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Boundedness of Some Commutators in Total Fofana Spaces

Pokou Nagacy
Laboratoire des Sciences et Technologies de l’Environnement, UFR Environnement, Université Jean

Lorougnon GUEDE, BP 150 Daloa, Côte d’Ivoire

pokounagacy@yahoo.com

Abstract. In this paper, we find necessary and sufficient conditions for the boundedness of the com-mutator of the Hardy-Littlewood maximal operator in total Fofana spaces. We also give in thesespaces the boundedness of some sublinear operators and their commutators.

1. Introduction and main results
Let d be a fixed positive integer and Rd the d-dimensional Euclidean space. The estimation ofcommutators plays an important role in studying the regularity of solutions of elliptic, parabolicand ultraparabolic partial differential equations of second order, and their boundedness can beused to characterize certain function spaces (see, for instance [3, 18,20]).We denote by L0(Rd) the complex vector space of equivalent classes (modulo equality Lebesguealmost everywhere) of Lebesgue measurable complex-valued functions on Rd . For f ∈ L0(Rd) and

x ∈ Rd , the Hardy-Littlewood maximal function is defined by the formula
Mf (x) = sup

r>0
|B(x, r)|−1

∫
B(x,r)

|f (y)| dy, (1.1)
where |B(x, r)| is the Lebesgue measure of the ball

B(x, r) = {y ∈ Rd : |x − y | < r}.

The maximal commutatorMb generated by the maximal operatorM and a locally integrable function
b is defined by

Mb(f )(x) = sup
r>0
|B(x, r)|−1

∫
B(x,r)

|b(x)− b(y)||f (y)|dy.

Furthermore the commutator generated by the operator M and a suitable function b is defined by
[b,M]f (x) = b(x)M(f )(x)−M(bf )(x).

Recall that the operators Mb and [b, M] essentially differ from each other since Mb is positiveand sublinear and [b,M] is neither positive nor sublinear. The operators M , [b,M] and Mb play
Received: 28 Jul 2024.
Key words and phrases. Total Fofana spaces; maximal operator; commutator; sublinear operators; BMO spaces.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 2an important role in real and harmonic analysis and applications (see for example [1, 2, 10, 15, 16]and the references therein). The boundedness of these operators on Lebesgue spaces has beenextended to several other spaces. For example, on Morrey spaces ( [4], [11]), modified Morreyspaces ( [14]), total Morrey spaces ( [13]), Fofana spaces ( [6]) to name but a few.Recently, P. Nagacy and B. A. Kpata [17] introduced the total Fofana spaces and established inthese spaces, under certain conditions, the boundedness of the Hardy-Littlewood maximal operators,the Riesz potential and fractional maximal operators.The main purpose of the present paper is to establish the boundedness of the commutators of theHardy-Littlewood maximal operator and some sublinear operators in total Fofana spaces. Beforestating our main results, let us start with some notations and basic definitions. Let 1 ≤ q, p ≤ ∞and r > 0. For f ∈ L0(Rd) we define
r ‖f ‖q,p :=

∥∥∥∥∥
[∫
Rd
|f χB(y,r)|q(x)dx

] 1
q

∥∥∥∥∥
pwith the Lp(Rd)-norm taken with respect to the variable y . We adopt the usual convention 1

∞ = 0.In 1988, Fofana [7] introduced the functions spaces (Lq, Lp)α (Rd), 1 ≤ q ≤ α ≤ p ≤ ∞, whichconsists of the set of all functions f ∈ L0(Rd) satisfying ‖f ‖q,p,α <∞, where
‖f ‖q,p,α = sup

r>0
rd(

1
α
− 1
q
− 1
p
)
r ‖f ‖q,p .

It is proved in [7] the following properties:
• for 1 ≤ q < α fixed and p going from α to ∞, the spaces (Lq, Lp)α (Rd) form a chain of distinctBanach spaces beginning with the Lebesgue space Lα(Rd) and ending by the classical Morreyspace Lq,d(1− q

α
)(Rd) = (Lq, L∞)α (Rd);

• there exists a constant C > 0 such that
‖f ‖q,p,α ≤ C ‖f ‖α . (1.2)

For an in-depth study of Fofana spaces, please consult the following references (see [6, 8, 9]).Let 1 ≤ q ≤ α, λ ≤ p ≤ ∞ and [r ]1 = min{1, r}, r > 0. The total Fofana spaces (Lq, Lp)α,λ(Rd)are defined by
(Lq, Lp)α,λ(Rd) =

{
f ∈ L0(Rd) : ‖f ‖(Lq ,Lp)α,λ(Rd ) <∞

}
where

‖f ‖(Lq ,Lp)α,λ(Rd ) = sup
r>0
[r ]
d( 1

α
− 1
q
− 1
p
)

1 [1/r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1 r ‖f ‖q,p .Note that(1) the space (Lq, Lp)α,λ(Rd) is a complex vector subspace of L0(Rd).(2) from the definition of (Lq, Lp)α,λ(Rd) spaces we deduce that the map L0(Rd) 3 f 7→
‖f ‖(Lq ,Lp)α,λ(Rd ) defines a norm on (Lq, Lp)α,λ(Rd).

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 3

(3) for 1 ≤ q ≤ α, λ <∞, the space (Lq, L∞)α,λ(Rd) is the total Morrey space Lq,d(1− q
α
),d(1− q

λ
)(Rd)defined in [13] and

Lq,d(1−
q
α
),d(1− q

λ
)(Rd) = Lq,d(1−

q
α
)(Rd) ∩ Lq,d(1−

q
λ
)(Rd)

with λ ≤ α.We define the space BMO(Rd) as the set of all locally integrable functions b with finite norm
‖b‖BMO(Rd ) = sup

r>0,x∈Rd
|B(x, r)|−1

∫
B(x,r)

|b(y)− bB(x,r)|dy

where bB(x,r) = |B(x, r)|−1 ∫
B(x,r)

b(y)dy.For a function b defined on Rd , we denote by
b−(x) :=

{
0 if b(x) ≥ 0
|b(x)| if b(x) < 0

and b+(x) = |b(x)| − b−(x). Obviously b+(x)− b−(x) = b(x).Our first result reads as follows
Theorem 1.1. Let 1 < q ≤ λ ≤ α < p <∞ such that 1q +

2
p <

1
α +

1
λ . Then the operator [b,M] is

bounded on (Lq, Lp)α,λ(Rd) if and only if b ∈ BMO(Rd) such that b− ∈ L∞(Rd).

The last two theorems concern the sublinear operators T satisfying the condition
|T (x)| ≤ C

∫
Rd

|f (y)|
|x − y |d dy x /∈ supp f , (1.3)

for any f ∈ L1(Rd) with compact support. We point out that the condition (1.3) was first introducedby Soria and Weiss [19]. This condition is satisfied by many operators such as the Hardy-Littewoodmaximal operator, Calderón-Zygmund singular integral operators, Bochner-Riesz operators at thecritical index, C. Fefferman’s singular multiplier. It is proved in [5] that T is bounded on Morreyspaces. It is also bounded on classical Fofana spaces (see [6]). In the setting of total Fofana spaces,we have the following result holds true.
Theorem 1.2. Let 1 < q ≤ λ,α < p < ∞ such that 1q +

2
p <

1
α +

1
λ . If T is sublinear operator

with is bounded on Lq and satisfies the condition (1.3) then T is also bounded on (Lq, Lp)α,λ(Rd).

If T is a linear operator and b ∈ BMO(Rd), we define the linear commutator [b, T ] by
[b, T ]f (x) = T (bf )(x)− b(x)T (f )(x) x ∈ Rd ,

with locally integrable functions f on Rd . It is also proved in [6] that [b, T ] is bounded on classicalFofana spaces and bounded on Morrey spaces in [5]. The next result shows the boundedness ontotal Fofana spaces of [b, T ].

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 4

Theorem 1.3. Let 1 < q ≤ λ,α < p <∞ such that 1q +
2
p <

1
α +

1
λ and b ∈ BMO(Rd). If a linear

operator T satisfies (1.3) and [b, T ] is bounded on Lq , then T is also bounded on (Lq, Lp)α,λ(Rd).

The remainder of this note is organized as follows: In Section 2 we recall some properties oftotal Fofana spaces. Section 3 is devoted to the proofs of Theorem 1.1 and section 4 deals withthe proofs of Theorem 1.2 and Theorem 1.3.
The letter C will be used for positive constants not depending on the relevant variables, andtheses constants may change from one occurrence to another. We propose the following abbreviation

A <∼ B for the inequalities A ≤ CB. If A <∼ B and B <∼ A, then we write A ≈ B.
2. Some properties of total Fofana spaces

The results of this section are proved in [17].The following result examines the relationship between total Fofana spaces and Fofana spaces.
Proposition 2.1. Let 1 ≤ q ≤ α, λ ≤ p ≤ ∞. Then

(Lq, Lp)α(Rd) ∩ (Lq, Lp)λ(Rd) ↪→ (Lq, Lp)α,λ(Rd)

and for f ∈ (Lq, Lp)α(Rd) ∩ (Lq, Lp)λ(Rd)

‖f ‖(Lq ,Lp)α,λ(Rd ) ≤ max{‖f ‖q,p,α , ‖f ‖q,p,λ}.

Proposition 2.2. Let 1 ≤ q ≤ λ ≤ α ≤ p ≤ ∞. Then

(Lq, Lp)α,λ(Rd) = (Lq, Lp)α(Rd) ∩ (Lq, Lp)λ(Rd)

and for f ∈ (Lq, Lp)α,λ(Rd)

‖f ‖(Lq ,Lp)α,λ(Rd ) = max{‖f ‖q,p,α , ‖f ‖q,p,λ}.

Total Fofana spaces are generalizations of classical Fofana spaces since Proposition 2.2 assertsthat (Lq, Lp)α,α(Rd) = (Lq, Lp)α(Rd).The family of spaces (Lq, Lp)α,λ(Rd) is increasing with respect to the p power. More precisely,we have the following.
Proposition 2.3. Let 1 ≤ q ≤ α, λ ≤ p1 ≤ p2 ≤ ∞. Then:

‖f ‖(Lq ,Lp2)α,λ(Rd ) <∼ ‖f ‖(Lq ,Lp1)α,λ(Rd ) , f ∈ L0(Rd)

and consequently, (Lq, Lp1)α,λ(Rd) ⊂ (Lq, Lp2)α,λ(Rd).

The following result states the boundedness property of M (the Hardy-Littlewood maximal op-erator) on total Fofana spaces.

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 5

Theorem 2.4. (1) Let 1 < q ≤ α, λ < p <∞ such that 1q +
2
p <

1
α +

1
λ . Then

‖Mf ‖(Lq ,Lp)α,λ(Rd ) <∼ ‖f ‖(Lq ,Lp)α,λ(Rd ) , f ∈ (Lq, Lp)α,λ(Rd).

(2) Let q = 1 < α,λ < p <∞. Then

‖Mf ‖(L1,∞,Lp)α,λ(Rd ) <∼ ‖f ‖(L1,Lp)α,λ(Rd ) , f ∈ (L1, Lp)α,λ(Rd),

where

‖f ‖(L1,∞,Lp)α,λ(Rd )

:= sup
r>0
[r ]
d( 1

α
−1− 1

p
)

1 [1/r ]
d(− 1

λ
+1+ 1

p
)

1

[∫
Rd

(∥∥f χB(y,r)∥∥∗1,∞)p dy] 1p
with ∥∥f χB(y,r)∥∥∗1,∞ = sup

r>0
r |{x ∈ B(y , r) : |f (x)| > r}| .

3. Proof of Theorem 1.1
For the proof of this theorem, we need some results.The following result (see [1, Corollary 1.11]) will be useful in the proof of Theorem 1.1.

Lemma 3.1. If b ∈ BMO(Rd), then there exists a positive constant C such that

Mbf (x) ≤ C ‖b‖BMO(Rd )M(Mf )(x)

for almost every x ∈ Rd and any locally integrable functions f on Rd .

Proposition 3.2. Let 1 < q ≤ α, λ < p <∞ such that 1q +
2
p <

1
α +

1
λ and b ∈ BMO(Rd). Then

Mb is bounded on (Lq, Lp)α,λ(Rd).

Proof. Let 1 < q ≤ α, λ < p < ∞ such that 1q + 2
p < 1

α +
1
λ , b ∈ BMO(Rd) and f ∈

(Lq, Lp)α,λ(Rd).By taking the (Lq, Lp)α,λ(Rd)-norm of both sides of the estimate in Lemma 3.1, we obtain
‖Mbf ‖(Lq ,Lp)α,λ(Rd ) <∼ ‖b‖BMO(Rd ) ‖M(Mf )‖(Lq ,Lp)α,λ(Rd ) .According to the first point of Theorem 2.4, we have

‖M(Mf )‖(Lq ,Lp)α,λ(Rd ) <∼ ‖Mf ‖(Lq ,Lp)α,λ(Rd ) <∼ ‖f ‖(Lq ,Lp)α,λ(Rd ) .We deduce that
‖Mbf ‖(Lq ,Lp)α,λ(Rd ) <∼ ‖b‖BMO(Rd ) ‖f ‖(Lq ,Lp)α,λ(Rd ) . (3.1)

�

Lemma 3.3. Let 1 ≤ q ≤ λ ≤ α <∞ and r > 0. Then

r−
d
q [r ]

d(− 1
α
+ 1
q
)

1 [1/r ]
d( 1

λ
− 1
q
)

1 max{r
d
α , r

d
λ } ≤ 2.

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 6

Proof. Let 1 ≤ q ≤ λ ≤ α <∞ and r > 0.Put C(r) = r −dq [r ]d(− 1α+ 1q )1 [1/r ]
d( 1

λ
− 1
q
)

1 max{r
d
α , r

d
λ }.

C(r) ≤

{
r−

d
α (r

d
α + r

d
λ ), 0 < r ≤ 1

r−
d
λ (r

d
α + r

d
λ ), r > 1

≤

{
1 + rd(−

1
α
+ 1
λ
), 0 < r ≤ 1

1 + rd(
1
α
− 1
λ
), r > 1.

Thus, C(r) ≤ 2 for all r > 0. �

Proof of Theorem 1.1. Let 1 < q ≤ λ ≤ α < p <∞ such that
1
q +

2
p <

1
α +

1
λ .

(1) Assume that b ∈ BMO(Rd) such that b− ∈ L∞(Rd) and f ∈ (Lq, Lp)α,λ(µ).Proceeding as in the proof of Theorem 4 in [13], we have
‖[b,M]f ‖(Lq ,Lp)α,λ(Rd ) ≤

∥∥Mbf + 2b
−Mf

∥∥
(Lq ,Lp)α,λ(Rd )

≤ ‖Mbf ‖(Lq ,Lp)α,λ(Rd )

+ 2
∥∥b−∥∥∞ ‖Mf ‖(Lq ,Lp)α,λ(Rd ) .

From (3.1) and the first point of Theorem 2.4, we deduce that
‖[b,M]f ‖(Lq ,Lp)α,λ(Rd ) <∼

(
‖b‖BMO(Rd ) +

∥∥b−∥∥∞) ‖f ‖(Lq ,Lp)α,λ(Rd ) .
(2) Conversely, assume that [b,M] is bounded on (Lq, Lp)α,λ(Rd).Let t > 0 and x ∈ Rd . Put B = B(x, t). Denote by MBf the local maximal function of f definedby:

MBf (x) = sup
B
′3x :B′⊂B

|B′ |−1
∫
B
′
|f (y)| dy.

Since χB ∈ Lα(Rd) ∩ Lλ(Rd), it follows from (1.2) that
χB ∈ (Lq, Lp)α (Rd)∩(Lq, Lp)λ (Rd). From Proposition 2.2, we deduce that χB ∈ (Lq, Lp)α,λ(Rd).Therefore, there exists a constant C > 0 such that

‖[b,M]χB‖(Lq ,Lp)α,λ(Rd ) ≤ C ‖χB‖(Lq ,Lp)α,λ(Rd ) .

We also have
|MB(b)− bχB| = |M(bχB)χB − bM(χB)χB|

≤ |M(bχB)− bM(χB)| = |[b,M]χB|.

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 7Applying Hölder’s inequality, Proposition 2.3 and Proposition 2.2, we get
|B|−1

∫
B

|b(z)−MB(b)(z)|dz

≤
(
|B|−1

∫
B

|b(z)−MB(b)(z)|qdz
) 1

q

≤ |B|−
1
q

(∫
B

|[b,M]χB(z)|qdz
) 1

q

<∼ t−
d
q [t]

d(− 1
α
+ 1
q
)

1 [1/t]
d( 1

λ
− 1
q
)

1 ‖[b,M]χB‖(Lq ,L∞)α,λ(Rd )

<∼ t−
d
q [t]

d(− 1
α
+ 1
q
)

1 [1/t]
d( 1

λ
− 1
q
)

1 ‖[b,M]χB‖(Lq ,Lp)α,λ(Rd )

<∼ t−
d
q [t]

d(− 1
α
+ 1
q
)

1 [1/t]
d( 1

λ
− 1
q
)

1 ‖χB‖(Lq ,Lp)α,λ(Rd )

<∼ t−
d
q [t]

d(− 1
α
+ 1
q
)

1 [1/t]
d( 1

λ
− 1
q
)

1 max{‖χB‖q,p,α , ‖χB‖q,p,λ}.

It follows from (1.2) that
|B|−1

∫
B

|b(z)−MB(b)(z)|dz <∼ t
− d
q [t]

d(− 1
α
+ 1
q
)

1 [1/t]
d( 1

λ
− 1
q
)

1 max{t
d
α , t

d
λ }.

So, by Lemma 3.3, we obtain
|B|−1

∫
B

|b(z)−MB(b)(z)|dz <∼ 2.Denote by
E := {y ∈ B : b(y) ≤ bB}, F := {y ∈ B : b(y) > bB}.Since ∫

E

|b(z)− bB|dz =
∫
F

|b(z)− bB|dz,in view of the inequality b(x) ≤ bB ≤ MB(b), for x ∈ E, we get
|B|−1

∫
B

|b(z)− bB|dz = 2|B|−1
∫
E

|b(z)− bB|dz

≤ 2|B|−1
∫
E

|b(z)−MB(b)(z)|dz

≤ 2|B|−1
∫
B

|b(z)−MB(b)(z)|dz <∼ 4.

By taking in the left hand side the supremum over all t > 0 and x ∈ Rd , we obtain
‖b‖BMO(Rd ) <∞.In order to show that b− ∈ L∞(Rd), note that MB(b) ≥ |b|. Hence

0 ≤ b− = |b| − b+ ≤ MB(b)− b+ ≤ MB(b)− b+ + b− = MB(b)− b.

Thus
(b−)B <∼ 2,

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 8and by the Lebesgue Differentiation Theorem we get
b−(x) <∼ 2for almost every x ∈ Rd . �

4. Proof of Theorem 1.2 and Theorem 1.3
We recall that the proofs of Theorem 1.2 and Theorem 1.3 are simply an adaptation of thosegiven in [5] (see also [6]).

Proof of Theorem 1.2. Let 1 < q ≤ λ,α < p <∞ such that 1q+ 2p < 1
α+

1
λ and f ∈ (Lq, Lp)α,λ(Rd).Fix y ∈ Rd and r > 0 we have

f = f χB(y,2r) +

∞∑
i=1

f χB(y,2i+1r)\B(y,2i r).

By the sublinearity of T and the condition (1.3) we obtain
|T f | <∼ |T (f χB(y,2r))|+

∞∑
i=1

|B(y , 2i+1r)|−1
∫
B(y,2i+1r)

|f (x)|dx

and therefore, an application of Hölder inequality leads to
|T f | <∼ |T (f χB(y,2r))|+

∞∑
i=1

|B(y , 2i+1r)|−
1
q ‖f χB(y,2i+1r)‖q.

Taking the Lq-norm of both sides on the ball B(y , r) and using the boundedness of T on Lq , weget
‖(T f )χB(y,r)‖q <∼ ‖f χB(y,2r)‖q +

∞∑
i=1

(2i)−
d
q ‖f χB(y,2i+1r)‖q.

Taking the Lp-norm of both sides with respect to y , it comes that
r ‖T f ‖q,p <∼ 2r ‖f ‖q,p +

∞∑
i=1

(2i)−
d
q
2i+1r ‖f ‖q,p .

On the one hand, we have,
2r ‖f ‖q,p =

[2r ]
d( 1

α
− 1
q
− 1
p
)

1 [1/2r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1

[2r ]
d( 1

α
− 1
q
− 1
p
)

1 [1/2r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1

2r ‖f ‖q,p

≤ [2r ]
d(− 1

α
+ 1
q
+ 1
p
)

1 [1/2r ]
d( 1

λ
− 1
q
− 1
p
)

1 ‖f ‖(Lq ,Lp)α,λ(Rd )

≤ (2[r ]1)
d(− 1

α
+ 1
q
+ 1
p
)

(
1

2
[1/r ]1

)d( 1
λ
− 1
q
− 1
p
)

‖f ‖(Lq ,Lp)α,λ(Rd ) .

Hence,
2r ‖f ‖q,p <∼ [r ]

d(− 1
α
+ 1
q
+ 1
p
)

1 [1/r ]
d( 1

λ
− 1
q
− 1
p
)

1 ‖f ‖(Lq ,Lp)α,λ(Rd ) . (4.1)

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 9On the other hand,
∞∑
i=1

(2i)−
d
q
2i+1r ‖f ‖q,p

=

∞∑
i=1

(2i)−
d
q
[2i+1r ]

d( 1
α
− 1
q
− 1
p
)

1 [1/2i+1r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1

[2i+1r ]
d( 1

α
− 1
q
− 1
p
)

1 [1/2i+1r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1

2i+1r ‖f ‖q,p

≤ ‖f ‖(Lq ,Lp)α,λ(Rd )
∞∑
i=1

(2i)−
d
q [2i+1r ]

d(− 1
α
+ 1
q
+ 1
p
)

1 [1/2i+1r ]
d( 1

λ
− 1
q
− 1
p
)

1

<∼ [r ]
d(− 1

α
+ 1
q
+ 1
p
)

1 [1/r ]
d( 1

λ
− 1
q
− 1
p
)

1 ‖f ‖(Lq ,Lp)α,λ(Rd )
∞∑
i=1

(2i)d(−
1
α
− 1
λ
+ 1
q
+ 2
p
).

Since − 1α − 1λ + 1q + 2p < 0, ∑∞i=1(2i)d(− 1α− 1λ+ 1q+ 2p ) <∞. Therefore,
∞∑
i=1

(2i)−
d
q
2i+1r ‖f ‖q,p (4.2)

<∼ [r ]
d(− 1

α
+ 1
q
+ 1
p
)

1 [1/r ]
d( 1

λ
− 1
q
− 1
p
)

1 ‖f ‖(Lq ,Lp)α,λ(Rd ) .

From (4.1) and (4.2), we deduce that
r ‖T f ‖q,p <∼ [r ]

d(− 1
α
+ 1
q
+ 1
p
)

1 [1/r ]
d( 1

λ
− 1
q
− 1
p
)

1 ‖f ‖(Lq ,Lp)α,λ(Rd ) .

It follows that
[r ]
d( 1

α
− 1
q
− 1
p
)

1 [1/r ]
d(− 1

λ
+ 1
q
+ 1
p
)

1 r ‖T f ‖q,p <∼ ‖f ‖(Lq ,Lp)α,λ(Rd ) . (4.3)
We obtain the desired result by taking the supremum over all r > 0 in the left hand side of (4.3). �
Proof of Theorem 1.3. Let 1 < q ≤ λ,α < p <∞ such that 1q + 2p < 1

α +
1
λ and b ∈ BMO(Rd).Let f be any element of f ∈ (Lq, Lp)α,λ(Rd). We recall that (Lq, Lp)α,λ(Rd) is a subspace of theMorrey space Lq,d(1− q

α
)(Rd). Proceeding as in the proof of [5, Theorem 2.2 ], we have that for all

y ∈ Rd and r > 0,
‖[b, T ]f χB(y,r)‖q

<∼ ‖f χB(y,2r)‖q

+

∞∑
i=1

(2i r)−d
[∫
B(y,r)

(∫
B(y,2i+1r)

|b(x)− b(z)||f (x)|dx
)q
dz

] 1
p

.

Therefore, using the John-Nirenberg theorem on BMO-functions (see [12, Corollary 7.1.8]), weobtain
‖[b, T ]f χB(y,r)‖q <∼ ‖f χB(y,2r)‖q + ‖b‖BMO(Rd )

∞∑
i=1

(2i)−
d
q ‖f χB(y,2i+1r)‖q.

Using the same argument as in the proof of Theorem 1.2, we end the proof. �

https://doi.org/10.28924/ada/ma.4.22


Eur. J. Math. Anal. 10.28924/ada/ma.4.22 10References
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https://doi.org/10.28924/ada/ma.4.22

	1. Introduction and main results
	2. Some properties of total Fofana spaces
	3.  Proof of Theorem 1.1
	4.  Proof of Theorem 1.2 and Theorem 1.3
	References

