# Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

Authors: Shengrong Wang; Jingshi Xu
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.


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## Body

## 1. Introduction

Denote byLloc1(Rn) the set of all complex-valued locally integrable functions on Rn. The Hardy operator was first considered in [1] as follows: (1)Hfx≔1x∫0xftdt,x≠0,for f∈Lloc1(R). In 1976, Faris [2] generalized it to the n dimensional Euclidean space as (2)Hfx≔1Ωnxn∫y<xfydy,x∈Rn∖0,where f∈Lloc1(Rn) and Ωn=πn/2/Γ(1+n/2) is the volume of the unit ball in Rn. In [3], Christ and Grafakos proved that the Hardy operator is bounded on Lp(Rn).Form∈N, the m-linear Hardy operator was defined in [4] as (3)Hf1,⋯fm≔1Ωmnxmn∫y1,⋯,ym<x∏i=1mfiyidy1⋯dym,x∈Rn∖0,for f1,…,fm in Lloc1(Rn). The 2-linear operator will be referred to as bilinear operator.A functionb belongs to BMO(Rn) (bounded mean oscillation), if b∈Lloc1(Rn) and (4)b∗≔supB1B∫Bbx-bBdx<∞,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.Letbi∈BMO(Rn) for i=1,…,m. Then the commutator generated by the m-linear Hardy operator and b1,…,bm is defined by (5)Hb→f1,…,fmx=∑i=1mHbiif1,…,fmx,where (6)Hbiif1,…,fmx=bixHf1,…,fmx-Hf1,…,fi-1,fibi,fi+1,…,fmx.When m=1, the operator is Hb(f)(x)=b(x)Hf(x)-H(bf)(x).Since the fundamental paper [5] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have attracted a great attention. Many constant exponent function spaces are generalized to variable exponent setting, such as Bessel potential spaces with a variable exponent [6], Besov and Trieble-Lizorkin spaces with variable exponents [7–12], Hardy spaces with variable exponents [13], Morrey spaces with variable exponents [14], Herz spaces with variable exponents [15], and Herz-Morrey spaces with variable exponents [16, 17]. These spaces have many applications, such as in the electrorheological fluid [18]; image restoration [19–23]; the Black-Scholes equation [24]; ordinary and partial differential equations [25–27] and references therein.The boundedness of the Hardy operator was considered in many function spaces, such as in variable Lebesgue spaces [28, 29], variable exponent Sobolev spaces [30]. The boundeness of commutators of the Hardy operator was obtained in λ-central BMO spaces [31], Herz spaces K˙p(·)α,q(Rn) with variable exponent p(·) [32]. In [33], Wu proved the boundedness of multilinear commutators of fractional Hardy operators on Herz-Morrey spaces MK˙p,qα,λ(Rn). In [34], Wu considered the boundedness for fractional Hardy type operator on Herz-Morrey spaces MK˙p,q(·)α,λ(Rn) with variable exponent q(·) but fixed α∈R and p∈(0,∞). Wu and Zhang obtained the boundedness of multilinear Hardy type operators on the product of Herz-Morrey spaces MK˙p,q(·)α,λ(Rn) with variable exponent q(·) and the boundedness of fractional Hardy type operators on Herz-Morrey spaces MK˙p,q(·)α,λ(Rn) with variable exponent q(·) in [35, 36]. Wu and Zhang considered the boundedness of commutators of the fractional Hardy operators on Herz-Morrey spaces MK˙p,q(·)α,λ(Rn) with variable exponent q(·) in [17]. Shu, Wang and Meng obtained the boundedness of commutators of Hardy type operators on Herz spaces K˙p(·),qα(·) with variable exponents α(·) and p(·) in [37]. Wu and Zhao proved the boundedness for variable fractional Hardy type operator on variable exponent Herz-Morrey spaces MK˙p,q(·)α(·),λ(Rn) in [38]. In [39], Xu and Yang introduced the Herz-Morrey-Hardy spaces HMK˙p(·),λα(·),q with variable exponents and established their characterization in terms of atom. Moreover, applying the characterization, they obtained the boundedness of some singular integral operators on these spaces.Motivated by the mentioned works, in this paper, we will consider the boundedness ofm-linear commutators generated by the m-linear Hardy operator and BMO functions on Herz spaces and Herz-Morrey spaces with variable exponents.


## 2. Main Results

To state our results, let us first recall some definitions and notations. LetΩ be a positive measurable subset of Rn, given a measurable function p(·):Ω→[1,∞). The Lebesgue space with variable exponent Lp(·)(Ω) is defined by (7)Lp·Ω≔fismeasurable:∫Rnfxλpxdx<∞forsomeλ>0.The Lebesgue space Lp(·)(Ω) becomes a Banach function space equipped with the norm (8)fLp·≔inf⁡λ>0:∫Ωfxλpxdx≤1.The space Llocp(·)(Rn) is the collection of all functions f such that fχK∈Lp(·)(Rn) for each compact subset K⊂Rn. Here and what follows, χS denotes the characteristic function of a measurable set S⊂Rn. Let p(·):Rn→(0,∞); we denote p-≔essinfx∈Rnp(x), p+≔esssupx∈Rnp(x). The set P(Rn) consists of all p(·) such that p->1 and p+<∞; P0(Rn) consists of all p(·) such that p->0 and p+<∞. If p(·)∈P0(Rn), then the space Lp(·) is similarly defined as above. p′(·) means that the conjugate exponent of p(·) that means 1/p(·)+1/p′(·)=1.The standard Hardy-Littlewood maximal operatorM is defined for Lloc1(Rn) function f by (9)Mfx=supB∋x1B∫Bfydy,∀x∈Rn,where B is a ball. We denote by B(Rn) the set of p(·)∈P(Rn) such that M is bounded on Lp(·)(Rn). The common sufficient conditions for variable exponent to be in B(Rn) are the following well known log-Hölder continuity, which introduced in [40–42].Definition 1.
Letα(·) be a real-valued measurable function on Rn.
(i) If there exists a constantC1 such that (10)αx-αy≤C1log⁡e+1/x-y,x,y∈Rn,x-y<12,then the function α(·) is called locally log-Hölder continuous.
(ii) If there exists a constantC2 such that (11)αx-α0≤C2log⁡e+1/x,∀x∈Rn,then the function α(·) is called log-Hölder continuous at the origin and denoted by α(·)∈P0log(Rn);
(iii) If there existα∞∈R and a positive constant C3 such that (12)αx-α∞≤C3log⁡e+x,∀x∈Rn,then the function α(·) is called log-Hölder continuous at infinity and denoted by α(·)∈P∞log(Rn).
(iv) Ifα(·) are both locally log-Hölder continuous and log-Hölder continuous at the infinity, then the function α(·) is called global log-Hölder continuous and denoted by α(·)∈Plog(Rn).To state the definitions of Herz space and Herz-Morrey space with variable exponents, we use the following notations. For eachk∈Z we denote Bk≔x∈Rn:x≤2k,Dk≔Bk∖Bk-1,χk≔χDk,χ~m=χm,m≥1,χ~0=χB0.Definition 2.
Let0<q≤∞, p(·)∈P(Rn), and α(·):Rn→R with α∈L∞(Rn).
(i) The homogeneous Herz spaceK˙p(·)α(·),q(Rn) is defined by (13)K˙p·α·,qRn≔f∈Llocp·Rn∖0:fK˙p·α·,q<∞,where (14)fK˙p·α·,q≔∑k=-∞∞2kα·fχkLp·q1/q<∞.
(ii) The inhomogeneous Herz spaceKp(·)α(·),q(Rn) is defined by (15)Kp·α·,qRn≔f∈Llocp·Rn:fKp·α·,q<∞,where (16)fKp·α·,q≔∑m=0∞2mα·fχ~mLp·q1/q<∞.Remark 3.
Obviously, if0<q1≤q2≤∞, then K˙p(·)α(·),q1(Rn)⊂K˙p(·)α(·),q2(Rn). If both α(·) and p(·) are constants, then K˙p(·)α(·),q(Rn)=K˙pα,q(Rn) is the classical Herz spaces in [43].To generalize the above spaces to variable exponentq(·), we need the notation of the variable mixed sequence space lq(·)(Lp(·)), which is defined as follows. Given a sequence of functions {fj}j∈Z, define the modular (17)ρlq·Lp·fjj≔∑j∈Zinf⁡λj>0:ρp·λj-1/q·fj≤1,where λ1/∞=1. If q+<∞ or q(·)≤p(·), the above can be written as (18)ρlq·Lp·fjj=∑v∈Zfj·q·Lp·/q·.The norm is (19)fjjlq·Lp·≔inf⁡μ>0:ρlq·Lp·fjμj≤1.Now, the spaceK˙p(·)α(·),q(·) is the collection of all functions f∈Llocp·(Rn∖0) such that (20)fK˙p·α·,q·Rn≔2α·kfχkk∈Zlq·Lp·<∞,and the space Kp(·)α(·),q(·) is the collection of all functions in f∈Llocp(·)(Rn) such that (21)fKp·α·,q·Rn≔2α·kfχ~kk∈N0lq·Lp·<∞.In [44], Drihem and Seghiri proved the following result.Lemma 4 (see [44, Proposition 1]).
Letα(·)∈L∞(Rn),p(·),q(·)∈P0(Rn). If α(·) and q(·) are log-Hölder continuous at infinity, then (22)Kp·α·,q·Rn=Kp·α∞,q∞Rn.Additionally, if α(·) and q(·) have a log decay at the origin, then (23)fK˙p·α·,q·≈∑k≤02kα0fχkLp·q01/q0+∑k>02kα∞fχkLp·q∞1/q∞.Definition 5.
Let0<q≤∞, p(·)∈P(Rn),0≤λ<∞, and α(·):Rn→R with α∈L∞(Rn).
(i) The homogeneous Herz-Morrey spaceMK˙p(·),λα(·),q(Rn) is defined by (24)MK˙p·,λα·,qRn≔f∈Llocp·Rn∖0:fMKp·,λα·,q<∞,where (25)fMK˙p·,λα·,q≔supL∈Z⁡2-Lλ∑k=-∞L2kα·fχkLp·q1/q<∞.
(ii) The inhomogeneous Herz spaceMKp(·),λα(·),q(Rn) is defined by (26)MKp·,λα·,qRn≔f∈Llocp·Rn∖0:fMKp·,λα·,q<∞,where (27)fMKp·,λα·,qRn≔supL∈N⁡2-Lλ∑m=-∞L2kα·fχ~mLp·q1/q<∞.Remark 6.
Ifα(·) is constant, then MK˙p(·),λα(·),q(Rn)=MK˙p(·),λα,q(Rn) was defined in [45]. If λ=0, then MK˙p(·),λα(·),q(Rn)=K˙p(·)α(·),q(Rn). If both α(·) and p(·) are constants and λ=0, then MK˙p(·),λα(·),q(Rn)=K˙pα,q(Rn) is the classical Herz space in [43].Definition 7.
Letp(·)∈P(Rn),q(·)∈P0log(Rn)∩P∞log(Rn),0≤λ<∞. Let α(·) be a bounded real-valued measurable function on Rn. The homogeneous Herz-Morrey space MK˙p(·),λα(·),q(·) and nonhomogeneous Herz-Morrey space MKp(·),λα(·),q(·) are defined, respectively, by (28)MK˙p·,λα·,q·≔f∈Llocp·Rn∖0:fMK˙p·,λα·,q·<∞,and (29)MKp·,λα·,q·≔f∈Llocp·Rn:fMKp·,λα·,q·<∞,where (30)fMK˙p·,λα·,q·≔supL∈Z⁡2-Lλ2α·kfχkk≤Llq·Lp·,and (31)fMKp·,λα·,q·≔supL∈N0⁡2-Lλ2α·kfχ~kk=0Llq·Lp·.By the previous definition, we obtain the following proposition.Proposition 8.
Letp(·),q(·)∈P0(Rn), λ∈[0,∞), and α(·)∈L∞(Rn).
(i) Ifα(·), q(·)∈P0log(Rn)∩P∞log(Rn), then for any f∈Llocp(·)(Rn∖0), (32)fMK˙p·,λα·,q·≈max⁡supL⩽0,L∈Z⁡2-Lλ2kα0fχkk≤Llq0Lp·,supL>0,L∈Z⁡2-Lλ2kα0fχkk<0lq0Lp·+2-Lλ2kα∞fχkk=0Llq∞Lp·,
(ii) Ifα(·), q(·)∈P∞log(Rn), then (33)MKp·,λα·,q·=MKp·,λα∞,q∞.Proof.
Obviously,(34)fMK˙p·,λα·,q·=max⁡supL⩽0,L∈Z⁡2-Lλ2kα·fχkk≤Llq·Lp·,supL>0,L∈Z⁡2-Lλ2kα·fχkk≤Llq·Lp·.When L≤0, from Lemma 4 we know that (35)2kα·fχkk≤Llq·Lp·≈2kα0fχkk≤Llq0Lp·.When L>0, from Lemma 4 again we also obtain that (36)2kα·fχkk<Llq·Lp·≈2kα0fχkk<0lq0Lp·+2kα∞fχkk=0Llq∞Lp·.Thus we proved (i). The proof of (ii) is similar.Lemma 9 (see [46, Proposition 2]).
Let0<q≤∞, p(·)∈P(Rn), and α(·)∈L∞(Rn). If α(·) is log-Hölder continuous both at the origin and at infinity, then (37)fMK˙p·,λα·,q≈max⁡supL≤0,L∈Z⁡2-Lλ∑k=-∞L2kα0qfχkLp·q1/qsupL>0,L∈Z⁡2-Lλ∑k=-∞-12kα0qfχkLp·q1/q+2-Lλ∑k=0L2kα∞qfχkLp·q1/q.Lemma 10 (see [45, Lemma 1 and (10)]).
Supposep(·)∈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, (38)χSLp·χBLp·≤CSBδ1,(39)χSLp′·χBLp′·≤CSBδ2.The following is our main results. Although the results are also hold for all multilinear cases, for brevity, we only consider the bilinear case.Theorem 11.
LetH be a bilinear Hardy operator; p1(·), p2(·)∈B(Rn) satisfying 1/p(·)=1/p1(·)+1/p2(·), α(·)∈L∞(Rn)∩P0log(Rn)∩P∞log(Rn),α(0)=α1(0)+α2(0), α∞=α1∞+α2∞, q(·)∈P0log(Rn)∩P∞log(Rn), 1/q(0)=1/q1(0)+1/q2(0), 1/q∞=1/q1∞+1/q2∞, λ=λ1+λ2, 0≤λ1,λ2<∞, δj1, δj2∈(0,1) are the constants in Lemma 10 for exponents pj(·). If for j=1, 2 with λj+nδj2>αj∞≥αj(0), then Hb→1 is bounded from MK˙p1(·),λ1α1(·),q1(·)(Rn)×MK˙p2(·),λ2α2(·),q2(·)(Rn) to MK˙p(·),λα(·),q(·)(Rn), where b→=(b1,b2),b1,b2∈BMO.Corollary 12.
LetH be a bilinear Hardy operator; p1(·), p2(·)∈B(Rn) satisfying 1/p(·)=1/p1(·)+1/p2(·), 0<q≤∞, α(·)∈L∞(Rn)∩P0log(Rn)∩P∞log(Rn),α(0)=α1(0)+α2(0), α∞=α1∞+α2∞, 1/q=1/q1+1/q2, δj1, δj2∈(0,1) are the constants in Lemma 10 for exponents pj(·). If for j=1,2 with nδj2>αj∞≥αj(0), then Hb→1 is bounded from K˙p1(·)α1(·),q1(Rn)×K˙p2(·)α2(·),q2(Rn) to K˙p(·)α(·),q(Rn), where b→=(b1,b2),b1,b2∈BMO.Corollary 13.
LetH be a bilinear Hardy operator; p1(·), p2(·)∈B(Rn) satisfying 1/p(·)=1/p1(·)+1/p2(·), α(·)∈L∞(Rn)∩P0log(Rn)∩P∞log(Rn),α(0)=α1(0)+α2(0), α∞=α1∞+α2∞, q(·)∈P0log(Rn)∩P∞log(Rn), 1/q(0)=1/q1(0)+1/q2(0), 1/q∞=1/q1∞+1/q2∞, δj1, δj2∈(0,1) are the constants in Lemma 10 for exponents pj(·). If for j=1,2 with nδj2>αj∞≥αj(0), then Hb→1 is bounded from K˙p1(·)α1(·),q1(·)(Rn)×K˙p2(·)α2(·),q2(·)(Rn) to K˙p(·)α(·),q(·)(Rn), where b→=(b1,b2),b1,b2∈BMO.Corollary 14.
LetH be a bilinear Hardy operator; p1(·), p2(·)∈B(Rn) satisfying 1/p(·)=1/p1(·)+1/p2(·), 0<q≤∞, α(·)∈L∞(Rn)∩P0log(Rn)∩P∞log(Rn),α(0)=α1(0)+α2(0), α∞=α1∞+α2∞, 1/q=1/q1+1/q2, λ=λ1+λ2, 0≤λ1,λ2<∞, δj1, δj2∈(0,1) are the constants in Lemma 10 for exponents pj(·). If for j=1,2 with λj+nδj2>αj∞≥αj(0), then Hb→1 is bounded from MK˙p1(·),λ1α1(·),q1(Rn)×MK˙p2(·),λ2α2(·),q2(Rn) to MK˙p(·),λα(·),q(Rn), where b→=(b1,b2),b1,b2∈BMO.


## 3. Proof of Theorem11

Since Corollaries12–14 are special case of Theorem 11, we only need to prove Theorem 11.To proceed, we need the following lemmas.Lemma 15 (see [45, Lemma 2]).
Ifp(·)∈B(Rn), then there exists a constant C>0 such that, for all balls B in Rn, (40)C-1≤1BχBLp·χBLp′·≤C.Lemma 16 (see [5, Theorem 2.1]).
Letp(·)∈P(Rn). Then for every f∈Lp(·)(Rn) and every g∈Lp′(·)(Rn), (41)∫Rnfxgxdx≤rpfLp·gLp′·,where rp=1+1/p+-1/p-.Lemma 17 (see [47, Theorem 2.3]).
Letp,p1,p2∈P0(Rn) such that 1/p(x)=1/p1(x)+p2(x). Then there exists a constant C depending only on p,p1 such that (42)fgLp·≤CfLp1·gLp2·holds for every f∈Lp1(·)(Rn) and g∈Lp2(·)(Rn).Lemma 18 (see [32, Theorem 3]).
Letp(·)∈B(Rn), t∈N and k>i(k,i∈N), then there exists a positive constant C such that, for b∈BMO(Rn), (43)C-1b∗t≤supB⊂Rn⁡1χBLp·Rnb-bBtχBLp·Rn≤Cb∗t,b-bBitχBkLp·Rn≤Ck-itb∗tχBkLp·Rn.Indeed, ift=1, then for all balls B and k,i∈Z with k>i, we have (44)supB⊂Rn⁡1χBLp·Rnb-bBχBLp·Rn≤Cb∗,b-bBiχBkLp·Rn≤Ck-ib∗χBkLp·Rn.From the definition of BMO, it is easy to know that (45)bBk-bBi≤k-ib∗fork>i.Lemma 19 (see [48, Proposition 1.2]).
Let0<p≤∞,δ>0. Then we have (46)∑j=-∞∞∑k=-∞∞2-k-jδakp1/p≤C∑j=-∞∞ajp1/pfor nonnegative sequences {aj}j=-∞∞. Here, when p=∞, it is understood that (46) stands for (47)supj∈Z⁡∑k=-∞∞2-k-jδak≤Csupj∈Z⁡aj.Proof of Theorem11.
Letf1∈MK˙p1(·),λ1α1(·),q1(·)(Rn), f2∈MK˙p2(·),λ2α2(·),q2(·)(Rn). Let (bi)Bk express the average of bi on the ball Bk for i=1,2, and k∈Z. If x∈Dk, then (48)Hb11f1,f2x≲2-2kn∫y1,y2<xf1y1f2y2b1x-b1y1dy1dy2≲2-2kn∫y1,y2<xf1y1f2y2b1x-b1Bkdy1dy2+2-2kn∫y1,y2<xf1y1f2y2b1y1-b1Bkdy1dy2≕I+II.Since 1/p=1/p1+1/p2, 1/p1+1/p1′=1 and 1/p2+1/p2′=1. Using Hölder’s inequality, we have (49)I≲2-2kn∫Bk∫Bkf1y1f2y2b1x-b1Bkdy1dy2≲2-2knb1x-b1Bk∫Bk∫Bkf1y1f2y2dy1dy2=c2-2knb1x-b1Bk∑i=-∞k∫Dif1y1dy1∑j=-∞k∫Djf2y2dy2≲2-2knb1x-b1Bk∑i=-∞kf1χiL1∑j=-∞kf2χjL1≲2-2knb1x-b1Bk∑i=-∞kf1χiLp1·χiLp1′·×∑j=-∞kf2χjLp2·χjLp2′·.By Lemma 18, Hölder’s inequality (Lemma 17), and Lemmas 10 and 15, we have (50)IχkLp·≲2-2knb1-b1Bk·χkLp·∑i=-∞kf1χiLp1·χiLp1′·×∑j=-∞kf2χjLp2·χjLp2′·≲2-2knb1∗χBkLp·∑i=-∞kf1χiLp1·χBiLp1′·×∑j=-∞kf2χjLp2·χBjLp2′·≲2-2knb1∗∑i=-∞kf1χiLp1·χBiLp1′·χBkLp1·×∑j=-∞kf2χjLp2·χBjLp2′·χBkLp2·≲b1∗∑i=-∞kf1χiLp1·×χBiLp1′·χBkLp1′·×∑i=-∞kf2χjLp2·×χBjLp2′·χBkLp2′·≲b1∗∑i=-∞k2-k-inδ12f1χiLp1·×∑j=-∞k2-k-jnδ22f2χjLp2·.Then we consider the term II. (51)II≲2-2kn∑i=-∞k∫Dif1y1b1y1-b1Bidy1×∑j=-∞k∫Djf2y2dy2+2-2kn∑i=-∞k∫Dif1y1b1Bk-b1Bidy1×∑j=-∞k∫Djf2y2dy2≕II1+II2.By Lemma 18 and Hölder’s inequality (Lemma 16), we have (52)II1≲2-2kn∑i=-∞kf1χiLp1·b1-b1BiχiLp1′·×∑j=-∞kf2χjLp2·χjLp2′·≲2-2knb1∗∑i=-∞kf1χiLp1·χBiLp1′·×∑j=-∞kf2χjLp2·χBjLp2′·.By Lemma 18 and Hölder’s inequality (Lemma 16), we have (53)II2≲2-2kn∑i=-∞kb1Bk-b1Bif1χiL1×∑j=-∞kf2χjL1≲2-2kn∑i=-∞kb1Bk-b1Bif1χiLp1·χiLp1′·×∑j=-∞kf2χjLp2·χjLp2′·≲2-2knb1∗∑i=-∞kk-if1χiLp1·χiLp1′·×∑j=-∞kf2χjLp2·χjLp2′·.By Hölder’s inequality (Lemma 16) and Lemmas 10 and 15, we have (54)II1χkLp·≲2-2knb1∗χBkLp1·∑i=-∞kf1χiLp1·χBiLp1′·×χBkLp2·∑j=-∞kf2χjLp2·χBjLp2′·≲b1∗∑i=-∞kf1χiLp1·χBiLp1′·χBkLp1′·×∑j=-∞kf2χjLp2·χBjLp2′·χBkLp2′·≲b1∗∑i=-∞k2-k-inδ12f1χiLp1·×∑j=-∞k2-k-jnδ22f2χjLp2·.
By Hölder’s inequality (Lemma16) and Lemmas 10 and 15, we get (55)II2χkLp·≲2-2knb1∗χBkLp·∑i=-∞kk-if1χiLp1·χBiLp1′·×∑j=-∞kf2χjLp2·χBjLp2′·≲2-2knb1∗χBkLp1·∑i=-∞kk-if1χiLp1·χBiLp1′·×χBkLp2·∑j=-∞kf2χjLp2·χBjLp2′·≲b1∗∑i=-∞kk-if1χiLp1·×χBiLp1′·χBkLp1′·×∑j=-∞kf2χjLp2·×χBjLp2′·χBkLp2′·≲b1∗∑i=-∞kk-i2-k-inδ12f1χiLp1·×∑j=-∞k2-k-jnδ22f2χjLp2·.Therefore, we have (56)Hb11f1,f2χkLp·≲b1∗∑i=-∞kk-i2-k-inδ12f1χiLp1·×∑j=-∞k2-k-jnδ22f2χjLp2·.
By Proposition8, we obtain that (57)Hb11f1,f2MK˙p·,λα·,q·≈max⁡supL≤0,L∈Z⁡2-Lλ2kα0Hb11f1,f2χkk≤Llq0Lp·,supL>0,L∈Z⁡2-Lλ2kα0Hb11f1,f2χkk<0lq0Lp·+2-Lλ2kα∞Hb11f1,f2χkk=0Llq∞Lp·≔max⁡E,F+G,where (58)E≔supL≤0,L∈Z⁡2-Lλ2kα0Hb11f1,f2χkk≤Llq0Lp·,F≔supL>0,L∈Z⁡2-Lλ2kα0Hb11f1,f2χkk<0lq0Lp·,G≔supL>0,L∈Z⁡2-Lλ2kα∞Hb11f1,f2χkk=0Llq∞Lp·.
Sinceα(0)=α1(0)+α2(0), 1/q(0)=1/q1(0)+1/q2(0), and λ=λ1+λ2, by (56) and Hölder’s inequality, we have (59)E=supL≤0,L∈Z⁡2-Lλ∑k=-∞L2kα0Hb11f1,f2χkLp·q01/q0≲b1∗supL≤0,L∈Z⁡2-Lλ1×∑k=-∞L∑i=-∞k2iα10f1χiLp1·k-i2i-knδ12-α10q101/q10×2-Lλ2∑k=-∞L∑j=-∞k2jα20f2χjLp2·2j-knδ22-α20q201/q20≕b1∗E1×E2.
Sincenδ12-α1(0)>0, by Lemma 19 we obtain that (60)E1≲supL≤0,L∈Z⁡2-Lλ1∑i=-∞k2iα1·f1χiLp1·q101/q10≲f1MK˙p1·,λ1α1·,q1·,where we wrote (k-i)2-|k-i|(nδ12-α1(0))≲2-|k-i|ε1 for some ε1∈(0,nδ12-α1(0)).
Sincenδ22-α2(0)>0, again by Lemma 19 we obtain that (61)E2≲supL≤0,L∈Z⁡2-Lλ2∑j=-∞k2jα22·f2χjLp2·q201/q20≲f2MK˙p2·,λ2α2·,q2·.Thus, we obtain (62)E≲b1∗f1MK˙p1·,λ1α1·,q1·f2MK˙p2·,λ2α2·,q2·.
Sinceα(0)=α1(0)+α2(0), 1/q(0)=1/q1(0)+1/q2(0), and λ=λ1+λ2, by (56) and Hölder’s inequality, we have (63)F=supL>0,L∈Z⁡2-Lλ∑k=-∞-12kα0Hb11f1,f2χkLp·q01/q0≲b1∗supL>0,L∈Z⁡2-Lλ1×∑k=-∞-1∑i=-∞k2iα10f1χiLp1·k-i2i-knδ12-α10q101/q10×2-Lλ2∑k=-∞-1∑j=-∞k2jα20f2χjLp2·2j-knδ22-α20q201/q20≕b1∗F1×F2.Here (64)F1≔supL>0,L∈Z⁡2-Lλ1×∑k=-∞-1∑i=-∞k2iα10f1χiLp1·k-i2i-knδ12-α10q101/q10,F2≔supL>0,L∈Z⁡2-Lλ2×∑k=-∞-1∑j=-∞k2jα20f2χjLp2·2j-knδ22-α20q201/q20.Since nδ12-α1(0)>0, by Lemma 19 we obtain that (65)F1≲supL>0,L∈Z⁡2-Lλ1∑i=-∞-12iα1·f1χiLp1·q101/q10≲f1MK˙p1·,λ1α1·,q1·,where we wrote (k-i)2-|k-i|(nδ12-α1(0))≲2-|k-i|ε1 for some ε1∈(0,nδ12-α1(0)).
Sincenδ22-α2(0)>0, by Lemma 19 we obtain that (66)F2≲supL>0,L∈Z⁡2-Lλ2∑j=-∞-12jα22·f2χjLp2·q201/q20≲f2MK˙p2·,λ2α2·,q2·.Therefore, we get (67)F≲b1∗f1MK˙p1·,λ1α1·,q1·f2MK˙p2·,λ2α2·,q2·.
Now we turn to estimateG. Since q∞=1/q1∞+1/q2∞, λ=λ1+λ2, by (56) and Hölder’s inequality, we have (68)G=supL>0,L∈Z⁡2-Lλ∑k=0L2kα∞Hb11f1,f2χkLp·q∞1/q∞≲b1∗supL>0,L∈Z⁡2-Lλ1×∑k=0L2kα1∞q1∞∑i=-∞kf1χiLp1·k-i2i-knδ12q1∞1/q1∞×2-Lλ2∑k=0L2kα2∞q2∞∑j=-∞kf2χjLp2·2j-knδ22q2∞1/q2∞≕b1∗G1×G2,where (69)G1≔supL>0,L∈Z⁡2-Lλ1∑k=0L2kα1∞q1∞∑i=-∞kf1χiLp1·k-i2i-knδ12q1∞1/q1∞,G2≔supL>0,L∈Z⁡2-Lλ2∑k=0L2kα2∞q2∞∑j=-∞kf2χjLp2·2j-knδ22q2∞1/q2∞.To go on, we need further preparation.
Ifi<0, by Proposition 8, we have (70)f1χiLp1·=2-iα102iα10q10f1χiLp1·q101/q10≲2-iα10∑l=-∞i2lα10q10f1χlLp1·q101/q10≲2iλ1-α102-iλ1∑l=-∞i2lα0f1χlLp1·q101/q10≲2iλ1-α10f1MK˙p1·,λ1α1·,q1·.Similarly, if j<0, we have (71)f2χjLp2·≲2jλ2-α20f2MK˙p2·,λ2α2·,q2·.If i≥0, we have (72)f1χiLp1·=2-iα1∞2iα1∞q1∞f1χiLp1·q1∞1/q1∞≲2-iα1∞∑l=-∞i2lα1∞q1∞f1χlLp1·q1∞1/q1∞≲2iλ1-α1∞2-iλ1∑l=-∞i2lα1∞f1χlLp1·q1∞1/q1∞≲2iλ1-α1∞f1MK˙p1·,λ1α1·,q1·.Similarly, if j≥0, we have (73)f2χjLp2·≲2jλ2-α2∞f2MK˙p2·,λ2α2·,q1·.
Sinceλ1+nδ12>α1∞≥α1(0) and by (70) and (72), we obtain that (74)G1≲f1MK˙p1·,λ1α1·,q1·supL>0,L∈Z⁡2-Lλ1×∑k=0L2kα1∞q1∞∑i=-∞-1k-i2i-knδ122iλ1-α10+∑i=0kk-i2i-knδ122iλ1-α1∞q1∞1/q1∞≲f1MK˙p1·,λ1α1·,q1·supL>0,L∈Z⁡2-Lλ1∑k=0L2kα1∞q1∞×∑i=-∞-1k-i2i-knδ12+λ1-α1∞2iα1∞-α102kλ1-α1∞+∑i=0kk-i2i-knδ12+λ1-α1∞2kλ1-α1∞q1∞1/q1∞≲f1MK˙p1·,λ1α1·,q1·supL>0,L∈Z⁡2-Lλ1×∑k=0L2kλ1q1∞∑i=-∞kk-i2i-knδ12-α1∞+λ1q1∞1/q1∞≲f1MK˙p1·,λ1α1·,q1·supL>0,L∈Z⁡2-Lλ1∑k=0L2kλ1q1∞1/q1∞≲f1MK˙p1·,λ1α1·,q1·.
Similarly, sinceλ2+nδ22>α2∞≥α2(0), by (71) and (73), we obtain that (75)G2≲f2MK˙p2·,λ2α2·,q2·supL>0,L∈Z⁡2-Lλ2×∑k=0L2kλ2q2∞∑j=-∞k2j-knδ22-α2∞+λ2q2∞1/q2∞≲f2MK˙p2·,λ2α2·,q2·supL>0,L∈Z⁡2-Lλ2∑k=0L2kλ2q2∞1/q2∞≲f2MK˙p2·,λ2α2·,q2·.Thus, we obtain (76)G≲b1∗f1MK˙p1·,λ1α1·,q1·f2MK˙p2·,λ2α2·,q2·.
Combining all the estimates forE,F,G together, we obtain (77)Hb11f1,f2MK˙p·,λα·,q·≲b1∗f1MK˙p1·,λ1α1·,q1·f2MK˙p2·,λ2α2·,q2·.Similarly, we have (78)Hb21f1,f2MK˙p·,λα·,q·≲b2∗f1MK˙p1·,λ1α1·,q1·f2MK˙p2·,λ2α2·,q2·.
Therefore, the proof of Theorem11 is completed.



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*Source: 7607893-2019-06-09.xml*