Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

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R E S E A R C H

Open Access

Some notes on commutators of the

fractional maximal function on variable

Lebesgue spaces

Pu Zhang

1*

_{, Zengyan Si}

2

_{and Jianglong Wu}

1

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Mudanjiang Normal University, Mudanjiang, P.R. China Full list of author information is available at the end of the article

Abstract

Let 0 <

α

<nandMαbe the fractional maximal function. The nonlinear commutator ofMαand a locally integrable functionbis given by [b,Mα](f) =bMα(f) –Mα(bf). In this paper, we mainly give some necessary and suﬃcient conditions for the

boundedness of [b,Mα] on variable Lebesgue spaces whenbbelongs to Lipschitz or

BMO(Rn_{) spaces, by which some new characterizations for certain subclasses of} Lipschitz andBMO(Rn_{) spaces are obtained.}

MSC: 42B25; 42B20; 42B35; 46E30

Keywords: Fractional maximal function; Nonlinear commutator; Variable Lebesgue space; Lipschitz space;BMOspace

1 Introduction and main results

LetT be the classical singular integral operator. In 1976, Coifman, Rochberg and Weiss [4] studied the commutator generated byT and a functionb∈BMO(Rn_{) as follows:}

[b,T](f)(x) =Tb(x) –b(·)f(·)(x) =b(x)T(f)(x) –T(bf)(x). (1.1)

A well-known result states that [b,T] is bounded onLp₍_Rn_{) for 1 <}_p_<_∞_{if and only if}

b∈BMO(Rn_{). The suﬃciency was obtained by Coifman, Rochberg and Weiss [}₄_{] and the}

necessity was proved by Janson [19]. Moreover, Janson also gave some characterizations of the Lipschitz spaceΛ˙β(Rn) (see Deﬁnition1.2below) via commutator [b,T] in [19] and proved that [b,T] is bounded fromLp₍_Rn_{) to}_Lq₍_Rn_{), for 1 <}_p_<_n_{/β, 1/}_p_{– 1/}_q₌_β/_n_and

0 <β< 1, if and only ifb∈ ˙Λβ(Rn) (see also Paluszyński [24]).

As usual, a cubeQ⊂Rn_{always has its sides parallel to the coordinate axes. Denote by}

|Q|the Lebesgue measure andχQthe characteristic function ofQ. Forf ∈L1_loc(Rn), we

write

fQ=

1

|Q|

Q

f(x)dx.

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For 0≤α<nandf ∈L1

loc(Rn), the fractional maximal functionMαis given by Mα(f)(x) =sup

Qx

1

|Q|1–α/n

Q

f(y)dy,

where the supremum is taken over all cubesQ⊂Rn_containing_x_{. When}_α_{= 0, we simply}

writeMinstead ofM0, which is exactly the Hardy–Littlewood maximal function.

Similar to (1.1), we can deﬁne two diﬀerent kinds of commutator of the fractional max-imal function as follows.

Deﬁnition 1.1 Let 0≤α<nandbbe a locally integrable function. The maximal com-mutator ofMαandbis given by

Mα,b(f)(x) =sup Qx

1

|Q|1–α/n

Q

b(x) –b(y)f(y)dy,

where the supremum is taken over all cubesQ⊂Rn_containing_x_.

The nonlinear commutator ofMαandbis deﬁned by

[b,Mα](f)(x) =b(x)Mα(f)(x) –Mα(bf)(x).

Whenα= 0, we simply denote byMb:=M0,band [b,M] := [b,M0].

We call [b,Mα] the nonlinear commutator because it is not even a sublinear operator, although the commutator [b,T] is a linear one. We would like to remark that the nonlinear commutator [b,Mα] and the maximal commutatorMα,bessentially diﬀer from each other.

For example,Mα,bis positive and sublinear, but [b,Mα] is neither positive nor sublinear. The mapping property of [b,Mα] has been extensively studied; see [1,2,8,12,13,16,23,

25–31], for instance. There are some applications of nonlinear commutators in analysis. For example, [b,M] can be used in studying the products of functions inH1_and_BMO_(see

[3], for instance).

In 1990, by using the real interpolation techniques, Milman and Schonbek [23] obtained a commutator result, by which they obtained theLp_{-boundedness of [}_b_,_M_{] and [}_b_,_M

α]

(0 <α<n) whenb∈BMO(Rn_{) and}_b_≥_{0. In 2000, Bastero, Milman and Ruiz [}₂_]

consid-ered the necessary and suﬃcient conditions for the boundedness of [b,M] inLp₍_Rn_{) when}

bbelongs toBMO(Rn_{). In 2009, Zhang and Wu [}₂₈_{] extended their results to commutators}

of the fractional maximal function. The results in [2] and [28] were extended to variable Lebesgue spaces in [29] and [30].

Recently, Zhang [26] studied the commutator [b,M] whenbbelongs to Lipschitz spaces. Some necessary and suﬃcient conditions for the boundedness of [b,M] on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in [27] and to the context of Orlicz spaces in [15,16] and [31].

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results also give aﬃrmative answers to the questions mentioned in [16] and [29] (see Re-marks1.4and1.5below, respectively). We would like to note that some of our results are new even in the case of Lebesgue spaces with constant exponents.

To state the results, we ﬁrst recall some deﬁnitions and notations.

Letγ ≥0, for a ﬁxed cubeQ0, the fractional maximal function with respect toQ0of a

locally integrable functionf is given by

Mγ,Q0(f)(x) = sup

Qx Q⊆Q0

1

|Q|1–γ/n

Q

f(y)dy,

where the supremum is taken over all cubesQsuch thatx∈Q⊆Q0.

Whenγ= 0, we simply writeMQ0instead ofM0,Q0.

Deﬁnition 1.2 For 0 <β< 1, we say a functionbbelongs to the Lipschitz spaceΛ˙β(Rn), denoted byb∈ ˙Λβ(Rn), if there exists a constantC> 0 such that for allx,y∈Rn,

b(x) –b(y)≤C|x–y|β.

The smallest such constantCis called theΛ˙βnorm ofband is denoted by b Λ˙β.

Deﬁnition 1.3 A locally integrable functionf is said to belong toBMO(Rn_{) if}

f BMO:=sup Q

1

|Q|

Q

f(x) –fQdx<∞,

where the supremum is taken over all cubesQinRn_.

For a functionbdeﬁned onRn_{, we denote by}

b–(x) = ⎧ ⎨ ⎩

0, ifb(x)≥0,

|b(x)|, ifb(x) < 0,

andb+₍_x_{) =}_|_b₍_x₎_|_–_b–₍_x_{). Obviously,}_b+₍_x_{) –}_b–₍_x_{) =}_b₍_x_).

Deﬁnition 1.4 Letp(·) :Rn_→_[1,_∞_{) be a measurable function. The variable Lebesgue}

space,Lp(·)₍_Rn_{), is deﬁned by}

Lp(·)Rn= f measurable:

Rn _|

f(x)| λ

p(x)

dx<∞for some constantλ> 0

.

The setLp(·)(Rn) becomes a Banach space with respect to the norm

f _Lp(·)₍_Rn₎=inf λ> 0 :

Rn _|

f(x)| λ

p(x) dx≤1

.

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Denote byP(Rn_{) the set of all measurable functions}_p₍_·_{) :}_Rn_→_[1,_∞_{) such that}

1 <p–:=ess inf

x∈Rn p(x) and p+: =ess sup

x∈Rn p(x) <∞,

and byB(Rn_{) the set of all}_p₍_·₎_∈_P₍_Rn_{) such that}_M_{is bounded on}_Lp(·)₍_Rn_).

Remark1.1 Ifp(·)∈B(Rn_{) and}_λ_{> 1, then}_λ_p₍_·₎_∈_B₍_Rn_{). See Remark 2.13 in [}₆_].

For notational convenience, we introduce a notationBγ₍_Rn_{) as follows.}

Deﬁnition 1.5 Let 0 <γ <n. We say that an ordered pair of variable exponents (p(·),

q(·))∈Bγ₍_Rn_{), if}_p₍_·₎_∈_P₍_Rn_{) with}_p

+<n/γ and 1/q(·) = 1/p(·) –γ/nwithq(·)(n–γ)/n∈

B(Rn_).

Remark1.2 The conditionq(·)(n–γ)/n∈B(Rn_{) is equivalent to saying that there exists}

aq0withn/(n–γ) <q0<∞such thatq(·)/q0∈B(Rn). Moreover,q(·)(n–γ)/n∈B(Rn)

impliesq(·)∈B(Rn). See Remark 2.13 in [6] for details. Our results can be stated as follows.

Theorem 1.1 Let0 <β< 1, 0 <α<n, 0 <α+β<n and let b be a locally integrable func-tion.Then the following statements are equivalent:

(1) b∈ ˙Λβ(Rn)andb≥0.

(2) [b,Mα]is bounded fromLp(·)(Rn)toLq(·)(Rn)for some(p(·),q(·))∈Bα+β(Rn).

(3) [b,Mα]is bounded fromLp(·)(Rn)toLq(·)(Rn)for all(p(·),q(·))∈Bα+β(Rn).

(4) There exists ans(·)∈B(Rn)such that

sup

Q

1

|Q|β/n

(b–|Q|–α/n_M

α,Q(b))χQ Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

<∞. (1.2)

(5) For alls(·)∈B(Rn₎_{we have}₍_1.2_).

Remark1.3 For the caseα= 0, the result was proved in [27]. Moreover, (1.2) gives a new characterization of nonnegative Lipschitz functions, compared with [27, Theorem 1.5].

For the case ofp(·) andq(·) being constants, we have the following result from Theo-rem1.1, which is new even for this case.

Corollary 1.1 Let0 <β< 1, 0 <α<n, 0 <α+β<n and let b be a locally integrable func-tion.Then the following statements are equivalent:

(1) b∈ ˙Λβ(Rn)andb≥0.

(2) [b,Mα]is bounded fromLp(Rn)toLq(Rn)for somepandqsuch that 1 <p<n/(α+β)and1/q= 1/p– (α+β)/n.

(3) [b,Mα]is bounded fromLp(Rn)toLq(Rn)for allpandqsuch that1 <p<n/(α+β) and1/q= 1/p– (α+β)/n.

(4) There existss∈[1,∞)such that

sup

Q

1

|Q|β/n

1

|Q|

Q

b(x) –|Q|–α/nMα,Q(b)(x) s

dx

1/s

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(5) For alls∈[1,∞)we have(1.3).

Remark1.4 The result was proved forα= 0 in [26, Theorem 1.4]. Corollary1.1improves the result of [16, Corollary 4.15] essentially and answers a question asked in [16, Re-mark 4.17] aﬃrmatively. Moreover, it was proved in [26, Theorem 1.4], see also Lemma2.2

below, thatb∈ ˙Λβ(Rn) andb≥0 if and only if

sup

Q

1

|Q|β/n

1

|Q|

Q

b(x) –MQ(b)(x) s

dx

1/s

<∞. (1.4)

Compared with (1.4), (1.3) gives a new characterization for nonnegative Lipschitz func-tions.

Theorem 1.2 Let0 <α<n and b be a locally integrable function.Then the following

state-ments are equivalent:

(1) b∈BMO(Rn₎_and_b–_∈_L∞₍_Rn₎_.

(2) [b,Mα]is bounded fromLp(·)(Rn)toLq(·)(Rn)for some(p(·),q(·))∈Bα(Rn).

(3) [b,Mα]is bounded fromLp(·)(Rn)toLq(·)(Rn)for all(p(·),q(·))∈Bα(Rn).

(4) There existss(·)∈B(Rn₎_{such that}

sup

Q

(b–|Q|–α/n_M

χQ Ls(·)₍_Rn₎

<∞. (1.5)

Remark1.5 The equivalence of (1), (2) and (3) was proved in [29]. Statements (4) and (5) give new necessary and suﬃcient conditions for the statements (1), (2) and (3). Especially, (1.5) gives a new characterization forb∈BMO(Rn_{) and}_b–_∈_L∞₍_Rn_{), which also answers}

a question asked in [29, Remark 4.1]. For the caseα= 0, the result was obtained in [30].

For the case ofp(·) andq(·) being constants, we have the following result by Theorem1.2.

Corollary 1.2 Let0 <α<n and b be a locally integrable function.Then the following

state-ments are equivalent:

(1) b∈BMO(Rn)andb–∈L∞(Rn).

(2) [b,Mα]is bounded fromLp(Rn)toLq(Rn)for somepandqsuch that1 <p<n/αand 1/q= 1/p–α/n.

(3) [b,Mα]is bounded fromLp(Rn)toLq(Rn)for allpandqsuch that1 <p<n/αand 1/q= 1/p–α/n.

(4) There existss∈[1,∞)such that

sup

Q

1

|Q|

Q

b(x) –|Q|–α/nMα,Q(b)(x) s

dx

1/s

<∞. (1.6)

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Remark1.6 It was shown in [2] and [28] that statements (1), (2) and (3) are equivalent to

sup

Q

1

|Q|

Q

b(x) –MQ(b)(x)sdx<∞, (1.7)

respectively. Compared with (1.7), (1.6) gives a new characterization.

Next, we give some necessary and suﬃcient conditions for the boundedness of the max-imal commutatorMα,bon variable Lebesgue spaces whenbbelongs to a Lipschitz space.

Theorem 1.3 Let0 <β< 1, 0 <α<n, 0 <α+β<n and let b be a locally integrable func-tion.Then the following statements are equivalent:

(1) b∈ ˙Λβ(Rn).

(2) Mα,bis bounded fromLp(·)(Rn)toLq(·)(Rn)for some(p(·),q(·))∈Bα+β(Rn).

(3) Mα,bis bounded fromLp(·)(Rn)toLq(·)(Rn)for all(p(·),q(·))∈Bα+β(Rn).

sup

Q

1

|Q|β/n

(b–bQ)χQ Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

<∞. (1.8)

Remark1.7 For the caseα= 0, similar results were given in [26] for Lebesgue spaces with constant exponents, and in [27] for the variable case.

Whenp(·) andq(·) are constants, we get the following result from Theorem1.3.

Corollary 1.3 Let0 <β< 1, 0 <α<n, 0 <α+β<n and b be a locally integrable function.

Then the following statements are equivalent: (1) b∈ ˙Λβ(Rn).

(2) Mα,bis bounded fromLp(Rn)toLq(Rn)for somepandqsuch that1 <p<n/(α+β)

and1/q= 1/p– (α+β)/n.

(3) Mα,bis bounded fromLp(Rn)toLq(Rn)for allpandqsuch that1 <p<n/(α+β)

and1/q= 1/p– (α+β)/n. (4) There existss∈[1,∞)such that

sup

Q

1

|Q|β/n

1

|Q|

Q

b(x) –bQ s

dx

1/s

<∞ (1.9)

(5) For alls∈[1,∞)we have(1.9).

Remark1.8 The equivalence of (1), (2) and (3) was proved in [26] (forα= 0) and in [16] (for 0 <α<n). The equivalence of (1), (4) and (5) is contained in Lemma2.1below.

Finally, for the case of completeness of this paper, we state a result similar to Theorem1.3

without proof, which can be deduced from [29] and [18].

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(1) b∈BMO(Rn₎_.

(2) Mα,bis bounded fromLp(·)(Rn)toLq(·)(Rn)for some(p(·),q(·))∈Bα(Rn).

(3) Mα,bis bounded fromLp(·)(Rn)toLq(·)(Rn)for all(p(·),q(·))∈Bα(Rn).

sup

Q

(b–bQ)χQ Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

<∞. (1.10)

Remark1.9 We note that Theorem1.4follows from [29] and [18] directly. Indeed, the equivalence of (1), (2) and (3) was proved in [29] Theorems 3.1 and 3.2 forα= 0 and 0 < α<n, respectively, and the equivalence of (1), (4) and (5) was obtained in [18, Lemma 3].

If p(·) andq(·) are constants, we have a result similar to Corollary1.3. We omit the details.

The remainder of this paper is organized as follows. In the next section, we give some lemmas that will be used later. In Sect.3, we prove Theorems1.1,1.2and1.3.

2 Preliminaries and lemmas

It is known that Lipschitz spaceΛ˙β(Rn) coincides with some Morrey–Companato space (see, e.g., [20]) and can be characterized by mean oscillation as in the following lemma, which is due to DeVore and Sharpley [9] and Janson, Taibleson and Weiss [20] (see also Paluszyński [24]).

Lemma 2.1 Let0 <β< 1and1≤q<∞.Deﬁne

˙

Λβ,q

Rn_:= _f_∈_L1 loc

Rn_: _f

˙

Λβ,q=sup

Q

1

|Q|β/n

1

|Q|

Q

_f(x) –fQ q

dx

1/q

<∞

.

Then,for all0 <β< 1and1≤q<∞,Λ˙β(Rn) =Λ˙β,q(Rn)with equivalent norms.

From the proof of Theorem 1.4 in [26], we can obtain the following characterization of nonnegative Lipschitz functions.

Lemma 2.2 Let0 <β< 1and b be a locally integrable function.Then the following state-ments are equivalent:

(1) Ifb∈ ˙Λβ(Rn)andb≥0.

(2) For all1≤s<∞,

sup

Q

1

|Q|β/n

1

|Q|

Q

b(x) –MQ(b)(x) s

dx

1/s

<∞. (2.1)

(3) (2.1)holds for some1≤s<∞.

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Ifb∈ ˙Λβ(Rn) andb≥0, then it follows from [26, Theorem 1.4] that (2.1) holds for all

s withn/(n–β) <s<∞. Applying Hölder’s inequality, we see that (2.1) also holds for

1≤s≤n/(n–β). So, the implication (1)⇒(2) is proven.

Lemma 2.3([2]) Let b be a locally integrable function.Then the following statements are

equivalent:

(1) b∈BMO(Rn)andb–∈L∞(Rn). (2) There existss∈[1,∞)such that

sup

Q

1

|Q|

Q

b(x) –MQ(b)(x) s

dx<∞. (2.2)

(3) For alls∈[1,∞)we have(2.2).

The following strong-type estimate for the fractional maximal function is well known, see [11] or [14] for details.

Lemma 2.4 Let0 <γ <n, 1 <p<n/γ and1/q= 1/p–γ/n.Then there exists a positive

constant C(n,γ,p)such that

Mγ(f)_Lq₍_Rn₎≤C(n,γ,p) f Lp₍_Rn₎.

As for the boundedness of the fractional maximal function on variable Lebesgue spaces, the following result was given in [6]; see Corollary 2.12 and Remark 2.13 in [6] for details.

Lemma 2.5 Let0 <γ<n,p(·)∈P(Rn₎_{with p}

+<n/γ and1/q(·) = 1/p(·) –γ/n.If q(·)(n–

γ)/n∈B(Rn_),_{then M}

γ is bounded from Lp(·)(Rn)to Lq(·)(Rn).

By Lemma2.4, if 0 <γ <n, 1 <p<n/γ andf ∈Lp(Rn), thenMγ(f)(x) <∞almost every-where. A similar result is also valid in variable Lebesgue spaces.

Lemma 2.6 Let 0 <γ <n, p(·)∈P(Rn₎_and _{1 <}_p

–≤p+ <n/γ.If f ∈Lp(·)(Rn), then Mγ(f)(x) <∞for almost every x∈Rn.

Proof Following the same procedure of the proof of [5, Proposition 3.15], we can achieve the desired result. Indeed, for anyf ∈Lp(·)₍_Rn_{), by Theorem 2.51 in [}₅_{] we can write}_f ₌

f1+f2, wheref1∈Lp+(Rn) andf2∈Lp–(Rn). ThenMγ(f)(x)≤Mγ(f1)(x) +Mγ(f2)(x). Noting

that 1 <p–≤p+<n/γ and 0 <γ <n, by Lemma2.4we see thatMγ(f1)(x) andMγ(f2)(x)

are ﬁnite almost everywhere. ThenMγ(f)(x) <∞for almost everyx∈Rn.

We also need some basic properties of variable Lebesgue spaces. Denote byp(·) the con-jugate index ofp(·). Obviously, ifp(·)∈P(Rn_{) then}_p₍_·₎_∈_P₍_Rn_{). The following lemma is}

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Lemma 2.7

(i) Letp(·)∈P(Rn₎_._{Then there exists a positive constant}_C_{such that for all}

f ∈Lp(·)₍_Rn₎_and_g_∈_Lp(·)₍_Rn₎_,

Rn

f(x)g(x)dx≤C f _Lp(·)₍_Rn₎ g _Lp(·)₍_Rn₎.

(ii) Letp(·),p1(·),p2(·)∈P(Rn)and1/p(·) = 1/p1(·) + 1/p2(·).Then there exists a

positive constantCsuch that for allf ∈Lp1(·)₍Rn₎_and_g∈_Lp2(·)₍Rn₎_, fg _Lp(·)₍_Rn₎≤C f _Lp1(·)₍_Rn₎ g _Lp2(·)₍_Rn₎.

Lemma 2.8([7]) Given p(·)∈P(Rn_),_{for all r}_{> 0}_{we have}

_|f|r_Lp(·)₍_Rn₎= f r_Lrp(·)₍_Rn₎.

Lemma 2.9([17]) Let p(·)∈B(Rn_),_{then there exists a constant C}_{> 0}_{such that}

1

|Q| χQ Lp(·)(Rn) χQ Lp(·)₍_Rn₎≤C

for all cubes Q inRn_.

Lemma 2.10([27]) Let0 <γ <n,p(·)∈P(Rn)with p+<n/γ and1/q(·) = 1/p(·) –γ/n.If q(·)(n–γ)/n∈B(Rn_),_{then there exists a constant C}_{> 0}_{such that}

χQ Lp(·)₍_Rn₎≤C|Q|γ/n χQ Lq(·)₍_Rn₎

for all cubes Q inRn.

Now, we give the following pointwise estimate for [b,Mα] whenb∈ ˙Λβ(Rn).

Lemma 2.11 Let0≤α<n, 0 <β< 1, 0 <α+β<n and let f be a locally integrable function.

If b∈ ˙Λβ(Rn)and b≥0,then,for any x∈Rnsuch that Mα(f)(x) <∞,we have _[_b_,_M_α_]_f₍_x₎_≤_b _Λ_˙

βMα+β(f)(x).

Proof For any ﬁxedx∈Rn_{such that}_M

α(f)(x) <∞, ifb≥0 andb∈ ˙Λβ(Rn) then [b,Mα](f)(x)=b(x)Mα(f)(x) –Mα(bf)(x)

=sup

Qx

1

|Q|1–α/n

Q

b(x)f(y)dy–sup

Qx

1

|Q|1–α/n

Q

b(y)f(y)dy

≤sup

Qx

1

|Q|1–α/n

Q

b(x) –b(y)f(y)dy

≤ b Λ˙β(Rn)sup

Qx

1

|Q|1–(α+β)/n

Q

f(y)dy

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Finally, we also need the following result.

Lemma 2.12([2,28]) Let0≤γ <n,Q be a cube inRn_,_{and f be a locally integrable}

func-tion.Then for all x∈Q,

Mγ(fχQ)(x) =Mγ,Q(f)(x)

and

Mγ(χQ)(x) =Mγ,Q(χQ)(x) =|Q|γ/n.

3 Proofs of Theorems1.1,1.2and1.3

To prove Theorem1.1, we ﬁrst prove the following lemma.

Lemma 3.1 Let0 <β< 1and0 <γ <n.If b is a locally integrable function and satisﬁes

sup

Q

1

|Q|β/n

(b–|Q|–γ/n_M

γ,Q(b))χQ Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎ ≤

C (3.1)

for some s(·)∈B(Rn),then b∈ ˙Λβ(Rn).

Proof Some ideas are taken from [2,28] and [29]. Reasoning as the proof of (4.4) in [29], see also the proof of Lemma 2.4 in [28], we have, for any cubeQ,

1

|Q|1+β/n

Q

b(x) –bQdx≤

2

|Q|1+β/n

Q

b(x) –|Q|–γ/nMγ,Q(b)(x)dx.

Indeed, for any cubeQ, letE={x∈Q:b(x)≤bQ}andF={x∈Q:b(x) >bQ}. It is easy

to check that the following equality is true (see [2] page 3331):

E

b(x) –bQdx=

F

b(x) –bQdx.

Noticing the obvious estimate

|bQ| ≤ |Q|–γ/nMγ,Q(b)(x) for anyx∈Q

andb(x)≤bQfor anyx∈E, we have

b(x)≤bQ≤ |bQ| ≤ |Q|–γ/nMγ,Q(b)(x) for anyx∈E.

Then, for anyx∈E,

b(x) –bQ≤b(x) –|Q|–γ/nMγ,Q(b)(x).

Therefore,

1

|Q|1+β/n

Q

|b(x) –bQ|dx=

1

|Q|1+β/n

E∪F

b(x) –bQdx

= 2

|Q|1+β/n

E

(11)

≤ 2 |Q|1+β/n

E

b(x) –|Q|–γ/nMγ,Q(b)(x)dx

≤_| 2

Q|1+β/n

Q

b(x) –|Q|–γ/n_M

γ,Q(b)(x)dx.

By Lemma2.7(i), (3.1) and Lemma2.9, we get

1

|Q|1+β/n

Q

b(x) –bQdx

≤_| C

Q|1+β/nb–|Q|

–γ/n_M

γ,Q(b)

χQ_Ls(·)₍_Rn₎ χQ Ls(·)₍_Rn₎

≤ C

|Q| χQ Ls(·)(Rn) χQ Ls(·)₍_Rn₎ ≤C.

So, the proof is completed by applying Lemma2.1.

Proof of Theorem1.1 Since the implications (3)⇒(2) and (5)⇒(4) follows readily, we only need to prove (1)⇒(3), (2)⇒(4), (4)⇒(1), and (3)⇒(5).

(1)⇒(3). Letb∈ ˙Λβ(Rn) andb≥0. We need to prove that [b,Mα] is bounded from

Lp(·)(Rn) toLq(·)(Rn) for all (p(·),q(·))∈Bα+β₍_Rn_{). For such} _p₍_·_{) and any}_f _∈_Lp(·)₍_Rn_{), it}

follows from Lemma2.6thatMα(f)(x) <∞for almost everyx∈Rn. By Lemma2.11, we have

[b,Mα]f(x)≤ b Λ˙βMα+β(f)(x).

Then, statement (3) follows from Lemma2.5.

(2)⇒(4). Let (p(·),q(·))∈Bα+β₍_Rn_{) be such that [}_b_,_M

α] is bounded fromLp(·)(Rn) to

Lq(·)₍_Rn_{). We will verify (}_1.2_{) for}_s₍_·_{) =}_q₍_·_{). For any ﬁxed cube}_Q_{and any}_x_∈_Q_{, it follows}

from Lemma2.12that

Mα(χQ)(x) =Mα,Q(χQ)(x) =|Q|α/n and Mα(bχQ)(x) =Mα,Q(b)(x).

Then, for anyx∈Q,

b(x) –|Q|–α/n_M

α,Q(b)(x)

=|Q|–α/nb(x)|Q|α/n–Mα,Q(b)(x)

=|Q|–α/nb(x)Mα(χQ)(x) –Mα(bχQ)(x)

=|Q|–α/n[b,Mα](χQ)(x).

Thus

b(x) –|Q|–α/nMα,Q(b)(x)

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Noting that [b,Mα] is bounded fromLp(·)(Rn) toLq(·)(Rn) with 1/q(·) = 1/p(·) – (α+β)/n and applying Lemma2.10, we have

b–|Q|–α/nMα,Q(b)

χQ_Lq(·)₍_Rn₎ ≤ |Q|–α/n_[_b_,_M

α](χQ)_Lq(·)₍_Rn₎ ≤C|Q|–α/n_[_b_,_M

α]_Lp(·)_→_Lq(·) χQ Lp(·)₍_Rn₎ ≤C|Q|β/n[b,Mα]_Lp(·)_→_Lq(·) χQ Lq(·)₍_Rn₎,

which gives (1.2) fors(·) =q(·) sinceQis arbitrary andCis independent ofQ. (4)⇒(1). By Lemma2.2, it suﬃces to prove

sup

Q

1

|Q|1+β/n

Q

b(x) –MQ(b)(x)dx<∞. (3.2)

For any ﬁxed cubeQ,

1

|Q|1+β/n

Q

b(x) –MQ(b)(x)dx

≤_| 1

Q|1+β/n

Q

b(x) –|Q|–α/nMα,Q(b)(x)dx

+ 1

|Q|1+β/n

Q

_|Q|–α/nMα,Q(b)(x) –MQ(b)(x)dx

:=I1+I2. (3.3)

ForI1, by statement (4) and applying Lemma2.7(i) and Lemma2.9, we have

I1≤ C

|Q|1+β/nb–|Q|

–α/n_M

α,Q(b)

χQ_Ls(·)₍_Rn₎ χQ _Ls(·)₍_Rn₎

≤_| C

Q|β/n

(b–|Q|–α/n_M

χQ Ls(·)₍_Rn₎ ≤C,

where the constantCis independent ofQ.

Next, we considerI2. Similar to the proof of Theorem 1.1 in [31], we can getI2≤C.

Now, we give the proof of this. For allx∈Q, it follows from Lemma2.12that

M(χQ)(x) =χQ(x) = 1 and M(bχQ)(x) =MQ(b)(x),

and

Mα(χQ)(x) =|Q|α/n and Mα(bχQ)(x) =Mα,Q(b)(x).

Then, for anyx∈Q,

_|Q|–α/nMα,Q(b)(x) –MQ(b)(x)

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≤ |Q|–α/nMα(bχQ)(x) –b(x)Mα(χQ)(x)

+b(x)M(χQ)(x) –M(bχQ)(x)

≤ |Q|–α/n|b|,Mα

(χQ)(x)+|b|,M

(χQ)(x). (3.4)

Sinces(·)∈B(Rn_{), statement (4) along with Lemma}_3.1_gives_b_{∈ ˙}_Λ

β(Rn), which implies |b| ∈ ˙Λβ(Rn). Thus, we can apply Lemma2.11to [|b|,Mα] and [|b|,M] since|b| ∈ ˙Λβ(Rn) and|b| ≥0. By Lemmas2.11and2.12, we have, for anyx∈Q,

_|b|,Mα

(χQ)(x)≤ b Λ˙βMα+β(χQ)(x)≤C b Λ˙β|Q|

(α+β)/n

and

_|b|,M(χQ)(x)≤ b Λ˙βMβ(χQ)(x)≤C b Λ˙β|Q| β/n_.

By (3.4), we have

I2=

1

|Q|1+β/n

Q

_|Q|–α/nMα,Q(b)(x) –MQ(b)(x)dx

≤_| C

Q|1+(α+β)/n

Q

_|_b_|_,_M_α

(χQ)(x)dx

+ C

|Q|1+β/n

Q

_|b|,M(χQ)(x)dx

≤C b Λ˙β.

Putting the above estimates forI1andI2into (3.3), we obtain (3.2).

(3)⇒(5). Assume statement (3) is true. Reasoning as in the proof of (2)⇒(4), we have

sup

Q

1

|Q|β/n

(b–|Q|–α/n_M

α,Q(b))χQ Lq(·)₍_Rn₎

χQ Lq(·)₍_Rn₎

<∞ (3.5)

for anyq(·) for which there exists ap(·) such that (p(·),q(·))∈Bα+β₍_Rn_).

For anys(·)∈B(Rn_{), choosing an} _r_>_n_/(_n_–_{β), we have}_rs₍_·₎₍_n_–_β_)/_n_∈_B₍_Rn_{) and}

rs(·)∈B(Rn_{) by Remark}_1.1_{. Set}_q₍_·_{) =}_rs₍_·_{) and deﬁne}_p₍_·_{) by 1/}_p₍_·_{) = 1/}_q₍_·_{) + (α}₊_β_)/_n_{. It}

is easy to check that (p(·),q(·))∈Bα+β₍_Rn_).

Noting that

1

s(·)= 1

rs(·)+ 1

rs(·)= 1

q(·)+ 1

rs(·),

it follows from Lemma2.7(ii), (3.5) and Lemma2.8that

1

|Q|β/n

(b–|Q|–α/n_M

χQ Ls(·)₍_Rn₎ ≤_| 1

Q|β/n

(b–|Q|–α/n_M

α,Q(b))χQ Lq(·)(Rn₎ χQ Lrs(·)₍_Rn₎

(14)

≤C χQ Lq(·)₍_Rn₎ χQ _Lrs(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

=

C χQ 1/_Lsr(·)₍_Rn₎ χQ 1/r

Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

=C,

which is what we want.

The proof of Theorem1.1is ﬁnished.

Remark3.1 The proof of (3)⇒(5) is also valid forβ= 0.

To prove Theorem1.2, we recall the following results obtained in [29].

Lemma 3.2

(1) Letp(·)∈B(Rn₎_._If₀_≤_b_∈_BMO₍_Rn₎_,_then_[_b_,_M_]_{is bounded from}_Lp(·)₍_Rn₎_{to itself}_.

(2) Let0 <γ <n,p(·)∈P(Rn₎_with_p

+<n/γ,1/q(·) = 1/p(·) –γ/nand q(·)/(n–γ)∈B(Rn₎_._If₀_≤_b_∈_BMO₍_Rn₎_,_then_[_b_,_M

α]is bounded fromLp(·)(Rn)to

Lq(·)₍_Rn₎_.

The following result can be deduced from the proof of Lemma 4.1 in [29].

Lemma 3.3 Let0 <γ <n.If b is a locally integrable function and satisﬁes

sup

Q

(b–|Q|–γ/n_M

γ,Q(b))χQ Ls(·)₍_Rn₎

χQ Ls(·)₍_Rn₎

<∞

for some s(·)∈B(Rn_),_{then b}_∈_BMO₍_Rn_).

Proof of Theorem 1.2 Since the equivalence of (1), (2) and (3) was given in [29, Theo-rem 1.1], the implication (2)⇒(4) follows from [29, Lemma 4.1] and (3)⇒(5) follows from Remark3.1, we only need to prove the implication (4)⇒(1).

For any ﬁxed cubeQ, it follows from (3.3) and (3.4) that

1

|Q|

Q

b(x) –MQ(b)(x)dx≤

1

|Q|

Q

b(x) –|Q|–α/nMα,Q(b)(x)dx

+ 1

|Q|1+α/n

Q

_|b|,Mα

(χQ)(x)dx

+ 1

|Q|

Q

_|b|,M(χQ)(x)dx

:=J1+J2+J3. (3.6)

ForJ1, by Lemma2.7(i), Lemma2.9and statement (4), we have

J1≤ C

|Q|b–|Q| –α/n_M

α,Q(b)

χQ_Ls(·)₍_Rn₎ χQ _Ls(·)₍_Rn₎ ≤C (b–|Q|–

α/n_M

χQ Ls(·)₍_Rn₎ ≤C,

(15)

Setq(·) =s(·)n/(n–α). By Remark1.1, we haveq(·)∈B(Rn_{) since}_s₍_·₎_∈_B₍_Rn_{). For a}

p(·) such that 1/q(·) = 1/p(·) –α/n, we havep(·)∈P(Rn_{) and}_p

+<n/α.

Noticing thats(·)∈B(Rn_{), statement (4) along with Lemma}_3.3_gives_b_∈_BMO₍_Rn_),

which implies|b| ∈BMO(Rn_{). Thus, we can apply Lemma}_3.2_{to [}_|_b_|_,_M

α] and [|b|,M] for the pair of exponentsp(·) andq(·) given as above and get

_|b|,M(χQ)_Lp(·)₍_Rn₎≤C χQ Lp(·)₍_Rn₎

and

_|b|,Mα

(χQ)_Lq(·)₍_Rn₎≤C χQ Lp(·)₍_Rn₎.

Then, it follows from Lemma2.7(i), Lemma2.10and Lemma2.9that

J2=

1

|Q|1+α/n

Q

_|b|,Mα

(χQ)(x)dx

≤_| C

Q|1+α/n|b|,Mα

(χQ)_Lq(·)₍_Rn₎ χQ _Lq(·)₍_Rn₎

≤_| C

Q|1+α/n χQ Lp(·)₍_Rn₎ χQ _Lq(·)₍_Rn₎

≤ C

|Q| χQ Lq(·)(Rn) χQ Lq(·)₍_Rn₎ ≤C.

Similarly, by Lemma2.7(i) and Lemma2.9, we have

J3=

1

|Q|

Q

_|b|,M(χQ)(x)dx

≤ C

|Q||b|,M

(χQ)_Lp(·)₍_Rn₎ χQ _Lp(·)₍_Rn₎ ≤_|C

Q| χQ Lp(·)(Rn) χQ Lp(·)(Rn₎ ≤C.

Putting the above estimates forJ1,J2andJ3into (3.6), we obtain

1

|Q|

Q

b(x) –MQ(b)(x)dx≤C,

which impliesb∈BMO(Rn_{) and}_b–_∈_L∞₍_Rn_{) by Lemma}_2.3_{, since the constant}_C_is

inde-pendent ofQ.

The proof of Theorem1.2is completed.

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(1)⇒(3). Ifb∈ ˙Λβ(Rn), then

Mα,b(f)(x) =sup Qx

1

|Q|1–α/n

Q

b(x) –b(y)f(y)dy

≤C b Λ˙βsup

Qx

1

|Q|1–(α+β)/n

Q

f(y)dy

=C b Λ˙βMα+β(f)(x).

This, together with Lemma2.5, shows thatMα,bis bounded fromLp(·)(Rn) toLq(·)(Rn).

(2)⇒(4). For any ﬁxed cubeQ, we have for allx∈Q,

|b(x) –bQ| ≤

1

|Q|

Q

b(x) –b(y)dy

= 1

|Q|

Q

b(x) –b(y)χQ(y)dy

≤ |Q|–α/nMα,b(χQ)(x).

Then, for allx∈Rn, b(x) –bQ

χQ(x)≤ |Q|–α/nMα,b(χQ)(x).

SinceMα,bis bounded fromLp(·)(Rn) toLq(·)(Rn), by Lemma2.10, we have

(b–bQ)χQ_Lq(·)₍_Rn₎≤ |Q|–α/nMα,b(χQ)_Lq(·)₍_Rn₎

≤C Mα,b Lp(·)_→_Lq(·)|Q|–α/n χQ Lp(·)₍_Rn₎ ≤C Mα,b Lp(·)_→_Lq(·)|Q|β/n χQ Lq(·)₍_Rn₎,

which gives (1.8) fors(·) =q(·) sinceQis arbitrary andCis independent ofQ.

The proof of Theorem1.3is ﬁnished.

Acknowledgements

The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments.

Funding

The ﬁrst author is supported by the National Natural Science Foundation of China (Grant Nos. 11571160, 11471176) and the Scientiﬁc Research Fund of Mudanjiang Normal University (No. MSB201201). The second author is supported by the Key Research Project for Higher Education in Henan Province (No. 19A110017).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Mudanjiang Normal University, Mudanjiang, P.R. China.}2_{School of Mathematics and}

Information Science, Henan Polytechnic University, Jiaozuo, P.R. China.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

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