# Commutators of Multilinear Calderón-Zygmund Operator and BMO Functions in Herz-Morrey Spaces with Variable Exponents

Authors: Canqin Tang; Qing Wu; Jingshi Xu
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We obtain the boundedness of a commutator generated by multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents.


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## Body

## 1. Introduction

In recent decades, variable exponent function spaces have attracted much attention. Since Kováčik and Rákosník [1] clarified fundamental properties of the variable Lebesgue and Sobolev spaces, there are many spaces studied, such as Besov and Trieble-Lizorkin spaces with variable exponents, Hardy spaces with variable exponents, Bessel potential spaces with a variable exponent, and Herz-Morrey spaces with variable exponents; see [2–13]. Recently, multilinear operators and their commutators in variable exponent function spaces are also intensively studied by a significant number of authors, such as multilinear commutators of multilinear singular integral with Lipschitz functions and BMO functions, respectively, in [14–19], multilinear commutators of BMO functions and multilinear singular integral operators with nonsmooth kernels in [20, 21], a vector-estimate of higher order commutators on Herz-Morrey spaces with variable exponent in [22], maximal multilinear commutators and maximal iterated commutators generated by multilinear operators and Lipschitz functions in [23], and weighted estimates for vector-valued commutators of multilinear operators in [24].Motivated by the above results, in this paper, we will consider the boundedness of a commutator generated by a multilinear Calderón-Zygmund operator and BMO functions on the variable Herz-Morrey spaces.To state the main result of this paper, we need to recall more notations.LetT be a multilinear singular integral operator which is initially defined on the m-fold product of the Schwartz space S(Rn). Its values are taken in the space of tempered distributions S′(Rn) such that, for x∉∩j=1msupp⁡fj, f→=(f1,…,fm),
(1)Tf→(x)=∫(Rn)m‍K(x,y1,…,ym)∏i=1m‍fi(yi)dy1⋯dym,
where f1,…,fm∈LC∞(Rn) (the space of compactly supported bounded functions). Here, the kernel K is a function in (Rn)m+1 away from the diagonal y0=y1=⋯=ym and satisfies the standard estimates
(2)|K(x,y1,…,ym)|≤A(|x-y1|+⋯+|x-ym|)-mn,|K(x,y1,…,ym)-K(x′,y1,…,ym)|≤A|x-x′|ϵ(|x-y1|+⋯+|x-ym|)mn+ϵ
provided that |x-x′|≤(1/2)max⁡{|x-y1|+⋯+|x-ym|}, and, for each 1≤i≤m,
(3)|K(x,y1,…,yi,…,ym)-K(x,y1,…,yi′,…,ym)|≤A|yi-yi′|ϵ(∑j=1m|x-yj|)mn+ϵ
provided that |yi-yi′|≤(1/2)max⁡{|x-y1|+⋯+|x-ym|}, where A and ϵ are positive constants.Such kernels are called them-linear Calderón-Zygmund kernels and the collection of such functions is denoted by m-CZK(A,ϵ) in [25].LetT be as in (1) with an m-CZK(A,ϵ) kernel. If, for some 1<q1, q2,…,qm<∞, T is bounded from Lq1×Lq2×⋯×Lqm to Lq with 1/q1+1/q2+⋯+1/qm=1/q, then we say T is an m-linear Calderón-Zygmund operator. Grafakos and Torres in [25] showed that if T is an m-linear Calderón-Zygmund operator, then T is bounded from Lq1×Lq2×⋯×Lqm to Lq for any 1<q1, q2,…,qm<∞ such that 1/q1+1/q2+⋯+1/qm=1/q. Then, Grafakos and Torres in [26] obtained weighted norm inequalities for multilinear Calderón-Zygmund operators.Ifb∈Lloc⁡1(Rn) (the set of all complex-valued locally integrable functions on Rn), set
(4)∥b∥*:=sup⁡B1|B|∫B‍|b(x)-bB|dx,
where the supremum is taken over all balls in Rn, bB is the mean of b on B, and what follows |E| is the Lebesgue measure of measurable set E in Rn. A function b is called bounded mean oscillation if ∥b∥*<∞ and BMO(Rn) is the set of all locally integrable functions b on Rn such that ∥b∥*<∞.Letb→=(b1,…,bm) and bj∈BMO(Rn) for 1≤j≤m. We will consider the commutator [b→,T], which is defined for suitable functions f1,…,fm by
(5)[b→,T]f→(x):=∑σ‍(-1)|σ′|bσ(x)Tbσ′f→(x);
where f→=(f1,…,fm), σ denotes a subset of {1,2,…,m},  σ′ is the complement of σ in {1,2,…,m}, |σ′| denotes the number of elements of σ′, bσ(x)=∏j∈σ‍bj(x), bσ′f→=(g1,…,gm), when j∈σ′, gj=bjfj, otherwise, gj=fj.Definition 1.
Letp(·):Rn→[1,∞) be a measurable function.(i)
The Lebesgue space with variable exponentLp(·)(Rn) is defined by
(6)Lp(·)(Rn):={fismeasurable:∫Rn‍(|f(x)|λ)p(x)dx<∞Lp(·)(Rn):=mforsomeλ>0{fismeasurable:∫Rn‍(|f(x)|λ)p(x)dx}.(ii)
The spaceLloc⁡p(·)(Rn) is defined by
(7)Lloc⁡p(·)(Rn):={f:fχK∈Lp(·)(Rn)forallcompactsubsetsK⊂Rn}, where, and in what follows, χS denotes the characteristic function of a measurable set S⊂Rn.
L
p
(
·
)
(
R
n
) is a Banach function space when equipped with the norm
(8)∥f∥Lp(·):=inf⁡{λ>0:∫Rn‍(|f(x)|λ)p(x)dx≤1}.
Lettingp(·):Rn→(0,∞), we denote
(9)p-:=essinf⁡x∈Rnp(x),p+:=essinf⁡x∈Rnp(x).
The set P(Rn) consists of all p(·) satisfying p->1 and p+<∞; P0(Rn) consists of all p(·) satisfying p->0 and p+<∞.Lp(·) can be similarly defined as mentioned above for p(·)∈P0(Rn). p′(·) means the conjugate exponent of p(·) that means 1/p(·)+1/p′(·)=1.
Letf∈Lloc⁡1(Rn). Then the standard Hardy-Littlewood maximal function of f is defined by
(10)Mf(x)=sup⁡x∈B1|B|∫B‍|f(y)|dy,∀x∈Rn,
where B is a ball and |B| is the volume of B. Let B(Rn) be the set of p(·)∈P(Rn) such that M is bounded on Lp(·)(Rn). For more on B(Rn), see [27, 28].Definition 2.
We say that a functionα(·):Rn→R is locally log-Hölder continuous if there exists a constant C such that
(11)|α(x)-α(y)|≤Clog⁡⁡(e+1/|x-y|),lllllllllllllllllllx,y∈Rn,|x-y|<12.
If
(12)|α(x)-α(0)|≤Clog⁡(e+1/|x|),∀x∈Rn,
then we say that α(·) is log-Hölder continuous at the origin. If there exists α∞∈R such that
(13)|α(x)-α∞|≤Clog⁡(e+|x|),∀x∈Rn,
then we say that α(·) is log-Hölder continuous at the infinity. If α(·) is both locally log-Hölder continuous and log-Hölder continuous at the infinity, then we say α(·) is global log-Hölder continuous.We denote byP0log⁡(Rn) and P∞log⁡(Rn) the class of all exponents p∈P(Rn) which is log-Hölder continuous at the origin and at the infinity, respectively.Next, we recall the definitions of Herz spaces and Herz-Morrey spaces with variable exponents. We use the following notations. For eachk∈Z, we define
(14)Bk≔{x∈Rn:|x|≤2k},Dk:=Bk∖Bk-1,χk:=χDk.Definition 3 (see [29, Definition  2]).
Letq∈(0,∞], p(·)∈P(Rn), and α(·):Rn→R with α∈L∞(Rn). The homogeneous Herz space K˙q,p(·)α(·)(Rn) is defined as the set of all f∈Lloc⁡p(·)(Rn∖{0}) such that
(15)∥f∥K˙q,p(·)α(·):={∑k∈Z‍∥2kα(·)fχk∥p(·)q}1/q<∞,
with the usual modifications when q=∞.Definition 4 (see [30, Definition  2]).
Let0<q<∞, p(·)∈P(Rn), 0≤λ<∞ and α(·):Rn→R with α∈L∞(Rn). The Herz-Morrey space MK˙q,p(·)α(·),λ(Rn) with variable exponents is defined by
(16)MK˙q,p(·)α(·),λ(Rn):={f∈Lloc⁡p(·)(Rn∖{0}):∥f∥MK˙q,p(·)α(·),λ(Rn)<∞},
where
(17)∥f∥MK˙q,p(·)α(·),λ(Rn):=sup⁡L∈Z2-Lλ(∑k=-∞L‍∥2kα(·)fχk∥Lp(·)(Rn)q)1/q.Ifα(·) is a constant, then MK˙q,p(·)α(·),λ(Rn)=MK˙q,p(·)α,λ(Rn) was defined in [30]. If λ=0, then MK˙q,p(·)α(·),λ(Rn)=K˙q,p(·)α(·)(Rn). If both α(·) and p(·) are constant and λ=0, then MK˙q,p(·)α(·),λ(Rn)=K˙q,pα(Rn) is the classical Herz space in [31].Lemma 5 (see [30, Lemma  1 and (10)]).
Ifp(·)∈B(Rn), then there exist constants δ1, δ2∈(0,1) and C>0 such that, for all balls B in Rn and all measurable subsets S⊂B,
(18)∥χS∥Lp(·)(Rn)∥χB∥Lp(·)(Rn)≤C(|S||B|)δ1,∥χS∥Lp′(·)(Rn)∥χB∥Lp′(·)(Rn)≤C(|S||B|)δ2.There is a position to state our result.Theorem 6.
Letbi∈BMO(Rn) and p(·), pi(·)∈B(Rn) satisfy 1/p(x)=1/p1(x)+1/p2(x)+⋯+1/pm(x), 1<pi-≤pi+<n/λi, (p(·)/p*)′∈B(Rn) for some 0<p*<p-, i=1,2,…,m. Let 0<qi<∞, 0≤λi<∞ and αi(·)∈L∞(Rn)∩P0log⁡(Rn)∩P∞log⁡(Rn) for i=1,2,…,m with
(19)λi-nδ1<αi-<αi+<nδ2,
where δ1,δ2∈(0,1) are the constants appearing in (18). Suppose that λ=∑i=1m‍λi, α(·)=∑i=1m‍αi(·), and 1/q=∑i=1m‍(1/qi). Then
(20)∥[b→,T](f→)∥MK˙q,p(·)α(·),λ(Rn)≤C∏i=1m‍∥bi∥*∥fi∥MK˙qi,pi(·)αi(·),λi(Rn)
with the constant C>0 independent of f→=(f1,f2,…,fm).Remark 7.
Letλi=0, and then the commutator [b→,T] is bounded from the product of variable exponents Herz spaces K˙q1,p1(·)α1(·)(Rn)×K˙q2,p2(·)α2(·)(Rn)×⋯×K˙qm,pm(·)αm(·)(Rn) to variable exponents Herz space K˙q,p(·)α(·)(Rn) when -nδ1<αi-<αi+<nδ2.
Finally, we point out thatC denotes a positive constant which may be different at different occurrences.


## 2. Proof of the Main Result

To give our proof, we need some lemmas.Lemma 8 (see [32, Proposition  2]).
Ifα(·)∈L∞(Rn)∩P0log⁡(Rn)∩P∞log⁡(Rn), 0≤λ<∞, p(·)∈P(Rn), and q∈(0,∞), then
(21)∥f∥MK˙q,p(·)α(·),λ(Rn)≈max⁡{sup⁡L>0,L∈Z22+2-Lλ(∑k=0L‍2kα∞q∥fχk∥Lp(·)(Rn)q)1/q]sup⁡L≤0,L∈Z2-Lλ(∑k=-∞L‍2kα(0)q∥fχk∥Lp(·)(Rn)q)1/q,sup⁡L>0,L∈Z22sup⁡L>0,L∈Z[2-Lλ(∑k=-∞-1‍2kα(0)q∥fχk∥Lp(·)(Rn)q)1/qsup⁡L>0,L∈Z22bbbllbbbb+2-Lλ(∑k=0L‍2kα∞q∥fχk∥Lp(·)(Rn)q)1/q]}.Lemma 9 (see [30, Lemma  2]).
Ifp(·)∈B(Rn), then there exists a constant C>0 such that, for all balls B in Rn,
(22)C-1≤1|B|∥χB∥Lp(·)(Rn)∥χB∥Lp′(·)(Rn)≤C.Lemma 1010 (see [29, Lemma  3]).
Letk be a positive integer. Then one has that, for all b∈BMO(Rn) and all i, j∈Z with j>i,
(23)C-1∥b∥*k≤sup⁡B:ball1∥χB∥Lp(·)(Rn)∥(b-bB)kχB∥Lp(·)(Rn)≤C∥b∥*k,∥(b-bBi)kχBj∥Lp(·)(Rn)≤C(j-i)k∥b∥*k∥χBj∥Lp(·)(Rn).In fact, ifk=1, then, from inequalities (23), for all balls B and i, j∈Z with j>i, we have
(24)∥(b-bB)χB∥Lp(·)(Rn)≤C∥b∥*∥χB∥Lp(·)(Rn),∥(b-bBi)χBj∥Lp(·)(Rn)≤C(j-i)∥b∥*∥χBj∥Lp(·)(Rn).Lemma 11 (see [1, Theorem  2.1] ).
Ifp(·)∈P(Rn), then, for all f∈Lp(·)(Rn) and all g∈Lp′(·)(Rn), one has
(25)∫Rn‍|f(x)g(x)|dx≤rp∥f∥Lp(·)(Rn)∥g∥Lp′(·)(Rn),
where rp:=1+1/p+-1/p-.Lemma 12 (see [14, Theorem 2.3]).
Letp,p1,p2∈P0(Rn) such that 1/p(x)=1/p1(x)+1/p2(x). Then there exists a constant Cp,p1 independent of the functions f and g such that
(26)∥fg∥Lp(·)(Rn)≤Cp,p1∥f∥Lp1(·)(Rn)∥g∥Lp2(·)(Rn)
holds for every f∈Lp1(·)(Rn) and g∈Lp2(·)(Rn).Lemma 1313 (see [14, Corollary  2.2]).
LetT be a 2-linear Calderón-Zygmund operator and let b1 and b2 be BMO functions. Let p(·)∈P0(Rn) such that there exists 0<p*<p- with (p(·)/p*)′∈B(Rn). If p1(·),p2(·)∈B(Rn) such that 1/p(x)=1/p1(x)+1/p2(x), then there exists a constant C independent of functions fi∈Lpi(·)(Rn) for i=1,2 such that
(27)∥[b1,b2,T](f1,f2)∥Lp(·)≤C∥b1∥*∥b2∥*∥f1∥Lp1(·)∥f2∥Lp2(·).Proof of Theorem6.
Although our method is suitable for any multilinear operator, for simplicity, we only consider 2-linear operators. Letb1 and b2 be BMO functions. Since the set of all bounded compactly supported functions is dense in Herz-Morrey spaces with variable exponents, we let f1 and f2 be bounded compactly supported functions; then, for x∈Rn, we write
(28)[b1,b2,T](f1,f2)(x)=b1(x)b2(x)T(f1,f2)(x)-b1(x)T(f1,b2f2)(x)-b2(x)T(b1f1,f2)(x)+T(b1f1,b2f2)(x).
We write
(29)fi(x)=∑li=-∞∞‍fi(x)χli(x)=:∑li=-∞∞‍fli(x),i=1,2.
Then, for each k∈Z, if li≤k-1, from Lemma 11, we obtain
(30)∫Rn‍[bi(x)-bi(yi)]fli(yi)dyi≤C[∫Rn‍(bi(x)-bBli)fli(yi)dyi+∫Rn‍(bBli-bi(yi))fli(yi)dyi]≤C∥fli∥Lpi(·)(Rn)(∥|bBli-bi|·χli∥Lpi′(·)(Rn)|bi(x)-bBli|·∥χli∥Lpi′(·)(Rn)≤C∥fli∥Lpi(·)(Rn)bbllbi+∥|bBli-bi|·χli∥Lpi′(·)(Rn)),
and by using inequalities (18), (22), and (24), it follows that
(31)∥∫Rn‍[bi(·)-bi(yi)]fli(yi)dyiχk∥Lpi(·)(Rn)≤C∥fli∥Lpi(·)(Rn)(∥|bi-bBli|χBk∥Lpi(·)(Rn)·∥χli∥Lpi′(·)(Rn)bbbbbbbllllibbbb+∥|bBli-bi|χBli∥Lpi′(·)(Rn)·∥χk∥Lpi(·)(Rn))≤C∥fli∥Lpi(·)(Rn)[(k-li)∥bi∥*∥χBk∥Lpi(·)(Rn)∥χli∥Lpi′(·)(Rn)bbbbbbbbbbbbibbbb+∥bi∥*∥χBli∥Lpi′(·)(Rn)∥χk∥Lpi(·)(Rn)]≤C∥bi∥*∥fli∥Lpi(·)(Rn)(k-li)∥χBk∥Lpi(·)(Rn)∥χBli∥Lpi′(·)(Rn)≤C∥bi∥*∥fli∥Lpi(·)(Rn)(k-li)∥χBk∥Lpi(·)(Rn)×∥χBk∥Lpi′(·)(Rn)∥χBli∥Lpi′(·)(Rn)∥χBk∥Lpi′(·)(Rn)≤C(k-li)2(li-k)nδ22kn∥bi∥*∥fli∥Lpi(·)(Rn).
Similarly, if li≥k+1, we have
(32)∥∫Rn‍[bi(·)-bi(yi)]fli(yi)dyiχk∥Lpi(·)(Rn)≤C∥fli∥Lpi(·)(Rn)(∥|bi-bBk|χBk∥Lpi(·)(Rn)·∥χli∥Lpi′(·)(Rn)bbbbbbbbbbbi+∥|bBk-bi|χBli∥Lpi′(·)(Rn)·∥χk∥Lpi(·)(Rn))≤C∥fli∥Lpi(·)(Rn)[∥χBli∥∥χBli∥∥bi∥*∥χBk∥Lpi(·)(Rn)∥χli∥Lpi′(·)(Rn)bbbbbbbbbbbbbbo+(li-k)∥bi∥*∥χBli∥Lpi′(·)(Rn)∥χk∥Lpi(·)(Rn)[∥bi∥*∥χBk∥Lpi(·)(Rn)∥χli∥Lpi′(·)(Rn)]≤C∥bi∥*∥fli∥Lpi(·)(Rn)(li-k)∥χBk∥Lpi(·)(Rn)∥χBli∥Lpi′(·)(Rn)≤C∥bi∥*∥fli∥Lpi(·)(Rn)(li-k)×∥χBk∥Lpi(·)(Rn)∥χBli∥Lpi(·)(Rn)∥χBli∥Lpi(·)(Rn)∥χBli∥Lpi′(·)(Rn)≤C(li-k)2(k-li)nδ12lin∥bi∥*∥fli∥Lpi(·)(Rn).
As for the case li=k, we get
(33)∥∫Rn‍[∫Rn‍bi(·)-bi(yi)]fli(yi)dyiχk∥Lpi(·)(Rn)≤C∥fli∥Lpi(·)(Rn)(∥|bBk-bi|χBli∥Lpi′(·)(Rn)∥∫Rn‍|bi-bBk|χBk∥Lpi(·)(Rn)·∥χli∥Lpi′(·)(Rn)bbbbbbbbbllbbbbbbi+∥|bBk-bi|χBli∥Lpi′(·)(Rn)·∥χk∫Rn‍∥Lpi(·)(Rn)∫Rn‍)≤C∥bi∥*∥fli∥Lpi(·)(Rn)∥χBk∥Lpi(·)(Rn)∥χBk∥Lpi′(·)(Rn)≤C2kn∥bi∥*∥fli∥Lpi(·)(Rn).
By Lemma8, we know
(34)∥[b1,b2,T](f1,f2)∥MK˙q,p(·)α(·),λ(Rn)≈max⁡{E,F},
where
(35)E:=sup⁡L≤0,L∈Z2-Lλ×((fl1,fl2)χk∑l1=-∞∞‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=-∞∞‍∑l2=-∞∞‍[b1,b2,T]mmmmmmmmmllllmmm×(fl1,fl2)χk∑l1=-∞∞‍∥Lp(·)(Rn)q∑k=-∞L‍)1/q,F:=sup⁡L>0,L∈Z[2-Lλ(∑k=-∞-1‍2kα(0)q∥[b1,b2,T](f1,f2)χk∥Lp(·)(Rn)q)1/qbbbbbbbbbb+2-Lλ(∑k=0L‍2kα∞qbbbbbbbbbbbblllllkbbbb×∥[b1,b2,T](f1,f2)χk∥Lp(·)(Rn)q∑k=-∞-1‍)1/q∑k=-∞-1‍].
Since the estimate of F is essentially similar to that of E, so it suffices to prove that E is bounded in Herz-Morrey spaces with variable exponents. It is easy to see that
(36)E≤C∑i=19‍Ii,
where
(37)I1:=sup⁡L≤0,L∈Z2-Lλ×((fl1,fl2)χk∑k=-∞L‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=-∞k-2‍∑l2=-∞k-2‍[b1,b2,T]dddddddddllllllldddddddd×(fl1,fl2)χk∑k=-∞L‍∥Lp(·)(Rn)q)1/q,I2:=sup⁡L≤0,L∈Z2-Lλ×(bbbbbbbbbbbbbbbbb×(fl1,fl2)χk∑l1=-∞k-2‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=-∞k-2‍∑l2=k-1k+1‍[b1,b2,T]bbbbbbbbllllllbbbllllbbbbbbb×(fl1,fl2)χk∑l1=-∞k-2‍∥Lp(·)(Rn)q)1/q,I3:=sup⁡L≤0,L∈Z2-Lλ×(bbbbbbbbbbbbbbbbbbb×(fl1,fl2)χk∑l1=-∞k-2‍∥∑k=-∞L‍2kα(0)q∥∑l1=-∞k-2‍∑l2=k+2∞‍[b1,b2,T]bbbbbbbbbbbllllllllllbbbbllbbbb×(fl1,fl2)χk∑l1=-∞k-2‍∥Lp(·)(Rn)q)1/q,I4:=sup⁡L≤0,L∈Z2-Lλ×((fl1,fl2)χk∑l2=-∞k-2‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=k-1k+1‍∑l2=-∞k-2‍[b1,b2,T]bbbbbbbbblllllllllllllbbbbbbbb×(fl1,fl2)χk∑l2=-∞k-2‍∥Lp(·)(Rn)q)1/q,I5:=sup⁡L≤0,L∈Z2-Lλbbbb×((fl1,fl2)χk∑l2=k-1k+1‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=k-1k+1‍∑l2=k-1k+1‍[b1,b2,T]bbbbbbbbbbllllllbbbllbbbbbb×(fl1,fl2)χk∑l2=k-1k+1‍∥Lp(·)(Rn)q)1/q,I6:=sup⁡L≤0,L∈Z2-Lλ×(∑k=-∞L‍2kα(0)q∥∑l1=k-1k+1‍∑l2=k+2∞‍[b1,b2,T]bbbbbbbbbllllllllllbbbbbbbbb×(fl1,fl2)χk∑l1=k-1k+1‍∥Lp(·)(Rn)q(fl1,fl2)χk∑l2=k-1k+1‍∥Lp(·)(Rn)q)1/q,I7:=sup⁡L≤0,L∈Z2-Lλ×((fl1,fl2)χk∑l2=-∞k-2‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=k+2∞‍∑l2=-∞k-2‍[b1,b2,T]bbbbbbbbblllllllbbbbbbbbbb×(fl1,fl2)χk∑l2=-∞k-2‍∥Lp(·)(Rn)q)1/q,I8:=sup⁡L≤0,L∈Z2-Lλ×((fl1,fl2)χk∑l2=k-1k+1‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=k+2∞‍∑l2=k-1k+1‍[b1,b2,T]bbbbbbbbbblllllllbbbbbbbbbb×(fl1,fl2)χk∑l2=k-1k+1‍∥Lp(·)(Rn)q)1/q,I9:=sup⁡L≤0,L∈Z2-Lλi×((fl1,fl2)χk∑l2=k+2∞‍∥Lp(·)(Rn)q∑k=-∞L‍2kα(0)q∥∑l1=k+2∞‍∑l2=k+2∞‍[b1,b2,T]bbbbbbbbbllllllbbbbbbbbbb×(fl1,fl2)χk∑l2=k+2∞‍∥Lp(·)(Rn)q)1/q.
Using the symmetry of f1 and f2, we only need to estimate I1, I2, I3, I5, I6, I9 because the estimates of I2, I3, and I6 are analogous to those of I4, I7, and I8, respectively. In what follows, we divide it into 6 steps.
Step
1. To estimate the term of I1, we note that li≤k-2 for i=1,2. Thus, for x∈Dk, yi∈Dli,
(38)|x-yi|≥||x|-|yi||>2k-1-2li≥2k-2.
Then, for x∈Dk, we get
(39)|K(x,y1,y2)|≤C(|x-y1|+|x-y2|)-2n≤C2-2kn.
Therefore,
(40)[b1,b2,T](fl1,fl2)(x)≤C2-2kn∏i=12‍∫Rn‍[bi(x)-bi(yi)]fli(yi)dyi.
By Lemma 12 and inequality (31), we obtain
(41)∥∑l1=-∞k-2‍∑l2=-∞k-2‍[b1,b2,T](fl1,fl2)χk∥Lp(·)(Rn)≤C∥2-2kn∏i=12‍(∑li=-∞k-2‍∫Rn‍[bi(·)-bi(yi)]fli(yi)dyi)χk∥Lp(·)(Rn)≤C2-2kn∏i=12‍∑li=-∞k-2‍∥∫Rn‍[bi(·)-bi(yi)]fli(yi)dyiχk∥Lpi(·)(Rn)≤C∏i=12‍∑li=-∞k-2‍(k-li)2(li-k)nδ2∥bi∥*∥fli∥Lpi(·)(Rn).
Since1/q=1/q1+1/q2, it follows that
(42)I1≤Csup⁡L≤0,L∈Z2-Lλ×{∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)∑k=-∞L‍2kα(0)q(∏i=12‍∑li=-∞k-2‍(k-li)2(li-k)nδ2∥bi∥*bbbbbbbbbbbbbbbbbbbb×∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)}1/q≤Csup⁡L≤0,L∈Z∏i=12‍2-Lλi∥bi∥*×{∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)qi∑k=-∞L‍(∑li=-∞k-2‍(k-li)2(li-k)nδ2+kαi(0)bbbbbbbbbbbbbbb×∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)qi}1/qi=:Csup⁡L≤0,L∈ZI11(L)I12(L),
where
(43)I1i(L):=2-Lλi∥bi∥*bbbbbbb×{∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)∑k=-∞L‍(∑li=-∞k-2‍(k-li)2(li-k)μi2liαi(0)bbbbbbbbbbllbbbbbb×∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)qi}1/qi.
Here, we denote μi=nδ2-αi(0) for short. It follows from condition (19) that μi>0. To continue calculations for (43), we consider the two cases 0<qi≤1 and 1<qi<∞.
If0<qi≤1, since, for a1, a2,…≥0,
(44)(∑h=1∞‍ah)qi≤∑h=1∞‍ahqi,
then
(45)I1i(L)≤C2-Lλi∥bi∥*{∑k=-∞L‍∑li=-∞k-2‍(k-li)qi2(li-k)μiqi2liαi(0)qibbbbbbbbbbbbbbbbbbbbbbb×∥fli∥Lpi(·)(Rn)qi∑li=-∞k-2‍}1/qi=C2-Lλi∥bi∥*{∑li=-∞L-2‍2liαi(0)qi∥fli∥Lpi(·)(Rn)qibbbbbbbbbbbbbbbbbbbb×∑k=li+2L‍(k-li)qi2(li-k)μiqi}1/qi≤C∥bi∥*2-Lλi{∑li=-∞L-2‍2liαi(0)qi∥fli∥Lpi(·)(Rn)qi}1/qi≤C∥bi∥*∥fli∥MK˙qi,pi(·)αi(·),λi(Rn).
If 1<qi<∞, by the Hölder inequality, we obtain
(46)I1i(L)=2-Lλi∥bi∥*bbb×{∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)∑k=-∞L‍(∑li=-∞k-2‍(k-li)2(li-k)μi2liαi(0)bbbbbbbbbbblllbbbbbbb×∥fli∥Lpi(·)(Rn)∑li=-∞k-2‍)qi}1/qi≤C2-Lλi∥bi∥*×{(∑li=-∞k-2‍(k-li)qi′2(li-k)μi(qi′/2))qi/qi′∑k=-∞L‍(∑li=-∞k-2‍2(li-k)μi(qi/2)2liαi(0)qi∥fli∥Lpi(·)(Rn)qi)bbbbbbbbbb×(∑li=-∞k-2‍(k-li)qi′2(li-k)μi(qi′/2))qi/qi′}1/qi≤C2-Lλi∥bi∥*×{∑k=-∞L‍(∑li=-∞k-2‍2(li-k)μi(qi/2)2liαi(0)qi∥fli∥Lpi(·)(Rn)qi)}1/qi=C2-Lλi∥bi∥*(∑li=-∞L-2‍2liαi(0)qibbbbbbbbbbbbbbbbbbb×∥fli∥Lpi(·)(Rn)qi∑k=li+2L‍2(li-k)μi(qi/2))1/qi≤C∥bi∥*2-Lλi(∑li=-∞L-2‍2liαi(0)qi∥fli∥Lpi(·)(Rn)qi)1/qi≤C∥bi∥*∥fi∥MK˙qi,pi(·)αi(·),λi(Rn).
Thus, for any 0<qi<∞,
(47)I1≤Csup⁡L≤0,L∈ZI11(L)I12(L)≤C∥b1∥*∥b2∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn)∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Step
2. To estimate I2, for x∈Dk, yi∈Dli, i=1,2 and l1≤k-2, k-1≤l2≤k+1, we have
(48)|x-y2|≥|x-y1|≥||x|-|y1||>2k-2.
Since l1≤k-2, it follows from inequality (31) that
(49)∥∑l1=-∞k-2‍∑l2=k-1k+1‍[b1,b2,T](fl1,fl2)χk∥Lp(·)(Rn)≤C2-2kn∑l1=-∞k-2‍∥∫Rn‍[b1(·)-b1(y1)∫Rn‍]fl1(y1)dy1∥Lp1(·)(Rn)×∑l2=k-1k+1‍∥∫Rn‍[b2(·)-b2(y2)]fl2(y2)dy2χk∥Lp2(·)(Rn)≤C(∑l1=-∞k-2‍(k-l1)2(l1-k)nδ2∥b1∥*∥fl1∥Lp1(·)(Rn))×2-kn∑l2=k-1k+1‍∥∫Rn‍[b2(·)-b2(y2)]fl2(y2)dy2χk∥Lp2(·)(Rn).
For k-1≤l2≤k+1, combining the above term and inequalities (32) and (33), we have
(50)I2≤Csup⁡L≤0,L∈Z2-Lλ∥b1∥*∥b2∥*×{∑k=-∞L‍2kα(0)q(∑l1=-∞k-2‍∑l2=k-1k+1‍(k-l1)2(l1-k)nδ2bbbbbbbbbbbllllllllbbbbbb×∥fl1∥Lp1(·)(Rn)2(l2-k)n∥fl2∥Lp2(·)(Rn)∑l1=-∞k-2‍)q}1/q≤Csup⁡L≤0,L∈Z2-Lλ1∥b1∥*×{∑k=-∞L‍(∑l1=-∞k-2‍(k-l1)2(l1-k)μ12l1α1(0)bbbbbbllbbbbbbb×∥fl1∥Lp1(·)(Rn)∑l1=-∞k-2‍)q1}1/q1×2-Lλ2∥b2∥*{∑k=-∞L‍(∑l2=k-1k+1‍2kα2(0)2(l2-k)nbbbbbbbbbbbllbbbbbbbbbbb×∥fl2∥Lp2(·)(Rn)∑l2=k-1k+1‍)q2}1/q2=:Csup⁡L≤0,L∈ZI21(L)I22(L).
Here, we used 2-nδ2<1 and 2(l2-k)n(1-δ1)<2(l2-k)n in the first inequality.
Obviously,(51)I21(L)=I11(L)≤C∥b1∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn).
Therefore, we only need to estimate I22(L):
(52)I22(L)=2-Lλ2∥b2∥*{∑l2=k-1k+1‍∑k=-∞L‍2kα2(0)q2bbbbbbbbbbllbbbbbbbb×(∑l2=k-1k+1‍2(l2-k)n∥fl2∥Lp2(·)(Rn))q2}1/q2≤C∥b2∥*2-Lλ2(∑k=-∞L‍2kα2(0)q2∥f2χk∥Lp2(·)(Rn)q2)1/q2≤C∥b2∥*∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Step
3. To estimate I3, for x∈Dk, yi∈Dli, i=1,2, and l1≤k-2,l2≥k+2, then we have
(53)|x-y1|≥|x|-|y1|>2k-2,|x-y2|≥|y2|-|x|>2l2-2.
Thus, for x∈Ck, we get
(54)[b1,b2,T](fl1,fl2)(x)≤C2-kn∫Rn‍[b1(x)-b1(y1)]fl1(y1)dy12-l2n×∫Rn‍[b2(x)-b2(y2)]fl2(y2)dy2.
From inequalities (31) and (32), we obtain
(55)∥∑l1=-∞k-2‍∑l2=k+2∞‍[b1,b2,T](fl1,fl2)χk∥Lp(·)(Rn)≤C∑l1=-∞k-2‍(k-l1)2(l1-k)nδ2∥b1∥*∥fl1∥Lp1(·)(Rn)×∑l2=k+2∞‍(l2-k)2(k-l2)nδ1∥b2∥*∥fl2∥Lp2(·)(Rn).
Consequently, it follows that
(56)I3≤Csup⁡L≤0,L∈Z2-Lλ∥b1∥*∥b2∥*×{∑k=-∞L‍2kα(0)q(∑l1=-∞k-2‍(k-l1)2(l1-k)nδ2∥fl1∥Lp1(·)(Rn))qbbbbbbbbbb×(∑l2=k+2∞‍(l2-k)2(k-l2)nδ1∥fl2∥Lp2(·)(Rn))q}1/q≤Csup⁡L≤0,L∈Z2-Lλ1∥b1∥*×{∑k=-∞L‍(∑l1=-∞k-2‍(k-l1)2(l1-k)μ12l1α1(0)bbbbbbbbbllbbbbbbb×∥fl1∥Lp1(·)(Rn)∑l1=-∞k-2‍)q1}1/q1×2-Lλ2∥b2∥*×{∑k=-∞L‍(∑l2=k+2∞‍(l2-k)bbbbbbbbblllllbbbbb×2(k-l2)(nδ1+α2(0))2l2α2(0)∥fl2∥Lp2(·)(Rn)∑l2=k+2∞‍)q2}1/q2=:Csup⁡L≤0,L∈ZI31(L)I32(L).
Note that
(57)I31(L)=I21(L)≤C∥b1∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn),
so we only compute I32(L). From inequality (19) and nδ1+α2(0)-λ2>0, we obtain
(58)I32(L)=2-Lλ2∥b2∥*{∑k=-∞L‍(∑l2=k+2∞‍(l2-k)2(k-l2)(nδ1+α2(0))2l2α2(0)bbbbbbbbbbbbbbbbbbbbbbbbb×∥f2χl2∥Lp2(·)(Rn)∑l2=k+2∞‍)q2}1/q2≤C2-Lλ2∥b2∥*×{×∥f2χm∥Lp2(·)(Rn)q2∑k=-∞L‍)1/q2]q2∑k=-∞L‍[∑l2=k+2∞‍(l2-k)2(k-l2)(nδ1+α2(0))2l2λ2bbbbbbbblllllllbbbb×2-l2λ2(∑m=-∞l2‍2mα2(0)q2bbbiibbbbbbbbllllllllllllllllbbbbbbbb×∥f2χm∥Lp2(·)(Rn)q2∑m=-∞l2‍)1/q2]q2}1/q2≤C2-Lλ2∥b2∥*{×∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn)∑k=-∞L‍)q2∑k=-∞L‍2kλ2q2bbbbbbbbbbbbbbbbbb×(∑l2=k+2∞‍(l2-k)2(k-l2)(nδ1+α2(0)-λ2)bbbbbbbbbbbbbblllllllbbbbbbbbb×∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn)∑l2=k+2∞‍)q2}1/q2≤C∥b2∥*2-Lλ2(∑k=-∞L‍2kλ2q2)1/q2∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn)≤C∥b2∥*∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Therefore,
(59)I3≤Csup⁡L≤0,L∈ZI31(L)I32(L)≤C∥b1∥*∥b2∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn)∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Step
4. It turns to estimate the term I5. Applying the Hölder inequality and Lemma 13, we have
(60)I5≤Csup⁡L≤0,L∈Z2-Lλ×{∑k=-∞L‍2kα(0)q(∑l1=k-1k+1‍∑l2=k-1k+1‍∥fl1[b1,b2,T]bbbbbbbbbbbbbbbbbbbbbbbbbbbbb×(fl1,fl2)∥Lp(·)(Rn)∑l2=k-1k+1‍)q}1/q≤Csup⁡L≤0,L∈Z2-Lλ{(∑l1=k-1k+1‍∑l2=k-1k+1‍∥b1∥*∥b2∥*∑k=-∞L‍2kα(0)qbbbbbbbbllbbbbbbbb×(∑l1=k-1k+1‍∑l2=k-1k+1‍∥b1∥*∥b2∥*bbbbbbbbbbblllllllllllllllllbbbbbbbb×∥fl1∥Lp1(·)(Rn)∥fl2∥Lp2(·)(Rn)∑l1=k-1k+1‍)q}1/q≤C∏i=12‍∥bi∥*sup⁡L≤0,L∈Z2-Lλi×{∑k=-∞L‍2kαi(0)qi(∑li=k-1k+1‍∥fli∥Lpi(·)(Rn))qi}1/qi≤C∏i=12‍∥bi∥*sup⁡L≤0,L∈Z2-Lλi×(∑k=-∞L‍2kαi(0)qi∥fiχk∥Lpi(·)(Rn)qi)1/qi≤C∥b1∥*∥b2∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn)×∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Step
5. Now it goes to the estimate of I6.
It is clear that, forx∈Dk and k-1≤l1≤k+1 and l2≥k+2,
(61)|[b1,b2,T](f1,f2)(x)|≤C2-kn∫Rn‍[b1(x)-b1(y1)]f1(y1)dy12-l2n×∫Rn‍[b2(x)-b2(y2)]f2(y2)dy2.
Therefore,
(62)∥∑l1=k-1k+1‍∑l2=k+2∞‍[b1,b2,T](fl1,fl2)χk∥Lp(·)(Rn)≤C∥b1∥*∥b2∥*×∑l1=k-1k+1‍∑l2=k+2∞‍2(l1-k)n∥fl1∥Lp1(·)(Rn)×(l2-k)2(k-l2)nδ1∥b2∥*∥fl2∥Lp2(·)(Rn),I6≤Csup⁡L≤0,L∈Z2-Lλ×{∑k=-∞L‍2kα(0)q(∑l1=k-1k+1‍∑l2=k+2∞‍∥[b1,b2,T]bbbbbbbbbbbbbbbbbbbbbbbibbb×(fl1,fl2)∥Lp(·)(Rn)∑l1=k-1k+1‍)q}1/q≤Csup⁡L≤0,L∈Z2-Lλ∥b1∥*∥b2∥*×{(∑l1=k-1k+1‍2kα1(0)2(l1-k)n∥fl1∥Lp1(·)(Rn))q1∑k=-∞L‍2kα(0)q(∑l1=k-1k+1‍∑l2=k+2∞‍2(l1-k)n∥fl1∥Lp1(·)(Rn)bbbbbbbbbllllllllllllllbbbbbbbb×(l2-k)2(k-l2)nδ1∥fl2∥Lp2(·)(Rn)∑l2=k+2∞‍)q}1/q≤Csup⁡L≤0,L∈Z2-Lλ1∥b1∥*×{∑k=-∞L‍(∑l1=k-1k+1‍2kα1(0)2(l1-k)n∥fl1∥Lp1(·)(Rn))q1}1/q1×2-Lλ2∥b2∥*{∑l2=k+2∞‍∑k=-∞L‍(∑l2=k+2∞‍(l2-k)2(k-l2)(nδ1+α2(0))bbbbbbbbblllllllbbbbbbbbb×2l2α2(0)∥fl2∥Lp2(·)(Rn)∑l2=k+2∞‍)q2}1/q2=:Csup⁡L≤0,L∈ZI61(L)I62(L).
Here, the estimate of I61(L) is similar to that of I22(L) and I62(L)=I32(L).
Step
6. Finally, we will finish the estimation of the last term I9. Note that l2≥k+2 and |x-yi|>2li-2 for x∈Dk, yi∈Dli, i=1,2: we get
(63)[b1,b2,T](fl1,fl2)(x)≤C2-l1n∫Rn‍[b1(x)-b1(y1)]fl1(y1)dy1·2-l2n×∫Rn‍[b2(x)-b2(y2)]fl2(y2)dy2.
Applying the Hölder inequality to the last integral, we obtain
(64)∥∑l1=k+2∞‍∑l2=k+2∞[b1,b2,T](fl1,fl2)χk∥Lp(·)(Rn)≤C∥b1∥*∥b2∥*×∑l1=k+2∞‍∑l2=k+2∞(l1-k)2(k-l1)nδ1∥fl1∥Lp1(·)(Rn)×(l2-k)2(k-l2)nδ1∥fl2∥Lp2(·)(Rn)≤C∏i=12‍∥bi∥*∑li=k+2∞‍(li-k)2(k-li)nδ1∥fli∥Lpi(·)(Rn).
Thus,
(65)I9≤Csup⁡L≤0,L∈Z2-Lλ×{∑k=-∞L‍2kα(0)q(∑l1=k+2∞‍∑l2=k+2∞‍∥fl1[b1,b2,T]bbbbbbbbbbbbbbbbbbbbbbbbbbbbbi×(fl1,fl2)∥Lp(·)(Rn)∑l1=k+2∞‍)q}1/q≤Csup⁡L≤0,L∈Z2-Lλ×{∑k=-∞L‍2kα(0)q(∏i=12‍∥bi∥*∑li=k+2∞‍(li-k)bbbbbbbbbbllllbbbbbbbbi×2(k-li)nδ1∥fli∥Lpi(·)(Rn)∏i=12‍)q}1/q≤Csup⁡L≤0,L∈Z∏i=12‍2-Lλi∥bi∥*×{∑k=-∞L‍(∑li=k+2∞‍(li-k)2(k-li)(nδ1+αi(0))bbbbbbbbblllllllbbbbb×2liαi(0)∥fli∥Lpi(·)(Rn)∑li=k+2∞‍)qi}1/qi=:Csup⁡L≤0,L∈ZI91(L)I92(L)≤C∥b1∥*∥b2∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn)×∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
Here, the estimate of I9i(L), i=1,2, is similar to that of I32(L).
Combing all estimates forIi, i=1,2,…,9 together, we get
(66)E≤C∥b1∥*∥b2∥*∥f1∥MK˙q1,p1(·)α1(·),λ1(Rn)∥f2∥MK˙q2,p2(·)α2(·),λ2(Rn).
This finishes the proof of Theorem6.



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*Source: 162518-2014-05-25.xml*