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Research article

Vector valued bilinear Calderón-Zygmund operators with kernels of Dini's type in variable exponents Herz-Morrey spaces

  • Received: 15 June 2023 Revised: 14 August 2023 Accepted: 21 August 2023 Published: 05 September 2023
  • MSC : 42B20, 42B25, 42B35

  • The main purpose of this paper is to establish the weighted boundedness result of vector valued bilinear ϖ(t)-type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, where ϖ being nondecreasing and ϖDini(1).

    Citation: Yanqi Yang, Qi Wu. Vector valued bilinear Calderón-Zygmund operators with kernels of Dini's type in variable exponents Herz-Morrey spaces[J]. AIMS Mathematics, 2023, 8(11): 25688-25713. doi: 10.3934/math.20231310

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  • The main purpose of this paper is to establish the weighted boundedness result of vector valued bilinear ϖ(t)-type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, where ϖ being nondecreasing and ϖDini(1).



    In 1985, Yabuta [1] proposed the definitions of ϖ(t)-type Calderón-Zygmund operators, he introduced certain ϖ(t)-type Calderón-Zygmund operators to facilitate his study of certain classes of pseudodifferential operators. After that, Maldonado and Naibo [2] established the weighted norm inequalities for the bilinear Calderón-Zygmund operators of type ϖ(t), and applied them to the study of para-products and bilinear pseudo-differential operators with mild regularity. In 2009, Lu and Zhang [3] established the a number of results concerning boundedness of multi-linear ϖ(t)-type Calderón-Zygmund operators. we recall the so-called ϖ(t)-type Calderón-Zygmund operators.

    Let ϖ(t): [0,)[0,) be a nondecreasing function with 0<ϖ(1)<. For α>0, we say that ϖDini(a) if

    |ϖ|Dini(α)=10ϖα(t)tdt<. (1.1)

    It is evident that for 0<α1<α2, there is Dini(α1)<Dini(α2). If ϖDini(1), then

    0ϖ(2j)10ϖ(t)tdt<,

    here and in what follows, for any quantities A and B, if there exists a constant C>0 such that ACB, we write AB. If AB and BA, we write AB.

    A measurable function K(,,) on Rn×Rn×Rn{(x,y1,y2):x=y1=y2} is said to be a bilinear ϖ(t)-type Calderón-Zygmund kernel if it satisfies: for all (x,y1,y2)Rn with xyi,i=1,2, if there exists a constant A>0 such that

    |K(x,y1,y2)|Aϖ(2i=1|xyi|)2n, (1.2)

    and for (x,y1,y2)(Rn)3 with xy1,y2, and

    |K(x,y1,y2)K(z,,y1,y2)|Aω(|xz|2i=1|xyi|)[2i=1|xyi|]2n. (1.3)

    whenever 2|xz|<max{|xy1|,|xy2|}.

    Definition 1.1. ([2]) Let ϖDini(1). One can say that Tϖ is a bilinear ϖ(t)-type operator with the kernel K satisfying (1.2) and (1.3), for all f1, f2Cc(Rn),

    Tϖ(f1,f2)(x)=RnRnK(x,y1,y2)f1(y1)f2(y2)dy1dy2,xsuppf1suppf2. (1.4)

    In the following, for each kZ, we define Bk={xRn:|x|2k}, Dk=BkBk1, χk=χDk, m1, ˜χ0=χB0.

    Given a function p(x)P(Rn), the space Lp(x)(Rn) is now defined by

    fLp()(Rn)=inf{η>0:Rn(|f(x)|η)p(x)dx1}.

    Denote P(Rn) to be the set of the all measurable functions p(x) with

    p=:essinfxRnp(x)>1

    and

    p+=:esssupxRnp(x)<,

    and B(Rn) to be the set of all functions p()P(Rn) satisfying the condition that the Hardy-littlewood maximal operator M is bounded on Lp()(Rn), P0(Rn) the set of all measurable functions p(x) with p>0 and p+<.

    The space Lp()loc(Rn) is defined by

    Lp()loc(Rn)={f:fχKLp()loc(Rn) for all compact subsets KRn},

    where and what follows, χS denotes the characteristic function of a measurable set SRn.

    Let p()P(Rn) and ω be a nonnegative measurable function on Rn. Then the weighted variable exponent Lebesgue space Lp()(ω) is the set of all complex-valued measurable functions f such that fωLp(). The space Lp()(ω) is a Banach space equipped with the norm

    fLp()(ω)=fωLp().

    Let fL1loc(Rn). Then the standard Hardy-Littlewood maximal function of f is defined by

    Mf(x)=supxB1|B|Bf(y)dy,xRn,

    where the supremum is taken over all balls containing x in Rn.

    Definition 1.2. ([4]) Let α() be a real-valued function on Rn.

    (ⅰ) For any x,yRn, |xy|<1/2, if

    |α(x)α(y)|1log(e+1/|xy|),

    then α() is said local log-Hölder continuous on Rn.

    (ⅱ) For all xRn, if

    |α(x)α(0)|1log(e+1/|x|),

    then α() is said log-Hölder continuous functions at origin, denote by Plog0(Rn) the set of all log-Hölder continuous at origin.

    (ⅲ) If there exists αR, for xRn, if

    |α(x)α|1log(e+|x|),

    then α() is said log-Hölder continuous at infinity, denote by Plog(Rn) the set of all log-Hölder continuous functions at infinity.

    (ⅳ) The function α() is global log-Hölder continuous if α() are both locally log-Hölder continuous and log-Hölder continuous at infinity. Denote by Plog(Rn) the set of all global log-Hölder continuous functions.

    Let ω be a weighted function on Rn, that is, ω is real-valued, non-negative and locally integrable. ω is said to be a Muckenhoupt A1 weight if

    Mω(x)ω(x)a.e.,xRn.

    For 1<p<, we say that ω is an Ap weight if

    supB(1|B|Bω(x)dx)(1|B|Bω(x)1pdx)p1<.

    Definition 1.3. ([5]) Let p()P(Rn). For some constant C, a weight ω is said to be an Ap() weight, if for all balls B in Rn such that

    1|B|ωχBLp()(Rn)ω1χBLp()(Rn)C.

    Lemma 1.1. ([5]) If p()Plog(Rn)P(Rn) and ωAp(), then for each fLp()(ω),

    (Mf)ωLp()fωLp(),

    Before give the definitions of the weighted Herz space and Herz-Morrey space with variable exponents, we also need the notation of the variable mixed sequence space q(Lp()), which was firstly defined in [6]. Let ω be a nonnegative measurable function. Given a sequence of functions {fj}jZ, we define the modular

    ρq(Lp()(ω))((fj)j)=jZinf{λj:Rn(|fj(x)ω(x)|λ1q(x)j)p(x)dx1},

    where λ1=1. If q+< or q()p(), the above can be written as

    ρq(Lp()(ω))((fj)j)=jZfjω|q()Lp()q().

    The norm is

    (fj)jρq(Lp()(ω))=inf{μ>0:ρq(Lp()(ω))((fjμ)j)1}.

    Definition 1.4. ([7]) Let p()P(Rn), qP0(Rn). Let α() be a bounded real-valued measurable function on Rn. The homogeneous weighted Herz space ˙Kα(),q()p()(ω) are defined by

    ˙Kα(),q()p()(ω)={fLp()loc(Rn{0},ω):f˙Kα(),q()p()(ω)<},

    where

    f˙Kα(),q()p()(ω)=(2jα()fχj)jρq(Lp()(ω)).

    Lemma 1.2. ([7]) Let α()L(Rn), p(),q()P0(Rn) and ω be a weight. If α() and q() are log-Hölder continuous at the origin, then T

    f˙Kα(),q()p()(ω)=f˙Kα,qp()(ω).

    Additionally, if α() and q() are log-Hölder continuous at the origin, then

    f˙Kα(),q()p()(ω)(k02kα(0)fχkq(0)Lp())1q(0)+(k>02kαfχkq(0)Lp())1q.

    Definition 1.5. ([8]) Let p(),q()P0(Rn), λ[0,1). Let α() be a bounded real-valued measurable function on Rn. The homogeneous weighted Herz-Morrey space M˙Kα(),q()p(),λ(ω) are defined by

    M˙Kα(),q()p(),λ(ω)={fLp()loc(Rn{0},ω):fM˙Kα(),q()p(),λ(ω)<},

    where

    fM˙Kα(),q()p(),λ(ω)=supLZ2Lλ(2kα()kfχk)kLρq(Lp()(ω)).

    Lemma 1.3. ([8]) Let p(),q()P0(Rn), ω be a weight, λ[0,) and αL(Rn). If α(), q()Plog0(Rn)Plog(Rn), then for any fLp()loc(Rn{0},ω),

    fM˙Kα(),q()p(),λ(ω)max{supL0,LZ2Lλ(2kα(0)fχk)kLlq0(Lp()(ω)),supL>0,LZ[2Lλ(2kα(0)fχk)kLρq0(Lp()(ω))+2Lλ(2kαfχk)Lk=0ρq0(Lp()(ω))]},

    where and hereafter, q0=q(0).

    Lemma 1.4. ([8]) If p()Plog(Rn)P(Rn) and ωAp(), then there exist constants δ1,δ2(0,1), such that for all balls B in Rn and all measurable subsets SB,

    χSLp()(ω)χBLp()(ω)(|S||B|)δ1,χSLp()(ω1)χBLp()(ω1)(|S||B|)δ2.

    Before proving the main results, we need the following lemmas.

    For δ>0, we denote [M(|f|δ)]1δ by Mδ. Let fL1loc(Rn). Then the sharp maximal function is defined by

    M#f(x)=supQ1QQ|f(y)fQ|dy,

    where the supremum is taken over all the cubes Q containing the point x, and where as usual fQ denotes the average of f on Q. we denote [M#(|f|δ)]1δ by M#δ.

    Lemma 2.1. ([3]) Let Tω be a bilinear ω(t)-type Calderón-Zygmund operator with ϖDini(1) and let 0<δ<12. Then, for any vector function f=(f1,f2), where each component is smooth and with compact support, the following inequality holds

    M#δ(Tω(f1,f2))(x)M(f1)(x)M(f2)(x).

    Lemma 2.2. ([9]) Let 0<p,δ< and ωA. There exists a positive constant C such that

    Rn[Mδf(x)]pω(x)dxRn[M#δf(x)]pω(x)dx

    for every function f such that the left hand side is finite.

    Lemma 2.3. ([10]) Let p(),p1(),p2()P0(Rn) such that 1p(x)=1p1(x)+1p2(x). Then for every fLp1()(Rn) and gLp2()(Rn), there exists

    fgLp()fLp1()gLp2()

    If pP(Rn), ω is a weight with ω=ω1×ω2, there exists

    fgLp()(ω)fLp1()(ω1)gLp2()(ω2).

    Lemma 2.4. ([11]) Let 0<p, δ>0. Then for non-negative sequence {aj}j=, there exists

    (j=(k=2|kj|δak)p)1p(j=apj)1p,

    when p=, above inequality stands for

    k=(2|kj|δak)supjZaj.

    Lemma 2.5. ([12]) Assume that for some p0(0,) and every ω0A, let F be a family of pairs of non-negative functions such that

    Rnf(x)p0ω0(x)dxRng0(x)p0ω0(x)dx,(f,g)F. (2.1)

    Then for all 0<p< and ω0A,

    Rnf(x)pω0(x)dxRng0(x)pω0(x)dx,(f,g)F.

    Furthermore, for every p,q(0,), ω0A, and sequences {(fj,gj)}F,

    (j=1(fj))qLp(ω0)(j=1(gj))qLp(ω0). (2.2)

    Lemma 2.6. ([8]) Assume that for some p0 and let F be a family of pairs of non-negative functions such that (2.1) holds. Let p()P0(Rn). If there exists sp such that ωsAp()s and M is bounded on L(p()s)(ωs). Then for every q(1,) and sequence {(fj,gj)}jNF

    (j=1(fj))qLp()(ω)(j=1(gj))qLp()(ω).

    Lemma 2.7. ([13]) Let p()P(Rn), and ω be a weight. If the maximal operator M is bounded both on Lp()(ω) and Lp()(ω1), q(1.), then

    (j=1(Mfj)q)1qLp()(ω)(j=1|fj|q)1qLp()(ω).

    Lemma 2.8. Let Tϖ be a bilinear Calderón-Zygmund operator with ϖDini(1) and p()P0 such that there exists sp such that ωsAp()s and M is bounded on L(p()s)(ωs). Suppose that ω=ω1×ω2 and ωiApi(),i=1,2. If piPlog(Rn)P(Rn)(i=1,2) satisfying

    1p(x)=1p1(x)+1p2(x)

    for xRn. Then for compactly supported bounded functions fj1,fj2Lp0(Rn), jN such that

    (j=1|Tϖ(fj1,fj2)|q)1qLp()(ω)2i=1(j=1|fji|qi)1qiLpi()(ωi),

    where qi(1,) for i=1,2 and

    1q=1q1+1q2.

    Proof of Lemma 2.8. Since fj1,fj2 are bounded functions with compact support, Tϖ(fj1,fj2)Lp(Rn) for every 0<p<. With Lemmas 2.1 and 2.2, Lu and Zhang [3] showed that for all ωA,

    Rn|Tϖ(f1,f2)(x)|pω(x)dxRn(Mf1(x)Mf2(x))pω(x)dx.

    Therefore, by Lemmas 2.5 and 2.6, we have

    (j=1|Tϖ(fj1,fj2)|q)1qLp()(ω)(j=1|Mfj1(x)Mfj2(x)|q)1qLp()(ω).

    Since

    1q=1q1+1q2,    1p=1p1+1p2

    and ω=ω1ω2, together with Hölders inequality, Lemmas 2.3 and 2.7, we have

    (j=1|Mfj1(x)Mfj2(x)|q)1qLp()(ω)2i=1(j=1|Mfji|qi)1qiLpi()(ωi)2i=1(j=1|fji|qi)1qiLpi()(ωi).

    We complete the proof of Lemma 2.8.

    Theorem 3.1. Let Tϖ be a bilinear ϖ-type Calderón-Zygmund operator with ϖDini(1), p1 and p2Plog(Rn)Plog(Rn) santisfying

    1p(x)=1p1(x)+1p2(x)

    and p()P0 such that there exists sp such that ωsAp()s and M is bounded on L(p()s)(ωs), where ω=ω1ω2 and ωiApi(), i=1,2. Suppose that

    α()L(Rn)Plog0(Rn)Plog(Rn),α(0)=α1(0)+α2(0),
    α=α1+α2,q()Plog0(Rn)Plog(Rn),
    1q(0)=1q1(0)+1q2(0),1q=1q1+1q2,
    λ=λ1+λ2,0λi<,δi1,δi2(0,1)

    are the constants in Lemma 1.4 for exponents pi() and weights ωi(i=1,2). Let ri(1,) and

    1r=1r1+1r2.

    If λinδi1<αi, αi(0)nδi2, then

    (j=1|Tϖ(fj1,fj2)|r)1rM˙Kα(),q()p(),λ(ω)2i=1(j=1|(fji)|ri)1riM˙Kαi(),qi()pi(),λi(ωi)

    for all fjiM˙Kαi(),qi()pi(),λi(ωi), jN, i=1,2.

    Proof of Theorem 3.1. We only consider bounded compact supported functions for the set of all bounded compactly supported functions is dense in weighted variable Lebesgue spaces (see [13]). Let fv1 and fv2 be bounded functions with compact support for vN and write

    fvi=l=fvilχl=l=fvil,i=1,2,vN.

    By Lemma 1.3, we have

    (v=1|Tϖ(fv1,fv2)|r)1rM˙Kα(),q()p(),λ(w)max{supL0,LZ2Lλ(2kα(0)(v=1|Tϖ(fv1,fv2)|r)1rχk)kLq0(Lp()(w))    supL>0,LZ[2Lλ(2kα(0)(v=1|Tϖ(fv1,fv2)|r)1rχk)k<0q0(Lp()(w))     +2Lλ(2kα(v=1|Tϖ(fv1,fv2)|r)1rχk)Lk=0q(Lp()(w))]}=max{E,F},

    where

    E=supL0,LZ2Lλ(2kα(0)(v=1|Tϖ(fv1,fv2)|r)1rχk)kLq0(Lp()(w)),F=supL>0,LZ{G+H},G=2Lλ(2kα(0)(v=1|Tϖ(fv1,fv2)|r)1rχk)k<0q0(Lp()(w)),H=2Lλ(2kα(v=1|Tϖ(fv1,fv2)|r)1rχk)Lk=0q(Lp()(w)).

    Since to estimate G is essentially similar to estimate E, it is suffice to obtain that E and H are bounded in Herz-Morrey space with variable exponents. It is easy to see that

    E9i=iEi,H9i=iHi,

    where

    E1=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k2l=k2j=Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E2=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k2l=k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E3=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k2l=j=k+2Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E4=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k+1l=k1k2j=Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E5=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k+1l=k1k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E6=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|k+1l=k1j=k+2Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E7=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1l=k+2k2j=Tϖ(fv1l,fv2j)r)1rχkq(0)Lp()(w))1q(0),E8=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|l=k+2k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),E9=supL0,LZ2Lλ(Lk=2kα(0)q(0)(v=1|l=k+2j=k+2Tϖ(fv1l,fv2j)|r)1rχkq(0)Lp()(w))1q(0),
    H1=2Lλ(Lk=02kαq(v=1|k2l=k2j=Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H2=2Lλ(Lk=02kαq(v=1|k2l=k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H3=2Lλ(Lk=02kαq(v=1|k2l=j=k+2Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H4=2Lλ(Lk=02kαq(v=1|k+1l=k1k2j=Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H5=2Lλ(Lk=02kαq(v=1|k+1l=k1k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H6=2Lλ(Lk=02kαq(v=1|k+1l=k1j=k+2Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H7:=2Lλ(Lk=02kαq(v=1|l=k+2k2j=Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H8:=2Lλ(Lk=02kαq(v=1|l=k+2k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q,H9:=2Lλ(Lk=02kαq(v=1|l=k+2j=k+2Tϖ(fv1l,fv2j)|r)1rχkqLp()(w))1q.

    We will use the following estimates. If lk1, by Hölder's inequality, Lemma 1.4 and Definition 1.3, we have

    2knRn(v=1|fvil|ri)1ridyiχkLpi()(wi)C2knχBkLpi()(wi)(v=1|fvi|ri)1riwiχlLpi()χlw1iLpi()C2kn|Bk|χBk1Lpi()(w1i)χBlLpi()(w1i)(v=1|fvi|ri)1riχlLpi()(wi)C2(lk)nδ2i(v=1|fvi|ri)1riχlLpi()(wi). (3.1)

    If l=k, then

    2knRn(v=1|fvil|ri)1ridyiχkLpi()(wiC2knχBkLpi()(wi)(v=1|fvi|ri)1riwiχlLpi()χlw1iLpi()C2knχBkLpi()(wi)χBlLpi()(w1i)(v=1|fvi|ri)1riχlLpi()(wi)(v=1|fvi|ri)1riχlLpi()(wi). (3.2)

    If lk+1, then

    2knRn(v=1|fvil|ri)1ridyiχkLpi()(wi)C2knχBkLpi()(wi)(v=1|fvi|ri)1riwiχlLpi()χlw1iLpi()C2knχBkLpi()(wi)χBlLpi()(wi)χBl1Lpi()(wi)×χBlLpi()(w1i)(v=1|fvi|ri)1riχlLpi()(wi)C2(lk)n(1δ1i)(v=1|fvi|ri)1riχlLpi()(wi). (3.3)

    Reverse the order of f1 and f2, it is obviously that the estimates of E2, E3 and E6 are similar to those of E4, E7 and E8, respectively. Thus We just need to estimate E1E3,E5,E6 and E9.

    For E1, since l, jk2, then for i=1,2,

    |xyi||x||yi|>2k12min{l,j}2k2,xDk, y1Dl, y2Dj.

    Therefore, for xDk, we have

    |K(x,y1,y2)|C(|xy1|+|xy2|)2nC22kn.

    Thus, for any xDkandl,jk2, we have

    |T(fv1l,fv2j)(x)|RnRn|fv1l(y1)||fv2j(y2)|(|xy1|+|xy2|)2ndy1dy222knRnRn|fv1l(y1)||fv2j(y2)|dy1dy2.

    Hence, together with the Hölder's and Minkowski's inequality, we have

    (v=1|k2l=k2j=Tϖ(fv1l,fv2j)|r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1k2j=2knRn|fv2j(y2)|dy2)r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1)r1)1r1χkLp1()(w1)×(v=1(k2j=2knRn|fv2j(y2)|dy2)r2)1r2χkLp2()(w2)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkLp1()(w1)×k2j=2knRn(v=1|fv2j(y2)|r2)1r2dy2χkLp2()(w2). (3.4)

    Since

    1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2,

    by Hölder's inequality, we have

    E1supL0,LZ2Lλ(Lk=2kα(0)q(0)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkq(0)Lp1()(w1)×k2j=2knRn(v=1|fv2j(y2)|r2)1r2dy2χkq(0)Lp2()(w2))1q(0)supL0,LZ2Lλ1×(Lk=2kα1(0)q1(0)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkq1(0)Lp1()(w1))1q1(0)×supL0,LZ2Lλ2×(Lk=2kα2(0)q2(0)k2j=2knRn(v=1|fv2j(y2)|r2)1r2dy2χkq2(0)Lp2()(w2))1q2(0)=E1,1×E1,2.

    For convenience's sake, we write

    E1,i=supL0,LZ2Lλi×{Lk=2kαi(0)qi(0)k2l=2knRn(v=1|fil(yi)|ri)1ridyiχkqi(0)Lpi()(wi)}1qi(0).

    For nδi2αi(0)>0, by (3.1) and Lemma 2.4 we have

    E1,isupL0,LZ2Lλi{Lk=2kαi(0)qi(0)Big.×(k2l=2(lk)nδi2(v=1|fvi|ri)1riχlLpi()(wi))qi(0)}1qi(0)=supL0,LZ2Lλi×{Lk=(k2l=2lαi(0)(v=1|fvi|ri)1riχlLpi()(wi)2(lk)(nδi2αi(0)))qi(0)}1qi(0)supL0,LZ2Lλi(L2l=2lαi(0)qi(0)(v=1|fvi|ri)1riχlqi(0)Lpi()(wi))1qi(0)(v=1|fvi|ri)1riM˙Kαi(),qi()pi(),λi(wi),

    where we write 2|kl|(nδi2αi(0))=2|kl|εi for εi=nδi2αi(0)>0, then we have

    E1(v=1|fv1|r1)1r1M˙Kα1(),q1()p1(),λ1(w1)(v=1|fv2|r2)1r2M˙Kα2(),q2()p2(),λ2(w2).

    To estimate E2, since lk2, k1jk+1, then we have

    |xy2||xy1||x||y1|2k2,xDk, y1Dl, y2Dj.

    Therefore, for xDk, we have

    |K(x,y1,y2)|C(|xy1|+|xy2|)2nC22kn.

    Thus, for any xDk,lk2,k1jk+1, we have

    |T(fv1l,fv2j)(x)|RnRn|fv1l(y1)||fv2j(y2)|(|xy1|+|xy2|)2ndy1dy222knRnRn|fv1l(y1)||fv2j(y2)|dy1dy2.

    Combining the Hölder's with and Minkowski's inequality, hence we obtain

    (v=1|k2l=k+1j=k1Tϖ(fv1l,fv2j)|r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1k+1j=k12knRn|fv2j(y2)|dy2)r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1)r)1rχkLp1()(w1)×(v=1(k+1j=k12knRn|fv2j(y2)|dy2)r)1rχkLp2()(w2)k2l=2knRn(v=1|fv1l(y1)|r)1rdy1χkLp1()(w1)×k+1j=k12knRn(v=1|fv2j(y2)|r)1rdy2χkLp2()(w2). (3.5)

    Since

    1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2,

    by Hölder's inequality, we have

    E2supL0,LZ2Lλ(Lk=2kα(0)q(0)k2l=2knRn(v=1|fv1l(y1)|r)1rdy1χkq(0)Lp1()(w1)×k+1j=k12knRn(v=1|fv2j(y2)|r)1rdy2χkq(0)Lp2()(w2))1q(0)supL0,LZ2Lλ1×(Lk=2kα1(0)q1(0)k2l=2knRn(v=1|fv1l(y1)|r)1rdy1χkq1(0)Lp1()(w1))1q1(0)×supL0,LZ2Lλ2(Lk=2kα2(0)q2(0)k+1j=k12knRn(v=1|fv2j(y2)|r)1rdy2χkq2(0)Lp2()(w2))1q2(0)=E2,1×E2,2.

    It is obvious that

    E2,1=E1,1(v=1|fv1|r1)1r1M˙Kα1(),q1()p1(),λ1(w1).

    Now we turn to estimate E2,2. By (3.1)–(3.3), we have

    E2,2supL0,LZ2Lλ2(Lk=2kα2(0)q2(0)k+1j=k12(jk)n(v=1|fv2|r2)1r2χjq2(0)Lp2()(w2))1q2(0)supL0,LZ2Lλ2(L+1k=2kα2(0)q2(0)(v=1|fv2|r2)1r2χkq2(0)Lp2()(w2))1q2(0)(v=1|fv2|r2)1r2M˙Kα2(),q2()p2(),λ2(w2),

    where we use 2nδ22<1 and 2(jk)n(1δ12)<2(jk)n<22n,j{k1,k,k+1} for (3.1) and (3.3) respectively. Thus, we obtain

    E2(v=1|fv1|r1)1r1M˙Kα1(),q1()p1(),λ1(w1)(v=1|fv2|r2)1r2M˙Kα2(),q2()p2(),λ2(w2).

    To estimate E3, since lk2 and jk+2, we have

    |xy1||x||y1|2k2,|xy2||y2||x|>2j2,xDk,y1Dl,y2Dj.

    Therefore, for any xDk,lk2,jk+2, we get

    |Tϖ(f1l,f2j)(x)|RnRn|fv1l(y1)||fv2j(y2)|(|xy1|+|xy2|)2ndy1dy22kn2jnRnRn|fv1l(y1)||fv2j(y2)|dy1dy2.

    Thus, by Hölder's inequality and Minkowski's inequality, we have

    (v=1|k2l=j=k+2T(fv1l,fv2j)|r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1j=k+22jnRn|fv2j(y2)|dy2)r)1rχkLp()(w)(v=1(k2l=2knRn|fv1l(y1)|dy1)r1)1r1χkLp1()(w1)×(v=1(j=k+22jnRn|fv2j(y2)|dy2)r2)1r2χkLp2()(w2)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkLp1()(w1)×j=k+22jnRn(v=1|fv2j(y2)|r2)1r2dy2χkLp2()(w2). (3.6)

    Since

    1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2,

    by Hölder's inequality, we have

    E3supL0,LZ2Lλ(Lk=2kα(0)q(0)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkq(0)Lp1()(w1)×j=k+22jnRn(v=1|fv2j(y2)|r2)1r2dy2χkq(0)Lp2()(w2))1q(0)supL0,LZ2Lλ1×(Lk=2kα1(0)q1(0)k2l=2knRn(v=1|fv1l(y1)|r1)1r1dy1χkq1(0)Lp1()(w1))1q1(0)×supL0,LZ2Lλ2×(Lk=2kα2(0)q2(0)j=k+22jnRn(v=1|fv2j(y2)|r2)1r2dy2χkq2(0)Lp2()(w2))1q2(0)=E3,1×E3,2.

    It is obvious that

    E3,1=E1,1(v=1|fv1|r1)1r1M˙Kα1(),q1()p1(),λ1(w1).

    Since nδ21+α2(0)>0, by (3.3), we obtain

    E3,2supL0,LZ2Lλ2(Lk=2kα2(0)q2(0)×(j=k+22(kj)nδ21(v=1|fv2|r2)1r2χjLp2()(w2))q2(0))1q2(0)supL0,LZ2Lλ2×(Lk=(Lj=k+22jα2(0)(v=1|fv2|r2)1r2χjLp2()(w2)2(kj)(nδ21+α2(0)))q2(0))1q2(0)+supL0,LZ2Lλ2×(Lk=(2kα2(0)0j=L+1(v=1|fv2|r2)1r2χjLp2()(w2)2(kj)nδ21)q2(0))1q2(0)+supL0,LZ2Lλ2×(Lk=(2kα2(0)j=1(v=1|fv2|r2)1r2χjLp2()(w2)2(kj)nδ21)q2(0))1q2(0)=I1+I2+I3.

    First, we consider I1. By Lemma 2.4, we have

    I1supL0,LZ2Lλ2×(Lk=(Lj=k+22jα2(0)(v=1|fv2|r2)1r2χjLp2()(w2)2(kj)(nδ21+α2(0)))q2(0))1q2(0)supL0,LZ2Lλ2(L+2j=2jα2(0)q2(0)(v=1|fv2|r2)1r2χjq2(0)Lp2()(w2))1q2(0)(v=1|fv2|r2)1r2M˙Kα2(),q2()p2(),λ2(w2),

    where we write 2|kj|(nδ21+α2(0))=2|kj|η2 for η2=nδ21+α2(0)>0. Next, we consider I_{2} . Since n\delta_{21}+\alpha_{2}(0)-\lambda_{2} > 0 , we obtain

    \begin{eqnarray*} I_{2}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = L+1}^{0}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &&\times2^{-j(n\delta_{21}+\alpha_{2}(0))}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}} \times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = L+1}^{0}2^{-j(n\delta_{21}+\alpha_{2}(0)-\lambda_{2})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}\sup\limits_{j\leq0}2^{-j\lambda_{2}}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{L\Big(-n\delta_{21}-\alpha_{2}(0)\Big)}\Big(\sum\limits_{k = -\infty}^{L}2^{k\Big(n\delta_{21}+\alpha_{2}(0)\Big)q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.\\ \end{eqnarray*}

    Then, we consider I_{3} . Since \delta_{21}+\alpha_{2}(0)-\lambda_{2} > 0 , we obtain

    \begin{eqnarray*} I_{3}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\Big.\Big. \times\sum\limits_{j = 1}^{\infty}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{-j(n\delta_{21}+\alpha_{2\infty})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}\sup\limits_{j\geq1}2^{-j\lambda_{2}}2^{j\alpha_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = 1}^{\infty}2^{-j(n\delta_{21}+\alpha_{2\infty}-\lambda_{2})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k(n\delta_{21}+\alpha_{2}(0))q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{L\Big(-\lambda_{2}+n\delta_{21}+\alpha_{2}(0)\Big)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*}

    Thus, we have

    E_{3}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    To estimate E_{5} , using Hölder's inequality and Lemma 2.8, we have

    \begin{eqnarray*} E_{5}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_{1l},f_{2j})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big(\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\Big.\Big. \times\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*}

    To estimate E_{6} , since k-1\leq l\leq k+1 and j\geq k+2 , we obtain

    |x-y_{1}| > 2^{k-2},\quad|x-y_{2}| > 2^{j-2},\quad x\in D_{k},\; y_{1}\in D_{l},\; y_{2}\in D_{j}.

    Thus, for any x\in D_{k}, k-1\leq l\leq k+1 and j\geq k+2 , we obtain

    \begin{eqnarray*} |T(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-kn}2^{-jn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*}

    Therefore, by Hölder's inequality and Minkowski's inequality, we obtain

    \begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k+2}^{\infty}T(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} (3.7)

    Since

    \frac{1}{q_(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2},

    by Hölder's inequality, we have

    \begin{eqnarray*} E_{6}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}}(\cdot)(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{6,1}\times E_{6,2}. \end{eqnarray*}

    By the interchange of f_{1} and f_{2} , we find the estimateof E_{6, 1} and E_{2, 2} are similar, and E_{6, 2} = E_{3, 2} . To estimate E_{9} , since l, j\geq k+2 , we get

    |x-y_{i}| > 2^{k-2},\quad x\in D_{k},\; y_{1}\in D_{l},\; y_{2}\in D_{j}.

    Therefore, for any x\in D_{k}, \ l, j\geq k+2 , we have

    \begin{eqnarray*} |T_{\varpi}(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-ln}2^{-jn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*}

    Thus, by Hölder's inequality and Minkowski's inequality, we have

    \begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k+2}^{\infty}T(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k+2}^{\infty}2^{-\ln}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} (3.8)

    Since

    \frac{1}{q_(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2},

    by Hölder's inequality, we have

    \begin{eqnarray*} E_{9}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}})^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{9,1}\times E_{9,2}. \end{eqnarray*}

    Obviously, the estimates of E_{9, i} are similar to those of E_{3, 2}(i = 1, 2) .

    All estimates for E_{i} i = 1, 2, \cdots, 9 considered, we have

    E\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    Finally, we estimate H . By the interchange of f_{1} and f_{2} , we see that the estimates of H_{2}, H_{3} and H_{6} are similar to those of H_{4}, H_{7} and H_{8} , respectively. Thus we just need to estimate H_{1} H_{3} , H_{5}, H_{6} and H_{9} .

    For the subsequent proof process, we need following further preparation. If l < 0 , by Lemma 1.3, we have

    \begin{eqnarray} \Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})} & = &2^{-l\alpha_{i}(0)}\Big(2^{l\alpha_{i}(0)q_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{-l\alpha_{i}(0)}\Big(\sum\limits_{t = -\infty}^{l}2^{t\alpha_{i}(0)q_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{t}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{l(\lambda-\alpha_{i}(0))}2^{-l\lambda}\Big(\sum\limits_{t = -\infty}^{l}\Big\|2^{t\alpha_{i}(0)}\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{t}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{l(\lambda-\alpha_{i}(0))}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray} (3.9)

    To estimate H_{1} , since

    l,j\leq k-2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}}

    and \lambda = \lambda_{1}+\lambda_{2} , by (3.4) and Hölder's inequality, we have

    \begin{eqnarray*} H_{1}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{1,1}\times H_{1,2}, \end{eqnarray*}

    where

    H_{1,i} = 2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}(y_{i})|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}.

    By (3.1), we obtain

    \begin{eqnarray*} H_{1,i}&\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{k-2}2^{(l-k)n\delta_{i2}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big.\Big.\\ &&+\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\&\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&+2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ & = &I_{4}+I_{5}. \end{eqnarray*}

    If q_{i\infty}\geq1 , since n\delta_{i2}-\alpha_{i\infty} > 0 and n\delta_{i2}-\alpha_{i}(0) > 0 , by the Minkowski's inequality and (3.9), we obtain

    \begin{eqnarray*} I_{4}& = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\{\sum\limits_{k = 0}^{L}\Big(2^{k\alpha_{i\infty}}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}2^{ln\delta_{i2}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\{\sum\limits_{k = 0}^{L}2^{-k\Big(n\delta_{i2}-\alpha_{i\infty}\Big)q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}2^{l\Big(n\delta_{i2}+\lambda_{i}-\alpha_{i}(0)\Big)}\\ &&\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray*}

    If q_{i\infty} < 1 , since n\delta_{i2}-\alpha_{i\infty} > 0 and n\delta_{i2}-\alpha_{i}(0) > 0 , by (3.9), we have

    \begin{eqnarray*} I_{4}&\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{(l-k)n\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{ln\delta_{i2}q_{i\infty}}\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}2^{-kn\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{\ln\delta_{i2}q_{i\infty}}\sum\limits_{k = 0}^{L}2^{-k\Big(n\delta_{i2}-\alpha_{i\infty}\Big)q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{\ln\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}2^{l\Big(n\delta_{i2}+\lambda_{i}-\alpha_{i}(0)\Big)q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray*}

    We consider I_{5} . Since n\delta_{i2}-\alpha_{i\infty} > 0 , by Lemma 2.4, we have

    \begin{eqnarray*} I_{5}& = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}\Big(\sum\limits_{l = 0}^{k}2^{l\alpha_{i\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)(n\delta_{i2}-\alpha_{i\infty})}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{l = 0}^{k}2^{l\alpha_{i\infty}q_{i\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}\Big(w_{i}\Big)}, \end{eqnarray*}

    where we write 2^{-|k-l|(n\delta_{i2}-\alpha_{i\infty})}\lesssim2^{-|k-l|\eta_{i}} for \eta_{i} = n\delta_{i2}-\alpha_{i\infty} .

    Thus, we get

    H_{1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    To estimate H_{2} , since

    l\leq k-2,\ \ \ \ k-1\leq j\leq k+1,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}}

    and \lambda = \lambda_{1}+\lambda_{2} , by (3.6) and Hölder's inequality, we have

    \begin{eqnarray*} H_{2}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}\Big(y_{1}\Big)|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{2,1}\times H_{2,2}. \end{eqnarray*}

    It isobvious that

    H_{2,1} = H_{1,1}\lesssim\|(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}})^{\frac{1}{r_{1}}}\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}.

    Now we estimate H_{2, 2} . Combining (3.1)–(3.3), we have

    \begin{eqnarray*} H_{2,2}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\sum\limits_{j = k-1}^{k+1}2^{(j-k)n}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = -1}^{L+1}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}, \end{eqnarray*}

    where we use 2^{-n\delta_{22}} < 1 and 2^{(j-k)n(1-\delta_{21})} < 2^{(j-k)n} for (3.6) and (3.8), respectively. Thus, we obtain

    H_{2}\lesssim\|(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}})^{\frac{1}{r_{1}}}\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    To estimate H_{3} , since

    l\leq k-2,\ \ \ \ j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}}

    and \lambda = \lambda_{1}+\lambda_{2} , together (3.6) with the Hölder's inequality, we have

    \begin{eqnarray*} H_{3}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\rho_{\infty}}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{3,1}\times H_{3,2}. \end{eqnarray*}

    It is easy to see that

    H_{3,1} = H_{1,1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}.

    Since n\delta_{21}+\alpha_{2\infty} > 0 , by (3.3), we obtain

    \begin{eqnarray*} H_{3,2}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big(\sum\limits_{j = k+2}^{\infty}2^{(k-j)n\delta_{21}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(\sum\limits_{j = k+2}^{L+2}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)\Big(n\delta_{21}+\alpha_{2\infty}\Big)}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &&+2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k\alpha_{2\infty}}\sum\limits_{j = L+3}^{\infty}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)n\delta_{21}}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &I_{6}+I_{7}. \end{eqnarray*}

    For I_{6} , by Lemma 2.4, we obtain

    \begin{eqnarray*} I_{6}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(\sum\limits_{j = k+2}^{L+2}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)\Big(n\delta_{21}+\alpha_{2\infty}\Big)}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{j = 0}^{L+2}2^{j\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)(w_{2})}}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha(\cdot),q_{2}(\cdot)}\Big(w_{2}\Big)}, \end{eqnarray*}

    where we write 2^{-|k-j|(n\delta_{21}+\alpha_{2\infty})} = 2^{-|k-j|\vartheta_{2}} for \vartheta_{2} = n\delta_{21}+\alpha_{2\infty} > 0.

    For I_{7} , since n\delta_{21}+\alpha_{2\infty}-\lambda_{2} > 0, we have

    \begin{eqnarray*} I_{7}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2\infty})}\sum\limits_{j = L+3}^{\infty}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})} \times2^{-j(n\delta_{21}+\alpha_{2\infty})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2}\infty}}\\ &\lesssim&\sup\limits_{j\geq1}2^{-j\lambda_{2}}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})} \times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2\infty})}\sum\limits_{j = L+3}^{\infty}2^{-j(n\delta_{21}+\alpha_{2\infty}-\lambda_{2})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}2^{-L\lambda_{2}+\Big(n\delta_{21}+\alpha_{2\infty}\Big)L-L\Big(n\delta_{21}+\alpha_{2\infty}-\lambda_{2}\Big)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*}

    Thus, we get

    H_{3}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    To estimate {H}_{5} , using Hölder's inequality and Lemma 2.8, we have

    \begin{eqnarray*} H_{5}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|T(f_{1l},f_{2j})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big(\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|\|_{L^{p_{1}(\cdot)}(w_{1})} \Big.\Big.\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*}

    To estimate H_{6} , since

    k-1\leq l\leq k+1,\ \ \ \ j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}}

    and \lambda = \lambda_{1}+\lambda_{2} , by (3.7) and Hölder's sinequality, we have

    \begin{eqnarray*} H_{6}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{6,1}\times H_{6,2}. \end{eqnarray*}

    By the interchange of f_{1} and f_{2} , we see that that of H_{6, 1} is similar to the estimate of H_{2, 2} and H_{6, 2} = H_{3, 2}.

    To estimate H_{9} , since

    l,j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}}

    and \lambda = \lambda_{1}+\lambda_{2} , by (3.8) and Hölder's inequality, we have

    \begin{eqnarray*} H_{9}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{9,1}\times H_{9,2}. \end{eqnarray*}

    Obviously, the estimates of H_{9, i} are similar to those of H_{3, 2} for i = 1, 2 , respectively.

    Taking all estimates for H_{i} together, i = 1, 2, \cdots, 9 , we obtain

    H\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.

    This completes the proof.

    On the basis of vector valued bilinear Calderón-Zygmund operators with kernels of Dini's type are bounded on variable Lebesgue spaces, with the help of properties of the \varpi(t) and space decomposition methods for variable exponents Herz-Morrey spaces. We establish the weighted boundedness result of vector valued bilinear \varpi(t) -type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, this is a new and meaningful result.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Doctoral Scientific Research Foundation of Northwest Normal University (202003101203), Young Teachers Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2021-03) and Open Foundation of Hubei Key Laboratory of Applied Mathematics (Hubei University) (HBAM202205).

    The authors declare that there are no conflicts of interest.



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