International Journal of Analysis and Applications

Volume 17, Number 4 (2019), 596-619

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-596

MULTILINEAR BMO ESTIMATES FOR THE COMMUTATORS OF MULTILINEAR

FRACTIONAL MAXIMAL AND INTEGRAL OPERATORS ON THE PRODUCT

GENERALIZED MORREY SPACES

FERİT GÜRBÜZ

Hakkari University, Faculty of Education, Department of Mathematics Education, Hakkari 30000, Turkey

Corresponding author: feritgurbuz@hakkari.edu.tr

Abstract. In this paper, we establish multilinear BMO estimates for commutators of multilinear frac-

tional maximal and integral operators both on product generalized Morrey spaces and product generalized

vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal

and singular integral operators.

1. Introduction and main results

The classical Morrey spaces Lp,λ have been introduced by Morrey in [21] to study the local behavior

of solutions of second order elliptic partial differential equations(PDEs). In recent years there has been an

explosion of interest in the study of the boundedness of operators on Morrey-type spaces. It has been obtained

that many properties of solutions to PDEs are concerned with the boundedness of some operators on Morrey-

type spaces. Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces

can not handle. Morrey spaces which we have been handling are called classical Morrey spaces(see [21]).

But, classical Morrey spaces are not totally enough to describe the boundedness properties. To this end, we

need to generalize parameters p and q, among others p, but this issue will exceed the scope of the article,

so we pass this part. Though we do not consider the direct applications of Morrey spaces to PDEs, Morrey

Received 2019-03-25; accepted 2019-04-22; published 2019-07-01.

2010 Mathematics Subject Classification. 42B20, 42B25, 42B35.

Key words and phrases. multi-sublinear fractional maximal operator; multilinear fractional integral operator; multilinear

commutator; generalized Morrey space; generalized vanishing Morrey space; multilinear BMO space.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

596

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-596


Int. J. Anal. Appl. 17 (4) (2019) 597

spaces can be applied to PDEs. Applications to the second order elliptic partial differential equations can

be found in [10] and [26].

We will say that a function f ∈ Lp,λ = Lp,λ (Rn) if

sup
x∈Rn,r>0


r−λ ∫

B(x,r)

|f (y)|p dy



1/p

< ∞. (1.1)

Here, 1 < p < ∞ and 0 < λ < n and the quantity of (1.1) is the (p,λ)-Morrey norm, denoted by ‖f‖Lp,λ. In

recent years, more and more researches focus on function spaces based on Morrey spaces to fill in some gaps

in the theory of Morrey type spaces (see, for example, [11–14, 16, 18, 23, 25, 29]). Moreover, these spaces are

useful in harmonic analysis and PDEs. But, this topic exceeds the scope of this paper. Thus, we omit the

details here. On the other hand, the study of the operators of harmonic analysis in vanishing Morrey space,

in fact has been almost not touched. A version of the classical Morrey space Lp,λ(Rn) where it is possible to

approximate by ”nice” functions is the so called vanishing Morrey space V Lp,λ(Rn) has been introduced by

Vitanza in [27] and has been applied there to obtain a regularity result for elliptic PDEs. This is a subspace

of functions in Lp,λ(Rn), which satisfies the condition

lim
r→0

sup
x∈Rn
0<t<r


t−λ ∫

B(x,t)

|f (y)|p dy



1/p

= 0,

where 1 < p < ∞ and 0 < λ < n for brevity, so that

V Lp,λ(Rn) =


f ∈ Lp,λ(Rn) : limr→0 supx∈Rn

0<t<r

t−
λ
p‖f‖Lp(B(x,t)) = 0


 .

For the properties and applications of vanishing Morrey spaces, see also [1].

After studying Morrey spaces in detail, researchers have passed to the concept of generalized Morrey

spaces. Firstly, motivated by the work of [21], Mizuhara [19] introduced generalized Morrey spaces Mp,ϕ.

Then, the generalized Morrey spaces Mp,ϕ with normalized norm is defined as follows:

Definition 1.1. (generalized Morrey space) Let ϕ(x,r) be a positive measurable function on Rn×(0,∞).

If 0 < p < ∞, then the generalized Morrey space Mp,ϕ ≡ Mp,ϕ(Rn) is defined by{
f ∈ Llocp (R

n) : ‖f‖Mp,ϕ = sup
x∈Rn,r>0

ϕ(x,r)−1|B(x,r)|−
1
p‖f‖Lp(B(x,r)) < ∞

}
.

Obviously, the above definition recover the definition of Lp,λ(Rn) if we choose ϕ(x,r) = r
λ−n
p , that is

Lp,λ (Rn) = Mp,ϕ (Rn) |
ϕ(x,r)=r

λ−n
p

.



Int. J. Anal. Appl. 17 (4) (2019) 598

Everywhere in the sequel we assume that inf
x∈Rn,r>0

ϕ(x,r) > 0 which makes the above spaces non-trivial,

since the spaces of bounded functions are contained in these spaces. We point out that ϕ(x,r) is a measurable

nonnegative function and no monotonicity type condition is imposed on these spaces.

In [12], [16], [18], [19] and [25], the boundedness of the maximal operator and Calderón-Zygmund operator

on the generalized Morrey spaces Mp,ϕ has been obtained, respectively.

For brevity, in the sequel we use the notations

Mp,ϕ (f; x,r) :=
|B(x,r)|−

1
p ‖f‖Lp(B(x,r))
ϕ(x,r)

,

and

MWp,ϕ (f; x,r) :=
|B(x,r)|−

1
p ‖f‖WLp(B(x,r))
ϕ(x,r)

.

In this paper, extending the definition of vanishing Morrey spaces [27], we introduce generalized vanishing

Morrey spaces V Mp,ϕ(Rn) with normalized norm in the following form.

Definition 1.2. (generalized vanishing Morrey space) The generalized vanishing Morrey space V Mp,ϕ(Rn)

is defined by {
f ∈ Mp,ϕ(Rn) : lim

r→0
sup
x∈Rn

Mp,ϕ (f; x,r) = 0

}
.

Everywhere in the sequel we assume that

lim
r→0

1

inf
x∈Rn

ϕ(x,r)
= 0, (1.2)

and

sup
0<r<∞

1

inf
x∈Rn

ϕ(x,r)
< ∞, (1.3)

which make the spaces V Mp,ϕ(Rn) non-trivial, because bounded functions with compact support belong to

this space. The spaces V Mp,ϕ(Rn) are Banach spaces with respect to the norm

‖f‖V Mp,ϕ ≡‖f‖Mp,ϕ = sup
x∈Rn,r>0

Mp,ϕ (f; x,r) .

The spaces V Mp,ϕ(Rn) are also closed subspaces of the Banach spaces Mp,ϕ(Rn), which may be shown by

standard means.

Furthermore, we have the following embeddings:

V Mp,ϕ ⊂ Mp,ϕ, ‖f‖Mp,ϕ ≤‖f‖V Mp,ϕ.

On the other hand, it is well known that, for the purpose of researching non-smoothness partial differ-

ential equation, mathematicians pay more attention to the singular integrals. Moreover, the fractional

type operators and their weighted boundedness theory play important roles in harmonic analysis and



Int. J. Anal. Appl. 17 (4) (2019) 599

other fields, and the multilinear operators arise in numerous situations involving product-like operations,

see [2, 3, 5–8, 14, 17, 20, 24] for instance.

First of all, we recall some basic properties and notations used in this paper.

Let Rn be the n-dimensional Euclidean space of points x = (x1, ...,xn) with norm |x| =
(∑n

i=1x
2
i

)1/2
and

corresponding m-fold product spaces (m ∈ N) be (Rn)m = Rn ×···×Rn. Let B = B(x,r) denotes open ball

centered at x of radius r for x ∈ Rn and r > 0 and Bc(x,r) its complement. Also |B(x,r)| is the Lebesgue

measure of the ball B(x,r) and |B(x,r)| = vnrn, where vn = |B(0, 1)|. We also denote by −→y = (y1, . . . ,ym),

d−→y = dy1 . . .dym, and by
−→
f the m-tuple (f1, ...,fm), m, n the nonnegative integers with n ≥ 2, m ≥ 1.

Let
−→
f ∈ Llocp1 (R

n) ×···×Llocpm (R
n). Then multi-sublinear fractional maximal operator M

(m)
α is defined

by

M(m)α

(−→
f
)

(x) = sup
t>0
|B (x,t)|

α
n


 m∏
i=1

1

|B (x,t)|

∫
B(x,t)

|fi (yi)|


d−→y , 0 ≤ α < mn.

From definition, if α = 0 then M
(m)
α is the multi-sublinear maximal operator M

(m) and also; in the case of

m = 1, M
(m)
α is the classical fractional maximal operator Mα.

The theory of multilinear Calderón-Zygmund singular integral operators, originated from the works of

Coifman and Meyer’s [4], plays an important role in harmonic analysis. Its study has been attracting a lot of

attention in the last few decades. A systematic analysis of many basic properties of such multilinear singular

integral operators can be found in the articles by Coifman-Meyer [4], Grafakos-Torres [7–9], Chen et al. [2],

Fu et al. [5].

Let T(m) (m ∈ N) be a multilinear operator initially defined on the m-fold product of Schwartz spaces

and taking values into the space of tempered distributions,

T(m) : S (Rn) ×···×S (Rn) → S (Rn) .

Following [7], recall that the m(multi)-linear Calderón-Zygmund operator T(m) (m ∈ N) for test vector
−→
f = (f1, . . . ,fm) is defined by

T(m)
(−→
f
)

(x) =

∫
(Rn)m

K (x,y1, . . . ,ym)

{
m∏
i=1

fi (yi)

}
dy1 · · ·dym, x /∈

m⋂
i=1

suppfi,

where K is an m-Calderón-Zygmund kernel which is a locally integrable function defined off the diagonal

y0 = y1 = · · · = ym on (Rn)
m+1

satisfying the following size estimate:

|K (x,y1, . . . ,ym)| ≤
C

|(x−y1, . . . ,x−ym)|
mn ,

for some C > 0 and some smoothness estimates, see [7–9] for details.

The result of Grafakos and Torres [7,9] shows that the multilinear Calderón-Zygmund operator is bounded

on the product of Lebesgue spaces.



Int. J. Anal. Appl. 17 (4) (2019) 600

Theorem 1.1. [7,9] Let T(m) (m ∈ N) be an m-linear Calderón-Zygmund operator. Then, for any numbers

1 ≤ p1, . . . ,pm,p < ∞ with 1p =
1
p1

+· · ·+ 1
pm

, T(m) can be extended to a bounded operator from Lp1×···×Lpm
into Lp, and bounded from L1 ×···×L1 into L 1

m
,∞.

On the other hand, the multilinear fractional type operators are natural generalization of linear ones.

Their earliest version was originated on the work of Grafakos [6] in 1992, in which he studied the multilinear

maximal function and multilinear fractional integral defined by

M(m)α

(−→
f
)

(x) = sup
t>0

1

rn−α

∫
|y|<r

∣∣∣∣∣
m∏
i=1

fi (x−θiy)

∣∣∣∣∣dy,
and

I(m)α

(−→
f
)

(x) =

∫
Rn

1

|y|n−α
m∏
i=1

fi (x−θiy) dy,

where θi (i = 1, . . . ,m) are fixed distinct are nonzero real numbers and 0 < α < n. We note that, if we simply

take m = 1 and θi = 1, then Mα and Iα are just the operators studied by Muckenhoupt and Wheeden in [22].

In this paper we deal with another kind of multilinear operator which was defined by Kenig and Stein [17]

for
−→
f = (f1, . . . ,fm), which is called multilinear fractional integral operator as follows

I(m)α

(−→
f
)

(x) =

∫
(Rn)m

1

|(x−y1, . . . ,x−ym)|
mn−α

{
m∏
i=1

fi (yi)

}
d−→y ,

whose kernel is

|K (x,y1, . . . ,ym)| = |(x−y1, . . . ,x−ym)|
−mn+α

, 0 < α < mn,

where f1, . . . ,fm : Rn → R are measurable and |(x−y1, . . . ,x−ym)| =
√

m∑
i=1

|x−yi|
2
.

They [17] proved that I
(m)
α (m ∈ N) is of strong type (Lp1 ×Lp2 ×···×Lpm,Lq) and weak type

(Lp1 ×Lp2 ×···×Lpm,Lq,∞). If we take m = 1, I
(m)
α (m ∈ N) is the classical fractional integral operator Iα.

Moreover, their main result (Theorem 1 in [17]) is the multi-version of well-known Hardy-Littlewood-Sobolev

inequality. Later, weighted inequalities for the multilinear fractional integral operators have been established

by Moen [20] and Chen-Xue [3], respectively. Yu and Tao [29] have also obtained the boundedness of the

operators I
(m)
α , T

(m) and M(m) (m ∈ N) on the product generalized Morrey spaces, respectively.

Now, we will give some properties related to the space of functions of Bounded Mean Oscillation, BMO,

which play a great role in the proofs of our main results, introduced by John and Nirenberg [15] in 1961.

This space has become extremely important in various areas of analysis including harmonic analysis, PDEs

and function theory. BMO-spaces are also of interest since, in the scale of Lebesgue spaces, they may be

considered and appropriate substitute for L∞. Appropriate in the sense that are spaces preserved by a wide

class of important operators such as the Hardy-Littlewood maximal function, the Hilbert transform and

which can be used as an end point in interpolating Lp spaces.



Int. J. Anal. Appl. 17 (4) (2019) 601

Let us recall the definition of the space of BMO(Rn).

Definition 1.3. [12, 16] The space BMO(Rn) of functions of bounded mean oscillation consists of locally

summable functions with finite semi-norm

‖b‖∗ ≡‖b‖BMO = sup
x∈Rn,r>0

1

|B(x,r)|

∫
B(x,r)

|b(y) − bB(x,r)|dy < ∞, (1.4)

where bB(x,r) is the mean value of the function b on the ball B(x,r). The fact that precisely the mean value

bB(x,r) figures in (1.4) is inessential and one gets an equivalent seminorm if bB(x,r) is replaced by an arbitrary

constant c :

‖b‖∗ ≈ sup
r>0

inf
c∈C

1

|B (x,r)|

∫
B(x,r)

|b (y) − c|dy.

Each bounded function b ∈ BMO. Moreover, BMO contains unbounded functions, in fact log|x| belongs to

BMO but is not bounded, so L∞(Rn) ⊂ BMO(Rn).

In 1961 John and Nirenberg [15] established the following deep property of functions from BMO.

Theorem 1.2. [15] If b ∈ BMO(Rn) and B (x,r) is a ball, then

∣∣{x ∈ B (x,r) : |b(x) − bB(x,r)| > ξ}∣∣ ≤ |B (x,r) |exp (− ξ
C‖b‖∗

)
, ξ > 0,

where C depends only on the dimension n.

By Theorem 1.2, we can get the following results.

Corollary 1.1. [12, 16] Let b ∈ BMO(Rn). Then, for any q > 1,

‖b‖∗ ≈ sup
x∈Rn,r>0


 1|B(x,r)|

∫
B(x,r)

|b(y) − bB(x,r)|pdy




1
p

(1.5)

is valid.

Corollary 1.2. [12, 16] Let b ∈ BMO(Rn). Then there is a constant C > 0 such that

∣∣bB(x,r) − bB(x,t)∣∣ ≤ C‖b‖∗(1 + ln t
r

)
for 0 < 2r < t, (1.6)

and for any q > 1, it is easy to see that

‖b− (b)B‖Lq(B) ≤ Cr
n
q ‖b‖∗

(
1 + ln

t

r

)
, (1.7)

where C is independent of b, x, r and t.

Now inspired by Definition 1.3, we can give the definition of multilinear BMO (= BMO). Indeed in this

paper we will consider a multilinear version (= multilinear BMO or BMO) of the BMO.



Int. J. Anal. Appl. 17 (4) (2019) 602

Definition 1.4. We say that
−→
b = (b1, . . . ,bm) ∈ BMO if

∥∥∥−→b ∥∥∥
BMO

= sup
x∈Rn,r>0

m∏
i=1

1

|B(x,r)|

∫
B(x,r)

∣∣∣bi (yi) − (bi)B(x,r)∣∣∣dyi < ∞,
where

(bi)B(x,r) =
1

|B(x,r)|

∫
B(x,r)

bi(yi)dyi.

Remark 1.1. Notice that (BMO)
m

is contained in BMO and we have

∥∥∥−→b ∥∥∥
BMO

≤
m∏
i=1

‖bi‖∗ ,

so

(BMO)
m ⊂ BMO

is valid.

We now make some conventions. Throughout this paper, we use the symbol A . B to denote that there

exists a positive consant C such that A ≤ CB. If A . B and B . A, we then write A ≈ B and say that

A and B are equivalent. For a fixed p ∈ [1,∞), p′ denotes the dual or conjugate exponent of p, namely,

p′ = p
p−1 and we use the convention 1

′ = ∞ and ∞′ = 1.

Remark 1.2. Let 0 < α < mn and 1 < pi < ∞ with 1p =
m∑
i=1

1
pi

, 1
qi

= 1
pi
− α

mn
, 1
q

=

m∑
i=1

1
qi

= 1
p
− α

n
and

−→
b = (b1, . . . ,bm) ∈ (BMO)

m
for i = 1, . . . ,m. Then, from Corollary 1.2, it is easy to see that

m∏
i=1

‖bi − (bi)B‖Lqi(B) ≤ C
m∏
i=1

|B(x,r)|
1
qi ‖bi‖∗

(
1 + ln

t

r

)
, (1.8)

and

m∏
i=1

‖bi − (bi)B‖Lqi(2B) ≤
m∏
i=1

(
‖bi − (bi)2B‖Lqi(2B) + ‖(bi)B − (bi)2B‖Lqi(2B)

)
.

m∏
i=1

|B(x,r)|
1
qi ‖bi‖∗

(
1 + ln

t

r

)
. (1.9)

On the other hand, Xu [28] has established the boundedness of the commutators generated by m-linear

Calderón-Zygmund singular integrals and RBMO functions with nonhomogeneity on the product of Lebesgue

space. Inspired by [2, 3, 7, 9, 24, 28], commutators T
(m)
−→
b

generated by m-linear Calderón-Zygmund operators

T(m) and bounded mean oscillation functions
−→
b = (b1, . . . ,bm) is given by

T
(m)
−→
b

(−→
f
)

(x) =

∫
(Rn)m

K (x,y1, . . . ,ym)

[
m∏
i=1

[bi (x) − bi (yi)] fi (yi)

]
d−→y ,



Int. J. Anal. Appl. 17 (4) (2019) 603

where K (x,y1, . . . ,ym) is a m-linear Calderón-Zygmund kernel, bi ∈ (BMO)
i
(Rn) for i = 1, . . . ,m. Note

that Tb is the special case of T
(m)
−→
b

with taking m = 1. Similarly, let bi (i = 1, . . . ,m) be a locally in-

tegrable functions on Rn, then the commutators generated by m-linear fractional integral operators and
−→
b = (b1, . . . ,bm) is given by

I
(m)

α,
−→
b

(−→
f
)

(x) =

∫
(Rn)m

1

|(x−y1, . . . ,x−ym)|
mn−α

[
m∏
i=1

[bi (x) − bi (yi)] fi (yi)

]
d−→y ,

where 0 < α < mn, and fi (i = 1, . . . ,m) are suitable functions.

The commutators of a class of multi-sublinear maximal operators corresponding to T
(m)
−→
b

and I
(m)

α,
−→
b

(m ∈ N)

above are, respectively, defined by

M
(m)
−→
b

(−→
f
)

(x) = sup
t>0


 m∏
i=1

1

|B (x,t)|

∫
B(x,t)

[|bi (x) − bi (yi)|] |fi (yi)|


d−→y ,

and

M
(m)

α,
−→
b

(−→
f
)

(x) = sup
t>0
|B (x,t)|

α
n


 m∏
i=1

1

|B (x,t)|

∫
B(x,t)

[|bi (x) − bi (yi)|] |fi (yi)|


d−→y , 0 ≤ α < mn.

The following result is known.

Lemma 1.1. [24] (Strong bounds of I
(m)
−→
b ,α

) Let 0 < αi < n, 1 < p1, . . . ,pm < ∞, 1p =
m∑
i=1

1
pi

, α =

m∑
i=1

αi and

1
q

= 1
p
− α

n
. Then there is C > 0 independent of

−→
f and

−→
b such that

∥∥∥I(m)−→
b ,α

(−→
f
)∥∥∥

Lq(Rn)
≤ C

m∏
i=1

‖bi‖∗‖fi‖Lpi(Rn) .

Using the idea in the proof of Lemma 3.2 in [13], we can obtain the following Corollary 1.3:

Corollary 1.3. (Strong bounds of M
(m)

α,
−→
b

) Under the assumptions of Lemma 1.1, the operator M
(m)

α,
−→
b

is

bounded from Lp1 (R
n) ×···Lpm(Rn) to Lq(Rn). Moreover, we have

∥∥∥M(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(Rn)
≤ C

m∏
i=1

‖bi‖∗‖fi‖Lpi(Rn) .

Proof. Set

Ĩ
(m)
−→
b ,α

(|f|) (x) =
∫

(Rn)m

1

|(x−y1, . . . ,x−ym)|
mn−α

[
m∏
i=1

[|bi (x) − bi (yi)|] |fi (yi)|

]
d−→y 0 < α < mn.



Int. J. Anal. Appl. 17 (4) (2019) 604

It is easy to see that Lemma 1.1 holds for Ĩ
(m)
−→
b ,α

. On the other hand, for any t > 0, we have

Ĩ
(m)
−→
b ,α

(|f|) (x) ≥
∫

(B(x,t))m

1

|(x−y1, . . . ,x−ym)|
mn−α

[
m∏
i=1

[|bi (x) − bi (yi)|] |fi (yi)|

]
d−→y

≥
1

tmn−α

∫
B(x,t)

[
m∏
i=1

[|bi (x) − bi (yi)|] |fi (yi)|

]
d−→y .

Taking supremum over t > 0 in the above inequality, we get

M
(m)

α,
−→
b

(−→
f
)

(x) ≤ C−1n,αĨ
(m)
−→
b ,α

(|f|) (x) Cn,α = |B (0, 1)|
mn−α
n . (1.10)

�

As a simple corollary of Lemma 1.1 and Corollary 1.3, we can obtain the following result.

Corollary 1.4. (Strong bounds of T
(m)
−→
b

and M
(m)
−→
b

) Let 1 < p1, . . . ,pm < ∞ and 0 < p < ∞ with 1p =
m∑
i=1

1
pi

.

Then there is C > 0 independent of
−→
f and

−→
b such that

∥∥∥T(m)−→
b

(−→
f
)∥∥∥

Lp(Rn)
≤ C

m∏
i=1

‖bi‖∗‖fi‖Lpi(Rn) ,

∥∥∥M(m)−→
b

(−→
f
)∥∥∥

Lp(Rn)
≤ C

m∏
i=1

‖bi‖∗‖fi‖Lpi(Rn) .

The purpose of this paper is to consider the mapping properties on Mp1,ϕ1 ×···×Mpm,ϕm and V Mp1,ϕ1 ×

···×V Mpm,ϕm for the commutators of multilinear fractional maximal and integral operators, respectively.

Similar results still hold for commutators of multilinear maximal and singular integral operators. Commu-

tators of multilinear fractional maximal and integral operators on product generalized Morrey spaces have

not also been studied so far and this paper seems to be the first in this direction. Now, let us state the main

results of this paper.

Theorem 1.3. Let 0 < α < mn and 1 ≤ pi < mnα with
1
p

=

m∑
i=1

1
pi

,

1
q

=
m∑
i=1

1
pi

+
m∑
i=1

1
qi
− α

n
and
−→
b ∈ (BMO)m (Rn) for i = 1, . . . ,m. Let functions ϕ,ϕi : Rn×(0,∞) → (0,∞)

(i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies the condition

∞∫
r

(
1 + ln

t

r

)m essinft<τ<∞
m∏
i=1

ϕi(x,τ)τ
n
p

t

n


1q−

m∑
i=1

1
qi


+1

dt ≤ Cϕ (x,r) , (1.11)

where C does not depend on x ∈ Rn and r > 0.



Int. J. Anal. Appl. 17 (4) (2019) 605

Then, I
(m)

α,
−→
b

and M
(m)

α,
−→
b

(m ∈ N) are bounded operators from product space Mp1,ϕ1 ×···×Mpm,ϕm to Mq,ϕ.

Moreover, we have ∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Mq,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖Mpi,ϕi , (1.12)

∥∥∥M(m)
α,
−→
b

(−→
f
)∥∥∥

Mq,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖Mpi,ϕi . (1.13)

Corollary 1.5. Let 1 < pi < ∞ and 0 < p < ∞ with 1p =
m∑
i=1

1
pi

and
−→
b ∈ (BMO)m (Rn) for i = 1, . . . ,m.

Let functions ϕ,ϕi : Rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies the condition

∞∫
r

(
1 + ln

t

r

)m essinft<τ<∞
m∏
i=1

ϕi(x,τ)τ
n
p

t
n
p
+1

dt ≤ Cϕ (x,r) ,

where C does not depend on x ∈ Rn and r > 0.

Then, T
(m)
−→
b

and M
(m)
−→
b

(m ∈ N) are bounded operators from product space Mp1,ϕ1 ×···×Mpm,ϕm to Mp,ϕ.

Moreover, we have ∥∥∥T(m)−→
b

(−→
f
)∥∥∥

Mp,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖Mpi,ϕi ,

∥∥∥M(m)−→
b

(−→
f
)∥∥∥

Mp,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖Mpi,ϕi .

Our another main result is the following.

Theorem 1.4. Let 0 < α < mn and 1 ≤ pi < mnα with
1
p

=

m∑
i=1

1
pi

,

1
q

=
m∑
i=1

1
pi

+
m∑
i=1

1
qi
− α

n
and
−→
b ∈ (BMO)m (Rn) for i = 1, . . . ,m. Let functions ϕ,ϕi : Rn×(0,∞) → (0,∞)

(i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies conditions (1.2)-(1.3) and

∞∫
r

(
1 + ln

t

r

)m m∏
i=1

ϕi(x,t)
t
n
p

t

n


1q−

m∑
i=1

1
qi


+1

dt ≤ C0ϕ (x,r) , (1.14)

where C0 does not depend on x ∈ Rn and r > 0,

lim
r→0

ln 1
r

inf
x∈Rn

ϕ(x,r)
= 0 (1.15)

and

cδ :=

∞∫
δ

(1 + ln |t|)m sup
x∈Rn

m∏
i=1

ϕi(x,t)
t
n
p

t

n


1q−

m∑
i=1

1
qi


+1

dt < ∞ (1.16)

for every δ > 0.



Int. J. Anal. Appl. 17 (4) (2019) 606

Then, I
(m)

α,
−→
b

and M
(m)

α,
−→
b

(m ∈ N) are bounded operators from product space V Mp1,ϕ1 ×···×V Mpm,ϕm to

V Mq,ϕ. Moreover, we have∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

V Mq,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖V Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖V Mpi,ϕi , (1.17)

∥∥∥M(m)
α,
−→
b

(−→
f
)∥∥∥

V Mq,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖V Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖V Mpi,ϕi . (1.18)

Corollary 1.6. Let 1 < pi < ∞ and 0 < p < ∞ with 1p =
m∑
i=1

1
pi

and
−→
b ∈ (BMO)m (Rn) for i = 1, . . . ,m.

Let functions ϕ,ϕi : Rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies conditions (1.2)-(1.3)

and
∞∫
r

(
1 + ln

t

r

)m m∏
i=1

ϕi(x,t)
t
n
p

t
n
p
+1
dt ≤ C0ϕ (x,r) ,

where C0 does not depend on x ∈ Rn and r > 0,

lim
r→0

ln 1
r

inf
x∈Rn

ϕ(x,r)
= 0

and

cδ :=

∞∫
δ

(1 + ln |t|)m sup
x∈Rn

m∏
i=1

ϕi(x,t)
t
n
p

t
n
p
+1
dt < ∞

for every δ > 0.

Then, T
(m)
−→
b

and M
(m)
−→
b

(m ∈ N) are bounded operators from product space V Mp1,ϕ1 ×···×V Mpm,ϕm to

V Mp,ϕ. Moreover, we have∥∥∥T(m)−→
b

(−→
f
)∥∥∥

V Mp,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖V Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖V Mpi,ϕi ,

∥∥∥M(m)−→
b

(−→
f
)∥∥∥

V Mp,ϕ
.
∥∥∥−→b ∥∥∥

BMO
‖fi‖V Mpi,ϕi .

m∏
i=1

‖bi‖∗‖fi‖V Mpi,ϕi .

The article is organized as follows. A key lemma is given and proved in Section 2. Section 3 will be

devoted to the proofs of the theorems (Theorems 1.3 and 1.4) stated above.

2. A Key Lemma

In order to prove the main results (Theorems 1.3 and 1.4), we need the following lemma.

Lemma 2.1. Let x0 ∈ Rn, 0 < α < mn and 1 ≤ pi < mnα with
1
p

=

m∑
i=1

1
pi

, 1
q

=
m∑
i=1

1
pi

+
m∑
i=1

1
qi
− α

n
and

−→
b ∈ (BMO)m (Rn) for i = 1, . . . ,m. Then the inequality

‖I(m)
α,
−→
b

(−→
f
)
‖Lq(B(x0,r)) .

m∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

(2.1)



Int. J. Anal. Appl. 17 (4) (2019) 607

holds for any ball B(x0,r) and for all
−→
f ∈ Llocp1 (R

n) ×···×Llocpm (R
n).

Proof. In order to simplify the proof, we consider only the situation when m = 2. Actually, a similar

procedure works for all m ∈ N. Thus, without loss of generality, it is sufficient to show that the conclusion

holds for I
(2)

α,
−→
b

(−→
f
)

= I
(2)
α,(b1,b2)

(f1,f2).

We just consider the case pi > 1 for i = 1, 2. For any x0 ∈ Rn, set B = B (x0,r) for the ball centered at

x0 and of radius r and 2B = B (x0, 2r). Indeed, we also decompose fi as fi (yi) = fi (yi) χ2B + fi (yi) χ(2B)c

for i = 1, 2. And, we write f1 = f
0
1 + f

∞
1 and f2 = f

0
2 + f

∞
2 , where f

0
i = fiχ2B, f

∞
i = fiχ(2B)c, for i = 1, 2.

Thus, we have

∥∥∥I(2)α,(b1,b2) (f1,f2)∥∥∥Lq(B(x0,r)) ≤
∥∥∥I(2)α,(b1,b2) (f01 ,f02)∥∥∥Lq(B(x0,r)) +

∥∥∥I(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥Lq(B(x0,r))
+
∥∥∥I(2)α,(b1,b2) (f∞1 ,f02)∥∥∥Lq(B(x0,r)) +

∥∥∥I(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥Lq(B(x0,r))
= F1 + F2 + F3 + F4.

Firstly, we use the boundedness of I
(2)
α,(b1,b2)

from Lp1 ×Lp2 into Lq (see Lemma 1.1) to estimate F1, and

we obtain

F1 =
∥∥∥I(2)α,(b1,b2) (f01 ,f02)∥∥∥Lq(B(x0,r)) .

2∏
i=1

‖bi‖∗‖fi‖Lpi(2B)

. r
n
q

2∏
i=1

‖bi‖∗‖fi‖Lpi(2B)

∞∫
2r

dt

t
n
q
+1

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
q
+1

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Secondly, for F2 =
∥∥∥I(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥Lq(B(x0,r)), we decompose it into four parts as follows:
F2 .

∥∥∥[(b1 −{b1}B)] [(b2 −{b2}B)] I(2)α (f01 ,f∞2 )∥∥∥
Lq(B(x0,r))

+
∥∥∥[(b1 −{b1}B)] I(2)α [f01 , (b2 −{b2}B) f∞2 ]∥∥∥

Lq(B(x0,r))

+
∥∥∥[(b2 −{b2}B)] I(2)α [(b1 −{b1}B) f01 ,f∞2 ]∥∥∥

Lq(B(x0,r))

+
∥∥∥I(2)α [(b1 −{b1}B) f01 , (b2 −{b2}B) f∞2 ]∥∥∥

Lq(B(x0,r))

≡ F21 + F22 + F23 + F24.



Int. J. Anal. Appl. 17 (4) (2019) 608

Let 1 < p1,p2 <
2n
α

, such that 1
p

= 1
p1

+ 1
p2

, 1
q

= 1
p
− α

n
, 1
r

= 1
q1

+ 1
q2

and 1
q

= 1
r

+ 1
q
. Then, using Hölder’s

inequality and by (1.8) we have

F21 =
∥∥∥(b1 − (b1)B) (b2 − (b2)B) I(2)α (f01 ,f∞2 )∥∥∥

Lq(B(x0,r))

. ‖(b1 − (b1)B) (b2 − (b2)B)‖Lr(B(x0,r))
∥∥∥I(2)α (f01 ,f∞2 )∥∥∥

Lq(B(x0,r))

. ‖b1 − (b1)B‖Lq1 (B(x0,r)) ‖b2 − (b2)B‖Lq2 (B(x0,r))

×r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
q
+1

.
2∏
i=1

‖bi‖∗|B(x0,r)|
1
q1

+ 1
q2 r

n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
q
+1

.
2∏
i=1

‖bi‖∗r
n
(

1
q1

+ 1
q2

)
r
n
(

1
p1

+ 1
p2
−α
n

) ∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t)) t
n
(

1
q1

+ 1
q2

)
dt

t
n
q
+1

=

2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
,

where in the second inequality we have used the following fact:

It is clear that |(x0 −y1, x0 −y2)|
2n−α ≥ |x0 −y2|

2n−α
. By Hölder’s inequality, we have

∣∣∣I(2)α (f01 ,f∞2 ) (x)∣∣∣ . ∫
Rn

∫
Rn

∣∣f01 (y1)∣∣ |f∞2 (y2)|
|(x−y1,x−y2)|

2n−αdy1dy2

.
∫
2B

|f1 (y1)|dy1
∫

(2B)c

|f2 (y2)|
|x0 −y2|

2n−αdy2

≈
∫
2B

|f1 (y1)|dy1
∫

(2B)c

|f2 (y2)|
∞∫

|x0−y2|

dt

t2n−α+1
dy2

. ‖f1‖Lp1 (2B) |2B|
1− 1

p1

∞∫
2r

‖f2‖Lp2 (B(x0,t)) |B (x0, t)|
1− 1

p2
dt

t2n−α+1

.

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
q
+1
,

where 1
p

= 1
p1

+ 1
p2

. Thus, the inequality

∥∥∥I(2)α (f01 ,f∞2 )∥∥∥
Lq(B(x0,r))

. r
n
q

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
q
+1

is valid.



Int. J. Anal. Appl. 17 (4) (2019) 609

On the other hand, for the estimates used in F22, F23, we have to prove the below inequality:

∣∣∣I(2)α [f01 , (b2 (·) − (b2)B) f∞2 ] (x)∣∣∣ . ‖b2‖∗
∞∫

2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
p1

+ 1
p2

)
+1−α

. (2.2)

To estimate F22, the following inequality

∣∣∣I(2)α [f01 , (b2 (·) − (b2)B) f∞2 ] (x)∣∣∣ . ∫
2B

|f1 (y1)|dy1
∫

(2B)c

|b2 (y2) − (b2)B| |f2 (y2)|
|x0 −y2|

2n−α dy2

is satisfied. It’s obvious that ∫
2B

|f1 (y1)|dy1 . ‖f1‖Lp1 (2B) |2B|
1− 1

p1 , (2.3)

and using Hölder’s inequality and by (1.6) and (1.7) we have

∫
(2B)c

|b2 (y2) − (b2)B| |f2 (y2)|
|x0 −y2|

2n−α dy2

.
∫

(2B)c

∣∣∣b2 (y2) − (b2)B(x0,r)∣∣∣ |f2 (y2)|

 ∞∫
|x0−y2|

dt

t2n−α+1


dy2

.

∞∫
2r

∥∥∥b2 (y2) − (b2)B(x0,t)∥∥∥Lq2 (B(x0,t)) ‖f2‖Lp2 (B(x0,t)) |B (x0, t)|1−
(

1
p2

+ 1
q2

)
dt

t2n−α+1

+

∞∫
2r

∣∣∣(b2)B(x0,t) − (b2)B(x0,r)∣∣∣‖f2‖Lp2 (B(x0,t)) |B (x0, t)|1− 1p2 dtt2n−α+1
. ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)2
|B (x0, t)|

1
q2 ‖f2‖Lp2 (B(x0,t)) |B (x0, t)|

1−
(

1
p2

+ 1
q2

)
dt

t2n−α+1

+ ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)
|B (x0, t)|‖f2‖Lp2 (B(x0,t)) |B (x0, t)|

1− 1
p2

dt

t2n−α+1

. ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)2
‖f2‖Lp2 (B(x0,t))

dt

t
n
(
1+ 1

p2

)
+1−α

. (2.4)

Hence, by (2.3) and (2.4), it follows that:

∣∣∣I(2)α [f01 , (b2 (·) − (b2)B) f∞2 ] (x)∣∣∣
. ‖b2‖∗‖f1‖Lp1 (2B) |2B|

1− 1
p1

∞∫
2r

(
1 + ln

t

r

)2
‖f2‖Lp2 (B(x0,t))

dt

t
n
(
1+ 1

p2

)
+1−α

. ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
p1

+ 1
p2

)
+1−α

.



Int. J. Anal. Appl. 17 (4) (2019) 610

This completes the proof of inequality (2.2). Thus, let 1 < τ < ∞, such that 1
q

= 1
q1

+ 1
τ

. Then, using

Hölder’s inequality and from (2.2) and (1.7), we get

F22 =
∥∥∥[(b1 −{b1}B)] I(2)α [f01 , (b2 −{b2}B) f∞2 ]∥∥∥

Lq(B(x0,r))

. ‖b1 − (b1)B‖Lq1 (B)
∥∥∥I(2)α [f01 , (b2 − (b2)B) f∞2 ]∥∥∥

Lτ(B)

.
2∏
i=1

‖bi‖∗ |B (x0,r)|
1
q1

+ 1
τ

×
∞∫

2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
p1

+ 1
p2

)
+1−α

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Similarly, F23 has the same estimate above, here we omit the details, thus the inequality

F23 =
∥∥∥[(b2 −{b2}B)] I(2)α [(b1 −{b1}B) f01 ,f∞2 ]∥∥∥

Lq(B(x0,r))

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1

is valid.

Now we turn to estimate F24. Similar to (2.2), we have to prove the following estimate for F24:

∣∣∣I(2)α [(b1 − (b1)B) f01 , (b2 − (b2)B) f∞2 ] (x)∣∣∣ ≤ 2∏
i=1

‖bi‖∗

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

(2.5)

Firstly, the following inequality

∣∣∣I(2)α [(b1 − (b1)B) f01 , (b2 − (b2)B) f∞2 ] (x)∣∣∣ . ∫
2B

|b1 (y1) − (b1)B| |f1 (y1)|dy1
∫

(2B)c

|b2 (y2) − (b2)B| |f2 (y2)|
|x0 −y2|

2n−α dy2

is valid.

It’s obvious that from Hölder’s inequality and (1.7)

∫
2B

|b1 (y1) − (b1)B| |f1 (y1)|dy1 . ‖b1‖∗ |B (x0,r)|
1− 1

p1 ‖f1‖Lp1 (2B) . (2.6)

Then, by (2.4) and (2.6) we have

∣∣∣I(2)α [(b1 − (b1)B) f01 , (b2 − (b2)B) f∞2 ] (x)∣∣∣ ≤ 2∏
i=1

‖bi‖∗

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.



Int. J. Anal. Appl. 17 (4) (2019) 611

This completes the proof of inequality (2.5). Therefore, by (2.5) we deduce that

F24 =
∥∥∥I(2)α [(b1 − (b1)B) f01 , (b2 − (b2)B) f∞2 ]∥∥∥

Lq(B)

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Considering estimates F21, F22, F23, F24 together, we get the desired conclusion

F2 =
∥∥∥I(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥Lq(B(x0,r)) .

2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Similar to F2, we can also get the estimates for F3,

F3 =
∥∥∥I(2)α,(b1,b2) (f∞1 ,f02)∥∥∥Lq(B(x0,r)) .

2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

At last, we consider the last term F4 =
∥∥∥I(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥Lq(B(x0,r)). We split F4 in the following way:

F4 . F41 + F42 + F43 + F44,

where

F41 =
∥∥∥(b1 − (b1)B) (b2 − (b2)B) I(2)α (f∞1 ,f∞2 )∥∥∥

Lq(B)
,

F42 =
∥∥∥(b1 − (b1)B) I(2)α [f∞1 , (b2 − (b2)B) f∞2 ]∥∥∥

Lq(B)
,

F43 =
∥∥∥(b2 − (b2)B) I(2)α [(b1 − (b1)B) f∞1 ,f∞2 ]∥∥∥

Lq(B)
,

F44 =
∥∥∥I(2)α [(b1 − (b1)B) f∞1 , (b2 − (b2)B) f∞2 ]∥∥∥

Lq(B)
.

Now, let us estimate F41, F42, F43, F44 respectively.

For the term F41, let 1 < τ < ∞, such that 1q =
(

1
q1

+ 1
q2

)
+ 1

τ
, 1
τ

= 1
p1

+ 1
p2
− α

n
. Then, by Hölder’s

inequality and (1.8), we get

F41 =
∥∥∥(b1 − (b1)B) (b2 − (b2)B) I(2)α (f∞1 ,f∞2 )∥∥∥

Lq(B)

. ‖b1 − (b1)B‖Lq1 (B) ‖b2 − (b2)B‖Lq2 (B)
∥∥∥I(2)α (f∞1 ,f∞2 )∥∥∥

Lτ(B)

.
2∏
i=1

‖bi‖∗ |B (x0,r)|
1
q1

+ 1
q2 r

n
τ

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
τ
+1

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
,

where in the second inequality we have used the following fact:



Int. J. Anal. Appl. 17 (4) (2019) 612

Noting that |(x0 −y1, x0 −y2)|
2n−α ≥ |x0 −y1|

n−α
2 |x0 −y2|

n−α
2 and by Hölder’s inequality, we get

∣∣∣I(2)α (f∞1 ,f∞2 ) (x)∣∣∣
.
∫
Rn

∫
Rn

∣∣f1 (y1) χ(2B)c∣∣ ∣∣f2 (y2) χ(B)c∣∣
|(x0 −y1,x0 −y2)|

2n−α dy1dy2

.
∫

(2B)c

∫
(2B)c

|f1 (y1)| |f2 (y2)|
|x0 −y1|

n−α
2 |x0 −y2|

n−α
2
dy1dy2

.
∞∑
j=1

2∏
i=1

∫
2j+1B\2jB

|fi (yi)|
|x0 −yi|

n−α
2
dyi

.
∞∑
j=1

2∏
i=1

(
2jr
)−n+α

2

∫
2j+1B

|fi (yi)|dyi

.
∞∑
j=1

(
2jr
)−2n+α 2∏

i=1

‖fi‖Lpi(2j+1B)
∣∣2j+1B∣∣1− 1pi

.
∞∑
j=1

2j+2r∫
2j+1r

(
2j+1r

)−2n+α−1 2∏
i=1

‖fi‖Lpi(2j+1B)
∣∣2j+1B∣∣1− 1pi dt

.
∞∑
j=1

2j+2r∫
2j+1r

2∏
i=1

‖fi‖Lpi(B(x0,t)) |B (x0, t)|
1− 1

pi
dt

t2n+1−α

.

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t)) |B (x0, t)|
2−
(

1
p1

+ 1
p2

)
dt

t2n+1−α

.

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
τ
+1
.

Moreover, for p1, p2 ∈ [1,∞) the inequality

∥∥∥I(2)α (f∞1 ,f∞2 )∥∥∥
Lτ(B(x0,r))

. r
n
τ

∞∫
2r

2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
τ
+1

(2.7)

is valid.

For the terms F42, F43, similar to the estimates used for (2.2), we have to prove the following inequality:

∣∣∣I(2)α [f∞1 , (b2 − (b2)B) f∞2 ] (x)∣∣∣ . ‖b2‖∗
∞∫

2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
p1

+ 1
p2

)
+1−α

. (2.8)



Int. J. Anal. Appl. 17 (4) (2019) 613

Noting that |(x0 −y1, x0 −y2)|
2n−α ≥ |x0 −y1|

n−α
2 |x0 −y2|

n−α
2 , we get

∣∣∣I(2)α [f∞1 , (b2 − (b2)B) f∞2 ] (x)∣∣∣
.
∫
Rn

∫
Rn

|b2 (y2) − (b2)B|
∣∣f1 (y1) χ(2B)c∣∣ ∣∣f2 (y2) χ(2B)c∣∣

|(x0 −y1,x0 −y2)|
2n−α dy1dy2

.
∫

(2B)c

∫
(2B)c

|b2 (y2) − (b2)B| |f1 (y1)| |f2 (y2)|
|x0 −y1|

n−α
2 |x0 −y2|

n−α
2

dy1dy2

.
∞∑
j=1

∫
2j+1B\2jB

|f1 (y1)|
|x0 −y1|

n−α
2
dy1

∫
2j+1B\2jB

|b2 (y2) − (b2)B| |f2 (y2)|
|x0 −y2|

n−α
2

dy2

.
∞∑
j=1

(
2jr
)−2n+α ∫

2j+1B

|f1 (y1)|dy1
∫

2j+1B

|b2 (y2) − (b2)B| |f2 (y2)|dy2.

On the other hand, it’s obvious that

∫
2j+1B

|f1 (y1)|dy1 ≤‖f1‖Lp1 (2j+1B)
∣∣2j+1B∣∣1− 1p1 , (2.9)

and using Hölder’s inequality and by (1.6) and (1.7)

∫
2j+1B

|b2 (y2) − (b2)B| |f2 (y2)|dy2

≤‖b2 − (b2)2j+1B‖Lq2 (2j+1B) ‖f2‖Lp2 (2j+1B)
∣∣2j+1B∣∣1−( 1p2 + 1q2 )

+ |(b2)2j+1B − (b2)B|‖f2‖Lp2 (2j+1B)
∣∣2j+1B∣∣1− 1p2

. ‖b2‖∗
∣∣2j+1B∣∣ 1q2 (1 + ln 2j+1r

r

)
‖f2‖Lp2 (2j+1B)

∣∣2j+1B∣∣1−( 1p2 + 1q2 )
+ ‖b2‖∗

(
1 + ln

2j+1r

r

)∣∣2j+1B∣∣‖f2‖Lp2 (2j+1B) ∣∣2j+1B∣∣1− 1p2
. ‖b2‖∗

(
1 + ln

2j+1r

r

)2 ∣∣2j+1B∣∣1− 1p2 ‖f2‖Lp2 (2j+1B) . (2.10)
Hence, by (2.9) and (2.10), it follows that:

∣∣∣I(2)α [f∞1 , (b2 − (b2)B) f∞2 ] (x)∣∣∣
.
∞∑
j=1

(
2jr
)−2n+α ∫

2j+1B

|f1 (y1)|dy1
∫

2j+1B

|b2 (y2) − (b2)B| |f2 (y2)|dy2

. ‖b2‖∗
∞∑
j=1

(
2jr
)−2n+α (

1 + ln
2j+1r

r

)2 ∣∣2j+1B∣∣2−( 1p1 + 1p2 ) 2∏
i=1

‖fi‖Lpi(2j+1B)



Int. J. Anal. Appl. 17 (4) (2019) 614

. ‖b2‖∗
∞∑
j=1

2j+2r∫
2j+1r

(
2j+1r

)−2n+α−1 (
1 + ln

2j+1r

r

)2 ∣∣2j+1B∣∣2−( 1p1 + 1p2 ) 2∏
i=1

‖fi‖Lpi(2j+1B) dt

. ‖b2‖∗
∞∑
j=1

2j+2r∫
2j+1r

(
1 + ln

2j+1r

r

)2 ∣∣2j+1B∣∣2−( 1p1 + 1p2 ) 2∏
i=1

‖fi‖Lpi(2j+1B)
dt

t2n−α+1

. ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)2
|B (x0, t)|

2−
(

1
p1

+ 1
p2

) 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t2n−α+1

. ‖b2‖∗

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
p1

+ 1
p2

)
+1−α

.

This completes the proof of (2.8).

Now we turn to estimate F42. Let 1 < τ < ∞, such that 1q =
1
q1

+ 1
τ

. Then, by Hölder’s inequality, (1.7)

and (2.8), we obtain

F42 =
∥∥∥(b1 − (b1)B) I(2)α [f∞1 , (b2 − (b2)B) f∞2 ]∥∥∥

Lq(B)

. ‖(b1 − (b1)B)‖Lq1 (B)
∥∥∥I(2)α [f∞1 , (b2 − (b2)B) f∞2 ]∥∥∥

Lτ(B)

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Similarly, F43 has the same estimate above, here we omit the details, thus the inequality

F43 =
∥∥∥(b2 − (b2)B) I(2)α [(b1 − (b1)B) f∞1 ,f∞2 ]∥∥∥

Lq(B)

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1

is valid.

Finally, to estimate F44, similar to the estimate of (2.8), we have

∣∣∣I(2)α [(b1 − (b2)B) f∞1 , (b2 − (b2)B) f∞2 ] (x)∣∣∣
.
∞∑
j=1

(
2jr
)−2n+α ∫

2j+1B

|b1 (y1) − (b1)B| |f1 (y1)|dy1
∫

2j+1B

|b2 (y2) − (b2)B| |f2 (y2)|dy2

.
2∏
i=1

‖bi‖∗

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.



Int. J. Anal. Appl. 17 (4) (2019) 615

Thus, we have

F44 =
∥∥∥I(2)α [(b1 − (b1)B) f∞1 , (b2 − (b2)B) f∞2 ]∥∥∥

Lq(B)

.
2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

By the estimates of F4j above, where j = 1,2,3,4, we know that

F4 =
∥∥∥I(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥Lq(B(x0,r)) .

2∏
i=1

‖bi‖∗r
n
q

∞∫
2r

(
1 + ln

t

r

)2 2∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t
n
(

1
q
−
(

1
q1

+ 1
q2

))
+1
.

Consequently, combining all the estimates for F1, F2, F3, F4, we complete the proof of Lemma 2.1. �

3. Proofs of the main results

Now we are ready to return to the proofs of Theorems 1.3 and 1.4.

3.1. Proof of Theorem 1.3.

Proof. To prove Theorem 1.3, we will use the following relationship between essential supremum and essential

infimum (
essinf
x∈E

f (x)

)−1
= esssup

x∈E

1

f (x)
, (3.1)

where f is any real-valued nonnegative function and measurable on E (see [30], page 143). Indeed, we

consider (1.12) firstly.

Since
−→
f ∈ Mp1,ϕ1 × ··· × Mpm,ϕm, by (3.1) and the non-decreasing, with respect to t, of the norm

m∏
i=1

‖fi‖Lpi(B(x,t)), we get

m∏
i=1

‖fi‖Lpi(B(x,t))

essinf
0<t<τ<∞

m∏
i=1

ϕi(x,τ)τ
n
p

≤ esssup
0<t<τ<∞

m∏
i=1

‖fi‖Lpi(B(x,t))

m∏
i=1

ϕi(x,τ)τ
n
p

≤ esssup
0<τ<∞,x∈Rn

m∏
i=1

‖fi‖Lpi(B(x,t))

m∏
i=1

ϕi(x,τ)τ
n
p

≤
m∏
i=1

‖fi‖Mpi,ϕi . (3.2)



Int. J. Anal. Appl. 17 (4) (2019) 616

For 1 < p1, . . . ,pm < ∞, since (ϕ1, . . . ,ϕm,ϕ) satisfies (1.11) and by (3.2), we have

∞∫
r

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

≤
∞∫
r

(
1 + ln

t

r

)m
m∏
i=1

‖fi‖Lpi(B(x,t))

essinf
t<τ<∞

m∏
i=1

ϕi(x,τ)τ
n
p

essinf
t<τ<∞

m∏
i=1

ϕi(x,τ)τ
n
p

t

n


1q−

m∑
i=1

1
qi


+1

dt

≤ C
m∏
i=1

‖fi‖Mpi,ϕi

∞∫
r

(
1 + ln

t

r

)m essinft<τ<∞
m∏
i=1

ϕi(x,τ)τ
n
p

t

n


1q−

m∑
i=1

1
qi


+1

dt

≤ C
m∏
i=1

‖fi‖Mpi,ϕi ϕ(x,r). (3.3)

Then by (2.1) and (3.3), we get

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Mq,ϕ
= sup
x∈Rn,r>0

ϕ (x,r)
−1 |B(x,r)|−

1
q

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(B(x,r))

.
m∏
i=1

‖bi‖∗ sup
x∈Rn,r>0

ϕ (x0,r)
−1
∞∫
r

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

.
m∏
i=1

‖bi‖∗‖fi‖Mpi,ϕi .

Thus we obtain (1.12).

The conclusion of (1.13) is a direct consequence of (1.10) and (1.12). Indeed, from the process proving

(1.12), it is easy to see that the conclusions of (1.12) also hold for Ĩ
(m)
−→
b ,α

. Combining this with (1.10), we can

immediately obtain (1.13), which completes the proof. �

3.2. Proof of Theorem 1.4.

Proof. Since the inequalities (1.17) and (1.18) hold by Theorem 1.3, we only have to prove the implication

lim
r→0

sup
x∈Rn

r−
n
p

m∏
i=1

‖fi‖Lpi(B(x,r))

m∏
i=1

ϕi(x,r)

= 0 implies lim
r→0

sup
x∈Rn

r−
n
q

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(B(x,r))

ϕ(x,r)
= 0. (3.4)

To show that

sup
x∈Rn

r−
n
q

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(B(x,r))

ϕ(x,r)
< ε for small r,



Int. J. Anal. Appl. 17 (4) (2019) 617

we use the estimate (2.1):

sup
x∈Rn

r−
n
q

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(B(x,r))

ϕ(x,r)
. sup
x∈Rn

m∏
i=1

‖bi‖∗

ϕ(x,r)

∞∫
r

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

.

We take r < δ0, where δ0 is small enough and split the integration:

r−
n
q

∥∥∥I(m)
α,
−→
b

(−→
f
)∥∥∥

Lq(B(x,r))

ϕ(x,r)
≤ C [Iδ0 (x,r) + Jδ0 (x,r)] , (3.5)

where δ0 > 0 (we may take δ0 < 1), and

Iδ0 (x,r) :=

m∏
i=1

‖bi‖∗

ϕ(x,r)

δ0∫
r

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

,

and

Jδ0 (x,r) :=

m∏
i=1

‖bi‖∗

ϕ(x,r)

∞∫
δ0

(
1 + ln

t

r

)m m∏
i=1

‖fi‖Lpi(B(x0,t))
dt

t

n


1q−

m∑
i=1

1
qi


+1

,

and r < δ0. Now we can choose any fixed δ0 > 0 such that

sup
x∈Rn

t−
n
p

m∏
i=1

‖fi‖Lpi(B(x,t))

m∏
i=1

ϕi(x,t)

<
ε

2CC0
, t ≤ δ0,

where C and C0 are constants from (1.14) and (3.5), which is possible since
−→
f ∈ V Mp1,ϕ1 ×···×V Mpm,ϕm.

This allows to estimate the first term uniformly in r ∈ (0,δ0):
m∏
i=1

‖bi‖∗ sup
x∈Rn

CIδ0 (x,r) <
ε

2
, 0 < r < δ0

by (1.14).

For the second term, writing 1 + ln t
r
≤ 1 + |ln t|+ ln 1

r
, by the choice of r sufficiently small because of the

conditions (1.15) we obtain

Jδ0 (x,r) ≤
cδ0 + c̃δ0 ln

1
r

ϕ(x,r)

m∏
i=1

‖bi‖∗‖fi‖V Mpi,ϕi ,

where cδ0 is the constant from (1.16) with δ = δ0 and c̃δ0 is a similar constant with omitted logarithmic

factor in the integrand. Then, by (1.15) we can choose r small enough such that

sup
x∈Rn

Jδ0 (x,r) <
ε

2
,

which completes the proof of (3.4).



Int. J. Anal. Appl. 17 (4) (2019) 618

For M
(m)

α,
−→
b

, we can also use the same method to obtain the desired result, but we omit the details.

Therefore, the proof of Theorem 1.4 is completed. �

Acknowledgement: This study has been given as the plenary talk by the author at the “The Inter-

national Workshop on Mathematical Methods in Engineering” (MME-2017), Çankaya University, Ankara,

Turkey, 27-29 April 2017.

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	1. Introduction and main results
	2. A Key Lemma
	3. Proofs of the main results
	3.1. Proof of Theorem 1.3.
	3.2. Proof of Theorem 1.4.

	References