Boundedness of rough singular integral operators and commutators on Morrey Herz spaces with variable exponents

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R E S E A R C H

Open Access

Boundedness of rough singular integral

operators and commutators on Morrey-Herz

spaces with variable exponents

Liwei Wang

1*

_{, Meng Qu}

2

_{and Lisheng Shu}

2

*_{Correspondence:}

wangliwei@ahpu.edu.cn

1_{Department of Applied}

Mathematics, School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, The People’s Republic of China

Full list of author information is available at the end of the article

Abstract

By decomposing functions, we establish some boundedness results for some rough singular integrals on the homogeneous Morrey-Herz spacesMK˙_qα_,_p(·₍),_·λ₎(Rn_{), where the} two main indices are variable. The corresponding results as regards their

commutators are also considered.

MSC: 42B20; 42B25

Keywords: variable exponent; Morrey-Herz spaces; commutators; singular integrals

1 Introduction

In recent years, function spaces with variable exponents have been intensively studied; see [–] for example. The origin of such spaces is the study of PDE with non-standard growth conditions, ﬂuid dynamics and image restoration; see [–]. By virtue of the fundamental work [] by Kováčik and Rákosník appearing in , the Lebesgue spaces and various other function spaces have been investigated in the variable exponent setting. Meanwhile, the boundedness of some classical operators, such as the Hardy-Littlewood maximal op-erator, singular integrals and commutators, has been proved on these spaces; see [–] and the references therein.

Herz spaces have been playing a central role in harmonic analysis. After they were in-troduced in [], the theory of these spaces had a remarkable development in part due to its useful applications. For instance, they are good substitutes of the ordinary Hardy spaces when considering the boundedness of non-translation invariant singular integral operators, they also appear in the characterization of multiplier on Hardy spaces and in the regularity theory for elliptic and parabolic equations in divergence form; see [–] for example.

One of the important problems on Herz spaces is the boundedness of sublinear opera-tors satisfying the size condition

Tf(x)

Rn |f(y)|

|x–y|ndy, x∈/suppf, (.)

for integrable and compactly supported functionsf. We mention that condition (.) was initially studied by Soria and Weiss [] and it is satisﬁed by several classical

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erators, such as Calderón-Zymund operators, the Carleson maximal operator and the Hardy-Littlewood maximal operator. Hernández, Li, Lu, and Yang [–] proved that if a sublinear operatorT is bounded onLp(Rn) and satisﬁes the size condition (.), then T is bounded on the homogeneous Herz spacesK˙pα,q(Rn) and on the non-homogeneous Herz spacesKpα,q(Rn). This result is extended to the generalized Herz spacesK˙_pα₍,_·q₎(Rn) and K_pα₍,_·q₎(Rn_{) with variable exponent}_{p(·) by Izuki []. In , Almeida and Drihem [] made} a further step and gave boundedness results forTonK˙_pα₍(_··₎),q(Rn_{) and}_Kα(·),q

p(·) (Rn), where the

exponentαis variable as well.

Denote bySn–_{the unit sphere in}_Rn_(n_≥_{) with normalized Lebesgue measure}_dσ_{. Let} ∈Ls_(Sn–_{) for some}_s_∈_(,_{∞] be homogeneous of degree zero. In this paper, we consider}

sublinear operators satisfying the size condition

Tf(x)

Rn

|(x–y)|

|x–y|n f(y)dy, x∈/suppf, (.)

for integrable and compactly supported functionsf and commutators deﬁned by

[b,T]f(x) :=b(x)Tf(x) –T(bf)(x), b∈BMO

Rn_. _(.)

Lu et al.ﬁrst proved the boundedness for both T and [b,T] on Herz spaces and

Morrey-Herz spaces with constant exponent; see [, ]. Motivated by the work of Lu [] and Almeida [], we shall generalize these results to the case of variable exponent and also consider the boundedness on Morrey-Herz spaces with variable exponent. Our approach is mainly based on some properties of variable exponent and BMO norms ob-tained by the author [].

For brevity, we denoteB:={y∈Rn:|x–y|<r}.fBstands for the integral average off onB,i.e. fB=_|_B_|

Bf(x)dx.p(·) means the conjugate exponent /p(·)+/p(·) = .Cdenotes a positive constant, which may have diﬀerent values even in the same line.f gmeans thatf ≤Cg, andf ≈gmeansf gf.

2 Preliminaries and lemmas

LetE⊂Rn_{with the Lebesgue measure}_|_E_|_{> ,}_p(_·_{) :}_E_→_[,_∞_{) be a measurable function.} The Lebesgue space with variable exponentLp(·)_{(E) is deﬁned by}

Lp(·)(E) =

f is measurable :

E

|f(x)| λ

p(x)

dx<∞for some constantλ> 

.

This is a Banach space with the Luxemburg-Nakano norm

f _Lp(·)₍_E₎=inf

λ>  :

E _|f_(x)|

λ p(x)

dx≤

.

The space with variable exponentLp_loc(·)(E) is deﬁned by

Lp_loc(·)(E) =f :f ∈Lp(·)(K) for all compact subsetsK⊂E.

We denote

p–=ess inf

p(x) :x∈E, p+=ess sup

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P(E) =p(·) :p–>  andp+<∞

,

and

B(E) =p(·)∈P(E) :Mis bounded onLp(·)(E),

where the Hardy-Littlewood maximal operatorMis deﬁned by

Mf(x) =sup

r>

r–n

B(x,r)∩E

f(y)dy.

A functionα(·) :Rn_→_R_{is called log-Hölder continuous at the origin (or has a log decay} at the origin), if there exists a constantClog>  such that

α(x) –α()≤ Clog

log(e+ /|x|)

for allx∈Rn_{. If, for some}_α

∞∈RandClog> , we have

_{α(x) –}α∞≤ Clog log(e+|x|)

for allx∈Rn_{, then}_α(_·_{) is called log-Hölder continuous at inﬁnity (or has a log decay at} inﬁnity).

ByP_log(Rn_{) and}_Plog

∞(Rn) we denote the class of all exponentsp∈P(Rn) which have

a log decay at the origin and at inﬁnity, respectively. It is worth noting that if p(·)∈

Plog

 (Rn)∩P

log

∞(Rn), then we havep(·)∈B(Rn); see [] or [].

LetBk={x∈Rn:|x| ≤k},Rk=Bk\Bk–andχk=χRk be the characteristic function of the setRkfork∈Z. Almeida and Direhem [] ﬁrst introduced the following Herz spaces with variable exponents.

Deﬁnition . Let  <q≤ ∞,p(·)∈P(Rn_{) and}_{α(·) :}_Rn_→_R_with_α_∈_L∞_(Rn_).

() The homogeneous Herz spaceK˙_pα₍(_··₎),q(Rn₎_{is deﬁned as the set of all}_f _∈_Lp(·)

loc(Rn\{})

such that

f _˙

K_p(·)α(·),q(Rn₎:=

k∈Z

kα(·)fχkq_Lp(·)₍_Rn₎

/q <∞.

() The non-homogeneous Herz spaceK_pα₍(_··₎),q(Rn)consists of allf∈Lp_loc(·)(Rn)such that

f _Kα(·),q

p(·) (Rn):= fχB Lp(·)(Rn)+

k≥

kα(·)fχk q Lp(·)(Rn₎

/q <∞,

with the usual modiﬁcation whenq=∞.

Deﬁnition . Let  <q≤ ∞, p(·)∈P(Rn_{), }_≤_λ_<_∞_and_α(_·_{) :}_Rn_→_R_with_α _∈ L∞(Rn_{). The homogeneous Morrey-Herz space}_M_K˙α(·),λ

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f ∈Lp_loc(·)(Rn_{\{}) such that}

f _MK˙_q,p(·)α(·),λ(Rn₎:=sup L∈Z

–Lλ

_L

k=–∞

kα(·)fχk q Lp(·)(Rn₎

/q <∞,

with the usual modiﬁcation whenq=∞.

Remark . It is easy to see that MK˙_qα_,(_p·₍),_·₎(Rn_{) =} _K˙α(·),q

p(·) (Rn). If α(·) is constant, then

MK˙_qα_,_p(·₍),_·λ₎(Rn_{) =}_M_K˙α,λ

q,p(·)(Rn), which is ﬁrst deﬁned by Izuki []. If bothα(·) andp(·) are

constant, thenMK˙_qα_,_p(·₍),_·₎(Rn_{) =}_K˙α,p

q (Rn) are classical Herz spaces in [], andMK˙_qα_,_p(·₍),_·λ₎(Rn) = MK˙α,λ

p,q(Rn) are classical Morrey-Herz spaces in [].

In [, ], Lu and Zhu obtained the following result:

Proposition . Let <q≤ ∞,p(·)∈P(Rn_{), }_≤_λ_<_∞_and_α_∈_L∞₍_Rn_)._If_α _is log-Hölder continuous both at origin and at inﬁnity,then

f _M_K_˙α(·),λ

q,p(·)(Rn)≈max

sup

L<,L∈Z –Lλ

_L

k=–∞

kα()q fχk q_Lp(·)₍_Rn₎ /q

,

sup

L≥,L∈Z

–Lλ _–

k=–∞

kα()q fχk q_Lp(·)₍_Rn₎ /q

+ –Lλ _L

k=

kα∞q fχk q_Lp(·)₍_Rn₎ /q

.

Before stating our main results, we introduce some key lemmas which will be used later.

Lemma .([]) (Generalized Hölder inequality) Let p(·)∈P(Rn_),_{if f} _∈_Lp(·)_(Rn₎_and g∈Lp(·)₍_Rn_),_then

Rn

f(x)g(x)dx≤rp f Lp(·)(Rn₎ g _Lp(·)₍_Rn₎,

where rp=  + /p–– /p+.

Lemma .([]) Let∈Ls_(Sn–_),_s_∈_[,_∞_]._{If a}_{> ,}_d_∈_(,_s]_and_–n₊(n–)d

s <ν<∞, then

|y|≤a|x|

|y|ν_(x_–_y)d_dy

/d

_Ls₍_Sn–₎|x|(ν+n)/d.

Lemma .([]) Let p(·)∈P(Rn_)._{If q}_∈_(p

+,∞)and _p₍_x₎ =_q₍_x₎ +_q (x∈Rn),then we

have

fg _Lp(·)₍_Rn₎ f _Lq(·)₍_Rn₎ g _Lq₍_Rn₎

for all measurable functions f and g.

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Lemma . Let p(·)∈B(Rn_),_{then we see that for all balls B in}_Rn_,



|B| χB Lp(·)(Rn) χB Lp(·)₍_Rn₎.

Lemma . Let p(·)∈B(Rn),then we see that for all balls B inRnand all measurable subsets S⊂B,

χS Lp(·)(Rn₎ χB Lp(·)(Rn₎

_|S| |B|

δ

, χS Lp _(·)

(Rn₎ χB Lp(·)(Rn₎

_|S| |B|

δ ,

whereδ,δare constants with <δ,δ< .

Lemma . Let b∈BMO(Rn_),_k_>_j_(k,_j_∈_N_),_{then we have}

sup

B⊂Rn  χB Lp(·)(Rn₎

(b–bB)χB_Lp(·)₍_Rn₎≈ b BMO

and

(b–bBj)χBkLp(·)(Rn₎(k–j) b BMO χBk Lp(·)(Rn₎.

Lemma . Letα∈L∞(Rn₎_{and r}

> .Ifαis log-Hölder continuous both at origin and

at inﬁnity,then

rα_(x)rα_(y)× ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(r r)

α+_, _{ <}_r

≤r/,

, r/ <r≤r,

(r r)

α–_, _r

> r,

for any x∈B(,r)\B(,r/)and y∈B(,r)\B(,r/),with the implicit constant not

de-pending on x,y,r,and r.

3 Main results and their proofs

In this section, we prove the boundedness of sublinear operatorsT satisfying the size

condition (.) and the commutators [b,T] deﬁned as in (.) on the homogeneous

Morrey-Herz spaceMK˙_qα_,(_p·₍),_·₎λ(Rn_{), respectively.}

Our main results in this paper can be stated as follows.

Theorem . Suppose that p(·)∈P_log(Rn₎_∩_P_∞log_(Rn₎_and_∈_Ls_(Sn–_),_s_{> (p}₎

+.Let <

λ<n,  <q≤ ∞, –/s<ν<∞and letα(·)∈L∞(Rn₎_{be log-Hölder continuous both at the} origin and at inﬁnity,such that

λ–nδ–ν–n/s<α–≤α+<nδ–ν–n/s, (.)

where <δ,δ< are the constants appearing in Lemma..Then every sublinear

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Theorem . Suppose that b∈BMO(Rn₎_{and let}_[b,_T

]be deﬁned as in(.).Under the

assumptions in Theorem.,if[b,T]is bounded on Lp(·)(Rn),then[b,T]is also bounded

on MK˙_qα_,(_p·₍),_·λ₎(Rn).

Remark . According to Remark ., Theorem . and Theorem . extend the corre-sponding results in [] to a more generalized function space. Moreover, comparing with [, ], our main results generalize the integral kernel to the case of∈Ls_(Sn–_).

Our proofs use partially some decomposition techniques already used in [] where the constant exponent case was studied. We consider only  <q<∞, the arguments are similar in the caseq=∞.

Proof of Theorem. Letf ∈MK˙_qα_,(_p·₍),_·λ₎(Rn_{). We decompose}

f(x) =

∞

j=–∞

f(x)χj(x) =

∞

j=–∞

fj(x).

Then we have

T(f)

q

MK˙_q,p(·)α(·),λ(Rn₎=sup L∈Z

–Lλq L

k=–∞

kα(·)_T

(f)χk q Lp(·)(Rn₎

sup

L∈Z –Lλq

L

k=–∞

kα(·)

_k_–

j=–∞

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

+sup

L∈Z –Lλq

L

k=–∞

kα(·)

_k_+

j=k–

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

+sup

L∈Z

–Lλq L

k=–∞

kα(·)

_∞

j=k+

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

=:U+U+U.

First we estimateU. Noting that ifx∈Rk,y∈Rjandj≤k– , then|x–y| ≈ |x| ≈k, by Lemma ., we get

kα(x)

_k_–

j=–∞

T(fj)(x)

χk(x) k–

j=–∞

–kn

Rj

kα(x)_(x_–_y)_f

j(y)dy·χk(x)

k–

j=–∞

–kn(k–j)α+

Rj

jα(y)(x–y)fj(y)dy·χk(x),

which combining with Lemma . yields

kα(·)

k–

j=–∞

_T_(f_j_)(·)χk(·)

Lp(·)(Rn₎

k–

j=–∞

–kn(k–j)α+_jα(·)_f

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We deﬁne a variable exponentp(·) by_p₍_x₎ =_p₍_x₎+_s, sinces> (p)+, by Lemma . and

Lemma ., we obtain

(x–·)χj(·)_Lp(·)₍_Rn₎

(x–·)χj(·)_Ls₍_Rn₎χj(·)_Lp(·)₍_Rn₎

–jν

Rj

|y|sν_(x_–_y)s_dy

/s

χBj(·)Lp(·)₍_Rn₎

–jνk(ν+n/s) _Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎|Bj|–/s, (.)

where the last inequality is based on the fact that χBj(·) Lp(·)₍_Rn₎≈ χBj(·) Lp(·)₍_Rn₎|Bj|–/s; see [], p., for the details.

From (.), (.), Lemma ., and Lemma ., we deduce that

kα(·)

k–

j=–∞

T(fj)(·)χk(·)

Lp(·)(Rn₎

k–

j=–∞

–kn(k–j)(α++ν+n/s)

Ls₍_Sn–₎jα(·)f_j

Lp(·)(Rn₎χBj(·)_Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎

k–

j=–∞

(k–j)(α++ν+n/s)

Ls₍_Sn–₎jα(·)f_j Lp(·)(Rn₎

χBj(·) Lp(·)₍_Rn₎ χBk(·) Lp(·)(Rn₎

k–

j=–∞

(j–k)(nδ–α+–ν–n/s)_jα(·)_f

j_Lp(·)₍_Rn₎. (.)

For convenience below we putσ=nδ–α+–ν–n/s. It follows from the condition (.)

thatσ> . Now we can distinguish two cases as follows: Case◦: If  <q≤, using the well-known inequality

_∞

j=

aj q

≤

∞

j=

aq_j (aj> ,j= , , . . .), (.)

we have

Usup

L∈Z –Lλq

L

k=–∞

k–

j=–∞

(j–k)σqjα(·)fj q Lp(·)(Rn₎

sup

L∈Z

–Lλq L–

j=–∞

_jα(·)_f

j q Lp(·)(Rn₎

L

k=j+

(j–k)σq

sup

L∈Z –Lλq

L–

j=–∞

jα(·)fj q Lp(·)(Rn₎

f q

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Case◦: If  <q<∞, by Hölder’s inequality, we have

Usup

L∈Z –Lλq

L

k=–∞

_k_–

j=–∞

(j–k)σq/jα(·)fjq_Lp(·)₍_Rn₎ _k_–

j=–∞

(j–k)σq/ q/q

sup

L∈Z

–Lλq L

k=–∞

_k_–

j=–∞

(j–k)σq/_jα(·)_f

j q Lp(·)(Rn₎

sup

L∈Z

–Lλq L–

j=–∞

jα(·)fjq_Lp(·)₍_Rn₎ L

k=j+

(j–k)σq/

sup

L∈Z

–Lλq L–

j=–∞

jα(·)_f

j q Lp(·)(Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

Next we estimateU. By Proposition . and the hypothesisTis bounded onLp(·)(Rn),

so that

U≈ max

sup

L<,L∈Z –Lλq

L

k=–∞

kα()

_k_+

j=k–

T(fj)(·)

χk(·) q

Lp(·)(Rn₎ ,

sup

L≥,L∈Z

–Lλq

–

k=–∞

kα()

_k_+

j=k–

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

+ –Lλq L

k=

kα∞

_k_+

j=k–

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

max

sup

L

k=–∞

kα()|fχk|q_Lp(·)₍_Rn₎,

sup

L≥,L∈Z

–Lλq

–

k=–∞

kα()|fχk|q_Lp(·)₍_Rn₎+ –L

λq L

k=

kα∞|fχk|q_Lp(·)₍_Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

ForU, once again by Proposition ., we have

U≈ max

sup

L

k=–∞

kα()

_∞

j=k+

T(fj)(·)

χk(·) q

Lp(·)(Rn₎ ,

sup

L≥,L∈Z

–Lλq

–

k=–∞

kα()

_∞

j=k+

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

+ –Lλq L

k=

kα∞

_∞

j=k+

T(fj)(·)

χk(·) q

Lp(·)(Rn₎

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To estimateI, observe that ifx∈Rk,y∈Rjandj≥k+ , then|x–y| ≈ |y| ≈j. We apply Lemma . and obtain

T(fj)(x)–jn

Rj

(x–y)fj(y)dy

–jn fj Lp(·)(Rn₎(x–·)χj(·)_Lp(·)₍_Rn₎. (.)

An application of (.), (.), and Lemma . gives

∞

j=k+

T(fj)(·)χk(·)

Lp(·)(Rn₎

∞

j=k+

–jn fj Lp(·)(Rn₎(x–·)χj(·)_Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎

∞

j=k+

(k–j)(ν+n/s) _Ls₍_Sn–₎ f_j _Lp(·)₍_Rn₎

χBk(·) Lp(·)(Rn₎ χBj(·) Lp(·)(Rn₎

∞

j=k+

(k–j)(nδ+ν+n/s) f

j Lp(·)(Rn₎. (.)

In the sequel, we putη=nδ+ν+n/sfor short. If  <q≤, in view ofη> , from (.),

(.), and (.), we conclude that

I sup

L<,L∈Z

–Lλq L

k=–∞

kα()q

∞

j=k+

(k–j)ηq _f j q_Lp(·)₍_Rn₎

sup

L

k=–∞

kα()q L–

j=k+

(k–j)ηq fj q_Lp(·)₍_Rn₎

+ sup

L

k=–∞

kα()q

∞

j=L

(k–j)ηq fj q_Lp(·)₍_Rn₎=:I+I.

ForI, noting thatη+α() >η+α–> , hence we have

I sup

L–

j=–∞

jα()q fj q_Lp(·)₍_Rn₎ j–

k=–∞

(k–j)(η+α())q

sup

L–

j=–∞

jα()q fj q_Lp(·)₍_Rn₎ f q

MK˙_q,p(·)α(·),λ(Rn₎.

ForI, sinceλ–η–α() <λ–η–α–< , we get

I sup

L

k=–∞

kα()q

∞

j=L

(k–j)ηq–jα()qjλq

×–jλq j

l=–∞

(10)

sup

_L

k=–∞

k(α()+η)q

_∞

j=L

j(λ–η–α())q

f q

MK˙_q,p(·)α(·),λ(Rn₎

sup

L<,L∈Z

–LλqL(α()+η)qL(λ–η–α())q f q

MK˙_q,p(·)α(·),λ(Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

If  <q<∞, we have

I sup

L

k=–∞

kα()q

_L

j=k+

(k–j)η fj Lp(·)(Rn₎ q

+ sup

L<,L∈Z

–Lλq L

k=–∞

kα()q

_∞

j=L+

(k–j)η _f

j Lp(·)(Rn₎ q

=:Ia+Ib.

ForIa, by Hölder’s inequality, we get

Ia sup L<,L∈Z

–Lλq L

k=–∞

L

j=k+

jα()q fj q_Lp(·)₍_Rn₎

(k–j)(η+α())q/

×

_L

j=k+

(k–j)(η+α())q/

q/q

sup

L<,L∈Z

–Lλq L

j=–∞

jα()q_f j q_Lp(·)₍_Rn₎

j–

k=–∞

(k–j)(η+α())q/

sup

L<,L∈Z

–Lλq L

j=–∞

jα()q _f j q_Lp(·)₍_Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

ForIb, as argued inI, we obtain

Ib sup L<,L∈Z

–Lλq L

k=–∞

_∞

j=L+

jα() fj Lp(·)(Rn₎(k–j)(η+α()+λ)/(k–j)(η+α()–λ)/ q

sup

L

k=–∞

∞

j=L+

jα()q fj q_Lp(·)₍_Rn₎

(k–j)(η+α()+λ)q/

×

_∞

j=L+

(k–j)(η+α()–λ)q/ q/q

sup

L

k=–∞

∞

j=L+

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×–jλq j

l=–∞

lα()q fl q_Lp(·)₍_Rn₎

sup

L

k=–∞

kλq

∞

j=L+

(k–j)(η+α()–λ)q/ f q

MK˙_q,p(·)α(·),λ(Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

Hence, we arrive at the desired inequality,

II+I f q

MK˙_q,p(·)α(·),λ(Rn₎.

We omit the estimation ofJsince it is essentially similar to that ofI. Consequently, the proof of Theorem . is complete. Proof of Theorem. Letf ∈MK˙_qα_,(_p·₍),_·λ₎(Rn_{). We decompose}

f(x) =

∞

j=–∞

f(x)χj(x) =

∞

j=–∞

fj(x).

Then we have

[b,T](f)

q

MK˙_q,p(·)α(·),λ(Rn₎=sup L∈Z

–Lλq L

k=–∞

kα(·)[b,T](f)χk q Lp(·)(Rn₎

sup

L∈Z

–Lλq L

k=–∞

kα(·)

_k_–

j=–∞

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

+sup

L∈Z

–Lλq L

k=–∞

kα(·)

_k_+

j=k–

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

+sup

L∈Z

–Lλq L

k=–∞

kα(·)

_∞

j=k+

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

=:V+V+V.

First we estimateV. Noting that|x–y| ≈ |x| ≈kforx∈Rk,y∈Rjandj≤k– , then, from Lemma . and Lemma ., it follows that

kα(x)

k–

j=–∞

[b,T](fj)(x)χk(x)

k–

j=–∞

–kn

Rj

kα(x)b(x) –b(y)(x–y)fj(y)dy·χk(x)

k–

j=–∞

–kn(k–j)α+

Rj

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k–

j=–∞

–kn(k–j)α+_{b(x) –}_b Bj

Rj

jα(y)(x–y)fj(y)dy

+

Rj jα(y)_b

Bj–b(y)(x–y)fj(y)dy . (.)

Similar to (.), we deﬁne a variable exponentp(·) by _p₍_x₎ =_p₍_x₎+_s, sinces> (p)+, by

Lemma ., Lemma ., and Lemma ., we get

bBj–b(·)

(x–·)χj(·)_Lp(·)₍_Rn₎

(x–·)χj(·)_Ls₍_Rn₎bBj–b(·)

χj(·)_Lp(·)₍_Rn₎

b BMOχBj(·)_Lp(·)₍_Rn₎(x–·)χj(·)_Ls₍_Rn₎

b BMO(k–j)(ν+n/s) Ls₍_Sn–₎χ_B_j(·) Lp(·)(Rn₎

(k–j)(ν+n/s)χBj(·)Lp(·)₍_Rn₎. (.)

Then by (.), Lemma ., (.), (.), Lemma ., and Lemma ., we obtain

kα(·)

k–

j=–∞

[b,T](fj)(·)χk(·)

Lp(·)(Rn₎

k–

j=–∞

–kn(k–j)α+_jα(·)_f

j_Lp(·)₍_Rn₎bBj–b(·)

χk(·)_Lp(·)₍_Rn₎(x–·)χj(·)_Lp(·)₍_Rn₎

+bBj–b(·)

(x–·)χj(·)_Lp(·)₍_Rn₎χk(·)_Lp(·)₍_Rn₎

k–

j=–∞

–kn(k–j)α+_jα(·)_f

j_Lp(·)₍_Rn₎

×(k–j) b BMO(k–j)(ν+n/s) Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎ + b BMO(k–j)(ν+n/s) Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎

k–

j=–∞

(k–j)–kn(k–j)α+_jα(·)_f

j_Lp(·)₍_Rn₎(k–j)(ν+n/s)χBj(·)Lp(·)(Rn₎χBk(·)Lp(·)(Rn₎

k–

j=–∞

(k–j)(k–j)(α++ν+n/s)_jα(·)_f

j_Lp(·)₍_Rn₎

χBj(·) Lp(·)₍_Rn₎ χBk(·) Lp(·)₍_Rn₎

k–

j=–∞

(k–j)(j–k)(nδ–α+–ν–n/s)_jα(·)_f

j_Lp(·)₍_Rn₎. (.)

By comparing (.) with (.), after applying the same arguments used in the estimation ofUin Theorem ., we can easily get, for all  <q<∞,

V f q

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Next we estimateV. Using Proposition . and the boundedness of [b,T] onLp(·)(Rn),

we derive the estimate

V≈ max

sup

L<,L∈Z

–Lλq L

k=–∞

kα()

_k_+

j=k–

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎ ,

sup

L≥,L∈Z

–Lλq

–

k=–∞

kα()

_k_+

j=k–

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

+ –Lλq L

k=

kα∞

_k_+

j=k–

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

max

sup

L

k=–∞

kα()|fχk| q Lp(·)(Rn₎,

sup

L≥,L∈Z

–Lλq

–

k=–∞

_kα()_|f_χ

k| q

Lp(·)(Rn₎+ –L

λq L

k=

_kα∞_|f_χ k|

q Lp(·)(Rn₎

f q

MK˙_q,p(·)α(·),λ(Rn₎.

ForV, we apply Proposition . again and obtain

V≈ max

sup

L<,L∈Z

–Lλq L

k=–∞

kα()

_∞

j=k+

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎ ,

sup

L≥,L∈Z

–Lλq

–

k=–∞

kα()

_∞

j=k+

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

+ –Lλq L

k=

kα∞

_∞

j=k+

[b,T](fj)(·)

χk(·) q

Lp(·)(Rn₎

=:max{G,H}. (.)

To estimate G, we note that ifx∈Rk,y∈Rjandj≥k+ , then|x–y| ≈ |y| ≈j. By Lemma ., we have

[b,T](fj)(x)–jn

Rj

b(x) –b(y)(x–y)fj(y)dy

–jnb(x) –bBk

Rj

(x–y)fj(y)dy

+

Rj

b(y) –bBk(x–y)fj(y)dy

–jn fj Lp(·)(Rn₎b(x) –bBk(x–·)χj(·)Lp(·)₍_Rn₎ +(x–·)b(·) –bBk

χj(·)_Lp(·)₍_Rn₎

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As argued for (.), we get actually

(x–·)b(·) –bBk

χj(·)_Lp(·)₍_Rn₎

(x–·)χj(·)_Ls₍_Rn₎b(·) –bBk

χj(·)_Lp(·)₍_Rn₎

(j–k) b BMOχBj(·)Lp(·)(Rn₎(x–·)χj(·)_Ls₍_Rn₎

(j–k) b BMO–jνk(ν+n/s) Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎|Bj|–/s

(j–k)(k–j)(ν+n/s)_χ

Bj(·)Lp(·)₍_Rn₎. (.)

From (.) and (.), it follows that

[b,T](fj)(·)χk(·)_Lp(·)₍_Rn₎

–jn fj Lp(·)(Rn₎

(k–j)(ν+n/s) _Ls₍_Sn–₎χ_B_j(·)

Lp(·)(Rn₎b(·) –bBk

χk(·)_Lp(·)₍_Rn₎

+ (j–k) b BMO(k–j)(ν+n/s) Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎

–jn fj Lp(·)(Rn₎

(k–j)(ν+n/s)

Ls₍_Sn–₎χ_B_j(·)

Lp(·)₍_Rn₎ b BMOχBk(·)Lp(·)(Rn₎ + (j–k) b BMO(k–j)(ν+n/s) Ls₍_Sn–₎χ_B

j(·)Lp(·)₍_Rn₎χBk(·)Lp(·)(Rn₎

(j–k) fj Lp(·)(Rn₎–jn(k–j)(ν+n/s)χ_B_j(·)

Lp(·)₍_Rn₎χBj(·)Lp(·)(Rn₎

χBk(·) Lp(·)(Rn₎ χBj(·) Lp(·)(Rn₎

(j–k)(k–j)(nδ+ν+n/s) _f

j Lp(·)(Rn₎. (.)

Hence we have

∞

j=k+

_[b,_T_](f_j_)(·)χk(·)

Lp(·)(Rn₎

∞

j=k+

(j–k)(k–j)(nδ+ν+n/s)_f

j Lp(·)(Rn₎. (.)

By comparing (.) with (.), as long as we repeat the same procedure as the estimation ofIin Theorem ., we can immediately get for all  <q<∞,

G f q

MK˙_q,p(·)α(·),λ(Rn₎.

We omit the estimation ofHsince it is essentially similar to that ofG. Consequently, the proof of Theorem . is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors worked jointly in drafting and approved the ﬁnal manuscript.

Author details

1_{Department of Applied Mathematics, School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000,}

The People’s Republic of China.2_{Department of Applied Mathematics, School of Mathematics and Computer Science,}

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Acknowledgements

The authors would like to thank the referee for his very valuable comments and suggestions. Meng Qu is supported by National Natual Science Foundation of China (11471033) and University NSR Project of Anhui Province (KJ2014A087).

Received: 3 June 2016 Accepted: 2 September 2016

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