)(t) =


tp(x) if p(x) ∈ (0,∞),
0 if p(x) = ∞ and t ∈ [0, 1],
∞ if p(x) = ∞ and t ∈ (1,∞].

The variable exponent modular associated to p(·) is defined by

ϱp(·)(f) :=

∫
Rn

ϕp(x)(|f(x)|)dx.

The variable exponent Lebesgue space Lp(·) := Lp(·)(Rn) is the family of (equivalence
classes of) functions f ∈ M(Rn) such that ϱp(·)(f/λ) is finite for some λ > 0.
Lp(·) is a quasi-Banach space equipped with the quasinorm

∥f∥p(·) := inf

{
µ > 0 : ϱp(·)

(
1

µ
f

)
≤ 1

}
.

• We say that a continuous function g : Rn → R is locally log-Hölder continuous, abbre-
viated g ∈ C log

loc (R
n), if there exists clog(g) ≥ 0 such that

|g(x)− g(y)| ≤
clog(g)

log(e + 1/|x− y|)
for all x, y ∈ Rn. (1)

The function g : Rn → R is said to be globally log-Hölder continuous, abbreviated
g ∈ C log(Rn), if it is locally log-Hölder continuous and there exists g∞ ∈ R and c∞(g) ≥ 0
such that

|g(x)− g∞| ≤ c∞(g)

log(e + |x|)
for all x ∈ Rn.

We define the following class of variable exponents

P log(Rn) :=

{
p ∈ P :

1

p
∈ C log(Rn)

}
.

We define
1

p∞
:= lim

|x|→∞

1

p(x)
and we use the convention

1

∞
= 0.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1719

2.3. Variable exponent Besov-Morrey spaces

We recall the definition of variable exponent Besov-Morrey spaces. We refer to the
papers [1], [18], [17] and [8], for further results on these spaces.

Definition 1. For p, u ∈ P(Rn) with 0 < p− ≤ p(x) ≤ u(x) ≤ ∞, the variable exponent
Morrey space Mp(·),u(·) := Mp(·),u(·)(Rn) consists of all functions f ∈ M(Rn) with finite
quasinorm

∥f∥Mp(·),u(·)
:= sup

x∈Rn, r>0
r

n
u(x)

− n
p(x)

∥∥fχB(x.r)

∥∥
Lp(·)

. (2)

Definition 2. Let p, q, u ∈ P(Rn) with p(x) ≤ u(x). Given a sequence (fν)ν ⊂ M(Rn),
we set

ϱℓq(·)(Mp(·),u(·)) ((fν)ν) :=
∑
ν≥0

sup
x∈Rn, r>0

inf
{
λ > 0 : ϱp(·)

(
r

n
u(x)

− n
p(x) fνχB(x.r)/λ

1
q(·)
)
≤ 1
}
.

(3)

Remark 1. When q+ <∞ or q+ = ∞ and p(x) ≥ q(x) we can simplify (3) to obtain

ϱℓq(·)(Mp(·),u(·)) ((fν)ν) :=
∑
ν≥0

sup
x∈Rn, r>0

∥∥∥ϕq(·) (r n
u(x)

− n
p(x) |fν |χB(x.r)

)∥∥∥
L p(·)

q(·)

Definition 3. Let p, q, u ∈ P(Rn) with p(x) ≤ u(x).The mixed Morrey-sequence space
ℓq(·)(Mp(·),u(·)) consists of all sequences (fν)ν ⊂ M(Rn) such that, ϱℓq(·)(Mp(·),u(·)) (µ(fν)) <

∞ for some µ > 0. For (fν)ν ∈ ℓq(·)(Mp(·),u(·)) we define

∥(fν)ν∥ℓq(·)(Mp(·),u(·))
:= inf

{
µ > 0 : ϱℓq(·)(Mp(·),u(·))

(
1

µ
(fν)

)
≤ 1

}
. (4)

Proposition 1. Let p, q, u ∈ P(Rn) with p(x) ≤ u(x). Let (fν)ν ∈ ℓq(·)(Mp(·),u(·))

(i) The functional ∥·∥ℓq(·)(Mp(·),u(·))
is a quasinorm in ℓq(·)(Mp(·),u(·)) and

∥(fν)ν∥tℓq(·)(Mp(·),u(·))
=
∥∥(|fν |t)ν∥∥ℓq(·)/t(Mp(·)/t,u(·)/t)

, ∀t > 0.

(ii) If fν0 = f for some f ∈Mp(·),u(·)(Rn) and ν0 ∈ N0, and fν = 0 for all ν ̸= ν0, then

∥(fν)ν∥ℓq(·)(Mp(·),u(·))
= ∥f∥Mp(·),u(·)

.

Theorem 1. The functional (4) defines a quasinorm in the vector space ℓq(·)(Mp(·),u(·))
for any p, q, u ∈ P(Rn) with p(x) ≤ u(x). Moreover, it induces a norm in the following
cases (each one understood for almost every x ∈ Rn):

(i) p(x) ≥ 1 and q ∈ [1,∞] is constant;

(ii) 1 ≤ q(x) ≤ p(x) ≤ u(x) ≤ ∞;



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1720

(iii) 1
p(x) +

1
q(x) ≤ 1.

Besov-Morrey spaces: To define Besov spaces based on Mp(·),u(·), let us first recall
the définition of a Littlewood-Paley partition of unity {φν}, ν ≥ 0.
The functions φν are defined as follows. Let φ0 ∈ C∞

0 (Rn) a real function such that φ0 ≡ 1
on B(0; 1) and suppφ0 ⊂ B(0; 2).
Set

φν(ξ) = φ0(2
−νξ)− φ0(2

−ν+1ξ) for all ν ∈ N.

Then φν is supported on the dyadic shell

Dν =
{
ξ ∈ Rn : 2ν−1 ≤ |ξ| ≤ 2ν+1

}
with Dν ∩Dµ = ∅ if |ν − µ| > 1. One has∑

ν≥0

φν = 1.

Then for all f ∈ S ′,

f =
∑
ν≥0

φνf.

The Littlewood-Paley partition of unity is used to define the Fourier multiplier φj(D) as
followed

φν(D)f(x) = F−1(φν · f̂)(x) =
∫
Rn

φν(ξ)f̂(ξ)e
ix·ξdξ.

Definition 4. Let {φν} be the Littlewood-Paley partition of unity. Let s ∈ C log
loc and

p, q, u ∈ P(Rn) such that 0 < p− ≤ p(x) ≤ u(x) ≤ ∞. The Besov-Morrey spaces

N s(·)
p(·),u(·),q(·) consists of all distributions f ∈ S ′(Rn) such that

∥f∥N s(·)
p(·),u(·),q(·)

:= ∥φ0(D)f∥Mp(·),u(·)
+

∥∥∥∥(2νs(·)φν(D)f
)
ν≥1

∥∥∥∥
ℓq(·)(Mp(·),u(·))

<∞. (5)

Remark 2.

(i) Let us notice that Besov-Morrey spaces N s(·)
p(·),u(·),q(·) are defined by the composite

ℓq(·)(Mp(·),u(·)) while Triebel-Lizorkin-Morrey spaces Es(·)
p(·),u(·),q(·) are defined byMp(·),u(·)

(
ℓq(·)

)
.

(ii) The case of the boundedness of non-regular PDOs on variable exponent Triebel-
Lizorkin-Morrey spaces has been studied in [4].

Proposition 2. Let s ∈ C log
loc , p ∈ P log(Rn) and q, u ∈ P(Rn) with p(x) ≤ u(x) and 1/q

locally log-Hölder continuous. Then it holds

S ↪→ N s(·)
p(·),u(·),q(·) ↪→ S ′.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1721

3. Basic tools

In the following, we present some results which will be useful in the last section.
First of all, we recall the η-functions defined on Rn by

ην,m(x) = 2nν (1 + 2ν |x|)−m , ν ∈ N0, m > 0.

Note that ην,m ∈ L1 for m > n and the corresponding L1-norm does not depend on ν.

The next lemmas can be found in [7](Lemma 19) and [9](Lemma 6.1.).

Lemma 1. Let α ∈ C log
loc (R

n) and let m ≥ 0, l ≥ clog(α), where clog is the constant from
(1) for α. Then

2να(x)ην,m+l(x− y) ≤ c2να(y)ην,m(x− y)

with c > 0 independent of x, y ∈ Rn and ν ∈ N0.

Lemma 2. Let t > 0, ν ∈ N0 and m > n. Then there exist c = c(t,m, n) such that for all
g ∈ S ′(Rn) with suppFg ⊂

{
ξ ∈ Rn : |ξ| ≤ 2ν+1

}
,

|g(x)| ≤ c
(
ην,m ∗ |g|t(x)

)1/t
, x ∈ Rn.

Lemma 3. Let p ∈ P log(Rn) and u, q ∈ P such that 1
q ∈ C log

loc with

1 ≤ p− ≤ p(x) ≤ u(x) ≤ ∞. Ifm > n+clog(1/q)+nmax
{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
,

then there exist c > 0 such that∥∥(ην,m ∗ fν)ν
∥∥
ℓq(·)(Mp(·),u(·))

≤ c ∥(fν)ν∥ℓq(·)(Mp(·),u(·))

for all sequences (fν)ν ⊂ ℓq(·)(Mp(·),u(·)).

Lemma 4. Let p ∈ P log(Rn) and u ∈ P with 1 ≤ p− ≤ p(x) ≤ u(x) ≤ ∞.

If m > n+ nmax
{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
.

Then there exists c > 0 such that

∥ην,m ∗ f∥Mp(·),u(·)
≤ ∥f∥Mp(·),u(·)

.

Lemma 5. Let p, u, q ∈ P(Rn) with p(x) ≤ u(x). Let δ > 0. For any sequence (gj)j∈N0

of non-negative measurable functions on Rn, let

Gν(x) :=
∞∑
j=0

2−|ν−j|δgj(x), x ∈ Rn, ν ∈ N0.

Then it holds
∥(Gν)ν∥ℓq(·)(Mp(·),u(·))

≲
∥∥∥(gj)j∥∥∥ℓq(·)(Mp(·),u(·))

.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1722

4. Boundedness of pseudo-differential operators

We will use symbols for which x-regularity is measured in Hölder-Zygmund spaces.

Definition 5. The function a(x, ξ) on Rn × Rn belongs to the symbol class Cℓ
∗S

m
1,δ, δ ∈

[0, 1], ℓ > 0 if it is smooth in ξ and satisfies the following estimates:
∥∥∥∂αξ a(·, ξ)∥∥∥

Cℓ
∗
≤ cα ⟨ξ⟩m−|α|+ℓδ∣∣∣∂αξ a(x, ξ)∣∣∣ ≤ c′α ⟨ξ⟩

m−|α|
(6)

where ⟨ξ⟩ stands for
(
1 + |ξ|2

)1/2
. A pseudo-differential operator on S with symbol

a ∈ Cℓ
∗S

m
1,δ is defined by

a(x,D)f(x) =
1

(2π)n

∫
Rn

eix·ξa(x, ξ)Ff(ξ)dξ , f ∈ S.

We write a(x,D) ∈ Cℓ
∗OPS

m
1,δ if a(x, ξ) belongs to the class Cℓ

∗S
m
1,δ.

To study the boundedness of a(x,D), we will resolve its symbol a into elementary
symbols. Therefore, the operator a(x,D) with symbol a can be resolved into ”elementary
operators” ak(x,D) with symbols ak. This idea has been exploited to establish bound-
edness of pseudo-differential operators with non-regular symbols in Sobolev spaces Hs,p

and Hölder-Zygmund spaces Cℓ
∗ (see [11], [2]). We will proceed in the same way with the

adjoint operator.

Definition 6. [13] We call elementary symbol in the class Cℓ
∗S

m
1,δ, δ ∈ [0, 1], ℓ > 0 an

expression of the form

a(x, ξ) =
∑
j≥0

σj(x)φj(ξ)

where φ0 is smooth supported on the ball B(0, 2), φj(ξ) = φ(2−jξ) and φ ∈ C∞
0 is sup-

ported on the dyadic shell D0 = {ξ ∈ Rn| 1/2 ≤ |ξ| ≤ 2}, while σj is a uniformly bounded
sequence such that

∥σj∥Cℓ
∗S

m
1,δ

≤ c2j(m+ℓδ).

Example 1. Let {φj} be a Littlewood-Paley partition of unity and {σj} a sequence uni-
formly bounded in Cℓ

∗.
Set

a(x,D)f(x) =
∑
j≥0

σj(2
jδx)φj(D)f(x) , f ∈ S

then a(x,D) ∈ Cℓ
∗OPS

0
1,δ .

Lemma 6. [13] Let f =
∑

j≥0 fj in S ′, with suppf̂j ⊂ B(0, A2j) for some A > 0. Then,
for ℓ > 0,

∥f∥Cℓ
∗
≤ c(A) sup

j≥0

{
2jℓ ∥fj∥L∞

}
. (7)



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1723

Let us establish the two following lemmas which play a fundamental role in the proof

of the boundedness of pseudo-differential operators on N s(·)
p(·),u(·),q(·).

Lemma 7. Let c1, c2 > 0, p ∈ P log(Rn), u, q ∈ P such that , 1q ∈ C log
loc with

1 ≤ p− ≤ p(x) ≤ u(x) ≤ ∞. Let {fk}k∈N0
be a sequence of tempered distributions such

that
suppFf0 ⊂ B(0, 2c2)

and
suppFfk ⊂

{
ξ ∈ Rn : c12

k−1 ≤ |ξ| ≤ c22
k+1
}

for k > 0.

Then ∥∥∥∥∥
∞∑
k=0

fk

∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

.

Proof. Using (5) we have∥∥∥∥∥
∞∑
k=0

fk

∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
{
2js(·)φj(D)

( ∞∑
k=1

fk

)}
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Since φj is supported on the dyadic shell Dj , while φ0 is supported on the ball B(0; 2),
there are N1, N2 ∈ N0 such that

∥∥∥∥∥
∞∑
k=0

fk

∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥φ0(D)

(
N1∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
2js(·)φj(D)

 j+N2∑
k=j−N1

fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

=

∥∥∥∥∥
N1∑
k=0

φ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

φ̌j ∗ fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Let us now estimate these two terms.

• Estimation of the term

∥∥∥∥∥
N1∑
k=0

φ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

One has
suppF (φ̌0 ∗ fk) ⊂ {ξ ∈ Rn : |ξ| ≤ 2} .

Then by lemma 2, |φ̌0 ∗ fk| ≲ |fk| . It follows that∥∥∥∥∥
N1∑
k=0

φ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

≲
N1∑
k=0

∥fk∥Mp(·),u(·)



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1724

=

N1∑
k=0

∥(0, . . . , fk, 0, . . .)∥ℓq(·)(Mp(·),u(·))
by proposition 1

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

.

• Estimation of the term

∥∥∥∥∥∥
2js(·)

j+N∑
k=j−N

φ̌j ∗ fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Since φ̌j ∗ fk ∈ S ′ and suppF (φ̌j ∗ fk) ⊂
{
ξ ∈ Rn : |ξ| ≤ 2j+1

}
,

then by lemma 2,

|φ̌j ∗ fk| ≲
(
ηj,m ∗ |fk|t

)1/t
for any m > n+ clog(s) + clog(

1
q ) + nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
and any t > 0.

Thus with t = 1,∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

φ̌j ∗ fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥


j+N2∑
k=j−N1

2js(·) (ηj,m ∗ |fk|)


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

By lemma 1, we can move 2js(·) inside the convolution and get

2js(·) (ηj,m ∗ |fk|) ≲ ηj,m−clog(s) ∗ 2
js(·)|fk|.

Let us notice that if m > n+clog(s)+clog(
1
q )+nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
then m− clog(s) verifies the hypothesis of the lemma 3. Therefore∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

φ̌j ∗ fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥


j+N2∑
k=j−N1

(
ηj,m−clog(s) ∗ 2

js(·)|fk|
)

j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≤
j+N2∑

k=j−N1

∥∥∥∥{(ηj,m−clog(s) ∗ 2
js(·)|fk|

)}
j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

By lemma 3 ,∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

φ̌j ∗ fk


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
N1+N2∑
k=0

∥∥∥∥(2js(·)fj+k−N1

)
j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

.

The two estimations yield the desired estimate. □



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1725

Lemma 8. Let c > 0, p ∈ P log(Rn) and u, q ∈ P such that 1
q ∈ C log

loc with 1 ≤ p− ≤
p(x) ≤ u(x) ≤ ∞. Let s ∈ C log

loc such that s− > 0. Let {fk}k∈N0
be a sequence of tempered

distributions such that

suppFfk ⊂
{
ξ ∈ Rn : |ξ| ≤ c2k+1

}
.

Then ∥∥∥∥∥
∞∑
k=0

fk

∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(Mp(·),u(·))

.

Proof. Using the hypothesis on Suppφj , there is N ∈ N0 such that∥∥∥∥∥
∞∑
k=0

fk

∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
2js(·)φj(D)

 ∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

(8)

(i) Let us first estimate the term

∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

=

∥∥∥∥∥
N∑
k=0

φ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

.

Since
suppF

(
ψ̌0 ∗ fk

)
⊂
{
ξ ∈ Rn : |ξ| ≤ 2k+1

}
,

By lemma 2 ∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

≲

∥∥∥∥∥
∞∑
k=0

ηk,m ∗ |fk|

∥∥∥∥∥
Mp(·),u(·)

,

for any m > n+ clog(s) + clog(
1
q ) + nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
.

Then∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

≲

∥∥∥∥∥
∞∑
k=0

2−ks−
(
ηk,m−clog(s) ∗ 2

ks(·)|fk|
)∥∥∥∥∥

Mp(·),u(·)

by lemma 1.

Using proposition 1, we have∥∥∥∥∥
∞∑
k=0

2−ks−
(
ηk,m−clog(s) ∗ 2

ks(·)|fk|
)∥∥∥∥∥

Mp(·),u(·)

=

∥∥∥∥∥∥
{ ∞∑

k=0

2−ks−
(
ηk,m−clog(s) ∗ 2

ks(·)fk

)}
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
∥∥∥(ηk,m−clog(s) ∗ 2

ks(·)fk

)
k

∥∥∥
ℓq(·)(Mp(·),u(·))

by lemma 5

Thus, by lemma 2, it follows that∥∥∥∥∥φ0(D)

( ∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1726

(ii) Now let us estimate the term

∥∥∥∥∥∥
2js(·)φj(D)

 ∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

∥∥∥∥∥∥
2js(·)φj(D)

 ∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

=

∥∥∥∥∥∥


∞∑
k=j−N

2js(·) (φ̌j ∗ fk)


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Since {
suppF (φ̌j ∗ fk) ⊂

{
ξ ∈ Rn : |ξ| ≤ 2j+1

}
suppF (φ̌j ∗ fk) ⊂

{
ξ ∈ Rn : |ξ| ≤ 2k+1

}
,

by lemma 2 {
2js(·) (φ̌j ∗ fk) ≲ 2js(·) (ηj,m ∗ |fk|)
2js(·) (φ̌j ∗ fk) ≲ 2js(·) (ηk,m ∗ |fk|)

for m > n+ clog(1/q) + clog(s) + nmax
{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
.

Therefore∥∥∥∥∥∥2js(·)φj(D)

 ∞∑
k=j−N

fk

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥


j∑
k=j−N

2js(·) (ηj,m ∗ |fk|)


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

+

∥∥∥∥∥∥


∞∑
k=j+1

2−(k−j)s(·)2ks(·) (ηk,m ∗ |fk|)


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

We are left now to estimate each terms on the right-hand side.
Using lemma 1 we can move 2νs(·) inside the convolution 2νs(·) (ην,m ∗ |fk|) and get

2νs(·) (ην,m ∗ |fk|) ≲
(
ην,m0 ∗ 2νs(·)|fk|

)
, ν = j or k where m0 = m− clog(s).

Thus

•

∥∥∥∥∥∥


j∑
k=j−N

2js(·) (ηj,m ∗ |fk|)


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
0∑

ℓ=−N

∥∥∥∥{(ηj,m0 ∗ 2js(·)|fj+ℓ|
)}

j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥(2js(·)fj)j
∥∥∥∥
ℓq(·)(Mp(·),u(·))

by lemma 3.

Also

•

∥∥∥∥∥∥


∞∑
k=j+1

2−(k−j)s(·)2ks(·) (ηk,m ∗ |fk|)


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1727

≲

∥∥∥∥∥∥


∞∑
k=j+1

2−|j−k|s(·)
(
ηk,m0 ∗ 2ks(·)|fk|

)
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥
{ ∞∑

k=0

2−|j−k|s−
(
ηk,m0 ∗ 2ks(·)|fk|

)}
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
∥∥∥(ηk,m0 ∗ 2ks(·)|fk|

)
k

∥∥∥
ℓq(·)(Mp(·),u(·))

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

by lemma 5 and 3.

. □

4.1. The main estimate

In this subsection, we will establish the boundedness of pseudo-differential operators

on N s(·)
p(·),u(·),q(·).

Theorem 2. Let a(x, ξ) ∈ Cℓ
∗S

m
1,δ where m ∈ R, δ ∈ [0, 1] and ℓ > 0. Let p ∈ P log(Rn)

and u, q ∈ P such that 1
q ∈ C log

loc with 1 ≤ p− ≤ p(x) ≤ u(x) ≤ ∞. Let s ∈ C log
loc such that

0 < s− ≤ s(x) < ℓ. Then

a(x,D) : N s(·)+m
p(·),u(·),q(·) −→ N s(·)

p(·),u(·),q(·)

is bounded.

Remark 3. The operator (1 −∆)
m
2 , m ∈ R is an isomorphism that composes well with

pseudo-differential operators (see[13] and [15]). Thus, it is enough to treat the case m = 0.
Therefore let us set a(x, ξ) ∈ Cℓ

∗S
0
1,δ.

The symbol reduction method due to Coifman and Meyer[3], makes it possible to be limited
to symbols of the form(see [13], [11], [2] and [16])

a(x, ξ) =
∑
j≥0

σj(x)φj(ξ)

where σj satisfies
∥σj∥Cℓ

∗
≤ c2jℓδ (9)

and ∥σj∥L∞ ≤ c (10)

with c depending on δ and ℓ but not on j and φj is exactly a Littlewood-Paley function.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1728

Proof. Let

a(x, ξ) =
∑
j≥0

σj(x)φj(ξ)

with the conditions given above. Let us decompose this symbol into three parts.

First of all we have σj(x) =
∞∑
k=0

φk(D)σj(x).

By multiplying each member by φj(ξ), we obtain σj(x)φj(ξ) =

( ∞∑
k=0

φk(D)σj(x)

)
φj(ξ).

and then a(x, ξ) =

∞∑
j=0

( ∞∑
k=0

φk(D)σj(x)

)
φj(ξ).

By setting akj = φk(D)σj we have

a(x, ξ) =
∞∑
j=0

( ∞∑
k=0

akj

)
φj(ξ). (11)

Now we rewrite (11) as a sum of three parts

a(x, ξ) =
∑
j≥0

j−4∑
k=0

akj(x) +

j+3∑
k=j−3

akj(x) +
∞∑

k=j+4

akj(x)

φj(ξ)

= a1(x, ξ) + a2(x, ξ) + a3(x, ξ)

(i) Let φj(D)f = fj . Then we define three ” elementary” pseudo-differential operators:

a1(x,D)f =

∞∑
j=0

(
j−4∑
k=0

akjfj

)
,

a2(x,D)f =
∞∑
j=0

 j+3∑
k=j−3

akjfj

,
a3(x,D)f =

∞∑
j=0

 ∞∑
k=j+4

akjfj

.
(ii) It remains to estimate each of these three pseudo-differential operators.

For this purpose, it’s necessary to estimate ∥akj∥L∞
.

Let us recall the quasinorm of Cℓ
∗: ∥φk(D)σj∥Cℓ

∗
= supk 2

kℓ ∥φk(D)σj∥L∞ .
Since

∥φk(D)σj∥Cℓ
∗
≤ c ∥σj∥Cℓ

∗
.

Then
sup
k

2kℓ ∥φk(D)σj∥L∞
≤ c ∥σj∥Cℓ

∗
.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1729

Using (9), we obtain
∥akj∥L∞ ≤ c2jℓδ2−kℓ. (12)

We are ready to estimate the pseudo-differential operators a1(x,D), a2(x,D) and a3(x,D).
• The estimation of a1(x,D) .
We have

F

(
j−4∑
k=0

akjfj

)
=

j−4∑
k=0

F (φk(D)σj) ∗ F (φj(D)f)

=

j−4∑
k=0

(ψkFσj) ∗ (ψjFf) .

Using the fact that supp(f ∗ g) ⊂ suppf+suppg for all compactly supported distributions
f, g ∈ S ′, we have

suppF

(
j−4∑
k=0

akjfj

)
⊂
{
ξ ∈ Rn : |ξ| ∼ 2j+1

}
.

Then lemma 7 yields

∥a1(x,D)f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

(
j−4∑
k=0

akjfj

)∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
2js(·)

j−4∑
k=0

akjfj

)
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

 sup
j∈N0

max{j−4,0}∑
k=0

∥akj∥L∞

∥∥∥∥(2js(·)φj(D)f
)
j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥(2js(·)φj(D)f
)
j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

It follows that
∥a1(x,D)f∥N s(·)

p(·),u(·),q(·)
≲ ∥f∥N s(·)

p(·),u(·),q(·)
.

• Estimation of a2(x,D)

For the second part ∥a2(x,D)f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

 j+3∑
k=j−3

akjfj

∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

,

Let us first observe that

F

 j+3∑
k=j−3

akjfj

 =

j+3∑
k=j−3

F (φk(D)σj) ∗ F (φj(D)f)



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1730

=

j+3∑
k=j−3

(φkFσj) ∗ (φjFf) .

Therefore F

 j+3∑
k=j−3

akjfj

 is supported on the ball B(0, 2j+4). By lemma 8,

∥a2(x,D)f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
2js(·)

j+3∑
k=j−3

akjfj


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≤ 2−m

∥∥∥∥∥∥
 j+3∑

k=j−3

∥akj∥L∞
2js(·)φj(D)f


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Now using (12) one has

j+3∑
k=j−3

∥akj∥L∞ ≲
3∑

k=−3

2−kℓ <∞ (with δ = 1)

and then

∥a2(x,D)f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥(2js(·)φj(D)f
)
j

∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲ ∥f∥N s(·)
p(·),u(·),q(·)

• Estimation of a3(x,D)

Since F

 ∞∑
k=j+4

akjfj

 is not supported on any ball or shell, we cannot directly use neither

lemma 7 nor lemma 8. However, in S ′ we can write

∞∑
j=0

∞∑
k=j+4

akjfj =
∞∑
k=4

k−4∑
j=0

akjfj .

We have

F

k−4∑
j=0

akjfj

 =

k−4∑
j=0

(ψkFaj) ∗ (ψjFf)

then

suppF

k−4∑
j=0

akjfj

 ⊂
{
ξ ∈ Rn : |ξ| ∼ 2k+1

}
.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1731

Then lemma 7 yields

∥a3(x,D)f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
k=4

k−4∑
j=0

akjfj

∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

≲

∥∥∥∥∥∥
2ks(·)

k−4∑
j=0

akjfj


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥
k−4∑

j=0

∥akj∥L∞ 2ks(·)φj(D)f


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

If we use (12) with δ = 1, we have

∥a3(x,D)f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
k−4∑

j=0

2jℓ2−kℓ2ks(·)φj(D)f


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

=

∥∥∥∥∥∥
k−4∑

j=0

2(k−j)(s(·)−ℓ)2js(·)φj(D)f


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≤

∥∥∥∥∥∥
k−4∑

j=0

2−|k−j||s−−ℓ|2js(·)φj(D)f


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≤

∥∥∥∥∥∥
 ∞∑

j=0

2−|k−j||s−−ℓ|2js(·)φj(D)f


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

By hypothesis we have |s− − ℓ| > 0. Therefore, by lemma 5∥∥∥∥∥∥
k−4∑

j=0

2−|k−j||s−−ℓ|2js(·)φj(D)f


j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲
∥∥∥(2js(·)φj(D)f

)
k

∥∥∥
ℓq(·)(Mp(·),u(·))

.

Then
∥a3(x,D)f∥N s(·)

p(·),u(·),q(·)
≲ ∥f∥N s(·)

p(·),u(·),q(·)
.

The proof is completed. □

4.2. Thet adjoint operator estimate

In this subsection, let us go further by studying the boundedness of the adjoint oper-
ator. Let the adjoint operator A∗ of the operator A defined by∫

(Af)gdx =

∫
fA∗gdx f, g ∈ S. (13)



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1732

Since an operator a(x,D) with symbol a(x, ξ) ∈ Cℓ
∗S

0
1,δ can be decomposed in the form

a(x,D) = a1(x,D) + a2(x,D) + a3(x,D),

then its adjoint a(x,D)∗ can be written as follow

a(x,D)∗ = a1(x,D)∗ + a2(x,D)∗ + a3(x,D)∗.

This method has already been used by Marschall in [11] and [12].

Let’s calculate aλ(x,D)∗ for λ = 1, 2, 3. For that, let aλ(x,D)f =
∑
j∈Iλ

∑
k∈I′λ

akjfj where

Iλ ⊂ N0, I
′
λ ⊂ N0.

Using (13),∫
(aλ(x,D)f(x)) ḡ(x)dx =

∫ ∑
j∈Iλ

∑
k∈I′λ

akjφj(D)f(x)g(x)dx

=

∫ ∑
j∈Iλ

∑
k∈I′λ

F−1(φjFf)akjg

 (x)dx.

Plancherel’s theorem yields

∫
(aλ(x,D)f(x)) ḡ(x)dx =

∫
f(x)

∑
j∈Iλ

∑
k∈I′λ

F−1 (φjF(akjg))

(x)dx.

Then, the adjoint of aλ(x,D) is

aλ(x,D)∗f =
∑
j∈Iλ

∑
k∈I′λ

φj(D) (akjf) =
∑
j∈Iλ

∑
k∈I′λ

φj(D)

(
akj

∞∑
k′=0

φk′(D)f

)
.

Thus

aλ(x,D)∗f =
∑
j∈Iλ

∑
k∈I′λ

φj(D)

(
akj

∞∑
k′=0

fk′

)
=
∑
j∈Iλ

∑
k∈I′λ

∞∑
k′=0

φj(D) (akjfk′) for λ = 1, 2, 3

(14)
One has

F {φj(D) (akjfk′)} = φjF
{
F−1(φkFσj) · F−1(φk′Ff)

}
.

The intersection of the supports of φj and F
{
F−1(φkFσj) · F−1(φk′Ff)

}
is empty if the

non-negative integer k′ dœs not verify the following cases (See [14] and [11]):
j − 3 ≤ k′ ≤ j + 3 and k = 0, . . . , j + 3

j − 3 ≤ k ≤ j + 3 and k′ = 0, . . . , j + 3

k ≥ j + 4, k′ ≥ j + 4 and |k′ − k| ≤ 3.

.



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1733

It follows that

a1(x,D)∗f =

∞∑
j=0

j−4∑
k=0

φj(D)

akj j+3∑
k′=j−3

fk′

 ,

a2(x,D)∗f =
∞∑
j=0

j+3∑
k=j−3

φj(D)

(
akj

j+6∑
k′=0

fk′

)
,

a3(x,D)∗f =
∞∑
j=0

∞∑
k=j+4

φj(D)

(
akj

k+3∑
k′=k−3

fk′

)
.

Theorem 3. Let a(x, ξ) ∈ Cℓ
∗S

m
1,δ where m ∈ R, δ ∈ [0, 1] and ℓ > 0. Let p ∈ P log(Rn)

and u, q ∈ P such that 1
q ∈ C log

loc with 1 ≤ p− ≤ p(x) ≤ u(x) ≤ ∞. Let s ∈ C log
loc such that

0 < s− ≤ s(x) < ℓ. Then

a(x,D)∗ : N s(·)
p(·),u(·),q(·) −→ N s(·)−m

p(·),u()·,q(·)

is bounded.

Proof. As for the proof of theorem 2, we will proceed by estimating the three above
operators. Let m = 0.
• The estimation of a1(x,D)∗

∥a1(x,D)∗f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

j−4∑
k=0

φj(D)

akj j+3∑
k′=j−3

fk′

∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

j−4∑
k=0

φj(D) (akjfj)

∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

.

Here fj := φ′
jf where φ′

j is a suitably chosen smooth function supported in the annulus

|ξ| ∼ 2j .
Moreover we have

suppF

{
j−4∑
k=0

φj(D) (akjfj)

}
⊂
{
ξ ∈ Rn : |ξ| ∼ 2j

}
.

By applying lemma 7 and lemma 1 we obtain,

∥a1(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
2js(·)φ̌j ∗

j−4∑
k=0

akjfj

)
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1734

≲

∥∥∥∥∥∥
(
2js(·)ηj,m ∗

∣∣∣∣∣
j−4∑
k=0

akjfj

∣∣∣∣∣
)

j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

for any m > n+ clog(s) + clog(
1
q ) + nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
Thus, by lemmas 2,

∥a1(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
ηj,m−Clog(s) ∗

j−4∑
k=0

∣∣∣akj2js(·)fj∣∣∣
)

j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

Then by lemma 3 ,

∥a1(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(

j−4∑
k=0

∣∣∣∥akj∥L∞ 2js(·)φ′
j(D)f

∣∣∣)
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

Therefore
∥a1(x,D)∗f∥N s(·)

p(·),u(·),q(·)
≲ ∥f∥N s(·)

p(·),u(·),q(·)
. (15)

• The estimation of a2(x,D)∗

∥a2(x,D)∗f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

j+3∑
k=j−3

φj(D)

(
akj

j+6∑
k′=0

fk′

)∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥∥
3∑

k=−3

∞∑
j=0

φj(D)

(
a(k+j)j

j+6∑
k′=0

fk′

)∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

≲

∥∥∥∥∥∥
∞∑
j=0

φj(D)

(
a(k+j)j

j+6∑
k′=0

fk′

)∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

One has

supp F

{
φj(D)

(
a(k+j)j

j+6∑
k′=0

fk′

)}
⊂ B(0, c2j+1).

By lemma 8 ,

∥a2(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
2js(·)φj(D)

(
a(k+j)j

j+6∑
k′=0

fk′

))
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥
(∥∥a(k+j)j

∥∥
L∞ 2js(·)ηj,m−Clog(s) ∗

∣∣∣∣∣
j+6∑
k′=0

fk′

∣∣∣∣∣
)

j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))



Mohamed Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (4) (2022), 1716-1737 1735

for any m > n+ clog(s) + clog(
1
q ) + nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
.

Then by lemma 3

∥a2(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
2js(·)

j+6∑
k′=0

φk′(D)f

)
j

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

Then
∥a2(x,D)∗f∥N s(·)

p(·),u(·),q(·)
≲ ∥f∥N s(·)

p(·),u(·),q(·)
. (16)

• The estimation of a3(x,D)∗

∥a3(x,D)∗f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
j=0

∞∑
k=j+4

φj(D)

(
akj

k+3∑
k′=k−3

fk′

)∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

=

∥∥∥∥∥∥
∞∑
k=4

k−4∑
j=0

φj(D) (akjfj)

∥∥∥∥∥∥
N s(·)

p(·),u(·),q(·)

.

Here fj := φ′
jf where φ′

j is a suitably chosen smooth function supported in the annulus

|ξ| ∼ 2j .
Moreover we have

suppF


k−4∑
j=0

φj(D) (akjfj)

 ⊂
{
ξ ∈ Rn : |ξ| ∼ 2k

}
.

Then by lemma 7 ,

∥a3(x,D)∗f∥N s(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
2ks(·)

k−4∑
j=0

φj(D) (akjfj)


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

For any m > n+ clog(s) + clog(
1
q ) + nmax

{
0, supx∈Rn

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
we have

∥a3(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
k−4∑

j=0

ηj,m−Clog(s) ∗
∣∣∣akj2ks(·)φ′

j(D)f
∣∣∣


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

≲

∥∥∥∥∥∥
k−4∑

j=0

∣∣∣akj2ks(·)φ′
j(D)f

∣∣∣


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))



REFERENCES 1736

≲

∥∥∥∥∥∥
k−4∑

j=0

∥akj∥L∞

∣∣∣2ks(·)φ′
j(D)f

∣∣∣


k

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

.

The rest is the same as that of a3(x,D) in the proof of theorem 2.
We obtain

∥a3(x,D)∗f∥N s(·)
p(·),u(·),q(·)

≲ ∥f∥N s(·)
p(·),u(·),q(·)

(17)

The three estimates (15) , (16) and (17) yield the desired estimate. □

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