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 ϖ∈Dini(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 ϖ∈Dini(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 ϖ∈Dini(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 ϖ∈Dini(1), then
∞∑0ϖ(2−j)≈∫10ϖ(t)tdt<∞, |
here and in what follows, for any quantities A and B, if there exists a constant C>0 such that A≤CB, we write A≲B. If A≲B and B≲A, we write A≈B.
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 x≠yi,i=1,2, if there exists a constant A>0 such that
|K(x,y1,y2)|≤Aϖ(2∑i=1|x−yi|)−2n, | (1.2) |
and for (x,y1,y2)∈(Rn)3 with x≠y1,y2, and
|K(x,y1,y2)−K(z,,y1,y2)|≤Aω(|x−z|∑2i=1|x−yi|)[2∑i=1|x−yi|]−2n. | (1.3) |
whenever 2|x−z|<max{|x−y1|,|x−y2|}.
Definition 1.1. ([2]) Let ϖ∈Dini(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, f2∈C∞c(Rn),
Tϖ(f1,f2)(x)=∫Rn∫RnK(x,y1,y2)f1(y1)f2(y2)dy1dy2,x∉suppf1∩suppf2. | (1.4) |
In the following, for each k∈Z, we define Bk={x∈Rn:|x|≤2k}, Dk=Bk∖Bk−1, χk=χDk, m≥1, ˜χ0=χB0.
Given a function p(x)∈P(Rn), the space Lp(x)(Rn) is now defined by
‖f‖Lp(⋅)(Rn)=inf{η>0:∫Rn(|f(x)|η)p(x)dx≤1}. |
Denote P(Rn) to be the set of the all measurable functions p(x) with
p−=:essinfx∈Rnp(x)>1 |
and
p+=:esssupx∈Rnp(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:fχK∈Lp(⋅)loc(Rn) for all compact subsets K∈Rn}, |
where and what follows, χS denotes the characteristic function of a measurable set S⊂Rn.
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 fω∈Lp(⋅). The space Lp(⋅)(ω) is a Banach space equipped with the norm
‖f‖Lp(⋅)(ω)=‖fω‖Lp(⋅). |
Let f∈L1loc(Rn). Then the standard Hardy-Littlewood maximal function of f is defined by
Mf(x)=supx∈B1|B|∫Bf(y)dy,∀x∈Rn, |
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,y∈Rn, |x−y|<1/2, if
|α(x)−α(y)|≲1log(e+1/|x−y|), |
then α(⋅) is said local log-Hölder continuous on Rn.
(ⅱ) For all x∈Rn, 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 α∞∈R, for x∈Rn, 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.,x∈Rn. |
For 1<p<∞, we say that ω is an Ap weight if
supB(1|B|∫Bω(x)dx)(1|B|∫Bω(x)1−p′dx)p−1<∞. |
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|‖ωχB‖Lp(⋅)(Rn)‖ω−1χB‖Lp′(⋅)(Rn)≤C. |
Lemma 1.1. ([5]) If p(⋅)∈Plog(Rn)∩P(Rn) and ω∈Ap(⋅), then for each f∈Lp(⋅)(ω),
‖(Mf)ω‖Lp(⋅)≲‖fω‖Lp(⋅), |
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 ℓq(Lp(⋅)), which was firstly defined in [6]. Let ω be a nonnegative measurable function. Given a sequence of functions {fj}j∈Z, we define the modular
ρℓq(Lp(⋅)(ω))((fj)j)=∑j∈Zinf{λj:∫Rn(|fj(x)ω(x)|λ1q(x)j)p(x)dx≤1}, |
where λ1∞=1. If q+<∞ or q(⋅)≤p(⋅), the above can be written as
ρℓq(Lp(⋅)(ω))((fj)j)=∑j∈Z‖fjω|q(⋅)‖Lp(⋅)q(⋅). |
The norm is
‖(fj)j‖ρℓq(Lp(⋅)(ω))=inf{μ>0:ρℓq(Lp(⋅)(ω))((fjμ)j)≤1}. |
Definition 1.4. ([7]) Let p(⋅)∈P(Rn), q∈P0(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)={f∈Lp(⋅)loc(Rn∖{0},ω):‖f‖˙Kα(⋅),q(⋅)p(⋅)(ω)<∞}, |
where
‖f‖˙Kα(⋅),q(⋅)p(⋅)(ω)=‖(2jα(⋅)fχj)j‖ρℓq(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∞,q∞p(⋅)(ω). |
Additionally, if α(⋅) and q(⋅) are log-Hölder continuous at the origin, then
‖f‖˙Kα(⋅),q(⋅)p(⋅)(ω)≈(∑k≤0‖2kα(0)fχk‖q(0)Lp(⋅))1q(0)+(∑k>0‖2kα∞fχk‖q(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)={f∈Lp(⋅)loc(Rn∖{0},ω):‖f‖M˙Kα(⋅),q(⋅)p(⋅),λ(ω)<∞}, |
where
‖f‖M˙Kα(⋅),q(⋅)p(⋅),λ(ω)=supL∈Z2−Lλ‖(2kα(⋅)kfχk)k≤L‖ρℓq(Lp(⋅)(ω)). |
Lemma 1.3. ([8]) Let p(⋅),q(⋅)∈P0(Rn), ω be a weight, λ∈[0,∞) and α∈L∞(Rn). If α(⋅), q(⋅)∈Plog0(Rn)∩Plog∞(Rn), then for any f∈Lp(⋅)loc(Rn∖{0},ω),
‖f‖M˙Kα(⋅),q(⋅)p(⋅),λ(ω)≈max{supL≤0,L∈Z2−Lλ‖(2kα(0)fχk)k≤L‖lq0(Lp(⋅)(ω)),supL>0,L∈Z[2−Lλ‖(2kα(0)fχk)k≤L‖ρℓq0(Lp(⋅)(ω))+2−Lλ‖(2kα∞fχk)Lk=0‖ρℓq0(Lp(⋅)(ω))]}, |
where and hereafter, q0=q(0).
Lemma 1.4. ([8]) If p(⋅)∈Plog(Rn)∩P(Rn) and ω∈Ap(⋅), then there exist constants δ1,δ2∈(0,1), such that for all balls B in Rn and all measurable subsets S⊂B,
‖χS‖Lp(⋅)(ω)‖χB‖Lp(⋅)(ω)≲(|S||B|)δ1,‖χS‖Lp′(⋅)(ω−1)‖χB‖Lp′(⋅)(ω−1)≲(|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 f∈L1loc(Rn). Then the sharp maximal function is defined by
M#f(x)=supQ1Q∫Q|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 ϖ∈Dini(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 ω∈A∞. There exists a positive constant C such that
∫Rn[Mδf(x)]pω(x)dx≤∫Rn[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 f∈Lp1(⋅)(Rn) and g∈Lp2(⋅)(Rn), there exists
‖fg‖Lp(⋅)≲‖f‖Lp1(⋅)‖g‖Lp2(⋅) |
If p∈P(Rn), ω is a weight with ω=ω1×ω2, there exists
‖fg‖Lp(⋅)(ω)≲‖f‖Lp1(⋅)(ω1)‖g‖Lp2(⋅)(ω2). |
Lemma 2.4. ([11]) Let 0<p≤∞, δ>0. Then for non-negative sequence {aj}∞j=−∞, there exists
(∞∑j=−∞(∞∑k=−∞2−|k−j|δak)p)1p≲(∞∑j=−∞apj)1p, |
when p=∞, above inequality stands for
∞∑k=−∞(2−|k−j|δak)≲supj∈Zaj. |
Lemma 2.5. ([12]) Assume that for some p0∈(0,∞) and every ω0∈A∞, let F be a family of pairs of non-negative functions such that
∫Rnf(x)p0ω0(x)dx≲∫Rng0(x)p0ω0(x)dx,(f,g)∈F. | (2.1) |
Then for all 0<p<∞ and ω0∈A∞,
∫Rnf(x)pω0(x)dx≲∫Rng0(x)pω0(x)dx,(f,g)∈F. |
Furthermore, for every p,q∈(0,∞), ω0∈A∞, and sequences {(fj,gj)}∈F,
‖(∞∑j=1(fj))q‖Lp(ω0)≲‖(∞∑j=1(gj))q‖Lp(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 s≤p− such that ωs∈Ap(⋅)s and M is bounded on L(p(⋅)s)′(ω−s). Then for every q∈(1,∞) and sequence {(fj,gj)}j∈N⊂F
‖(∞∑j=1(fj))q‖Lp(⋅)(ω)≲‖(∞∑j=1(gj))q‖Lp(⋅)(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′(⋅)(ω−1), q∈(1.∞), then
‖(∞∑j=1(Mfj)q)1q‖Lp(⋅)(ω)≲‖(∞∑j=1|fj|q)1q‖Lp(⋅)(ω). |
Lemma 2.8. Let Tϖ be a bilinear Calderón-Zygmund operator with ϖ∈Dini(1) and p(⋅)∈P0 such that there exists s≤p− such that ωs∈Ap(⋅)s and M is bounded on L(p(⋅)s)′(ω−s). Suppose that ω=ω1×ω2 and ωi∈Api(⋅),i=1,2. If pi∈Plog(Rn)∩P(Rn)(i=1,2) satisfying
1p(x)=1p1(x)+1p2(x) |
for x∈Rn. Then for compactly supported bounded functions fj1,fj2∈Lp0(Rn), j∈N such that
‖(∞∑j=1|Tϖ(fj1,fj2)|q)1q‖Lp(⋅)(ω)≲2∏i=1‖(∞∑j=1|fji|qi)1qi‖Lpi(⋅)(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 ω∈A∞,
∫Rn|Tϖ(f1,f2)(x)|pω(x)dx≲∫Rn(Mf1(x)Mf2(x))pω(x)dx. |
Therefore, by Lemmas 2.5 and 2.6, we have
‖(∞∑j=1|Tϖ(fj1,fj2)|q)1q‖Lp(⋅)(ω)≲‖(∞∑j=1|Mfj1(x)Mfj2(x)|q)1q‖Lp(⋅)(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)1q‖Lp(⋅)(ω)≲2∏i=1‖(∞∑j=1|Mfji|qi)1qi‖Lpi(⋅)(ωi)≲2∏i=1‖(∞∑j=1|fji|qi)1qi‖Lpi(⋅)(ωi). |
We complete the proof of Lemma 2.8.
Theorem 3.1. Let Tϖ be a bilinear ϖ-type Calderón-Zygmund operator with ϖ∈Dini(1), p1 and p2∈Plog(Rn)∩Plog(Rn) santisfying
1p(x)=1p1(x)+1p2(x) |
and p(⋅)∈P0 such that there exists s≤p− such that ωs∈Ap(⋅)s and M is bounded on L(p(⋅)s)′(ω−s), where ω=ω1ω2 and ωi∈Api(⋅), 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+λ2,0≤λi<∞,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 λi−nδi1<αi∞, αi(0)≤nδi2, then
‖(∞∑j=1|Tϖ(fj1,fj2)|r)1r‖M˙Kα(⋅),q(⋅)p(⋅),λ(ω)≲2∏i=1‖(∞∑j=1|(fji)|ri)1ri‖M˙Kαi(⋅),qi(⋅)pi(⋅),λi(ωi) |
for all fji∈M˙Kαi(⋅),qi(⋅)pi(⋅),λi(ωi), j∈N, 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 v∈N and write
fvi=∞∑l=−∞fvilχl=∞∑l=−∞fvil,i=1,2,v∈N. |
By Lemma 1.3, we have
‖(∞∑v=1|Tϖ(fv1,fv2)|r)1r‖M˙Kα(⋅),q(⋅)p(⋅),λ(w)≈max{supL≤0,L∈Z2−Lλ‖(2kα(0)(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)k≤L‖ℓq0(Lp(⋅)(w)) supL>0,L∈Z[2−Lλ‖(2kα(0)(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)k<0‖ℓq0(Lp(⋅)(w)) +2−Lλ‖(2kα∞(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)Lk=0‖ℓq∞(Lp(⋅)(w))]}=max{E,F}, |
where
E=supL≤0,L∈Z2−Lλ‖(2kα(0)(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)k≤L‖ℓq0(Lp(⋅)(w)),F=supL>0,L∈Z{G+H},G=2−Lλ‖(2kα(0)(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)k<0‖ℓq0(Lp(⋅)(w)),H=2−Lλ‖(2kα∞(∞∑v=1|Tϖ(fv1,fv2)|r)1rχk)Lk=0‖ℓq∞(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
E≲9∑i=iEi,H≲9∑i=iHi, |
where
E1=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k−2∑l=−∞k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E2=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k−2∑l=−∞k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E3=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k−2∑l=−∞∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E4=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k+1∑l=k−1k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E5=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k+1∑l=k−1k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E6=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|k+1∑l=k−1∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E7=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1∣∞∑l=k+2k−2∑j=−∞Tϖ(fv1l,fv2j)r)1rχk‖q(0)Lp(⋅)(w))1q(0),E8=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|∞∑l=k+2k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0),E9=supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖(∞∑v=1|∞∑l=k+2∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q(0)Lp(⋅)(w))1q(0), |
H1=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k−2∑l=−∞k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H2=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k−2∑l=−∞k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H3=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k−2∑l=−∞∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H4=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k+1∑l=k−1k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H5=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k+1∑l=k−1k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H6=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|k+1∑l=k−1∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H7:=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|∞∑l=k+2k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H8:=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|∞∑l=k+2k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞,H9:=2−Lλ(L∑k=02kα∞q∞‖(∞∑v=1|∞∑l=k+2∞∑j=k+2Tϖ(fv1l,fv2j)|r)1rχk‖q∞Lp(⋅)(w))1q∞. |
We will use the following estimates. If l≤k−1, by Hölder's inequality, Lemma 1.4 and Definition 1.3, we have
‖2−kn∫Rn(∞∑v=1|fvil|ri)1ridyiχk‖Lpi(⋅)(wi)≤C2−kn‖χBk‖Lpi(⋅)(wi)‖(∞∑v=1|fvi|ri)1riwiχl‖Lpi(⋅)‖χlw−1i‖Lp′i(⋅)≤C2−kn|Bk|‖χBk‖−1Lp′i(⋅)(w−1i)‖χBl‖Lp′i(⋅)(w−1i)‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi)≤C2(l−k)nδ2i‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi). | (3.1) |
If l=k, then
‖2−kn∫Rn(∞∑v=1|fvil|ri)1ridyiχk‖Lpi(⋅)(wi≤C2−kn‖χBk‖Lpi(⋅)(wi)‖(∞∑v=1|fvi|ri)1riwiχl‖Lpi(⋅)‖χlw−1i‖Lp′i(⋅)≤C2−kn‖χBk‖Lpi(⋅)(wi)‖χBl‖Lp′i(⋅)(w−1i)‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi)≤‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi). | (3.2) |
If l≥k+1, then
‖2−kn∫Rn(∞∑v=1|fvil|ri)1ridyiχk‖Lpi(⋅)(wi)≤C2−kn‖χBk‖Lpi(⋅)(wi)‖(∞∑v=1|fvi|ri)1riwiχl‖Lpi(⋅)‖χlw−1i‖Lp′i(⋅)≤C2−kn‖χBk‖Lpi(⋅)(wi)‖χBl‖Lpi(⋅)(wi)‖χBl‖−1Lpi(⋅)(wi)×‖χBl‖Lp′i(⋅)(w−1i)‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi)≤C2(l−k)n(1−δ1i)‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(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 E1–E3,E5,E6 and E9.
For E1, since l, j≤k−2, then for i=1,2,
|x−yi|≥|x|−|yi|>2k−1−2min{l,j}≥2k−2,x∈Dk, y1∈Dl, y2∈Dj. |
Therefore, for x∈Dk, we have
|K(x,y1,y2)|≤C(|x−y1|+|x−y2|)−2n≤C2−2kn. |
Thus, for any x∈Dkandl,j≤k−2, we have
|T(fv1l,fv2j)(x)|≲∫Rn∫Rn|fv1l(y1)||fv2j(y2)|(|x−y1|+|x−y2|)2ndy1dy2≲2−2kn∫Rn∫Rn|fv1l(y1)||fv2j(y2)|dy1dy2. |
Hence, together with the Hölder's and Minkowski's inequality, we have
‖(∞∑v=1|k−2∑l=−∞k−2∑j=−∞Tϖ(fv1l,fv2j)|r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1k−2∑j=−∞2−kn∫Rn|fv2j(y2)|dy2)r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1)r1)1r1χk‖Lp1(⋅)(w1)×‖(∞∑v=1(k−2∑j=−∞2−kn∫Rn|fv2j(y2)|dy2)r2)1r2χk‖Lp2(⋅)(w2)≲‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖Lp1(⋅)(w1)×‖k−2∑j=−∞2−kn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖Lp2(⋅)(w2). | (3.4) |
Since
1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2, |
by Hölder's inequality, we have
E1≲supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖q(0)Lp1(⋅)(w1)×‖k−2∑j=−∞2−kn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖q(0)Lp2(⋅)(w2))1q(0)≲supL≤0,L∈Z2−Lλ1×(L∑k=−∞2kα1(0)q1(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖q1(0)Lp1(⋅)(w1))1q1(0)×supL≤0,L∈Z2−Lλ2×(L∑k=−∞2kα2(0)q2(0)‖k−2∑j=−∞2−kn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖q2(0)Lp2(⋅)(w2))1q2(0)=E1,1×E1,2. |
For convenience's sake, we write
E1,i=supL≤0,L∈Z2−Lλi×{L∑k=−∞2kαi(0)qi(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fil(yi)|ri)1ridyiχk‖qi(0)Lpi(⋅)(wi)}1qi(0). |
For nδi2−αi(0)>0, by (3.1) and Lemma 2.4 we have
E1,i≲supL≤0,L∈Z2−Lλi{L∑k=−∞2kαi(0)qi(0)Big.×(k−2∑l=−∞2(l−k)nδi2‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi))qi(0)}1qi(0)=supL≤0,L∈Z2−Lλi×{L∑k=−∞(k−2∑l=−∞2lαi(0)‖(∞∑v=1|fvi|ri)1riχl‖Lpi(⋅)(wi)2(l−k)(nδi2−αi(0)))qi(0)}1qi(0)≲supL≤0,L∈Z2−Lλi(L−2∑l=−∞2lαi(0)qi(0)‖(∞∑v=1|fvi|ri)1riχl‖qi(0)Lpi(⋅)(wi))1qi(0)≲‖(∞∑v=1|fvi|ri)1ri‖M˙Kαi(⋅),qi(⋅)pi(⋅),λi(wi), |
where we write 2−|k−l|(nδi2−αi(0))=2−|k−l|εi for εi=nδi2−αi(0)>0, then we have
E1≲‖(∞∑v=1|fv1|r1)1r1‖M˙Kα1(⋅),q1(⋅)p1(⋅),λ1(w1)‖(∞∑v=1|fv2|r2)1r2‖M˙Kα2(⋅),q2(⋅)p2(⋅),λ2(w2). |
To estimate E2, since l≤k−2, k−1≤j≤k+1, then we have
|x−y2|≥|x−y1|≥|x|−|y1|≥2k−2,x∈Dk, y1∈Dl, y2∈Dj. |
Therefore, for x∈Dk, we have
|K(x,y1,y2)|≤C(|x−y1|+|x−y2|)−2n≤C2−2kn. |
Thus, for any x∈Dk,l≤k−2,k−1≤j≤k+1, we have
|T(fv1l,fv2j)(x)|≲∫Rn∫Rn|fv1l(y1)||fv2j(y2)|(|x−y1|+|x−y2|)2ndy1dy2≲2−2kn∫Rn∫Rn|fv1l(y1)||fv2j(y2)|dy1dy2. |
Combining the Hölder's with and Minkowski's inequality, hence we obtain
‖(∞∑v=1|k−2∑l=−∞k+1∑j=k−1Tϖ(fv1l,fv2j)|r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1k+1∑j=k−12−kn∫Rn|fv2j(y2)|dy2)r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1)r)1rχk‖Lp1(⋅)(w1)×‖(∞∑v=1(k+1∑j=k−12−kn∫Rn|fv2j(y2)|dy2)r)1rχk‖Lp2(⋅)(w2)≲‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r)1rdy1χk‖Lp1(⋅)(w1)×‖k+1∑j=k−12−kn∫Rn(∞∑v=1|fv2j(y2)|r)1rdy2χk‖Lp2(⋅)(w2). | (3.5) |
Since
1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2, |
by Hölder's inequality, we have
E2≲supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r)1rdy1χk‖q(0)Lp1(⋅)(w1)×‖k+1∑j=k−12−kn∫Rn(∞∑v=1|fv2j(y2)|r)1rdy2χk‖q(0)Lp2(⋅)(w2))1q(0)≲supL≤0,L∈Z2−Lλ1×(L∑k=−∞2kα1(0)q1(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r)1rdy1χk‖q1(0)Lp1(⋅)(w1))1q1(0)×supL≤0,L∈Z2−Lλ2(L∑k=−∞2kα2(0)q2(0)‖k+1∑j=k−12−kn∫Rn(∞∑v=1|fv2j(y2)|r)1rdy2χk‖q2(0)Lp2(⋅)(w2))1q2(0)=E2,1×E2,2. |
It is obvious that
E2,1=E1,1≲‖(∞∑v=1|fv1|r1)1r1‖M˙Kα1(⋅),q1(⋅)p1(⋅),λ1(w1). |
Now we turn to estimate E2,2. By (3.1)–(3.3), we have
E2,2≲supL≤0,L∈Z2−Lλ2(L∑k=−∞2kα2(0)q2(0)‖k+1∑j=k−12(j−k)n(∞∑v=1|fv2|r2)1r2χj‖q2(0)Lp2(⋅)(w2))1q2(0)≲supL≤0,L∈Z2−Lλ2(L+1∑k=−∞2kα2(0)q2(0)‖(∞∑v=1|fv2|r2)1r2χk‖q2(0)Lp2(⋅)(w2))1q2(0)≲‖(∞∑v=1|fv2|r2)1r2‖M˙Kα2(⋅),q2(⋅)p2(⋅),λ2(w2), |
where we use 2−nδ22<1 and 2(j−k)n(1−δ12)<2(j−k)n<22n,j∈{k−1,k,k+1} for (3.1) and (3.3) respectively. Thus, we obtain
E2≲‖(∞∑v=1|fv1|r1)1r1‖M˙Kα1(⋅),q1(⋅)p1(⋅),λ1(w1)‖(∞∑v=1|fv2|r2)1r2‖M˙Kα2(⋅),q2(⋅)p2(⋅),λ2(w2). |
To estimate E3, since l≤k−2 and j≥k+2, we have
|x−y1|≥|x|−|y1|≥2k−2,|x−y2|≥|y2|−|x|>2j−2,x∈Dk,y1∈Dl,y2∈Dj. |
Therefore, for any x∈Dk,l≤k−2,j≥k+2, we get
|Tϖ(f1l,f2j)(x)|≲∫Rn∫Rn|fv1l(y1)||fv2j(y2)|(|x−y1|+|x−y2|)2ndy1dy2≲2−kn2−jn∫Rn∫Rn|fv1l(y1)||fv2j(y2)|dy1dy2. |
Thus, by Hölder's inequality and Minkowski's inequality, we have
‖(∞∑v=1|k−2∑l=−∞∞∑j=k+2T(fv1l,fv2j)|r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1∞∑j=k+22−jn∫Rn|fv2j(y2)|dy2)r)1rχk‖Lp(⋅)(w)≲‖(∞∑v=1(k−2∑l=−∞2−kn∫Rn|fv1l(y1)|dy1)r1)1r1χk‖Lp1(⋅)(w1)×‖(∞∑v=1(∞∑j=k+22−jn∫Rn|fv2j(y2)|dy2)r2)1r2χk‖Lp2(⋅)(w2)≲‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖Lp1(⋅)(w1)×‖∞∑j=k+22−jn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖Lp2(⋅)(w2). | (3.6) |
Since
1q(0)=1q1(0)+1q2(0)andλ=λ1+λ2, |
by Hölder's inequality, we have
E3≲supL≤0,L∈Z2−Lλ(L∑k=−∞2kα(0)q(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖q(0)Lp1(⋅)(w1)×‖∞∑j=k+22−jn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖q(0)Lp2(⋅)(w2))1q(0)≲supL≤0,L∈Z2−Lλ1×(L∑k=−∞2kα1(0)q1(0)‖k−2∑l=−∞2−kn∫Rn(∞∑v=1|fv1l(y1)|r1)1r1dy1χk‖q1(0)Lp1(⋅)(w1))1q1(0)×supL≤0,L∈Z2−Lλ2×(L∑k=−∞2kα2(0)q2(0)‖∞∑j=k+22−jn∫Rn(∞∑v=1|fv2j(y2)|r2)1r2dy2χk‖q2(0)Lp2(⋅)(w2))1q2(0)=E3,1×E3,2. |
It is obvious that
E3,1=E1,1≲‖(∞∑v=1|fv1|r1)1r1‖M˙Kα1(⋅),q1(⋅)p1(⋅),λ1(w1). |
Since nδ21+α2(0)>0, by (3.3), we obtain
E3,2≲supL≤0,L∈Z2−Lλ2(L∑k=−∞2kα2(0)q2(0)×(∞∑j=k+22(k−j)nδ21‖(∞∑v=1|fv2|r2)1r2χj‖Lp2(⋅)(w2))q2(0))1q2(0)≲supL≤0,L∈Z2−Lλ2×(L∑k=−∞(L∑j=k+22jα2(0)‖(∞∑v=1|fv2|r2)1r2χj‖Lp2(⋅)(w2)2(k−j)(nδ21+α2(0)))q2(0))1q2(0)+supL≤0,L∈Z2−Lλ2×(L∑k=−∞(2kα2(0)0∑j=L+1‖(∞∑v=1|fv2|r2)1r2χj‖Lp2(⋅)(w2)2(k−j)nδ21)q2(0))1q2(0)+supL≤0,L∈Z2−Lλ2×(L∑k=−∞(2kα2(0)∞∑j=1‖(∞∑v=1|fv2|r2)1r2χj‖Lp2(⋅)(w2)2(k−j)nδ21)q2(0))1q2(0)=I1+I2+I3. |
First, we consider I1. By Lemma 2.4, we have
I1≲supL≤0,L∈Z2−Lλ2×(L∑k=−∞(L∑j=k+22jα2(0)‖(∞∑v=1|fv2|r2)1r2χj‖Lp2(⋅)(w2)2(k−j)(nδ21+α2(0)))q2(0))1q2(0)≲supL≤0,L∈Z2−Lλ2(L+2∑j=−∞2jα2(0)q2(0)‖(∞∑v=1|fv2|r2)1r2χj‖q2(0)Lp2(⋅)(w2))1q2(0)≲‖(∞∑v=1|fv2|r2)1r2‖M˙Kα2(⋅),q2(⋅)p2(⋅),λ2(w2), |
where we write 2−|k−j|(nδ21+α2(0))=2−|k−j|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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