# Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent

Authors: Amjad Hussain; Guilian Gao
Journal: ISRN Mathematical Analysis
 (2014)

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## Abstract

The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators from products of
variable exponent Herz spaces to variable exponent Herz spaces.


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## Body

## 1. Introduction

In recent years, the interest in multilinear analysis for studying boundedness properties of multilinear integral operators has grown rapidly. The subject was founded by Coifman and Meyer [1] in their seminal work on singular integral operators like Calderón commutators and pseudosdifferential operators having multiparameter function input. Subsequently, many authors including Christ and Journé [2], Kenig and Stein [3], and Grafakos and Torres [4] have substantially added to the exiting theory.LetK be a locally integrable function defined away from the diagonal x=y1=⋯=ym in (ℝn)m+1, which for C>0 satisfies the estimates
(1)|K(x,y1,…,ym)|≤C(|x-y1|+⋯+|x-ym|)mn,
and for ϵ>0,
(2)|K(x,y1,…,ym)-K(x′,y1,…,ym)|≤C|x-x′|ϵ(∑j=1m|x-yj|)mn+ϵ,
whenever |x-x′|≤(1/2)max⁡{|x-y1|,…,|x-ym|}, and
(3)|K(x,y1,…,yi,…,ym)-K(x′,y1,…,yi,…,ym)|≤C|yi-yi′|ϵ(∑j=1m|x-yj|)mn+ϵ,
whenever |yi-yi′|≤(1/2)max⁡{|x-y1|,…,|x-ym|} for all 1≤i≤m. Then K is called m-linear Calderón-Zygmund kernel. In this paper, we consider an m-linear singular integral operator T associated with the kernel K, which is initially defined on product of the Schwartz space 𝒮(ℝn) and takes its values in the space of tempered distribution 𝒮′(ℝn) such that
(4)T(f1,…,fm)(x)=∫(ℝn)m‍K(x,y1,…,ym)f1(y1)⋯fm(ym)dy1⋯dym,
for x∉⋂j=1m‍supp⁡fj, where f1,…,fm∈LC∞(ℝn), the space of compactly supported bounded functions. If T is bounded from Lp1×⋯×Lpm to Lp with 1<p1⋯pm<∞ and 1/p1+⋯+1/pm=1/p, then we say that T is an m-linear Calderón-Zygmund operator. It has been proved in [4] that T is a bounded operator on product of Lebesgue spaces and endpoint weak estimates hold. For the boundedness of T and its commutators on the product of Herz-type spaces we refer the reader to see [5, 6] and [7], respectively.In the last few decades, however, a number of research papers have appeared in the literature which study the boundedness of integral operators, including the maximal function, singular operators, and fractional integral and commutators on function spaces with nonstandard growth conditions. Such kind of spaces is named as variable exponent function spaces which include variable exponent Lebesgue, Sobolev, Lorentz, Orlicz, and Herz-type function spaces. Among them the most fundamental and widely explored space is the Lebesgue spaceLp(x) with the exponent p depending on the point x of the space. We will describe it briefly in the next section; however, we refer to the book [8] and the survey paper [9] for historical background and recent developments in the theory of Lp(x) spaces. Despite the progress made, the problems of boundedness of multilinear singular integral operators and their commutators on Lp(x) spaces remain open. Recently, Huang and Xu [10] proved the boundedness of such integral operators on the product of variable exponent Lebesgue space. Motivated by their results, in this paper we will study the multilinear singular integral on Herz space with variable exponent. Similar results for the boundedness of commutators generated by these operators and BMO functions are also provided.Herz-type spaces are an important class of function spaces in harmonic analysis. In [11, 12], Izuki independently introduced Herz space with variable exponent p, by keeping the remaining two exponents α and q as constants. Variability of alpha was recently considered by Almeida and Drihem [13] in proving the boundedness results for some classical operators on such spaces. More recently, Samko [14] introduced the generalized Herz-type spaces where all the three exponents were allowed to vary. In this paper, we will study the multilinear singular operators on variable exponent Herz space K˙p(·)α,q introduced in [12].Throughout this paper,C denotes a positive constant which may change from one occurrence to another. The next section contains some basic definitions and the main results of this study. Finally, the last section includes the proofs of main results along with some supporting lemmas.


## 2. Main Results

LetΩ be a measurable subset of ℝn with Lebesgue measure |Ω|>0. Given a measurable function p(·):Ω→[1,∞], then for some λ>0 we define the variable exponent Lebesgue space as
(5)Lp(·)(Ω):={fismeasurable:∫Ω‍(|f(x)|λ)p(x)dx<∞},
and the space Lloc⁡p(·)(Ω) as
(6)Lloc⁡p(·)(Ω):={f:f∈Lp(·)(K)∀compactsubsetK⊂Ω}.
The Lebesgue space Lp(·)(Ω) becomes a Banach function space when equipped with the norm
(7)∥f∥Lp(·)(Ω):=inf⁡{λ>0:∫Ω‍(|f(x)|λ)p(x)dx≤1}.Given a locally integrable functionf on Ω, the Hardy-Littlewood maximal operator M is defined by
(8)Mf(x):=sup⁡r>0r-n∫B(x,r)∩Ω‍|f(y)|dy,
where B(x,r):={y∈ℝn:|x-y|<r}. We denote
(9)p-:=essinf⁡{p(x):x∈Ω},p+:=esssup⁡{p(x):x∈Ω}.
We also define
(10)𝒫(Ω):={p(·):p->1,p+<∞},ℬ(Ω):={Lp(·)p(·):p(·)∈𝒫(Ω),sssssssdsssMisboundedon⁡Lp(·)(Ω)}.Cruz-Uribe et al. [15] and Nekvinda [16] independently proved the following sufficient conditions for the boundedness of M on Lp(·)(Ω).Proposition 1.
LetΩ be an open set. If p(·)∈𝒫(Ω) satisfies
(11)|p(x)-p(y)|≤C-log⁡⁡(|x-y|),if|x-y|≤12,|p(x)-p(y)|≤Clog⁡⁡(e+|x|),if|y|≥|x|,
then one has p(·)∈ℬ(Ω).Define𝒫0(Ω) to be the set of measurable functions p:Ω→(0,∞) such that
(12)p-:=essinf⁡{p(x):x∈Ω}>0,p+:=esssup⁡{p(x):x∈Ω}<∞.
Given p∈𝒫0(Ω), one can define the space Lp(·) as above. Since we will not use it in the this paper, we omit the details here and refer the reader to see [10, 17].LetBk:={x:|x|≤2k}, Ak=Bk∖Bk-1 and χk=χAk be the characteristic function of the set Ak for k∈ℤ.Definition 2.
Forα∈ℝ, 0<q≤∞ and p(·)∈𝒫(ℝn), the homogeneous Herz space with variable exponent Kp(·)α,q(ℝn) is defined by
(13)K˙p(·)α,q(ℝn):={Lloc⁡p(·)(ℝn∖{0}):∥f∥K˙p(·)α,q(ℝn)<∞},
where
(14)∥f∥K˙p(·)α,q(ℝn):={∑k=-∞∞‍2kαq∥fχk∥Lp(·)(ℝn)q}1/q.In the sequel, unless stated otherwise, we will work on the whole spaceℝn and will not mention it. Taking b1 and b2 in BMO, Huang and Xu in [10] define the three commutator operators for suitable functions f and g. One of them is the operator
(15)[b1,b2,T](f,g)(x)=b1(x)b2(x)T(f,g)(x)-b1(x)T(f,b2g)(x)-b2(x)T(b1f,g)(x)+T(b1f,b2g)(x).
As corollaries of their main results they give the following estimates.Theorem A.
LetT be 2-linear Calderón-Zygmund operator and p(·)∈𝒫0(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 the functions f,fh∈Lp1(·), g,gh∈Lp2(·) and h∈ℕ such that
(16)∥T(f,g)∥Lp(·)≤C∥f∥Lp1(·)∥g∥Lp2(·),∥(∑h=1∞‍|T(fh,gh)|r)1/r∥Lp(·)≤C∥(∑h=1∞‍|fh|r1)1/r1∥Lp1(·)×∥(∑h=1∞‍|gh|r2)1/r2∥Lp2(·)
hold, where 1<rl<∞ for l=1,2 and 1/r=1/r1+1/r2.Theorem B.
LetT be 2-linear Calderón-Zygmund operator, b1,b2∈BMO(ℝn), and p(·)∈𝒫0(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 the functions f,fh∈Lp1(·), g,gh∈Lp2(·) and h∈ℕ such that
(17)∥[b1,b2,T](f,g)∥Lp(·)≤C∥b1∥*∥b2∥*∥f∥Lp1(·)∥g∥Lp2(·),∥(∑h=1∞‍|[b1,b2,T](fh,gh)|r)1/r∥Lp(·)≤∥(∑h=1∞‍|fh|r1)1/r1∥Lp1(·)∥(∑h=1∞‍|gh|r2)1/r2∥Lp2(·)×C∥b1∥*∥b2∥*
hold, where 1<rl<∞ for l=1,2 and 1/r=1/r1+1/r2.Motivated by these results, here we give the following two theorems.Theorem 3.
LetT be 2-linear Calderón-Zygmund operator and p(·)∈𝒫(ℝn). Furthermore, let p1(·),p2(·)∈ℬ(ℝn), 0<ql<∞, -nδpl<αl<nδpl′, l=1,2, where δpl,δpl′>0 are constants defined in the next section such that 1/p(x)=1/p1(x)+1/p2(x), 1/q=1/q1+1/q2, and α=α1+α2. If T is bounded from Lp1(·)×Lp2(·) to Lp(·), then
(18)∥T(f,g)∥K˙p(·)α,q≤C∥f∥K˙p1(·)α1,q1∥g∥K˙p2(·)α2,q2,∥(∑h=1∞‍|T(fh,gh)|r)1/r∥K˙p(·)α,q≤C∥(∑h=1∞‍|fh|r1)1/r1∥K˙p1(·)α1,q1×∥(∑h=1∞‍|gh|r2)1/r2∥K˙p2(·)α2,q2
hold for all f,fh∈K˙p1(·)α1,q1, g,gh∈K˙p2(·)α2,q2, where 1<rl<∞ for l=1,2 and 1/r=1/r1+1/r2.Theorem 4.
LetT be 2-linear Calderón-Zygmund operator, b1,b2∈BMO(ℝn), and p(·)∈𝒫(ℝn). Furthermore, let p1(·),p2(·)∈ℬ(ℝn), 0<ql<∞, -nδpl<αl<nδpl′, l=1,2, where δpl,δpl′>0 are constants defined in the next section, such that 1/p(x)=1/p1(x)+1/p2(x), 1/q=1/q1+1/q2, α=α1+α2. If [b1,b2,T] is bounded from Lp1(·)×Lp2(·) to Lp(·), then
(19)∥[b1,b2,T](f,g)∥K˙p(·)α,q≤C∥b1∥*∥b2∥*∥f∥K˙p1(·)α1,q1∥g∥K˙p2(·)α2,q2,∥(∑h=1∞‍|[b1,b2,T](fh,gh)|r)1/r∥K˙p(·)α,q≤∥(∑h=1∞‍|fh|r1)1/r1∥K˙p1(·)α1,q1∥(∑h=1∞‍|gh|r2)1/r2∥K˙p2(·)α2,q2×C∥b1∥*∥b2∥*
hold for all f,fh∈K˙p1(·)α1,q1, g,gh∈K˙p2(·)α2,q2, where 1<rl<∞ for l=1,2 and 1/r=1/r1+1/r2.


## 3. Proofs of the Main Results

In this section, we will prove main results stated in the last section. The ideas of these proofs mainly come from [5, 7]. We use the notation p′(x) to denote the conjugate index of p(x). Here we give some lemmas which will be helpful in proving Theorems 3 and 4.Lemma 5 (see [10, 18], generalized Hölder's inequality).
Letp,p1,p2∈𝒫(ℝn).(a)
Iff∈Lp(·), g∈Lp′(·), then one has
(20)∥fg∥L1≤Cp∥f∥Lp(·)∥g∥Lp′(·),
where Cp=1+1/p--1/p+.(b)
Iff∈Lp1(·), g∈Lp2(·) and 1/p(x)=1/p1(x)+1/p2(x), then there exists a constant Cp,p1 such that
(21)∥fg∥Lp(·)≤Cp,p1∥f∥Lp1(·)∥g∥Lp2(·)
holds, where Cp,p1=(1+1/(p1)--1/(p1)+)1/p-.Lemma 6 (see [12]).
Ifp∈ℬ(ℝn), then there exist a constant C>0 such that for all balls B in ℝn,
(22)∥χB∥Lp(·)∥χB∥Lp′(·)≤C|B|.Lemma 7 (see [12]).
Ifp∈ℬ(ℝn), then there exists constants δ,C>0 such that for all balls B in ℝn and all measurable subsets S⊂B,
(23)∥χB∥Lp(·)∥χS∥Lp(·)≤C|B||S|,∥χS∥Lp(·)∥χB∥Lp(·)≤C(|S||B|)δ.Lemma 8 (see [19, Remark 1]).
Ifpl∈ℬ(ℝn), l=1,2, then by Proposition 1 one has pl′∈ℬ(ℝn). Therefore, applying Lemma 7, one can take positive constants δpl,δpl′>0 such that
(24)∥χS∥Lpl(·)∥χB∥Lpl(·)≤C(|S||B|)δpl,∥χS∥Lpl′(·)∥χB∥Lpl′(·)≤C(|S||B|)δpl′,
for all balls B in ℝn and all measurable subsets S⊂B.Recently, Izuki [19] established a relationship between Lebesgue space with variable exponent and BMO space which can be stated in the form of the following lemma.Lemma 9.
One has that for allb∈BMO(ℝn) and all i,j∈ℤ with j>i,
(25)C-1∥b∥*≤sup⁡B:ball1∥χB∥Lp(·)∥(b-bB)χB∥Lp(·)≤C∥b∥*∥(b-bBi)χBj∥Lp(·)≤C(j-i)∥b∥*∥χBj∥Lp(·).The next lemma is the generalized Minkowski's inequality and is useful in proving vector valued inequalities.Lemma 10 (see [19]).
If1<r<∞, then there exists a constant C>0 such that for all sequences of functions {fh}h=1∞ satisfying ∥∥{fh}h∥ℓr∥L1ℝn<∞,
(26){∑h=1∞‍(∫ℝn‍|fh(y)|dy)r}1/r≤C∫ℝn‍{∑h=1∞‍|fh(y)|r}1/rdy.Proof of Theorem3.
In order to make computations easy, first we have to prove the following inequality:(27)2kα∥T(fχi,gχj)χk∥Lp(·)≤CD1(k,i)2iα1∥fχi∥Lp1(·)×D2(k,j)2jα2∥gχj∥Lp2(·),
where for k,i∈ℤ, l=1,2,
(28)Dl(k,i)={2(k-i)(αl-nδpl′),ifi≤k-2,1,ifk-1≤i≤k+1,2(k-i)(αl+nδpl),ifi≥k+2.
If k-1≤i≤k+1, k-1≤j≤k+1, then we have 2k~2i~2j; hence by the Lp(·) boundedness of T, we obtain
(29)2kα∥T(fχi,gχj)χk∥Lp(·)≤C2iα1∥fχi∥Lp1(·)2jα2×∥gχj∥Lp2(·).
In the other cases, we see that |x-y1|+|x-y2|~2max⁡(k,i,j), for x∈Ak, y1∈Ai, y2∈Aj. Thus by the generalized Hölder's inequality,
(30)|T(fχi,gχj)(x)χk(x)|≤C2-2max⁡⁡(k,i,j)n∥fχi∥1∥gχj∥1·χk(x)≤C2-2max⁡⁡(k,i,j)n∥fχi∥Lp1(·)∥χi∥Lp1′(·)×∥gχj∥Lp2(·)∥χj∥Lp2′(·)·χk(x).
Applying Lemma 5, we have
(31)∥T(fχi,gχj)χk∥Lp(·)≤C2-2max⁡⁡(k,i,j)n∥fχi∥Lp1(·)∥χi∥Lp1′(·)×∥gχj∥Lp2(·)∥χj∥Lp2′(·)∥χk∥Lp(·)≤C2-max⁡(k,i)n∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥χBk∥Lp1(·)×2-max⁡(k,j)n∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2(·).
Now, for i≤k-2, j≤k-2, by Lemmas 6 and 8, it is easy to show that
(32)∥T(fχi,gχj)χk∥Lp(·)≤C∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥χBk∥Lp1′(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2′(·)≤C∥fχi∥Lp1(·)2-(k-i)nδp1′∥gχj∥Lp2(·)2-(k-j)nδp2′.
By a similar argument for i≥k+2, j≥k+2, we get
(33)∥T(fχi,gχj)χk∥Lp(·)≤C∥fχi∥Lp1(·)∥χBk∥Lp1(·)∥χBi∥Lp1(·)∥gχj∥Lp2(·)∥χBk∥Lp2(·)∥χBj∥Lp2(·)≤C∥fχi∥Lp1(·)2(k-i)nδp1∥gχj∥Lp2(·)2(k-j)nδp2.
In view of (29)–(33), (27) is obvious. Now by Minkowski’s inequality and (27), we get
(34)2kα∥T(f,g)χk∥Lp(·)≤2kα∥∑i=-∞∞∑j=-∞∞‍T(fχi,gχj)χk∥Lp(·)≤∑i=-∞∞‍∑j=-∞∞‍2kα∥T(fχi,gχj)χk∥Lp(·)≤C∑i=-∞∞‍D1(k,i)2iα1∥fχi∥Lp1(·)×∑j=-∞∞‍D2(k,j)2jα2∥gχj∥Lp2(·).
Since 1/q=1/q1+1/q2, then by definition
(35)∥T(f,g)∥K˙p(·)α,q={∑k=-∞∞‍2kαq∥T(f,g)χk∥Lp(·)q}1/q≤{∑k=-∞∞(∑j=-∞∞‍)q∑i=-∞∞‍D1(k,i)2iα1∥fχi∥Lp1(·)dddddddddd×∑j=-∞∞‍D2(k,j)2jα2∥gχj∥Lp2(·))q}1/q≤C{∑k=-∞∞‍(∑i=-∞∞‍D1(k,i)2iα1∥fχi∥Lp1(·))q1}1/q1×{∑k=-∞∞‍(∑j=-∞∞‍D2(k,j)2jα2∥gχj∥Lp2(·))q2}1/q2:=I1×I2.
It remains to show thatI1≤C∥f∥K˙p1(·)α1,q1 and I2≤C∥g∥K˙p2(·)α2,q2. By symmetry, we only approximate I1. If 0<q1<1, then by the well-known inequality (∑|ai|)q1≤∑|ai|q1 and the inequality
(36)∑i=-∞∞‍D1(k,i)γ+∑j=-∞∞‍D2(k,j)γ<∞,foranyγ>0,
we have
(37)I1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1∑k=-∞∞‍D1(k,i)q1}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1}1/q1=C∥f∥K˙p1(·)α1,q1.
Ifq1>1, Hölder's inequality and inequality (36) yield
(38)I1≤C{∑k=-∞∞‍∑i=-∞∞‍{∑i=-∞∞‍}q1/q1′D1(k,i)q1/22iα1q1∥fχi∥Lp1(·)q1sss×{∑i=-∞∞‍D1(k,i)q1′/2}q1/q1′}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1∑k=-∞∞‍D1(k,i)q1/2}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1}1/q1=C∥f∥K˙p1(·)α1,q1.
Therefore, for 0<q1<∞(39)I1≤C∥f∥K˙p1(·)α1,q1.
By symmetry, for 0<q2<∞ we have
(40)I2≤C∥g∥K˙p2(·)α2,q2.
Finally, we obtain
(41)∥T(f,g)∥K˙p(·)α,q≤C∥f∥K˙p1(·)α1,q1∥g∥K˙p2(·)α2,q2.
By virtue of Lemma 10 and the fact that 1/r=1/r1+1/r2, it is easy to show that
(42)∥(∑h=1∞‍|T(fh,gh)|r)1/r∥K˙p(·)α,q≤C∥(∑h=1∞‍|fh|r1)1/r1∥K˙p1(·)α1,q1×∥(∑h=1∞‍|gh|r2)1/r2∥K˙p2(·)α2,q2
holds for all fh∈K˙p1(·)α1,q1, gh∈K˙p2(·)α2,q2, where 1<rl<∞ for l=1,2.
Thus the proof of Theorem3 is complete.Proof of Theorem4.
Similar to the proof of Theorem3, for the case k-1≤i≤k+1, k-1≤j≤k+1, we use Lp(·) boundedness of [b1,b2,T] to obtain
(43)2kα∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤C∥b1∥*∥b2∥*2iα1∥fχi∥Lp1(·)2jα2∥gχj∥Lp2(·).
For other possibilities we have|x-y1|+|x-y2|~2max⁡(k,i,j) for x∈Ak, y1∈Ai, y2∈Aj. Thus, we consider the following two cases.
Case I  (
i
≤
k
-
2, j≤k-2). We denote (bl)Bi by bli, where
(44)(bl)Bi=1|Bi|∫Bi‍bl(x)dx,l=1,2
and consider the following decomposition:
(45)[b1,b2,T](fχi,gχj)(x)=(b1(x)-b1i)(b2(x)-b2j)T(fχi,gχj)(x)-(b1(x)-b1i)T(fχi,(b2(·)-b2j)gχj)(x)-(b2(x)-b2j)T((b1(·)-b1i)fχi,gχj)(x)+T((b1(·)-b1i)fχi,(b2(·)-b2j)gχj)(x)=L1(x)+L2(x)+L3(x)+L4(x).
Thus,
(46)∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤∑m=14‍∥(Lm)χk∥Lp(·)=∑m=14‍Jm.
Now, we will estimate each Jm(m=1,…,4), separately. Applying Lemma 5, we have
(47)|L1(x)|≤C2-2kn|b1(x)-b1i||b2(x)-b2j|×∥fχi∥L1∥gχj∥L1≤C2-2kn∥fχi∥Lp1(·)∥χi∥Lp1′(·)∥gχj∥Lp2(·)∥χj∥Lp2′(·)×|b1(x)-b1i||b2(x)-b2j|.
Therefore, by virtue of generalized Hölder's inequality and Lemmas 6, 8, and 9, we get
(48)J1=∥(L1)χk∥Lp(·)≤C2-2kn∥fχi∥Lp1(·)∥χi∥Lp1′(·)∥gχj∥Lp2(·)∥χj∥Lp2′(·)×∥(b1(x)-b1i)(b2(x)-b2j)χk∥Lp(·)≤C2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1′(·)×∥(b1(x)-b1i)χBk∥Lp1(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)×∥(b2(x)-b2j)χBk∥Lp2(·)≤C∥b1∥*∥b2∥*2-2kn(k-i)∥fχi∥Lp1(·)×∥χBi∥Lp1′(·)∥χBk∥Lp1(·)×(k-j)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2(·)≤C∥b1∥*∥b2∥*(k-i)∥fχi∥Lp1(·)×∥χBi∥Lp1′(·)∥χBk∥Lp1′(·)(k-j)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2′(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)(k-i)2-(k-i)nδp1′×∥gχj∥Lp2(·)(k-j)2-(k-j)nδp2′.
Similarly, using Lemma 9, we approximate L2 as
(49)|L2|≤C2-2kn|b1(x)-b1i|∥fχi∥L1∥(b2(·)-b2j)gχj∥L1≤C2-2kn|b1(x)-b1i|∥fχi∥Lp1(·)∥χi∥Lp1′(·)×∥gχj∥Lp2(·)∥(b2(·)-b2j)χj∥Lp2′(·)≤C∥b2∥*2-2kn|b1(x)-b1i|∥fχi∥Lp1(·)×∥χBi∥Lp1′(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·).
Therefore, in view of Lemmas 6, 9, and 8, we have
(50)J2=∥(L2)χk∥Lp(·)≤C∥b2∥*2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1′(·)×∥(b1(x)-b1i)χk∥Lp(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1′(·)×(k-i)∥χBk∥Lp(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*(k-i)∥fχi∥Lp1(·)×∥χBi∥Lp1′(·)∥χBk∥Lp1′(·)∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2′(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)(k-i)2-(k-i)nδp1′×∥gχj∥Lp2(·)2-(k-j)nδp2′.
By symmetry, the estimate for J3 is similar to that for J2; therefore,
(51)J3≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)2-(k-i)nδp1′×∥gχj∥Lp2(·)(k-j)2-(k-j)nδp2′.
Finally, it remains to estimate J4. For that, we use Lemma 9 to write
(52)|L4|=|T((b1(·)-b1i)fχi,(b2(·)-b2j)gχj)(x)|≤C2-2kn∥(b1(·)-b1i)fχi∥L1∥(b2(·)-b2j)gχj∥L1≤C2-2kn∥fχi∥Lp1(·)∥(b1(·)-b1i)χi∥Lp1′(·)∥gχj∥Lp2(·)×∥(b2(·)-b2j)χj∥Lp2′(·)≤C2-2kn∥fχi∥Lp1(·)∥(b1(·)-b1i)χBi∥Lp1′(·)∥gχj∥Lp2(·)×∥(b2(·)-b2j)χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥gχj∥Lp2(·)×∥χBj∥Lp2′(·).
Thus, by Hölder's inequality and Lemmas 6 and 8, we obtain
(53)J4=∥(L4)χk∥Lp(·)≤C∥b1∥*∥b2∥*2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1(·)∥gχj∥Lp2(·)×∥χBj∥Lp2(·)∥χk∥Lp(·)≤C∥b1∥*∥b2∥*2-2kn∥fχi∥Lp1(·)∥χBi∥Lp1(·)∥χBk∥Lp1(·)×∥gχj∥Lp2(·)∥χBj∥Lp2(·)∥χBk∥Lp2(·)≤C∥b1∥*∥b2∥*(k-i)∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥χBk∥Lp1′(·)×∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2′(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)2-(k-i)nδp1′∥gχj∥Lp2(·)2-(k-j)nδp2′.
Combining the estimates for J1, J2, J3, and J4, we have
(54)∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)×(k-i)2-(k-i)nδp1′∥gχj∥Lp2(·)×(k-j)2-(k-j)nδp2′.
Case II
(
i
≥
k
+
2, j≥k+2). We denote (bl)Bk by blk, where (bl)Bk=(1/|Bk|)∫Bkbl(x)dx,l=1,2. In this case, we consider the following decomposition:
(55)[b1,b2,T](fχi,gχj)(x)=(b1(x)-b1k)(b2(x)-b2k)T(fχi,gχj)(x)-(b1(x)-b1k)T(fχi,(b2(·)-b2k)gχj)(x)-(b2(x)-b2k)T((b1(·)-b1k)fχi,gχj)(x)+T((b1(·)-b1k)fχi,(b2(·)-b2k)gχj)(x)=L~1(x)+L~2(x)+L~3(x)+L~4(x).
Thus,
(56)∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤∑m=14‍∥(L~m)χk∥Lp(·)=∑m=14‍J~m.
Let us first compute J~1. As in the proof of Theorem 3, in this case we estimate L~1 as
(57)|L~1|≤C|b1(x)-b1k||b2(x)-b2k|2-in∥fχi∥L12-jn×∥gχj∥L1≤C|b1(x)-b1k||b2(x)-b2k|2-in∥fχi∥Lp1(·)×∥χi∥Lp1′(·)2-jn∥gχj∥Lp2(·)∥χj∥Lp2′(·).
By virtue of generalized Hölder's inequality and Lemmas 6–9, we obtain
(58)J~1=∥(L~1)χk∥Lp(·)≤C2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)×∥(b1(x)-b1k)(b2(x)-b2k)χk∥Lp(·)≤C2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥(b1(x)-b1k)χk∥Lp1(·)×2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥(b2(x)-b2k)χk∥Lp2(·)≤C∥b1∥*∥b2∥*2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥χBk∥Lp1(·)×2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)∥χBk∥Lp2(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)∥χBk∥Lp1(·)∥χBi∥Lp1(·)∥gχj∥Lp2(·)∥χBk∥Lp2(·)∥χBj∥Lp2(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)2(k-i)nδp1∥gχj∥Lp2(·)2(k-j)nδp2.
Next, we approximate J~2. Using Lemma 9, it is easy to see that
(59)|L~2|≤C2-2in|b1(x)-b1k|∥fχi∥L1∥(b2(·)-b2k)gχj∥L1≤C2-in|b1(x)-b1k|∥fχi∥Lp1(·)∥χi∥Lp1′(·)×2-jn∥gχj∥Lp2(·)∥(b2(·)-b2k)χj∥Lp2′(·)≤C∥b2∥*2-in|b1(x)-b1k|∥fχi∥Lp1(·)∥χBi∥Lp1′(·)×(j-k)2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·).
Thus, in view of Lemmas 6, 9, and 8, we get
(60)J~2=∥(L~2)χk∥Lp(·)≤C∥b2∥*2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥(b1(x)-b1k)χk∥Lp(·)×(j-k)2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)∥χBk∥Lp(·)×(j-k)2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)∥χBk∥Lp1(·)∥χBi∥Lp1(·)(j-k)×∥gχj∥Lp2(·)∥χBk∥Lp2(·)∥χBj∥Lp2(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)2(k-i)nδp1∥gχj∥Lp2(·)×(j-k)2(k-j)nδp2.
By symmetry, the estimate for J~3 is similar to that for J~2; thus
(61)J~3≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)(i-k)2(k-i)nδp1×∥gχj∥Lp2(·)2(k-j)nδp2.
Lastly, it remains to compute J~4. For this purpose we use generalized Höder's inequality and lemmas 6 and 9 to obtain
(62)|L~4|=|T((b1(·)-b1k)fχi,(b2(·)-b2k)gχj)(x)|≤C2-in∥(b1(·)-b1k)fχi∥L12-jn×∥(b2(·)-b2k)gχj∥L1≤C2-in∥fχi∥Lp1(·)∥(b1(·)-b1k)χi∥Lp1′(·)×2-jn∥gχj∥Lp2(·)∥(b2(·)-b2k)χj∥Lp2′(·)≤C∥b1∥*∥b2∥*(i-k)2-in∥fχi∥Lp1(·)∥χBi∥Lp1′(·)×(j-k)2-jn∥gχj∥Lp2(·)∥χBj∥Lp2′(·)≤C∥b1∥*∥b2∥*(i-k)∥fχi∥Lp1(·)∥χBi∥Lp1(·)(j-k)∥gχj∥Lp2(·)∥χBj∥Lp2(·).
Thus, an application of Lemma 8 yields
(63)J~4=∥(L~4)χk∥Lp(·)≤C∥b1∥*∥b2∥*(i-k)∥fχi∥Lp1(·)∥χi∥Lp1(·)×(j-k)∥gχj∥Lp2(·)∥χj∥Lp2(·)∥χk∥Lp(·)≤C∥b1∥*∥b2∥*(i-k)∥fχi∥Lp1(·)∥χBk∥Lp1(·)∥χBi∥Lp1(·)(j-k)×∥gχj∥Lp2(·)∥χBk∥Lp2(·)∥χBj∥Lp2(·)≤C∥b1∥*∥b2∥*(i-k)∥fχi∥Lp1(·)2(k-i)nδp1×(j-k)∥gχj∥Lp2(·)2(k-j)nδp2.
Combining the estimates for J~1, J~2, J~3, and J~4, we have
(64)∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤C∥b1∥*∥b2∥*∥fχi∥Lp1(·)(i-k)2(k-i)nδp1×∥gχj∥Lp2(·)(j-k)2(k-j)nδp2.
In view of (43), (54), and (64), we arrive at
(65)2kα∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤C∥b1∥*∥b2∥*E1(k,i)2iα1∥fχi∥Lp1(·)×E2(k,j)2jα2∥gχj∥Lp2(·),
where for k,i∈ℤ, l=1,2,
(66)El(k,i)={(k-i)2(k-i)(αl-nδpl′),ifi≤k-2,1,ifk-1≤i≤k+1,(i-k)2(k-i)(αl+nδpl),ifi≥k+2.
Under the assumption -nδpl<αl<nδpl′, it is easy to see that
(67)∑i=-∞∞‍E1(k,i)γ+∑j=-∞∞‍E2(k,j)γ<∞,foranyγ>0.
Now by the Minkowski's inequality and (65), we get
(68)2kα∥[b1,b2,T](f,g)χk∥Lp(·)≤2kα∥∑i=-∞∞‍∑j=-∞∞‍[b1,b2,T](fχi,gχj)χk∥Lp(·)≤∑i=-∞∞‍∑j=-∞∞‍2kα∥[b1,b2,T](fχi,gχj)χk∥Lp(·)≤C∥b1∥*∥b2∥*∑i=-∞∞‍E1(k,i)2iα1∥fχi∥Lp1(·)×∑j=-∞∞‍E2(k,j)2jα2∥gχj∥Lp2(·).
Finally, by definition and the fact that 1/q=1/q1+1/q2, we obtain
(69)∥[b1,b2,T](f,g)∥K˙p(·)α,q={∑k=-∞∞‍2kαq∥[b1,b2,T](f,g)χk∥Lp(·)q}1/q≤C∥b1∥*∥b2∥*×{∑k=-∞∞‍{∑i=-∞∞‍E1(k,i)2iα1∥fχi∥Lp1(·)}q1}1/q1×{∑k=-∞∞‍{∑j=-∞∞‍E2(k,j)2jα2∥gχj∥Lp2(·)}q2}1/q2:=∥b1∥*∥b2∥*(I~1×I~2).
It is enough to show that I~1≤C∥f∥K˙p1(·)α1,q1 and I~2≤C∥g∥K˙p2(·)α2,q2. By symmetry, we only give estimates for I~1. For 0<q1<∞, by inequality (∑|ai|)q1≤∑|ai|q1 and inequality (67), we have
(70)I~1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1∑k=-∞∞‍E1(k,i)q1}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1}1/q1=C∥f∥K˙p1(·)α1,q1.
For q1>1, by Hölder's inequality and inequality (67), we obtain
(71)I~1≤C{{∑i=-∞∞‍}q1/q1′∑k=-∞∞‍∑i=-∞∞‍E1(k,i)q1/22iα1q1∥fχi∥Lp1(·)q1ddddddddd×{∑i=-∞∞‍E1(k,i)q1′/2}q1/q1′}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1∑k=-∞∞‍E1(k,i)q1/2}1/q1≤C{∑i=-∞∞‍2iα1q1∥fχi∥Lp1(·)q1}1/q1=C∥f∥K˙p1(·)α1,q1.
Hence, for 0<q1<∞(72)I~1≤C∥f∥K˙p1(·)α1,q1.
By symmetry, for 0<q2<∞, we have
(73)I~2≤C∥g∥K˙p2(·)α2,q2.
Therefore,
(74)∥[b1,b2,T](f,g)∥K˙p(·)α,q≤C∥b1∥*∥b2∥*∥f∥K˙p1(·)α1,q1∥g∥K˙p2(·)α2,q2.
By a similar procedure one can prove that
(75)∥(∑h=1∞‍|[b1,b2,T](fh,gh)|r)1/r∥K˙p(·)α,q≤∥(∑h=1∞‍|fh|r1)1/r1∥K˙p1(·)α1,q1∥(∑h=1∞‍|gh|r2)1/r2∥K˙p2(·)α2,q2×C∥b1∥*∥b2∥*
holds for all fh∈K˙p1(·)α1,q1, gh∈K˙p2(·)α2,q2, where 1<rl<∞ for l=1,2 and 1/r=1/r1+1/r2. Thus, we finish the proof of Theorem 4.Remark 11.
Note that whenαl≥nδpl′, then the results of Theorems 3 and 4 are no longer true. It needs to replace the variable exponent Herz space K˙p(·)α,q(ℝn) with HK˙p(·)α,q(ℝn) and the Herz-type Hardy spaces with the variable exponent recently introduced in [20]. The boundedness of multilinear singular integral operators from HK˙p(·)α,q(ℝn) to K˙p(·)α,q(ℝn) is still an interesting question that needs to be answered.Remark 12.
Although we considered the 2-linear case, the method can be extended for anym-linear case without any essential difficulty.



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