EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 47-63
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Boundedness of non regular pseudo-differential
operators on variable exponent Triebel-Lizorkin-Morrey

spaces

Mohamed Congo1,∗, Marie Françoise Ouedraogo1

1 Laboratoire de Théorie des Nombres, Algèbre, Géométrie Algébrique, Topologie Algébrique
et Applications(TN-AGATA). UFR Sciences Exactes et Appliquées/ Université Joseph
KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Abstract. In this paper, we study the boundedness of non regular pseudo-differential operators

on variable exponent Besov-Morrey spaces Es(·)
p(·),u(·),q(·) with symbols a(x, ξ) belonging to Cℓ

∗S
m
1,δ.

For these symbols x-regularity is measured in Hölder-Zygmund spaces.

2020 Mathematics Subject Classifications: 42B35,46E30,35S05

Key Words and Phrases: Pseudo-differential operators, Non regular symbols, Variable exponent
Triebel-Lizorkin-Morrey spaces

1. Introduction

Pseudo-differential calculus is a well-established tool for the analysis of partial differ-
ential equations, especially non-linear ones. Indeed, in [16] one can find many applications
of the calculus of non regular pseudo-differential operators to non-linear differential equa-
tions. The boundedness of these operators has been extensively addressed in several works.
For boundedness on Lebesgue spaces, Besov spaces, Triebel-Lizorkin spaces and Sobolev
spaces, we refer to [2], [6], [12] and [13].
The boundedness of pseudo-differential operators in Triebel-Lizorkin-Morrey spacses with
constant exponents denoted Es

p,u,q was studied by Yoshihiro Sawano in [15].
Our focus in this paper concerns the boundedness of pseudo-differential operators on

Triebel-Lizorkin-Morrey spaces with variable exponents denoted Es(·)
p(·),u(·),q(·) ( see [4]) with

symbols in the class Cℓ
∗S

m
1,δ.

The results of this paper are certainly relevant because they generalize those of [15].
Our approach is as follows. To treat the boundedness of these operators with non-regular
symbols belonging to Cℓ

∗S
m
1,δ we use elementary symbols as it was done in [2], [12], [14]

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4200

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

http://www.ejpam.com 47 © 2022 EJPAM All rights reserved.



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 48

and [15].
Indeed, the symbol reduction method, due to Coifman and Meyer[6], makes it possible to
be limited to symbols a(x, ξ) ∈ Cℓ

∗S
m
1,δ of the form a(x, ξ) =

∑
j≥0 σj(x)ψj(ξ)(see [14] and

[2]). Then, we rewrite the symbol as a sum of three parts, a ”low-high”, a ”high-high”, and
a ”high-low” part. Thus, the operator a(x,D) with symbol a can be resolved into three
operators a1(x,D), a2(x,D) and a3(x,D) with symbols a1, a2 and a3. Now it remains to
study the boundedness of each elementary operators.
We structure this paper in 4 sections as follows. In Section 2 we give the preliminaries,
where we recall the definitions of Morrey spaces and Besov-Morrey spaces with variable
exponents. In Section 3, we recall necessary tools for the proofs of the lemmas and the
main result that we give in Section 4.

2. Preliminaries

We denote by Rn the n-dimensional real Euclidean space, N the collection of all natural
numbers and N0 = N ∪ {0}. Z stands for the set of all integer numbers. 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. 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(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 of
a tempered distribution f is denoted by Ff or f̂ while its inverse transform is denoted by
F−1f or f̌ .

2.1. Variable exponents

For more information on the results of this paragraph, see [11] and [7].
• By P(Rn) we denote 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.



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 49

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 write g ∈ P log(Rn) if 0 < g− ≤ g(x) ≤ g+ ≤ +∞ with
1

g
∈ C log(Rn).

We define
1

g∞
:= lim

|x|→+∞

1

g(x)
and we use the convention

1

∞
= 0.

2.2. Variable exponent Triebel-Lizorkin-Morrey spaces

We refer to the papers [4], [18], [3], [5], [17] and [9], for further results on Triebel-
Lizorkin-Morrey spaces and variable exponent Triebel-Lizorkin-Morrey spaces.

• Morrey 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)

By the definition of the Lp(·) quasinorm, (2) can also be written as

∥f∥Mp(·),u(·)
:= sup

x∈Rn, r>0
inf

{
λ > 0 : ϱ

(
r

n
u(x)

− n
p(x)

f

λ
χB(x.r)

)
≤ 1

}
.

Definition 2. Let p, q, u ∈ P(Rn) with p(x) ≤ u(x). The mixed space Mp(·),u(·)(ℓq(·))
consists of all sequences (fν)ν ⊂ M(Rn) such that,

∥(fν)ν∥Mp(·),u(·)(ℓq(·))
:=

∥∥∥∥∥∥
(

+∞∑
ν=0

|fν(·)|q(·)
)1/q(·)

∥∥∥∥∥∥
Mp(·),u(·)

< +∞. (3)



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 50

Remark 1. [4] Note that ∥·∥Mp(·),u(·)(ℓq(·))
defined a quasinorm on Mp(·),u(·)(ℓq(·)). It is a

norm when min(p−, q−) ≥ 1.

Proposition 1. Let f and g be two measurable functions with 0 ≤ f(x) ≤ g(x) for a.e.
x ∈ Rn. Then it holds

∥f∥Mp(·),u(·)(ℓq(·))
≤ ∥g∥Mp(·),u(·)(ℓq(·))

.

Proposition 2. Let p, q, u ∈ P(Rn) with p(x) ≤ u(x) and 0 < t < +∞.
Let (fν)ν ⊂ M(Rn) ∥∥(|fν |t)ν∥∥

M p(·)
t ,

u(·)
t

(
ℓ q(·)

t

) = ∥(fν)ν∥tMp(·),u(·)(ℓq(·))

with the usual modification every time q(x) = +∞.

•Triebel-Lizorkin-Morrey spaces.
We first recall a Littlewood-Paley partition of unity {ψν}, ν ≥ 0.
The functions ψν are defined as follows. Let ψ0 ∈ C∞

0 (Rn) such that ψ0 ≡ 1 on B(0; 1)
and suppψ0 ⊂ B(0; 2).
Set

ψν(ξ) = ψ0(2
−νξ)− ψ0(2

−ν+1ξ) for all ν ∈ N.

Then ψν is supported on the dyadic shell

Dν =
{
ξ ∈ Rn : 2ν−1 ≤ |ξ| ≤ 2ν+1

}
.

If f ∈ S ′, then

f =
∑
ν≥0

ψνf.

The Fourier multiplier ψj(D) with symbol ψj is defined as

ψν(D)f(x) = F−1(ψν · f̂)(x) =
∫
Rn

ψν(ξ)f̂(ξ)e
ix·ξdξ.

Definition 3. Let {ψν} be the usual Littlewood-Paley partition of unity. Let s : Rn →
R, p, q ∈ P log(Rn) and u ∈ P(Rn) such that 0 < p− ≤ p(x) ≤ u(x) ≤ supu < +∞
and q−, q+ ∈ (0,+∞). The Triebel-Lizorkin-Morrey spaces Es(·)

p(·),u(·),q(·) consists of all

distributions f ∈ S ′(Rn) such that

∥f∥Es(·)
p(·),u(·),q(·)

:= ∥ψ0(D)f∥Mp(·),u(·)
+

∥∥∥∥(2νs(·)ψν(D)fν

)
ν≥1

∥∥∥∥
Mp(·),u(·)(ℓq(·))

< +∞. (4)

Remark 2. [4](remark4.4) Note that ∥·∥Es(·)
p(·),u(·),q(·)

defined a quasinorm on Es(·)
p(·),u(·),q(·). It

is a norm when min(p−, q−) ≥ 1.



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 51

3. Basic tools

In this section we present some useful results for the last section. At First, we recall
the η-functions defined 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 following lemma is from [8](Lemma19) and [10](Lemma6.1)

Lemma 1. Let α ∈ C log
loc (R

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

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

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

The following lemma is from [10](lemma A.6).

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

{
ξ ∈ Rn : |ξ| ≤ 2ν+1

}
, We have

|g(x)| ≤ c
(
ην,m ∗ |g|t(x)

)1/t
, x ∈ Rn.

The following lemma is from[4](theorem3.3).

Lemma 3. Let p, q ∈ P log(Rn) and u ∈ P(Rn) such that 1 < p− ≤ p(x) ≤ u(x) ≤ supu <
+∞ and q−, q+ ∈ (1,+∞). If

m > n+ nmax

{
0, sup

x∈Rn

(
1

p(x)
− 1

u(x)

)
− 1

p∞

}
,

then there exists c > 0 such that for all sequences (fν)ν ⊂Mp(·),u(·)(ℓq(·)).∥∥(ην,m ∗ fν)ν
∥∥
Mp(·),u(·)(ℓq(·))

≤ c ∥(fν)ν∥Mp(·),u(·)(ℓq(·))
.

The following lemma is from[1](Corollary 4.8.)

Lemma 4. Let p ∈ P log(Rn) and u ∈ P with 1 < p− ≤ p(x) ≤ u(x) ≤ supu < +∞. 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(·)
≤ c ∥f∥Mp(·),u(·)

.

The following lemma is from[4](Lemma 3.7).



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 52

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, we denote

Gν(x) :=

+∞∑
j=0

2−|ν−j|δgj(x), x ∈ Rn, ν ∈ N0.

Then it holds ∥(Gν)ν∥ℓq(·)(Mp(·),u(·))
≤ c(δ, q)

∥∥∥(gj)j∥∥∥ℓq(·)(Mp(·),u(·))
where

c(δ, q) = max

∑
ν∈Z

2−|ν|δ,

[∑
ν∈Z

2−|ν|δq−
]1/q− .

4. Boundedness of pseudo-differential operators

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

Definition 4. [14] 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
∗S

m
1,δ

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

(5)

In (5), ⟨ξ⟩ stand for
(
1 + |ξ|2

)1/2
.

A pseudo-differential operator on Es(·)
p(·),u(·),q(·) 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 ∈ Es(·)
p(·),u(·),q(·).

Definition 5. We call elementary symbol in the class Cℓ
∗S

m
1,δ, δ ∈ [0, 1], ℓ > 0 an expres-

sion of the form

a(x, ξ) =
∑
j≥0

aj(x)ψj(ξ)

where ψ0 is smooth supported on the ball B(0, 2), ψj(ξ) = ψ(2−jξ) and ψ ∈ C∞
0 is sup-

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

∥aj∥Cℓ
∗S

m
1,δ

≤ c2j(m+ℓδ).

Since a(x,D) and ψj(D) do not commute, to study boundedness of a(x,D), the sym-
bol reduction method due to Coifman and Meyer[6] makes it possible to be limited to
elementary symbols.
Therefore, the operator a(x,D) with symbol a can be resolved into ”elementary opera-
tors” ak(x,D) with symbols ak. This idea has been exploited to establish continuity of
pseudo-differential operators with non-regular symbols in inhomogeneous Sobolev spaces
Hs,p and Hölder-Zygmund spaces Cℓ

∗ ( see [12] and [2]).



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 53

Lemma 6. [14] 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∞

}
. (6)

The following lemmas plays a fundamental role in the proof of the boundedness of

pseudo-differential operators on Es(·)
p(·),u(·),q(·).

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

0 < p− ≤ p(x) ≤ u(x) ≤ supu < ∞ and q−, q+ ∈ (0,+∞). 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

∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

.

Proof. Let {ψj} be the Littlewood-Paley partition of unity defined above. By hypoth-
esis, ψj , j ≥ 1 are supported on the dyadic shell Dj , while ψ0 is supported on the ball
B(0; 2). Hence, there is N1, N2 ∈ N0 such that

ψ0(D)

(
+∞∑
k=0

fk

)
= ψ0(D)

(
N1∑
k=0

fk

)

and ψj(D)

(
+∞∑
k=0

fk

)
= ψj(D)

 j+N2∑
k=j−N1

fk


Then∥∥∥∥∥

+∞∑
k=0

fk

∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

=

∥∥∥∥∥
N1∑
k=0

ψ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥N1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

(7)

• Let us first estimate

∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

Since ψ̌j ∗ fk ∈ S ′ and suppF
(
ψ̌j ∗ fk

)
⊂
{
ξ ∈ Rn : |ξ| ≤ 2j+1

}
, then, by lemma 2,

|ψ̌j ∗ fk| ≲
(
ηj,m ∗ |fk|t

)1/t
, k = j −N1, . . . , j +N2.



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 54

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

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
and any t > 0.

Thus∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥N1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥


j+N2∑
k=j−N1

2js(·)
(
ηj,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

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

2js(·)
(
ηj,m ∗ |fk|t

)1/t
≲
(
ηj,m−clog(s) ∗ 2

js(·)t|fk|t
)1/t

.

Then ∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥N1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥


j+N2∑
k=j−N1

(
ηj,m−clog(s) ∗ 2

js(·)t|fk|t
)1/t

j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

=

∥∥∥∥∥∥


j+N2∑
k=j−N1

(
ηj,m−clog(s) ∗ 2

js(·)t|fk|t
)

j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

.

With t ∈ (0,min {1, p−, q−}), lemma 4 yields∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥N1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲
N1+N2∑
k=0

∥∥∥∥(2js(·)t|fj+k−N1 |t
)
j

∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

.

Then

∥∥∥∥∥∥
2js(·)

j+N2∑
k=j−N1

ψ̌j ∗ fk


j≥1

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

.

• Now we estimate the first term .
Since suppF

(
ψ̌0 ∗ fk

)
⊂ {ξ ∈ Rn : |ξ| ≤ 2}, then by lemma 2,

∣∣ψ̌0 ∗ fk
∣∣ ≲ |fk| .

Thus ∥∥∥∥∥
N1∑
k=0

ψ̌0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

≲
N1∑
k=0

∥fk∥Mp(·),u(·)

=

N1∑
k=0

∥(0, . . . , fk, 0, . . .)∥ℓq(·)(Mp(·),u(·))



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 55

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
ℓq(·)(Mp(·),u(·))

.

The proof is completed. □

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

0 < p− ≤ p(x) ≤ u(x) ≤ supu < +∞, s− > 0 and q−, q+ ∈ (0,+∞). Let {fk}k∈N0
be a

sequence of tempered distributions such that

suppFfk ⊂ B(0, c2k+1)

Then ∥∥∥∥∥
+∞∑
k=0

fk

∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

Proof. In view of the hypothesis on Suppψj , there is N ∈ N0 such that∥∥∥∥∥
+∞∑
k=0

fk

∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

=

∥∥∥∥∥ψ0(D)

(
+∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

+

∥∥∥∥∥∥
2js(·)ψj(D)

 +∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

.

(8)

(i) At first we estimate

∥∥∥∥∥∥
2js(·)ψj(D)

 +∞∑
k=j−N

fk


j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

.

We have∥∥∥∥∥∥
2js(·)ψj(D)

 +∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
ℓq(·)(Mp(·),u(·))

=

∥∥∥∥∥∥


+∞∑
k=j−N

2js(·)
(
ψ̌j ∗ fk

)
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

Since {
suppF

(
ψ̌j ∗ fk

)
⊂
{
ξ ∈ Rn : |ξ| ≤ 2j+1

}
suppF

(
ψ̌j ∗ fk

)
⊂
{
ξ ∈ Rn : |ξ| ≤ 2k+1

}
,

by lemma 2 , {
2js(·)

(
ψ̌j ∗ fk

)
≲ 2js(·)

(
ηj,m ∗ |fk|t

)1/t
2js(·)

(
ψ̌j ∗ fk

)
≲ 2js(·)

(
ηk,m ∗ |fk|t

)1/t
.

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

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
and t > 0.

Therefore∥∥∥∥∥∥
2js(·)ψj(D)

 +∞∑
k=j−N

fk


j≥N

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥


j∑
k=j−N

2js(·)
(
ηj,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 56

+

∥∥∥∥∥∥


+∞∑
k=j+1

2−(k−j)s(·)2ks(·)
(
ηk,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

Let us estimate each one of the two terms on the right-hand side.

Using lemmas 1 we can move 2νs(·) inside the convolution 2νs(·)
(
ην,m ∗ |fk|t

)1/t
. And we

have

2νs(·)
(
ην,m ∗ |fk|t

)1/t
≲
(
ην,m0 ∗ 2νs(·)t|fk|t

)1/t
, ν = j or k where m0 = m− clog(s)

Thus

•

∥∥∥∥∥∥


j∑
k=j−N

2js(·)
(
ηj,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

=

∥∥∥∥∥∥


j∑
k=j−N

ηj,m0 ∗ 2js(·)t|fk|t


j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

≲
0∑

k=−N

∥∥∥∥{ηj,m0 ∗ 2js(·)t|fk+j |t
}
j

∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

.

For t ∈ (0,min {p−, q−}), lemma 3 yields

0∑
k=−N

∥∥∥∥{ηj,m0 ∗ 2js(·)t|fk+j |t
}
j

∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

≲
0∑

k=−N

∥∥∥∥{2js(·)t|fk+j |t
}
j

∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

.

Then∥∥∥∥∥∥


j∑
k=j−N

2js(·)
(
ηj,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

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

.

And

•

∥∥∥∥∥∥


+∞∑
k=j+1

2−(k−j)s(·)2ks(·)
(
ηk,m ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥


+∞∑
k=j+1

2−|j−k|s(·)
(
ηk,m0 ∗ 2ks(·)t|fk|t

)
j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

≲

∥∥∥∥∥∥


+∞∑
k=j+1

2−|j−k|s−
(
ηk,m0 ∗ 2ks(·)t|fk|t

)
j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 57

≲

∥∥∥∥∥∥
{

+∞∑
k=0

2−|j−k|s−
(
ηk,m0 ∗ 2ks(·)t|fk|t

)}
j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

By lemma 5 ,∥∥∥∥∥∥
{

+∞∑
k=0

2−|j−k|s−
(
ηk,m0 ∗ 2ks(·)t|fk|t

)}
j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(
ℓ q(·)

t

) ≲
∥∥∥(ηk,m0 ∗ 2ks(·)t|fk|t

)
k

∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)
.

For t ∈ (0,min p−, q−), lemma 3 yields∥∥∥(ηk,m0 ∗ 2ks(·)t|fk|t
)
k

∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)
≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

.

Then∥∥∥∥∥∥


+∞∑
k=j+1

2−(k−j)s(·)2ks(·)
(
ηk,m0 ∗ |fk|t

)1/t
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

.

(ii) Now we estimate

∥∥∥∥∥ψ0(D)

(
+∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

.

Since
suppF

(
ψ̌0 ∗ fk

)
⊂
{
ξ ∈ Rn : |ξ| ≤ 2k+1

}
,

Then ∥∥∥∥∥ψ0(D)

(
+∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

=

∥∥∥∥∥
N∑
k=0

ψ0 ∗ fk

∥∥∥∥∥
Mp(·),u(·)

≲

∥∥∥∥∥
∞∑
k=0

(
ηk,m ∗ |fk|t

)1/t∥∥∥∥∥
Mp(·),u(·)

,

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

(
1

p(x) −
1

u(x)

)
− 1

p∞

}
by lemma 2.

Then by lemma 1∥∥∥∥∥ψ0(D)

(
+∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

≲

∥∥∥∥∥
∞∑
k=0

2−ks−
(
ηk,m−clog(s) ∗ 2

ks(·)t|fk|t
)∥∥∥∥∥

M p(·)
t ,

u(·)
t

=

∥∥∥∥∥∥
{

+∞∑
k=0

2−ks−
(
ηk,m−clog(s) ∗ 2

ks(·)t|fk|t
)}

j

∥∥∥∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 58

Thus∥∥∥∥∥ψ0(D)

(
+∞∑
k=0

fk

)∥∥∥∥∥
Mp(·),u(·)

≲
∥∥∥(ηk,m−clog(s) ∗ 2

ks(·)t|fk|t
)
k

∥∥∥
M p(·)

t ,
u(·)
t

(ℓ q(·)
t

)

≲
∥∥∥(2ks(·)fk)

k

∥∥∥
Mp(·),u(·)(ℓq(·))

by lemma 5 and lemma 2.

The proof is completed. □

Theorem 1. Let a(x, ξ) ∈ Cℓ
∗S

m
1,δ where m ∈ R, δ ∈ [0, 1] and ℓ > 0. Let 1 ≤ p− ≤ p(x) ≤

u(x) ≤ supu < +∞ and q−, q+ ∈ [1,+∞). Let s ∈ C log
loc such that 0 < s− ≤ s+ < ℓ. Then

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

p(·),u(·),q(·)

is bounded.

Proof. We recall that the symbol reduction method, due to Coifman and Meyer[6],
makes it possible to be limited to symbols a(x, ξ) ∈ Cℓ

∗S
m
1,δ of the form (see [14] and [2])

a(x, ξ) =
∑
j≥0

σj(x)ψj(ξ)

where σj satisfies

∥σj∥Cℓ
∗
≤ c2j(m+ℓδ) (9)

and ∥σj∥L∞
≤ c (10)

with c depending on δ and ℓ but not on j. And ψj is exactly a Littlewood-Paley function.
We have

σj(x) =
+∞∑
k=0

ψj(D)σj(x).

Then σj(x)ψj(ξ) =

(
+∞∑
k=0

ψj(D)σj(x)ψj(ξ)

)
.

Therefore a(x, ξ) =

+∞∑
j=0

(
+∞∑
k=0

ψk(D)σj(x)

)
ψj(ξ).

Set akj = ψk(D)σj . Then

a(x, ξ) =

+∞∑
j=0

(
+∞∑
k=0

akj

)
ψj(ξ). (11)



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 59

(i) At first, it’s necessary to estimate ∥akj∥L∞
.

We recall the quasinorm of Cℓ
∗: ∥ψk(D)σj∥Cℓ

∗
= supk 2

kℓ ∥ψk(D)σj∥L∞
.

Since ∥ψk(D)σj∥Cℓ
∗
≤ c ∥σj∥Cℓ

∗
.

Then sup
k

2kℓ ∥ψk(D)σj∥L∞
≤ c ∥σj∥Cℓ

∗
.

Using (9), we obtain
∥akj∥L∞

≤ c2j(m+ℓδ)2−kℓ. (12)

Note that (1 − ∆)
m
2 , m ∈ R is an isomorphism that composes well with pseudo-

differential operators (see[14] and [15]). Therefore, it is enough to examine the casem = 0.
If m = 0 then

∥akj∥L∞
≤ c2jℓδ2−kℓ (13)

(ii) Now we rewrite the symbol 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)

ψj(ξ)

= a1(x, ξ) + a2(x, ξ) + a3(x, ξ)

where

a1(x,D)f =
+∞∑
j=0

(
j−4∑
k=0

akjψj(D)f

)
,

a2(x,D)f =

+∞∑
j=0

 j+3∑
k=j−3

akjψj(D)f

 ,

a3(x,D)f =

+∞∑
j=0

 ∞∑
k=j+4

akjψj(D)f

 .

•We have

F

(
j−4∑
k=0

akjfj

)
=

j−4∑
k=0

F (ψk(D)σj) ∗ F (ψj(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 : c12

j−1 ≤ |ξ| ≤ c22
j+1
}
with c1, c2 > 0.

Then lemma 7 yields



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 60

∥a1(x,D)f∥Es(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
+∞∑
j=0

(
j−4∑
k=0

akjψj(D)f

)∥∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
(
2js(·)

j−4∑
k=0

akjψj(D)f

)
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥
(

j−4∑
k=0

∥σj∥L∞
2js(·)ψj(D)f

)
j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥(2js(·)ψj(D)f
)
j

∥∥∥∥
Mp(·),u(·)(ℓq(·))

.

Then
∥a1(x,D)f∥Es(·)

p(·),u(·),q(·)
≲ ∥f∥Es(·)

p(·),u(·),q(·)
.

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

=

∥∥∥∥∥∥
+∞∑
j=0

 j+3∑
k=j−3

akjfj

∥∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

,

we observe that

F

 j+3∑
k=j−3

akjfj

 =

j+3∑
k=j−3

F (ψk(D)σj) ∗ F (ψj(D)f)

=

j+3∑
k=j−3

(ψkFσj) ∗ (ψjFf) .

Then F

 j+3∑
k=j−3

akjfj

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

By lemma 8,

∥a2(x,D)f∥Es(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
2js(·)

j+3∑
k=j−3

akjfj


j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≤ 2−m

∥∥∥∥∥∥
 j+3∑

k=j−3

∥akj∥L∞
2js(·)ψj(D)f


j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

.



M. Congo, M. F. Ouedraogo / Eur. J. Pure Appl. Math, 15 (1) (2022), 47-63 61

One have

j+3∑
k=j−3

∥akj∥L∞
≲

3∑
k=−3

2−kℓ < +∞ (with δ = 1).

Then

∥a2(x,D)f∥Es(·)
p(·),u(·),q(·)

≲

∥∥∥∥(2js(·)ψj(D)f
)
j

∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲ ∥f∥Es(·)
p(·),u(·),q(·)

.

• Now let us estimate last part. Since F

 +∞∑
k=j+4

akjfj

 is not supported on any ball

or shell, we cannot directly use neither lemma7 nor lemma8.
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) .

We have suppF

(
j−4∑
k=0

akjfj

)
⊂
{
ξ ∈ Rn| c12

j−1 ≤ |ξ| ≤ c22
j+1
}
with c1, c2 > 0.

Thus we can use lemma 7.

∥a3(x,D)f∥Es(·)
p(·),u(·),q(·)

=

∥∥∥∥∥∥
+∞∑
k=4

k−4∑
j=0

akjfj

∥∥∥∥∥∥
Es(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
2ks(·)

k−4∑
j=0

akjfj


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲

∥∥∥∥∥∥
k−4∑

j=0

∥akj∥L∞
2ks(·)ψj(D)f


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

.

If we use(13) with δ = 1, we have

∥a3(x,D)f∥Es(·)
p(·),u(·),q(·)

≲

∥∥∥∥∥∥
k−4∑

j=0

2jℓ2−kℓ2ks(·)ψj(D)f


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))



REFERENCES 62

=

∥∥∥∥∥∥
k−4∑

j=0

2(k−j)(s(·)−ℓ)2js(·)ψj(D)f


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≤

∥∥∥∥∥∥
k−4∑

j=0

2−|k−j||s−−ℓ|2js(·)ψj(D)f


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≤

∥∥∥∥∥∥
+∞∑

j=0

2−|k−j||s−−ℓ|2js(·)ψj(D)f


k

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

.

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

j=0

2−|k−j||s−−ℓ|2js(·)ψj(D)f


j

∥∥∥∥∥∥
Mp(·),u(·)(ℓq(·))

≲
∥∥∥(2js(·)ψj(D)f

)
k

∥∥∥
Mp(·),u(·)(ℓq(·))

.

Then
∥a3(x,D)f∥Es(·)

p(·),u(·),q(·)
≲ ∥f∥Es(·)

p(·),u(·),q(·)
.

The proof is completed. □

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