©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 16doi: 10.28924/ada/ma.4.16
Hardy-Littlewood-Sobolev Theorem for Bourgain-Morrey Spaces and Approximation

Nouffou Diarra
Laboratoire de Mathématiques et Applications, UFR Mathématiques et Informatique, Université Félix

Houphouët Boigny Abidjan-Cocody, 22 BP 582 Abidjan 22, Côte d’Ivoire
Correspondence: nouffoud@yahoo.fr

Abstract. In this paper, we establish an extension of the Hardy-Littlewood-Sobolev theorem to thesetting of the Bourgain-Morrey space Mα
q,p(Rd) (1 ≤ q, p, α ≤ ∞), which theory goes back toBourgain in 1991. We also prove that Mα
q,p(Rd) is included in the closure of the Lebesgue space

Lα in the Morrey-type space F(q, p, α), which arises naturally in 2015 in the study of boundednessproperties of fractional integral operators. Therefore, we establish in Mα
q,p some approximationresults by compactly supported and/or regular functions. As an application of these results, weobtain an explicit solution in [Lp(Rd)]d of the equation divF = f whenever f is inMα

q,p , with d ≥ 3,
1 ≤ q ≤ α < d and 1

p
= 1

α
− 1

d
.

1. Introduction
Let d be a fixed positive integer. Rd is equipped with its usual Hilbert space structure and theeuclidean norm of any element x of Rd is denoted by |x |.Recall that the classical Lebesgue space Lq := Lq(Rd), with q ∈ [1,∞] , is defined to be theset of all measurable complex functions f on Rd such that

‖f ‖q :=

[∫
Rd
|f (x)|q dx

] 1
q

<∞

with the usual modification made when q =∞. In what follows, |E| and χE denote the Lebesguemeasure and the characteristic function of any measurable set E ⊂ Rd , respectively. Lqloc denotesthe set of all measurable complex functions f on Rd such that f χK ∈ Lq for any bounded measurablesubset K of Rd .For 1 ≤ q, α ≤ ∞, the Morrey spaceMα
q :=Mα

q (Rd) is defined as the set of all elements f of
Lqloc for which

‖f ‖Mα
q

:= sup
x∈Rd , r>0

|Q(x, r)|
1
α
− 1
q
∥∥f χQ(x,r)

∥∥
q
<∞,

Received: 20 Mar 2024.
Key words and phrases. Bourgain-Morrey spaces; Morrey-type space; Maximal operator; Hardy-Littlewood-Sobolevtheorem; Approximation; Divergence equation. 1

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https://doi.org/10.28924/ada/ma.4.16
https://orcid.org/0000-0003-1123-506X


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 2where
Q(x, r) =

d∏
j=1

[
xj −

r

2
, xj +

r

2

)
, x = (x1, x2, ..., xd) ∈ Rd and 0 < r <∞.

Morrey spaces were introduced in 1938 by C. Morrey [11] in order to study both the regularityproblem of solutions for quasi-linear elliptic partial differential equations and the calculus of vari-ations. Note that, for 1 ≤ q ≤ α ≤ ∞, Lα is included in Mα
q and the inclusion is proper when

q < α < ∞. Moreover, Morrey spaces describe local regularity of functions more precisely thanLebesgue spaces. However, some nice and useful properties of Lα are not shared by Mα
q when

1 ≤ q < α < ∞. For example, in this case, the set of all compactly supported and/or regularelements is not dense in Mα
q . Because of this unpleasant issue, several distinguished linear sub-spaces of Morrey spaces have been considered for their easy use in Harmonic Analysis, specificallyin boundedness problem of classical operators.The present paper focuses on Bourgain-Morrey spacesMα

q,p with 1 ≤ q, α, p ≤ ∞. Recall that,a special case of these spaces was first introduced by Bourgain [2] in 1991 in order to study theStein-Tomas estimate. Later on, Bourgain-Morrey spaces have been used fruitfully in the studyof Fourier restriction, multipliers problems and partial differential equations, and in the proof ofrefinements of Strichartz inequality (see [8–10] and the references therein). They are defined asfollows.
Definition 1.1. Let 1 ≤ q, α, p ≤ ∞. The Bourgain-Morrey space Mα

q,p := Mα
q,p(Rd) is defined

as the set of all f ∈ Lqloc for which

‖f ‖Mα
q,p

:=

∥∥∥∥{|Qk,m| 1
α
− 1
q
∥∥f χQk,m∥∥q}(k,m)∈Zd×Z

∥∥∥∥
`p
<∞,

where the sets

Qk,m =

d∏
j=1

[
kj2

m, (kj + 1) 2m
)
, k = (k1, k2, ..., kd) ∈ Zd , m ∈ Z

are the usual dyadic cubes of Rd and for any sequence {ai}i∈I included in C,

‖{ai}i∈I‖`p :=


(∑
i∈I
|ai |p

) 1
p

if p <∞

sup
i∈I
|ai | if p =∞.

It is well known that, when 1 ≤ q < α < p ≤ ∞, Lα is properly included in Mα
q,p, which is alinear subspace of Mα

q . Actually we have{
Lα ⊂Mα

q,p ⊂Mα
q,p1
⊂Mα

q,∞ =Mα
q , 1 ≤ q < α < p ≤ p1 ≤ ∞.

Mα
q,p ⊂Mα

q1,p , 1 ≤ q1 ≤ q ≤ α ≤ p ≤ ∞.
(1)

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 3Many useful results, well known for Lebesgue or Morrey spaces, have been extended to thesetting of Bourgain-Morrey spaces (see [8, 9]). For instance, the boundedness of some classicaloperators on these spaces has been investigated by Hatano et al. [8].Let 0 < γ < 1. Define the fractional integral operator Iγ as
Iγf (x) =

∫
Rd
|x − y |d(γ−1)f (y)dy

when the above integral makes sense.Recall that the Hardy-Littlewood-Sobolev theorem of fractional integration is one of the mostimportant tools in the study of partial differential equations. It reads as follows.
Theorem 1.2. [14] Let 0 < γ < 1

α ≤ 1 and 1
p = 1

α − γ. Then there is a real number Aα,γ such
that:

(i) when 1 < α

‖Iγf ‖p ≤ Aα,γ‖f ‖α , f ∈ Lα (2)
(ii) when α = 1

‖f ‖∗ 1
1−γ ,∞

≤ A1,γ‖f ‖1 , f ∈ L1. (3)
Recall that, for q ∈ [1,∞) , ‖ · ‖∗q,∞ denotes the quasi-norm of the weak-Lebesgue WLq definedby

WLq =

{
f ∈ L1

loc : ‖f ‖∗q,∞ = sup
λ>0

λ
∣∣{x ∈ Rd : |f (x)| > λ}

∣∣ 1
q <∞

}
.

The first aim of the present paper is to establish an extension of the above useful theorem tothe setting of Bourgain-Morrey spaces. Note that, our result refines that of Hatano et al., whichstates that fractional integral operators map Bourgain-Morrey spaces into the same type spaces(see [8, Theorem 4.4]).Another Morrey-type space considered in this paper is the space F(q, p, α) (1 ≤ q, α, p ≤ ∞),which arises naturally in the study of boundedness properties of fractional integral operators.It has been introduced in 2015 by Fofana et al. [7]. Note that, recently in 2020, the space
F(q, p, α) has been studied also in [15], where it is called the Riesz-Morrey space and denoted by
RMp,q, 1

p
− 1
α

(
Rd
). It is defined as follows.

Definition 1.3. Let 1 ≤ q, p, α ≤ ∞. The space F(q, p, α) := F(q, p, α)(Rd) is defined as the set
of all f ∈ Lqloc for which ‖f ‖F(q,p,α) is finite, where

‖f ‖F(q,p,α) =


sup
{Qi}∈P

∥∥∥{|Qi | 1
α
− 1
q ‖f χQi‖q

}
i∈I

∥∥∥
`p

if p <∞

sup
Q∈Q
|Q|

1
α
− 1
q ‖f χQi‖q if p =∞,

with
• Q =

{
Q(x, r) : (r, x) ∈ (0,∞)× Rd

}
• P =

{
{Qi}i∈I ⊂ Q : I is countable and Qi ∩Qj = ∅ if i 6= j

}
.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 4It is well known that the space F(q, p, α) is a linear subspace of Lqloc and a Banach space, whenendowed with ‖ · ‖F(q,p,α). F(q, p, α) is nontrivial if and only if q ≤ α ≤ p (see [7]). Moreover,when 1 ≤ q1 ≤ q ≤ α ≤ p ≤ p1, the following inclusion and equality relations hold:
{

Lα = F(q, α, α) ⊂ F(q, p, α) ⊂ F(q, p1, α) ⊂ F(q,∞, α) =Mα
q

F(q, p, α) ⊂ F(q1, p, α).
(4)

Note that (4) shows that the spaces F(q, p, α) provide a bridge connecting both Lebesgue spacesand Morrey spaces. Many results, well known for Lebesgue or Morrey spaces, have been extendedin the framework of these spaces (see [5, 7]). Furthermore, the relations (1) and (4) point out thatthe spaces Mα
q,p and F(q, p, α) satisfy almost the same inclusion relations. We also observe thatthe norm structures of these two spaces are very similar. Thus, a natural question is that, what isthe link between the spaces Mα

q,p and F(q, p, α) ?The second aim of this paper is to study the above mentioned question. We succeeded in provingthatMα
q,p is continuously included in F(q, p, α) and, when p <∞,Mα

q,p is included in the closureof Lα in F(q, p, α). Therefore, we also establish inMα
q,p some approximation results by compactlysupported and/or regular functions.As an application of the above mentioned results, we obtain an explicit solution in (Lp)d of theequation divF = f whenever f is in Mα

q,p , with d ≥ 3, 1 ≤ q ≤ α < d and 1
p = 1

α −
1
d .The remainder of the paper is organized as follows. Section 2 contains a more detailed presen-tation of our main results. Section 3 deals with some preliminary results on Mα

q,p . In Section 4we prove the inclusion of Mα
q,p in F(q, p, α) and also approximation results. Section 5 is devotedto prove our main theorem showing the action of fractional integral operators on Mα

q,p . Section 6contains an application to the divergence equation div F = f .Finally, let us make some conventions on notations used in this paper.
• C∞ denotes the set of all infinitely differentiable functions on Rd and C∞c stands for the set ofall elements of C∞ with compact support in Rd .
• Let φ be a fixed nonnegative element of C∞ such that its support is included in the unit cube
[0, 1]d and satisfying ∫

Rd
φ(x)dx = 1. For any integer n ≥ 1, we denote by φn the dilation definedby

φn(x) = ndφ(nx) , x ∈ Rd .

• Let ω be a fixed element of C∞ satisfying χQ(0,1) ≤ ω ≤ χQ(0,2). For any integer n ≥ 1, ωn isdefined by
ωn(x) = ω

(x
n

)
, x ∈ Rd .

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 52. Statement of the main results
For 0 < γ < 1, the fractional integral operator Iγ is known to be closely related to the fractionalmaximal operator Mγ defined by

Mγf (x) = sup
Q3x
|Q|γ−1

∫
Q

|f (y)|dy, f ∈ L1
loc , x ∈ Rd ,

where the supremum is taken over all cubes Q in Rd containing x .Our first result reads as follows.
Theorem 2.1. Let us assume that 0 < γ < 1

α ≤ 1 and 1
p = 1

α − γ. Then, for any element f of
Mα

1,p , we have

‖Mγf ‖p ≤ 2
d
(

2− 1
p

)
3d(2−γ) ‖f ‖Mα

1,p
. (5)

Note that Theorem 2.1 refines [8, Corollary 4.5]. As an immediate consequence of this theorem,we obtain the following result which refines [8, Theorem 4.4] and is our most significant result.
Theorem 2.2. Let us assume that 0 < γ < 1

α ≤ 1 and 1
p = 1

α−γ. Then there exists a real constant
C > 0 such that, for any element f of Mα

1,p , we have

‖Iγf ‖p ≤ C ‖f ‖Mα
1,p
. (6)

Since Lα ⊂ Mα
q,p ⊂ Mα

1,p when 1 ≤ q < α < p, Theorem 2.2 provides an extension of theHardy-Littlewood-Sobolev theorem (Theorem 1.2) to the setting of Bourgain-Morrey spaces.From (1) and (6), we have
Lα ⊂Mα

1,p ⊂ B(γ, p) , 0 < γ <
1

α
< 1 and 1

p
=

1

α
− γ, (7)

where
B(γ, p) =

{
f ∈ L1

loc : Iγ(|f |) ∈ Lp
}
, 0 < γ < 1 ≤ p ≤ ∞.

We recall that, for 1 ≤ q, p, α ≤ ∞, the space F(q, p, α) arises naturally in the search of acharacterization of the set B(γ, p) in [7], where it is established that
B(γ, p) ⊂ F(1, p, α)c ⊂ F(1, p, α) ⊂ WB(γ, p) , 0 < γ <

1

α
≤ 1 and 1

p
=

1

α
− γ, (8)

with
F(q, p, α)c =

{
f ∈ F(q, p, α) : lim

y→0
‖f − f (· − y)‖F(q,p,α) = 0

}
and

WB(γ, p) =
{
f ∈ L1

loc : Iγ(|f |) ∈ WLp
}
, 0 < γ < 1 ≤ p ≤ ∞.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 6Note that, it is proved in [7] that F(q, p, α)c is the closure of Lα in F(q, p, α) if p <∞. It is clearthat the inclusion relations (7) and (8) yield what follows
Mα

1,p ⊂ F(1, p, α)c , 0 < γ <
1

α
< 1 and 1

p
=

1

α
− γ. (9)

In the present paper we prove, without the use of fractional integral operators, the followingextension of the relation (9).
Theorem 2.3. Let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞. Then

‖f ‖F(q,p,α) ≤ 3
d
(

1+ 1
α
− 1
q

+ 1
p

)
2
d
(

1
q
− 1
p

)
‖f ‖Mα

q,p
, f ∈ L1

loc (10)
and therefore Mα

q,p is continuously included in F(q, p, α). Moreover, if p < ∞ then Mα
q,p is

included in F(q, p, α)c.

As done in [4, 6] for some special subspaces of the Morrey-type space (Lq, lp)α, usually calledthe Fofana space and closely related to F(q, p, α), we investigate in Bourgain-Morrey spacesapproximation by smooth functions. We shall prove what follows.
Theorem 2.4. Let 1 ≤ q ≤ α ≤ p <∞ and f be an element of L1

loc. Then the following assertions
are equivalent :

(i) f belongs to Mα
q,p,

(ii) lim
n→∞

‖f − f ∗ φn‖Mα
q,p

= 0, where f ∗ φn is the convolution product of f and φn,
(iii) f belongs to the closure in Mα

q,p of the set

C∞Mα
q,p

=
{
g ∈ C∞ : ∂βg ∈Mα

q,p for any β in Nd
}
,

where ∂βg stands for the derivative of order β of g.

Note that Theorem 2.4 implies that both C∞ ∩Mα
q,p and C∞Mα

q,p
are dense inMα

q,p if p <∞. Asa consequence of this theorem, we obtain the following approximation result.
Theorem 2.5. Let 1 ≤ q ≤ α ≤ p <∞ and f be any element of Mα

q,p . Then

lim
n→∞

‖f − (f ωn) ∗ φn‖Mα
q,p

= 0.

It is easy to see that, for any integer n ≥ 1, (f ωn) ∗ φn belongs to C∞c . Consequently, Theorem2.5 implies that C∞c ∩Mα
q,p is dense in Mα

q,p if p <∞.Let us consider the divergence equation
divF = f , f ∈ L1

loc. (11)
To our knowledge, for a given p in [1,∞), the characterization of the class of functions f forwhich the equation (11) has a solution F = (Fj)1≤j≤d in (Lp)d is still an open problem. However,Phuc and Torres proved that (see [13, Theorem 3.2]), for d

d−1 < p < ∞, the equation (11) has a

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 7solution in (Lp)d if and only if f belongs to the set B( 1
d , p). This result combined with (7) showsthat, for 1 < α < d and 1

p = 1
α−

1
d , a sufficient condition for the solvability in (Lp)d of the equation(11) is that f belongs to Mα

1,p . Moreover, an application of Theorem 2.2 and Theorem 2.5 allowsus to obtain an explicit solution of the equation (11) in (Lp)d , as shown below.
Theorem 2.6. Let us assume that d ≥ 3, 1 ≤ q ≤ α < d , 1

p = 1
α −

1
d and f is an element ofMα

q,p .
Then there exists a real constant cd such that F =

(
cd Rj

(
I 1
d
f
))

1≤j≤d
is a solution in (Lp)d of

the equation (11), where Rj (1 ≤ j ≤ d) stands for the Riesz transform defined by

Rjϕ(x) =
Γ
(
d+1

2

)
π
d+1

2

lim
ε→0+

∫
|x−y |≥ε

xj − yj
|x − y |d+1

ϕ(y)dy , x ∈ Rd , ϕ ∈ Lp.

3. Preliminaries
This section is devoted to prove some preliminary results.

3.1. Equivalent norms on Mα
q,p. We begin this subsection by recalling the definition of classicaldyadic grids.

Definition 3.1. A dyadic grid is a countable collection D of cubes of Rd which are dyadic translates
and dilations of the unit cube [0, 1)d . More precisely, D may be characterized as follows :
(i) if Q ∈ D then its side-length `(Q) = 2m for some m ∈ Z
(ii) if Q,P ∈ D then Q ∩ P ∈ {∅, Q, P}
(iii) for each m ∈ Z, the family Dm = {Q ∈ D / `(Q) = 2m} form a partition of Rd .

Example 3.2. • The standard dyadic grid D0 is defined by

D0 =
{

2m
(

[0, 1)d + k
)
/ m ∈ Z, k ∈ Zd

}
.

• Each of the following 3d collections of cubes in Rd

Dt =
{

2m
(

[0, 1)d + k + t
)
/ m ∈ Z, k ∈ Zd

}
, t ∈ {−1/3, 0, 1/3}d

is a dyadic grid.

The following property holds (see [3, Theorem 3.1] and its proof).
Proposition 3.3. For every cube Q of Rd , there exists an element t of {−1/3, 0, 1/3}d and a cube
Qt of Dt such that Q is included in Qt and `(Qt) ≤ 3 `(Q).

Let us introduce the following definition.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 8

Definition 3.4. Let 1 ≤ q ≤ α ≤ p ≤ ∞. For any dyadic grid D and any element f of Lqloc, we
define

‖f ‖Mα
q,p(D) =

∥∥∥∥∥
{
|Q|

1
α
− 1
q

(∫
Q

|f (x)|qdx
) 1

q

}
Q∈D

∥∥∥∥∥
`p

.

We shall prove that, for any t ∈ {−1/3, 0, 1/3}d , the norms ‖ · ‖Mα
q,p(Dt) and ‖ · ‖Mα

q,p
areequivalent. In order to do this, we establish the following preparatory lemma.

Lemma 3.5. Let 1 ≤ q ≤ α ≤ p ≤ ∞, D and D′ are two dyadic grids. Then for any element f of
Lqloc, we have  ‖f ‖Mα

q,p(D) ≤ 2
d
(

1
q
− 1
p

)
‖f ‖Mα

q,p(D′) if p <∞
‖f ‖Mα

q,∞(D) ≤ 2d‖f ‖Mα
q,∞(D′).

Proof. Let f be any element of Lqloc and fix m ∈ Z.We recall that both the families Dm = {Q ∈ D / `(Q) = 2m} and D′m = {Q′ ∈ D′ / `(Q′) = 2m}form partitions of Rd . Moreover, it is easy to see that, for any element Q of Dm, the subset
{Q′ ∈ D′ / Q ∩Q′ 6= ∅} of D′ has at most 2d elements.a) Suppose that p <∞. We have(∫

Q

|f (x)|qdx
) p

q

=

 ∑
Q′∈D′m

∫
Q∩Q′

|f (x)|qdx

 p
q

≤ 2
d
(

1− q
p

)
p
q

∑
Q′∈D′m

(∫
Q∩Q′

|f (x)|qdx
) p

q

.

Consequently∑
Q∈Dm

[
|Q|

1
α
− 1
q

(∫
Q

|f (x)|qdx
) 1

q

]p
= 2

d m
(

1
α
− 1
q

)
p
∑
Q∈Dm

(∫
Q

|f (x)|qdx
) p

q

≤ 2
d
(
p
q
−1
)

2
d m

(
1
α
− 1
q

)
p
∑
Q∈Dm

∑
Q′∈D′m, Q∩Q′ 6=∅

(∫
Q∩Q′

|f (x)|qdx
) p

q

= 2
d
(
p
q
−1
)

2
d m

(
1
α
− 1
q

)
p
∑
Q′∈D′m

∑
Q∈Dm

(∫
Q∩Q′

|f (x)|qdx
) p

q

≤ 2
d
(
p
q
−1
)

2
d m

(
1
α
− 1
q

)
p
∑
Q′∈D′m

(∫
Q′
|f (x)|qdx

) p
q

= 2
d
(
p
q
−1
) ∑
Q′∈D′m

[∣∣Q′∣∣ 1
α
− 1
q

(∫
Q′
|f (x)|qdx

) 1
q

]p
and so

‖f ‖Mα
q,p(D) ≤ 2

d
(

1
q
− 1
p

)
‖f ‖Mα

q,p(D′).

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 9b) Suppose that p =∞. Then, for any Q ∈ Dm, we have
|Q|

1
α
− 1
q

(∫
Q

|f (x)|qdx
) 1

q

= |Q|
1
α
− 1
q

 ∑
Q′∈D′m

∫
Q∩Q′

|f (x)|qdx

 1
q

≤ |Q|
1
α
− 1
q

∑
Q′∈D′m

(∫
Q∩Q′

|f (x)|qdx
) 1

q

=
∑
Q′∈D′m

∣∣Q′∣∣ 1
α
− 1
q

(∫
Q∩Q′

|f (x)|qdx
) 1

q

≤
∑

Q′∈D′m, Q∩Q′ 6=∅

∣∣Q′∣∣ 1
α
− 1
q

(∫
Q′
|f (x)|qdx

) 1
q

≤ 2d‖f ‖Mα
q,∞(D′)

and so
‖f ‖Mα

q,∞(D) ≤ 2d‖f ‖Mα
q,∞(D′).The proof is complete. �

The above lemma leads to the following corollary.
Corollary 3.6. Let 1 ≤ q ≤ α ≤ p ≤ ∞ and t be in {−1/3, 0, 1/3}d . Then for any element f of
Lqloc, we have 2

d
(

1
p
− 1
q

)
‖f ‖Mα

q,p
≤ ‖f ‖Mα

q,p(Dt) ≤ 2
d
(

1
q
− 1
p

)
‖f ‖Mα

q,p
if p <∞

2−d‖f ‖Mα
q,∞ ≤ ‖f ‖Mα

q,∞(Dt) ≤ 2d‖f ‖Mα
q,∞ .3.2. Continuity of the translation operator in Mα

q,p. This subsection deals with the continuity ofthe translation operator in Bourgain-Morrey spaces. We shall use in the sequel the followingproperties.
Proposition 3.7. [8] Let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞.
1) If α <∞, then there exists C1 > 0 such that for all y ∈ Rd and f ∈Mα

q,p , we have

‖f (· − y)‖Mα
q,p
≤ C1 ‖f ‖Mα

q,p
.

2) If q < α < p < ∞ then the set L∞c of all compactly supported bounded functions is dense in
Mα

q,p .
3) If q < α < p <∞ or p =∞ then there exists C2 > 0 such that for any element f of Mα

q,p , we
have

‖f ‖Mα
q,p
≤ C2 ‖f ‖α.

A classical property of Lebesgue spaces reads as follows.
Lemma 3.8. If 1 ≤ α <∞ and f is in Lα then we have

lim
y→0
‖f − f (· − y)‖α = 0.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 10This result can be extended to the setting of Bourgain-Morrey spaces and this extension willplay a key role in the proofs of our results.
Proposition 3.9. Let 1 ≤ q ≤ α ≤ p <∞ and f be any element of Mα

q,p . Then

lim
y→0
‖f − f (· − y)‖Mα

q,p
= 0.

Proof. If q = α or α = p then Mα
q,p = {0} and therefore, we have nothing to prove. Thus wesuppose that q < α < p. By Point 2) of Proposition 3.7, there exists a sequence (fn)n≥1 of elementsof L∞c such that lim

n→∞
‖fn − f ‖Mα

q,p
= 0. Moreover, according to Point 1) of Proposition 3.7, thereexists C1 > 0 such that

‖(fn − f )(· − y)‖Mα
q,p
≤ C1 ‖fn − f ‖Mα

q,p
, y ∈ Rd , n ≥ 1. (∗)

Let ε > 0 be a fixed real number. There exists an integer Nε such that
‖fNε − f ‖Mα

q,p
<

ε

2(1 + C1)
. (∗∗)

From (∗), (∗∗), Point 1) and Point 3) of Proposition 3.7 we have
‖f − f (· − y)‖Mα

q,p
≤ ‖f − fNε‖Mα

q,p
+ ‖fNε − fNε(· − y)‖Mα

q,p
+ ‖(fNε − f )(· − y)‖Mα

q,p

≤ ‖f − fNε‖Mα
q,p

+ ‖fNε(· − y)− fNε‖Mα
q,p

+ C1 ‖fNε − f ‖Mα
q,p

≤ (1 + C1) ‖f − fNε‖Mα
q,p

+ ‖fNε(· − y)− fNε‖Mα
q,p

<
ε

2
+ C2 ‖fNε(· − y)− fNε‖α .

According to Lemma 3.8, for any y ∈ Rd such that 0 < |y | < 1, we have
‖fNε(· − y)− fNε‖α <

ε

2C2and therefore we obtain
‖f − f (· − y)‖Mα

q,p
< ε.

This ends the proof. �

4. Inclusion and approximation results
4.1. Inclusion of Mα

q,p in F(q, p, α). This subsection is devoted to prove exclusively Theorem 2.3.
Proof of Theorem 2.31) • We recall that Mα

q,∞ =Mα
q = F(q,∞, α). Therefore, we have nothing to prove if p =∞.

• If p <∞ and α ∈ {q, p} then Mα
q,p = {0}. Thus the result is obvious.

• Assume that 1 ≤ q < α < p <∞.Let f be in L1
loc and {Qi : i ∈ I} be a disjoint family of cubes of Rd .

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 11

a) Let us consider an element i of I. We can associate to i an element t of {−1/3, 0, 1/3}d andan element R(i , t) of Dt such that Qi ⊂ R(i , t) and `(R(i , t)) ≤ 3 `(Qi) (see Proposition 3.3). Wehave
|Qi |

1
α
− 1
q

(∫
Qi

|f (x)|qdx
) 1

q

≤
(
|Qi |
|R(i , t)|

) 1
α
− 1
q

|R(i , t)|
1
α
− 1
q

(∫
R(i ,t)

|f (x)|qdx
) 1

q

≤ 3
d
(

1
α
− 1
q

)
|R(i , t)|

1
α
− 1
q

(∫
R(i ,t)

|f (x)|qdx
) 1

q

.

b) Let us fix t in {−1/3, 0, 1/3}d and set
Rt =

{
R ∈ Dt : ∃ i ∈ I such that R(i , t) = R

}
.

Note that, for all R ∈ Rt , we have
∀ i ∈ I, R = R(i , t) =⇒ `(Qi) ≤ `(R) ≤ 3`(Qi) =⇒ |Qi | ≤ |R| ≤ 3d |Qi |∑
i∈I, R=R(i ,t)

|Qi | ≤ |R|.

This shows that the cardinality of the set {i ∈ I : R = R(i , t)} does not exceed 3d .c) We have∑
i∈I

(
|Qi |

1
α
− 1
q

(∫
Qi

|f (x)|qdx
) 1

q

)p

=
∑

t∈{−1/3,0,1/3}d

∑
R∈Rt

∑
i :R=R(i ,t)

(
|Qi |

1
α
− 1
q

(∫
Qi

|f (x)|qdx
) 1

q

)p

≤
∑

t∈{−1/3,0,1/3}d

∑
R∈Rt

∑
i :R=R(i ,t)

(
3
d
(

1
α
− 1
q

)
p|R|

1
α
− 1
q

(∫
R

|f (x)|qdx
) 1

q

)p
(by Point a))

≤ 3
d
(

1
α
− 1
q

)
p

3d
∑

t∈{−1/3,0,1/3}d

∑
R∈Rt

(
|R|

1
α
− 1
q

(∫
R

|f (x)|qdx
) 1

q

)p
( by Point b) )

≤ 3
d
(

1
α
− 1
q

)
p

3d
∑

t∈{−1/3,0,1/3}d

∑
R∈Dt

(
|R|

1
α
− 1
q

(∫
R

|f (x)|qdx
) 1

q

)p
( because of Rt ⊂ Dt).

Therefore[∑
i∈I

(
|Qi |

1
α
− 1
q

(∫
Qi

|f (x)|qdx
) 1

q

)p] 1
p

≤ 3
d
(

1
α
− 1
q

+ 1
p

) ∑
t∈{−1/3,0,1/3}d

‖f ‖Mα
q,p(Dt).

Since the above inequality is true for all disjoint family {Qi : i ∈ I} of cubes of Rd , we have
‖f ‖F(q,p,α) ≤ 3

d
(

1
α
− 1
q

+ 1
p

) ∑
t∈{−1/3,0,1/3}d

‖f ‖Mα
q,p(Dt).

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 12Therefore, Corollary 3.6 implies that
‖f ‖F(q,p,α) ≤ 3

d
(

1+ 1
α
− 1
q

+ 1
p

)
2
d
(

1
q
− 1
p

)
‖f ‖Mα

q,pand consequently Mα
q,p is continuously included in F(q, p, α).2) Assume that p <∞ and let f be any element of Mα

q,p . By Point 1), f is in F(q, p, α) and, forany y ∈ Rd , we have
‖f − f (· − y)‖F(q,p,α) ≤ 3

d
(

1+ 1
α
− 1
q

+ 1
p

)
2
d
(

1
q
− 1
p

)
‖f − f (· − y)‖Mα

q,p
.

Therefore, Proposition 3.9 implies that
lim
y→0
‖f − f (· − y)‖F(q,p,α) = 0

and consequently f belongs to F(q, p, α)c. Thus, we obtain the desired result. �
4.2. Approximation inMα

q,p. In this subsection, we investigate approximation of elements of Bourgain-Morrey spaces by smooth functions. We shall use the following result.
Proposition 4.1. [8] Let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞ with α < ∞. Then there exists
C > 0 such that for all g ∈ L1 and f ∈Mα

q,p , we have

‖g ∗ f ‖Mα
q,p
≤ C ‖g‖1 ‖f ‖Mα

q,p
.

Propositions 3.7, 3.9 and 4.1 allow us to prove Theorem 2.4.
Proof of Theorem 2.4

• (i)⇒ (i i) Assume that f ∈Mα
q,p and n is a nonegative integer. For almost every x ∈ Rd ,

f (x)− f ∗ φn(x) =

∫
Rd
f (x)φ(u)du −

∫
Rd
f (x − y)ndφ(ny)dy

=

∫
Rd
f (x)φ(u)du −

∫
Rd
f
(
x −

u

n

)
φ(u)du

=

∫
Rd

[
f (x)− f

(
x −

u

n

)]
φ(u)du.

Therefore, for any dyadic cube Qk,m (
(k,m) ∈ Zd × Z

)
, the Minkowski inequality implies that,∥∥(f − f ∗ φn)χQk,m

∥∥
q
≤
∫
Rd

∥∥∥[f − f (· − u
n

)]
χQk,m

∥∥∥
q
φ(u)du

and so
‖f − f ∗ φn‖Mα

q,p
≤
∫
Rd

∥∥∥f − f (· − u
n

)∥∥∥
Mα

q,p

φ(u)du.

According to Proposition 3.9, we have
lim
n→∞

∥∥∥f − f (· − u
n

)∥∥∥
Mα

q,p

φ(u) = 0 , u ∈ Rd .

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 13Furthermore, by Minkowski’s inequality and Point 1) of Proposition 3.7, we have∥∥∥f − f (· − u
n

)∥∥∥
Mα

q,p

φ(u) ≤ (1 + C1) ‖f ‖Mα
q,p
φ(u) , u ∈ Rd , n ≥ 1.

Thus, an application of the dominated convergence theorem gives
lim
n→∞

‖f − f ∗ φn‖Mα
q,p

= 0.

• (i i)⇒ (i i i) Assume that the assertion (i i) holds.Let us fix an integer n ≥ 1 and β ∈ Nd . Since φn ∈ C∞c , f ∗φn belongs to C∞ and by Proposition4.1, f ∗ φn is in Mα
q,p . Furthermore, it is well known that ∂β (f ∗ φn) = f ∗ ∂βφn and by notingthat ∂βφn ∈ L1, Proposition 4.1 implies that ∂β (f ∗ φn) belongs to Mα

q,p . Thus f ∗ φn belongs to
C∞Mα

q,p
and since, by hypothesis,

lim
n→∞

‖f − f ∗ φn‖Mα
q,p

= 0,

we can conclude that f belongs to the closure in Mα
q,p of C∞Mα

q,p
.

• (i i i)⇒ (i) Since C∞Mα
q,p

is a subset of Mα
q,p, it is obvious that its closure in Mα

q,p is included in
Mα

q,p and therefore the claim follows. The proof is complete. �
We recall the following well known result in Lebesgue spaces.

Lemma 4.2. [1] If 1 ≤ α <∞ and f is in Lα then we have

lim
n→∞

‖f χEn‖α = 0,

where (En)n≥1 is a nonincreasing sequence of measurable subsets of Rd satisfying

∣∣∣∣∣∣⋂n≥1

En

∣∣∣∣∣∣ = 0.

The next proposition shows that an analogous result holds for Bourgain-Morrey spaces.
Proposition 4.3. Let 1 ≤ q ≤ α ≤ p < ∞, f be any element of Mα

q,p and (En)n≥1 be a

nonincreasing sequence of measurable subsets of Rd satisfying

∣∣∣∣∣∣⋂n≥1

En

∣∣∣∣∣∣ = 0. Then

lim
n→∞

‖f χEn‖Mα
q,p

= 0.

Proof. If q = α or α = p then Mα
q,p = {0} and therefore we have nothing to prove. Hence wesuppose that 1 ≤ q < α < p <∞. By Point 2) of Proposition 3.7, there exists a sequence (fn)n≥1of elements of L∞c such that

lim
n→∞

‖fn − f ‖Mα
q,p

= 0.Let ε > 0 be a fixed real number. From what precedes, there exists an integer Nε such that
‖fNε − f ‖Mα

q,p
<
ε

2
.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 14This and Point 3) of Proposition 3.7 imply that, for any n ≥ 1,

‖f χEn‖Mα
q,p
≤ ‖(f − fNε)χEn‖Mα

q,p
+ ‖fNεχEn‖Mα

q,p
≤ ‖f − fNε‖Mα

q,p
+ ‖fNεχEn‖Mα

q,p

≤
ε

2
+ C2 ‖fNεχEn‖α .Since fNε ∈ Lα, Lemma 4.2 implies that there exists an integer N0 ≥ 1 such that

n ≥ N0 =⇒ ‖fNεχEn‖α <
ε

2C2
.

Therefore
n ≥ N0 =⇒ ‖f χEn‖Mα

q,p
< ε.This provides the desired result. �

[1, Proposition 3.6] asserts that Proposition 4.3 is equivalent to the following dominated con-vergence theorem.
Proposition 4.4. Let 1 ≤ q ≤ α ≤ p <∞ and f be any element ofMα

q,p . If (fn)n≥1 is a sequence
of measurable functions satisfying |fn| ≤ |f | for all n ≥ 1 and lim

n→∞
fn = g almost everywhere, for

some measurable function g, then

lim
n→∞

‖fn − g‖Mα
q,p

= 0.

Proposition 4.4 yields obviously what follows.
Lemma 4.5. Let 1 ≤ q ≤ α ≤ p <∞. Then for any element f of Mα

q,p , we have

lim
n→∞

∥∥f − f χQ(0,n)

∥∥
Mα

q,p
= 0.

We are now ready to prove Theorem 2.5.
Proof of Theorem 2.5For any integer n ≥ 1, we have, by Proposition 4.1,

‖f − (f ωn) ∗ φn‖Mα
q,p
≤ ‖f − f ∗ φn‖Mα

q,p
+ ‖(f − f ωn) ∗ φn‖Mα

q,p

≤ ‖f − f ∗ φn‖Mα
q,p

+ C ‖f − f ωn‖Mα
q,p
‖φn‖1

≤ ‖f − f ∗ φn‖Mα
q,p

+ C ‖f − f ωn‖Mα
q,p
.

Notice that, for any integer n ≥ 1, |f − f ωn| ≤ ∣∣f − f χQ(0,n)

∣∣ and therefore we obtain
‖f − (f ωn) ∗ φn‖Mα

q,p
≤ ‖f − f ∗ φn‖Mα

q,p
+ C

∥∥f − f χQ(0,n)

∥∥
Mα

q,p
.

Thus, it follows from Theorem 2.4 and Lemma 4.5 that
lim
n→∞

‖f − (f ωn) ∗ φn‖Mα
q,p

= 0.

This finishes the proof. �

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 155. Fractional operators in Mα
q,pThis section is devoted to prove Theorem 2.1 and Theorem 2.2. In order to do this we need somepreparatory lemmas.Let 0 < γ < 1 and D be a dyadic grid. The dyadic fractional maximal operator MDγ is definedby

MDγ f (x) = sup

{
|Q|γ−1

∫
Q

|f (y)|dy / Q ∈ D, x ∈ Q
}
, f ∈ L1

loc , x ∈ Rd .

The following lemma is a consequence of Proposition 3.3.
Lemma 5.1. Let 0 < γ < 1. For any element f of L1

loc we have :

Mγf (x) ≤ 3d(1−γ) max
t∈{−1/3,0,1/3}d

MD
t

γ f (x) , x ∈ Rd .

Proof. Let us consider an element (f , x) of L1
loc × Rd and a cube Q of Rd containing x . ByProposition 3.3, there exist an element t of {−1/3, 0, 1/3}d and a cube Qt of Dt such that Q isincluded in Qt and `(Qt) ≤ 3 `(Q). Thus, we have

|Q|γ−1

∫
Q

|f (y)|dy = `(Q)d(γ−1)

∫
Q

|f (y)|dy ≤
[

1

3
`(Qt)

]d(γ−1) ∫
Q

|f (y)|dy

≤ 3d(1−γ)|Qt |γ−1

∫
Qt

|f (y)|dy ≤ 3d(1−γ)MD
t

γ f (x).

Consequently
Mγf (x) ≤ 3d(1−γ) max

t∈{−1/3,0,1/3}d
MD

t

γ f (x).

The proof is complete. �

Recall that the density of L∞c in Mα
q,p (see Point 2) of Proposition 3.7) has been proved in [8].Here, we improve this result which will play a key role in the proof of Lemma 5.3.

Lemma 5.2. Let 1 ≤ q ≤ α ≤ p <∞ and f be any element ofMα
q,p . Then there exists a sequence

(fn)n≥1 of elements of L∞c ∩Mα
q,p such that (|fn|)n≥1 ↑ |f | almost everywhere and

lim
n→∞

‖f − fn‖Mα
q,p

= 0.

Proof. Let us set, for any integer n ≥ 1,
fn = sgn(f ) min

(
|f |, nχQ(0,2n)

)
,

where, for any x ∈ Rd ,
sgn(f )(x) =

{
f (x)
|f (x)| if f (x) 6= 0

0 if f (x) = 0.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 16It is easy to see that (fn)n≥1 is a sequence of elements of L∞c ∩Mα
q,p satisfying (|fn|)n≥1 ↑ |f |almost everywhere and  |fn| ≤ |f | , n ≥ 1

lim
n→∞

fn = f almost everywhere.Therefore, an application of Proposition 4.4 leads to
lim
n→∞

‖f − fn‖Mα
q,p

= 0.

This ends the proof. �

As a consequence of Lemma 5.2, the following result holds true.
Lemma 5.3. Let us assume that 0 < γ < 1

α ≤ 1 and 1
p = 1

α − γ. Then for any dyadic grid D and
any element f of Mα

1,p , we have

‖MDγ f ‖p ≤ 2 ‖f ‖Mα
1,p
.

Proof. If α = 1 then Mα
q,p = {0} and so the result is obvious. Thus, we assume that α > 1.Let f be any element of Mα

1,p and D be a dyadic grid.1) Assume that f also belongs to L∞.a) We have, for all cube Q of Rd ,
|Q|γ−1

∫
Q

|f (y)|dy ≤ |Q|γ‖f ‖∞

and
|Q|γ−1

∫
Q

|f (y)|dy = |Q|γ−
1
α |Q|

1
α
−1

∫
Q

|f (y)|dy = |Q|γ−
1
α ‖f ‖Mα

1
.

Consequently,
lim

`(Q)→∞
|Q|γ−1

∫
Q

|f (y)|dy = 0 (∗)

and, for all Q ∈ Q,
|Q|γ−1

∫
Q

|f (y)|dy ≤

{
‖f ‖∞ if `(Q) ≥ 1

‖f ‖Mα
1

if `(Q) ≤ 1.

Thus, for all x ∈ Rd ,
Mγf (x) ≤ M with M = max

(
‖f ‖∞, ‖f ‖Mα

1

)
.

Since MDγ f ≤Mγf , we have, for any x ∈ Rd ,
MDγ f (x) ≤ M. (∗∗)

b) Assume that f 6= 0. By (∗∗), we have, for all x ∈ Rd ,
MDγ f (x) ∈ (0,M]. (∗ ∗ ∗)

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 17(i) Let us consider an integer j ≥ 0 and set Ej =
{
x ∈ Rd : MDγ f (x) ∈

(
2−j−1M, 2−jM

]}
Dj =

{
Q ∈ D : |Q|γ−1

∫
Q |f (y)|dy ∈

(
2−j−1M, 2−jM

]}
.

We have, for all x ∈ Rd ,
x ∈ Ej ⇐⇒ ∃Qx ∈ Dj : x ∈ Qx ⇐⇒ x ∈

⋃
Q∈Dj

Q.

Thus Ej =
⋃
Q∈Dj

Q.
Note that, by (∗), sup

{
`(Q) : Q ∈ Dj

}
<∞ and let us denote by ∆j the set of maximal elements(for the inclusion) of Dj . It is easy to see that

⋃
Q∈∆j

Q =
⋃
Q∈Dj

Q = Ej

∀ Q′, Q′′ ∈ ∆j , Q
′ 6= Q′′ =⇒ |Q′ ∩Q′′| = 0.Moreover, by the definition of Ej , we have, for all x ∈ Q ∈ ∆j ,

|Q|γ−1

∫
Q

|f (y)|dy ≤MDγ f (x) ≤ 2−jM < 2 |Q|γ−1

∫
Q

|f (y)|dy

and therefore, for all Q ∈ ∆j ,∫
Q

[
MDγ f (x)

]p
dx ≤ 2p |Q|

[
|Q|γ−1

∫
Q

|f (y)|dy
]p

= 2p
[
|Q|

1
α
−1

∫
Q

|f (y)|dy
]p
.

So we obtain ∫
Ej

[
MDγ f (x)

]p
dx =

∑
Q∈∆j

∫
Q

[
MDγ f (x)

]p
dx

≤ 2p
∑
Q∈∆j

[
|Q|

1
α
−1

∫
Q

|f (y)|dy
]p
. (∗ ∗ ∗∗)

(ii) By (∗ ∗ ∗), we have ⋃
j≥0

Ej = Rd . Meanwhile, the definition of Ej(j ≥ 0) shows that
∀ j ′, j ′′ ∈ N, with j ′ 6= j ′′,Ej ′ ∩ Ej ′′ = ∅.

Hence we have ∫
Rd

[
MDγ f (x)

]p
dx =

∑
j≥0

∫
Ej

[
MDγ f (x)

]p
dx

and therefore, by (∗ ∗ ∗∗),
‖MDγ f ‖p ≤ 2

∑
j≥0

∑
Q∈∆j

[
|Q|

1
α
−1

∫
Q

|f (y)|dy
]p 1

p

. (∗ ∗ ∗ ∗ ∗)

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 18Note that, if j ′ and j ′′ are two integer such that j ′′ > j ′ ≥ 0 then
∀ (Q′, Q′′) ∈ ∆j ′ × ∆j ′′ , Q

′ 6= Q′′

and so {
Qk,m : Qk,m ∈ ∆j ′

}
∩
{
Qk,m : Qk,m ∈ ∆j ′′

}
= ∅, k ∈ Zd , m ∈ Z.Therefore∑

j≥0

∑
Q∈∆j

[
|Q|

1
α
−1

∫
Q

|f (y)|dy
]p

=
∑
m∈Z

∑
k∈Zd :Qk,m∈

⋃
j≥0

∆j

[
|Qk,m|

1
α
−1

∫
Qk,m

|f (y)|dy

]p

≤ ‖f ‖pMα
1,p
.

This inequality combined with (∗ ∗ ∗ ∗ ∗) gives
‖MDγ f ‖p ≤ 2 ‖f ‖Mα

1,p
.

2) By Lemma 5.2, there exists a sequence (fn)n≥1 of elements of L∞c ∩Mα
q,p such that (|fn|)n≥1 ↑ |f |almost everywhere and lim

n→∞
‖f − fn‖Mα

q,p
= 0. Thus, the result obtained in Point 1) implies that

∥∥MDγ fn∥∥p ≤ 2 ‖fn‖Mα
1,p
≤ 2 ‖f ‖Mα

1,p
, n ≥ 1

0 ≤
(
MDγ fn

)
n≥1
↑MDγ f and so (∥∥MDγ fn∥∥p)n≥1

↑
∥∥MDγ f ∥∥pand therefore

‖MDγ f ‖p ≤ 2 ‖f ‖Mα
1,p
.The proof is complete. �

Now we prove Theorem 2.1 thanks to Lemma 5.1, Lemma 5.3 and Corollary 3.6.
Proof of Theorem 2.1Let f be in Mα

1,p . By Lemma 5.1 and Lemma 5.3, we have
‖Mγf ‖p ≤ 3d(1−γ)

∥∥∥∥∥ max
t∈{−1/3,0,1/3}d

MD
t

γ f

∥∥∥∥∥
p

≤ 3d(1−γ)
∑

t∈{−1/3,0,1/3}d

∥∥∥MDtγ f ∥∥∥
p

≤ 3d(1−γ)
∑

t∈{−1/3,0,1/3}d
2 ‖f ‖Mα

1,p(Dt) .

Note that the hypotheses imply that p <∞ and so Corollary 3.6 leads to
‖Mγf ‖p ≤ 3d(1−γ)2

d
(

2− 1
p

)
‖f ‖Mα

1,p
]
(
{−1/3, 0, 1/3}d

)
= 2

d
(

2− 1
p

)
3d(2−γ) ‖f ‖Mα

1,p
.

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 19The proof is complete. �
As an immediate consequence of Theorem 2.1 we can now prove Theorem 2.2.
Proof of Theorem 2.2Let f be in Mα

1,p . Note that the hypotheses imply that p <∞. By [12, Theorem 1], we have
‖Iγf ‖p ≤ D ‖Mγf ‖p ,where D is a real constant not depending on f . Therefore, Theorem 2.1 provides the desiredinequality. �

6. Application
Theorem 2.2, Theorem 2.5 and the boundedness properties of Riesz tranforms in Lebesgue spaceslead to the following result, which contains Theorem 2.6.

Proposition 6.1. Let us assume that d ≥ 3, 1 ≤ q ≤ α < d , 1
p = 1

α −
1
d and f is an element of the

Bourgain-Morrey space Mα
q,p . Then

1) for 1 ≤ j ≤ d , the function Fj = Rj

(
I 1
d
f
)

belongs to Lp

2) there exists a real constant cd depending only on d and such that F =
(
cd Fj

)
1≤j≤d is a solution

in (Lp)d of the equation (11).
Proof. Note that, the hypotheses imply that 1 ≤ q ≤ α < p <∞.1) Since Mα

q,p ⊂ Mα
1,p (see (1)), Theorem 2.2 implies that I 1

d
f ∈ Lp . Furthermore, it is wellknown that the Riesz tranform Rj is bounded on Lp , for 1 ≤ j ≤ d . Therefore, we deduce that

Fj = Rj

(
I 1
d
f
) belongs to Lp .2) a) Let ϕ be any element of C∞c . For 1 ≤ j ≤ d , the boundedness properties of Rj and I 1

d
showthat ψj = Rj

(
I 1
d
ϕ
) belongs to ⋂

r> d
d−1

Lr . Since d
d−1 < 2, there exists a real number r such that

d
d−1 < r < 2 and ψj ∈ Lr . Therefore, we can use the Fourier transform to obtain cd d∑

j=1

∂jψj = ϕ,
where cd is a real constant depending only on d (see [14, Formula (17), p.125]).b) Fix an integer n ≥ 1 and set fn = (f ωn) ∗ φn. Since fn ∈ C∞c , the result of Point a) implies thatdiv Fn = fn, where Fn = (Fnj )1≤j≤d with

Fnj = cdRj

(
I 1
d
fn

)
∈
⋂

r> d
d−1

Lr , 1 ≤ j ≤ d.

• According to Theorem 2.5, (fn)n≥1 converges to f in Mα
1,p .

• For 1 ≤ j ≤ d , the boundedness properties of Rj and I 1
d

imply that (Fnj )n≥1
converges to

cdFj = cdRj

(
I 1
d
f
) in Lp .

https://doi.org/10.28924/ada/ma.4.16


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 20Therefore, for any element ϕ of C∞c , we have∫
Rd

div F (x)ϕ(x)dx = −
d∑
j=1

∫
Rd
cdFj(x) ∂jϕ(x)dx

= lim
n→∞

− d∑
j=1

∫
Rd
Fnj (x) ∂jϕ(x)dx


= lim
n→∞

∫
Rd

 d∑
j=1

∂jFnj (x)

ϕ(x)dx

= lim
n→∞

∫
Rd

div Fn(x)ϕ(x)dx

= lim
n→∞

∫
Rd
fn(x)ϕ(x)dx =

∫
Rd
f (x)ϕ(x)dx.

Hence, div F = f . Thus, we obtain the desired result. �

Acknowledgement. The author would like to express his deep thanks to Professor Ibrahim Fofanafor his helpful assistance with this note.
References

[1] C. Bennett, R.C. Sharpley, Interpolation of operators, Academic Press, New York, (1988).[2] J. Bourgain, On the restriction and multiplier problems in R3, in: Geometric Aspects of Functional Analysis (1989-90), in: Lecture Notes in Math., Springer, Berlin, 1469, (1991), 179-191.[3] D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, in: Advanced Courses ofMathematical Analysis VI, World Scientific, Universidad de Málaga, Spain, 2017: pp. 25–85. https://doi.org/
10.1142/9789813147645_0002.[4] N. Diarra, I. Fofana, Characterization of some closed linear subspaces of Morrey spaces and approximation, Adv.Pure Appl. Math. 14 (2023) 41–72. https://doi.org/10.21494/ISTE.OP.2023.0980.[5] N. Diarra, I. Fofana, Complex interpolation of some Banach spaces including Morrey spaces, Khayyam J. Math.(2024).[6] M. Dosso, I. Fofana, M. Sanogo, On some subspaces of Morrey–Sobolev spaces and boundedness of Riesz integrals,Ann. Polon. Math. 108 (2013) 133–153. https://doi.org/10.4064/ap108-2-2.[7] I. Fofana, F.R. Faléa, B.A. Kpata, A class of subspaces of Morrey spaces and norm inequalities on Riesz potentialoperators, Afr. Mat. 26 (2015) 717–739. https://doi.org/10.1007/s13370-014-0241-3.[8] N. Hatano, T. Nogayama, Y. Sawano, D.I. Hakim, Bourgain–Morrey spaces and their applications to boundednessof operators, J. Funct. Anal. 284 (2023), 109720. https://doi.org/10.1016/j.jfa.2022.109720.[9] S. Masaki, J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018) 283–326. https://doi.org/10.1016/j.
anihpc.2017.04.003.[10] S. Masaki, J. Segata, Refinement of Strichartz estimates for Airy equation and application, RIMS KôkyûrokuBessatsu, B80 (2020) 11–25. http://hdl.handle.net/2433/260656.[11] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43(1938) 126–166.

https://doi.org/10.28924/ada/ma.4.16
https://doi.org/10.1142/9789813147645_0002
https://doi.org/10.1142/9789813147645_0002
https://doi.org/10.21494/ISTE.OP.2023.0980
https://doi.org/10.4064/ap108-2-2
https://doi.org/10.1007/s13370-014-0241-3
https://doi.org/10.1016/j.jfa.2022.109720
https://doi.org/10.1016/j.anihpc.2017.04.003
https://doi.org/10.1016/j.anihpc.2017.04.003
http://hdl.handle.net/2433/260656


Eur. J. Math. Anal. 10.28924/ada/ma.4.16 21

[12] B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192(1974) 261–274. https://doi.org/10.1090/S0002-9947-1974-0340523-6.[13] N.C. Phuc, M. Torrès, Characterizations of the existence and removable singularities of divergence-measure vectorfields, Indiana Univ. Math. J., 57 (2008) 1573–1597. https://www.jstor.org/stable/24902998.[14] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton,New Jersey (1970).[15] J. Tao, D. Yang, W. Yuan, A bridge connecting Lebesgue and Morrey spaces via Riesz norms, Banach J. Math. Anal.15 (2021) 20. https://doi.org/10.1007/s43037-020-00106-6.

https://doi.org/10.28924/ada/ma.4.16
https://doi.org/10.1090/S0002-9947-1974-0340523-6
https://www.jstor.org/stable/24902998
https://doi.org/10.1007/s43037-020-00106-6

	1. Introduction
	2. Statement of the main results
	3. Preliminaries
	3.1. Equivalent norms on Mq,p
	3.2. Continuity of the translation operator in Mq,p

	4. Inclusion and approximation results
	4.1. Inclusion of Mq,p in F(q,p,)
	4.2. Approximation in Mq,p

	5. Fractional operators in Mq,p
	6. Application
	References