# Boundedness of Multilinear Calderón-Zygmund Operators on Grand Variable Herz Spaces

Authors: Hammad Nafis; Humberto Rafeiro; Muhammad Asad Zaighum
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on product of grand variable Herz spaces. These results generalize the boundedness of multilinear Calderón-Zygmund operators on product of variable exponent Lebesgue spaces and variable Herz spaces.


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## Body

## 1. Introduction

There has been increased interest in the study of multilinear singular integral operators in recent years. The class of multilinear singular integrals with standard Calderón-Zygmund kernels provides the foundation and starting point of research of the theory. Such a class of multilinear Calderón-Zygmund operators was introduced and first studied by Coifman and Meyer [1–3] and later by Grafakos and Torres [4]. For the boundedness and other properties of multilinear fractional integrals, we refer to, e.g., [5–8].Variable Lebesgue spaces were introduced in [9], but stayed under the radar for a considerable amount of time. Apart from some previous sporadic episodes, the research boom on such spaces can be traced back to the foundational paper [10]. Since then, these spaces have attracted much attention of mathematicians, not only because of their connection with harmonic analysis but also due to their usefulness in application to a wide range of problems, see, e.g., [11]. The standard references to the general theory of variable Lebesgue spaces are [12, 13].The classical definition of Herz spaces was introduced in [14]. Many studies can be found related to these spaces and its variations, which include variable Herz spaces, continual Herz spaces, and Herz spaces with variable smoothness and integrability. For details, see [15–21] and references therein.Grand Lebesgue spaces on bounded sets, which proved to be useful in application to partial differential equations, were introduced in [22, 23]. In the last years, various operators of harmonic analysis have been intensively studied on grand spaces, see for instance [20, 24–32]. Grand Lebesgue sequence spaces were introduced recently in [33], where several operators of harmonic analysis were studied, e.g., maximal, convolutions, Hardy, Hilbert, and fractional operators.In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz spaces which were introduced in [34]. The present paper is organized in the following way. Apart from the introduction, in Section 2, we recall some definitions and results related to variable exponent spaces. Section 3 contains some details about multilinear Calderón-Zygmund kernels and the proof of the main result.Notations.
(i)
ℕ is the set of natural numbers and ℕ0≔ℕ∪0(ii)
ℤ is the set of integers(iii)
ℤ− is the set of negative integers(iv)
n,m¯≔n,n+1,n+2,⋯,m−1,m for n,m∈ℤ and n<m(v)
Bx,r is the ball of radius r center at the point x(vi)
Bk≔x∈ℝn:x≤2k for all k∈ℤ(vii)
Rt,τ≔Bτ\Bt=x:t<∣x∣<τ is a spherical layer(viii)
Rk≔Bk\Bk−1(ix)
χk≔χRk,χt,τ≔χRt,τ(x)
f≲g means that f≤Cg and f≃g means that f≲g≲f(xi)
constants (often different constant in the same chain of inequalities) will mainly be denoted byc or C


## 2. Function Spaces with Variable Exponent

In this section, we recall definitions and results related to variable exponent Lebesgue spaces, variable Herz spaces, and grand variable Herz spaces.


### 2.1. Lebesgue Space with Variable Exponent

For the current section, we refer to [10–13, 35] unless and until stated otherwise. Let q be a real-valued measurable function on ℝn with values in 1,∞. For X⊂ℝn, we suppose that
(1)1≤q−X≤qx≤q+X<∞,whereq−X≔essinfx∈Xqxandq+X≔esssupx∈Xqx. By Lq·ℝn, we denote the space of measurable function f on ℝn such that
(2)Iq·f=∫ℝnfxqxdx<∞.It is a Banach space, see [13, 35], endowed with norm:
(3)fq·=infη>0:Iq·fη≤1.Byq′, we denote the conjugate exponent of q, defined by q′x=qx/qx−1. In the sequel, we uselog condition:(4)qx−qy≤A−lnx−y,x−y≤12,x,y∈ℝn,where A=Aq>0 does not depend on x,y;decay condition at 0:(5)qx−q0≤Alnx,x≤12,holds for some q0∈1,∞; anddecay condition at∞: there exists a number q∞∈1,∞, such that
(6)qx−q∞≤Alne+x,where A=Aq does not depend on x.Given a functionf∈Lloc1ℝn, the Hardy-Littlewood maximal operator M is defined by
(7)Mfx≔supr>0r−n∫Bx,rfydyx∈ℝn.We adopt the following notations:(i)
Llocq·ℝn≔f:f∈Lq·KforallcompactsubsetsK⊂ℝn(ii)
Pℝn consists of all measurable functions q satisfying q−>1 and q+<∞(iii)
P0logℝn and P∞logℝn denote the classes of q∈Pℝn which satisfy (5) and (6), respectively(iv)
Bℝn is the set of all q∈Pℝn for which M is bounded on Lq·ℝnFor the following lemma, we refer to, e.g., [12].Lemma 1 (Generalized Hölder’s Inequality).
Givenp,q∈Pℝn, define r∈Pℝn by 1/rx=1/px+1/qx. Then, there exists a constant c such that for all f∈Lp·ℝn and g∈Lq·ℝn, fg∈Lr·ℝn and
(8)fgr·≤cfp·gq·.The following lemma appears in [21].Lemma 2.
LetD>1 and q∈P0logℝn∩P∞logℝn. Then
(9)1c0rn/q0≤χRr,Drq·≤c0rn/q0,for0<r≤1,(10)1c∞rn/q∞≤χRr,Drq·≤c∞rn/q∞,forr≤1,where c0≥1 and c∞≥1 depend on D, but do not depend on r.


### 2.2. Herz Spaces with Variable Exponent

The classical Herz spaces were first introduced in [14]. We recall the definition of variable exponent Herz spaces.Definition 3.
Let1<p<∞, α∈ℝ, and q∈Pℝn. The homogeneous Herz space K̇q·α,p is defined by
(11)K̇q·α,pℝn=f∈Llocq·ℝn\0:fK̇q·α,p<∞,where
(12)fK̇q·α,pℝn=∑k∈ℤ2kαfχkq·p1/p.
The nonhomogeneous Herz spaceKq·α,p is defined by
(13)Kq·α,pℝn=f∈Llocq·ℝn:fKq·α,p<∞,where
(14)fKq·α,pℝn=∑k∈ℕ02kαfχkq·p1/p+fχB0,1q·.
For the boundedness of integral operators on Herz type spaces, we refer to, e.g., [18, 19, 36].


### 2.3. Grand Lebesgue Sequence Space

In this section, we recall the definition of grand Lebesgue sequence space. For the following definition and statements, see [33]. In what follows, X stands for one of the sets ℤn,ℤ,ℕ, and ℕ0.Definition 4.
Let1≤p<∞ and θ>0. The grand Lebesgue sequence space lp,θ is defined by the norm
(15)xlp,θX≔supε>0εθ∑k∈Xxkp1+ε1/p1+ε=supε>0εθ/p1+εxlp1+εX.


### 2.4. Grand Variable Herz Space

Following [34], we now introduce the grand variable Herz spaces.Definition 5.
Letα∈L∞ℝ, 1≤p<∞, q:ℝn⟶1,∞, and θ>0. The homogeneous grand variable Herz space is defined by
(16)K̇q·α·,p,θℝn=f∈Llocq·ℝn\0:fK̇q·α·,p,θℝn<∞,where
(17)fK̇q·α·,p,θℝn=supε>0εθ∑k∈ℤ2kα·fχkq·p1+ε1/p1+ε=supε>0εθ/p1+εfK̇q·α·,p1+εℝn.The following lemma, see [15], is helpful to estimate the norm of characteristics functions.Lemma 6.
Letq∈P∞logℝn and let R=B0,r\B0,r/2. If R≤2−n, then
(18)χRq·≃R1/qx≃R1/q∞,with the implicit constants independent of r and x∈R.
The left-hand side equivalence remains true for everyR>0 if we assume, additionally, that q∈P0logℝn∩P∞logℝn.


## 2.1. Lebesgue Space with Variable Exponent

For the current section, we refer to [10–13, 35] unless and until stated otherwise. Let q be a real-valued measurable function on ℝn with values in 1,∞. For X⊂ℝn, we suppose that
(1)1≤q−X≤qx≤q+X<∞,whereq−X≔essinfx∈Xqxandq+X≔esssupx∈Xqx. By Lq·ℝn, we denote the space of measurable function f on ℝn such that
(2)Iq·f=∫ℝnfxqxdx<∞.It is a Banach space, see [13, 35], endowed with norm:
(3)fq·=infη>0:Iq·fη≤1.Byq′, we denote the conjugate exponent of q, defined by q′x=qx/qx−1. In the sequel, we uselog condition:(4)qx−qy≤A−lnx−y,x−y≤12,x,y∈ℝn,where A=Aq>0 does not depend on x,y;decay condition at 0:(5)qx−q0≤Alnx,x≤12,holds for some q0∈1,∞; anddecay condition at∞: there exists a number q∞∈1,∞, such that
(6)qx−q∞≤Alne+x,where A=Aq does not depend on x.Given a functionf∈Lloc1ℝn, the Hardy-Littlewood maximal operator M is defined by
(7)Mfx≔supr>0r−n∫Bx,rfydyx∈ℝn.We adopt the following notations:(i)
Llocq·ℝn≔f:f∈Lq·KforallcompactsubsetsK⊂ℝn(ii)
Pℝn consists of all measurable functions q satisfying q−>1 and q+<∞(iii)
P0logℝn and P∞logℝn denote the classes of q∈Pℝn which satisfy (5) and (6), respectively(iv)
Bℝn is the set of all q∈Pℝn for which M is bounded on Lq·ℝnFor the following lemma, we refer to, e.g., [12].Lemma 1 (Generalized Hölder’s Inequality).
Givenp,q∈Pℝn, define r∈Pℝn by 1/rx=1/px+1/qx. Then, there exists a constant c such that for all f∈Lp·ℝn and g∈Lq·ℝn, fg∈Lr·ℝn and
(8)fgr·≤cfp·gq·.The following lemma appears in [21].Lemma 2.
LetD>1 and q∈P0logℝn∩P∞logℝn. Then
(9)1c0rn/q0≤χRr,Drq·≤c0rn/q0,for0<r≤1,(10)1c∞rn/q∞≤χRr,Drq·≤c∞rn/q∞,forr≤1,where c0≥1 and c∞≥1 depend on D, but do not depend on r.


## 2.2. Herz Spaces with Variable Exponent

The classical Herz spaces were first introduced in [14]. We recall the definition of variable exponent Herz spaces.Definition 3.
Let1<p<∞, α∈ℝ, and q∈Pℝn. The homogeneous Herz space K̇q·α,p is defined by
(11)K̇q·α,pℝn=f∈Llocq·ℝn\0:fK̇q·α,p<∞,where
(12)fK̇q·α,pℝn=∑k∈ℤ2kαfχkq·p1/p.
The nonhomogeneous Herz spaceKq·α,p is defined by
(13)Kq·α,pℝn=f∈Llocq·ℝn:fKq·α,p<∞,where
(14)fKq·α,pℝn=∑k∈ℕ02kαfχkq·p1/p+fχB0,1q·.
For the boundedness of integral operators on Herz type spaces, we refer to, e.g., [18, 19, 36].


## 2.3. Grand Lebesgue Sequence Space

In this section, we recall the definition of grand Lebesgue sequence space. For the following definition and statements, see [33]. In what follows, X stands for one of the sets ℤn,ℤ,ℕ, and ℕ0.Definition 4.
Let1≤p<∞ and θ>0. The grand Lebesgue sequence space lp,θ is defined by the norm
(15)xlp,θX≔supε>0εθ∑k∈Xxkp1+ε1/p1+ε=supε>0εθ/p1+εxlp1+εX.


## 2.4. Grand Variable Herz Space

Following [34], we now introduce the grand variable Herz spaces.Definition 5.
Letα∈L∞ℝ, 1≤p<∞, q:ℝn⟶1,∞, and θ>0. The homogeneous grand variable Herz space is defined by
(16)K̇q·α·,p,θℝn=f∈Llocq·ℝn\0:fK̇q·α·,p,θℝn<∞,where
(17)fK̇q·α·,p,θℝn=supε>0εθ∑k∈ℤ2kα·fχkq·p1+ε1/p1+ε=supε>0εθ/p1+εfK̇q·α·,p1+εℝn.The following lemma, see [15], is helpful to estimate the norm of characteristics functions.Lemma 6.
Letq∈P∞logℝn and let R=B0,r\B0,r/2. If R≤2−n, then
(18)χRq·≃R1/qx≃R1/q∞,with the implicit constants independent of r and x∈R.
The left-hand side equivalence remains true for everyR>0 if we assume, additionally, that q∈P0logℝn∩P∞logℝn.


## 3. Boundedness of Multilinear Calderón-Zygmund Operators

Consider the multilinear operatorT of the form
(19)Tf1,⋯,fmx=∫ℝnmKx,y1,⋯,ymf1y1⋯fmymdy1⋯dym,where x∉⋂j=1msptfj and f1,⋯,fm∈Lc∞ℝn, the space of compactly supported functions. Let Kx,y1,⋯,ym be a locally integrable function defined away from the diagonal x=y1=⋯=ym∈ℝnm+1, which satisfies the size estimate
(20)Kx,y1,⋯,ym≤cx−y1+x−y2+⋯+x−ymmn,for some c>0 and all x,y1,⋯,ym∈ℝnm+1 with x≠yj for some j. For smoothness, assume that for some ε>0,
(21)Kx,y1,⋯,ym−Kx′,y1,⋯,ym≤cx−x′εx−y1+⋯+x−ymmn+ε,provided that x−x′≤1/2maxx−y1,⋯,x−ym and
(22)Kx,y1,⋯,yj,⋯,ym−Kx,y1,⋯,yj′,⋯,ym≤cyj−yj′εx−y1+⋯+x−ymmn+ε,whenever yj−yj′≤1/2maxx−y1,⋯,x−ym for all j∈1,m¯.Such kernels are called Calderón-Zygmund kernels and the class of all functions satisfying (20), (21), and (22) with parameters m,c, and ε will be denoted by m-CZKc,ε, compare [4]. We say that T be as in (19) is an m-linear Calderón-Zygmund operator, if
(i)
The related kernel belongs tom-CZKc,ε class(ii)
T is bounded from Lq1×⋯×Lqm to Lq for some 1<q1,q2,⋯,qm<∞, and 1/q=1/q1+⋯+1/qmGrafakos and Torres [4] proved the boundedness of T from Lq1×⋯×Lqm to Lq,∞ for some 1<q1,q2,⋯,qm<∞, and 1/q=1/q1+⋯+1/qm, and from L1×⋯×L1 to L1/m,∞.The boundedness of the multilinear Calderón-Zygmund operatorT on variable exponent Lebesgue spaces was proved in [37], as stated below.Lemma 7.
Letqi∈Bℝn, i∈1,m¯, q∈P0logℝn with 1/qx=1/q1x+⋯+1/qmx, and q/s′∈Bℝn for some 0<s<q−. Then, the m-linear Calderón-Zygmund operator T is bounded on the product of variable exponent Lebesgue spaces. Moreover,
(23)Tf⟶q·≲∏i=1mfiqi·,with the constant C independent of f⟶=f1,⋯,fm.We now state and prove the boundedness of multilinear Calderón-Zygmund operator on grand variable Herz spaces.Theorem 8.
Letqi∈Bℝn, i∈1,m¯, q,qi∈P0logℝn∩P∞logℝn such that qi0≤qi∞, 1/qx=1/q1x+⋯+1/qmx,1<qi−≤qi+<∞, and q/s′∈Bℝn for some 0<s<q−. Let θ>0, 1<pi<∞, and αi∈L∞ℝ be log-Hölder continuous both at the origin and at infinity for i∈1,m¯ with
(24)−nqi∞<αi0≤αi∞<n1−1qi0.
Suppose thatαx=∑i=1mαix and 1/p=∑i=1m1/pi. Then, the m-linear Calderón-Zygmund operator T is bounded on the product of grand variable Herz spaces. Moreover,
(25)Tf⟶K̇q·α·,p,θℝn≤∏i=1mfiK̇qi·αi·,pi,θℝn,with the constant C>0 independent of f⟶=f1,⋯,fm.Proof.
We restrict ourselves tom=2, the general case following in a similar manner. Defining fφx≔f1xχRφx and fσx≔f2xχRσx, we decompose the component functions of f⟶ as
(26)f1x=∑φ∈ℤfφx,f2x=∑σ∈ℤfσx.
For future usage, we divideℤ into the following sets
(27)Lk≔n∈ℤ∣n≤k−2,Mk≔n∈ℤ∣k−1≤n≤k+1,Nk≔n∈ℤ∣n≤k+2,and, for X and Y arbitrary subsets of ℤ, we define
(28)νkαX,Y≔2ka·∑φ∈X∑σ∈YTfφ,fσχkq·,andνkX,Y≔νk0X,Y.
From Definition5 and (26), we have
(29)Tf1,f2K̇q·α·,p,θℝn=supε>0εθ∑k∈ℤ2kα·Tf1,f2χkq·p1+ε1/p1+ε≤supε>0εθ∑k∈ℤ∑φ∈ℤ∑σ∈ℤ2kα·Tfφ,fσχkq·p1+ε1/p1+ε≲∑j=19Ij,where
(30)Ij≔supε>0εθ∑k∈ℤλjp1+ε1/p1+ε,(31)λ1=νkαLk,Lk,λ2=νkαLk,Mk,λ3=νkαLk,Nk,λ4=νkαMk,Lk,λ5=νkαMk,Mk,λ6=νkαMk,Nk,λ7=νkαNk,Lk,λ8=νkαNk,Mk,λ9=νkαNk,Nk.
It is necessary to estimateI1,I2,I3,I5,I6, and I9, since I4, I7, and I8 can be obtained in a similar manner as I2, I3, and I6, respectively. Estimation for I1: splitting ℤ=ℤ−∪ℕ0 and by the asymptotic 2kαx≃2kα0 (x∈Rk and k<0) and 2kαx≃2kα∞ (x∈Rk and k≥0), we get
(32)I1≲supε>0εθ∑k∈ℤ−2kα0p1+ενkLk,Lkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkLk,Lkp1+ε1/p1+ε≕I11+I12.
Forφ,σ∈Lk, x∈Rk, y1∈Rφ, and y2∈Rσ, we have
(33)x−y1≥x−y1≥2k−1−2φ≥2k−1−2k−2≳2k,(34)x−y2≥x−y2≥2k−1−2σ≥2k−1−2k−2≳2k,yielding
(35)Tfφ,fσx≤∫ℝn∫ℝnfφy1fσy2∣x−y1∣+∣x−y2∣2ndy1dy2≲2−2kn∫ℝnfφy1dy1∫ℝnfσy2dy2.
From estimate (35), Hölder’s inequality, 1/qx=1/q1x+1/q2x, and Lemma 2, we obtain
(36)νkLk,Lk≲∑φ∈Lk∑σ∈Lk2−2knfφq1·χφq1′·fσq2·χσq2′·χkq·≲∑φ∈Lk∑σ∈Lk2−2knfφq1·2φn/q1′0fσq2·2φn/q2′02kn/q0≲∑φ∈Lk2k−φn/q10−nfφq1·∑σ∈Lk2k−σn/q20−nfσq2·.
Taking into account the previous estimate forνkLk,Lk, the equality 1p=1p1+1p2, and Hölder’s inequality, we have
(37)I11≲supε>0εθ∑k∈ℤ−∑φ∈Lk2k−φn/q10−n+kα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ−∑σ∈Lk2k−σn/q20−n+kα20fσq2·p21+ε1/p21+ε≕I111I112.
By Hölder’s inequality, Fubini’s theorem for series,2−p11+ε<2−p1, and defining b1≔n/q10−n+α10<0, we obtain
(38)I111≲supε>0εθ∑k∈ℤ−∑φ∈Lk2b1k−φp11+ε22φα10p11+εfφq1·p11+ε×∑φ∈Lk2b1k−φp11+ε′2p11+εp11+ε′1/p11+ε≲supε>0εθ∑φ=−∞−12φα10p11+εfφq1·p11+ε∑k=φ+2−12b1k−φp11+ε21/p11+ε≲supε>0εθ∑φ∈Lk2φα10p11+εfφq1·p11+ε1/p11+ε≲f1K̇q1·α1·,p1,θℝn.
The estimateI112≲f2K̇q2·α2·,p2,θℝn is obtained, mutatis mutandis, via the estimation for I111. With this estimates at hand, we obtain
(39)I11≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
To estimateI12, we split as follow
(40)I12≲supε>0εθ∑k∈ℕ02kα∞p1+ε∑φ∈ℤ−∑σ∈ℤ−Tfφ,fσχkq·p1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ε∑φ=0k−2∑σ=0k−2Tfφ,fσχkq·p1+ε1/p1+ε≕I121+I122.
The estimateI122 follows in similar manner as in I11 with simply replaced αi0 by αi∞ and used the fact qi0≤qi∞.
For estimateI121, by Hölder’s inequality, Lemma 2, and the inequality qi0≤qi∞, we obtain
(41)νkℤ−,ℤ−≲∑φ∈ℤ−∑σ∈ℤ−2−2knfφq1·χφq1′·fσq2·χσq2′·χkq·≲∑φ∈ℤ−∑σ∈ℤ−2−2knfφq1·2φnq1′0fσq2·2σnq2′02knq∞≲∑φ∈ℤ−2k−φnq10−nfφq1·∑σ∈ℤ−2k−σnq20−nfσq2·.
From the estimate ofνkℤ−,ℤ−, the equality 1/p=1/p1+1/p2, and Hölder’s inequality, we have
(42)I121≲supε>0εθ∑k∈ℕ0∑φ∈ℤ−2k−φnq10−n+kα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈ℤ−2k−σnq20−n+kα2∞fσq2·p21+ε1/p21+ε≕A1A2.
Invoking the Hölder inequality and definingξ1≔n/q10−n+α1∞<0, we have
(43)A1≲supε>0εθ∑k∈ℕ0∑φ∈ℤ−2k−φξ1p11+ε22φα1∞p11+εfφq1·p11+ε×∑φ∈ℤ−2k−φξ1p11+ε′2p11+εp11+ε′1/p11+ε≲supε>0εθ∑k∈ℕ02kξ1p11+ε2∑φ∈ℤ−2−φξ1p11+ε22φα1∞p11+εfφq1·p11+ε1/p11+ε≲supε>0εθ∑φ∈ℤ−2φα1∞p11+εfφq1·p11+ε1/p11+ε≲f1K̇q1·α1·,p1,θℝn.
Similar estimate, with the corresponding changes, is obtained forA2, from which we obtain I12≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn. Hence
(44)I1≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Estimation forI2: as in the case of I1, we obtain the following estimate
(45)I2≲supε>0εθ∑k∈ℤ−2kα0p1+ενkLk,Mkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkLk,Mkp1+ε1/p1+ε≕I21+I22.
Notice that, forx∈Rk, y1∈Rφ, y2∈Rσ, φ∈Lk, and σ∈Mk, we have
(46)x−y1+x−y2≥x−y1≥x−y1≳2k,from which, taking Lemma 2 into consideration and elementary computations, we obtain
(47)νkLk,Mk≤∑φ∈Lk∑σ∈Mk2k−φnq10−nfφq1·2σ−kn2k−σnq20fσq2·,k∈ℤ−.
From the estimate forνkLk,Mk and Hölder’s inequality, we get
(48)I21≲supε>0εθ∑k∈ℤ−∑φ∈Lk2k−φnq10−n+kα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ∑σ∈Mk2kα202σ−knfσq2·p21+ε1/p21+ε≕I211I212.
Notice thatI211=I111. For the estimate I212, we reason as follows
(49)I212≲supε>0εθ∑k∈ℤ−2kα20p21+ε∑σ∈Mk2σ−knfσq2·p21+ε1/p21+ε≲supε>0εθ∑k∈ℤ−2kα20p21+εf2χkq2·p21+ε1/p21+ε≲f2K̇q2·α2·,p2,θℝn.
The termI22 is estimated by
(50)I22≲supε>0εθ∑k∈ℕ02kα∞p1+ενkℤ−,Mkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενk0,k−2¯,Mkp1+ε1/p1+ε≕I221+I222.
To estimateI221, by Hölder’s inequality, Lemma 2, and the inequality q10⩽q1∞, we have
(51)νkℤ−,Mk≲∑φ∈ℤ−∑σ∈Mk2k−φn/q10−nfφq1·2σ−kn2k−σn/q2∞fσq2·.
Thus using (51) and by Hölder’s inequality, we have
(52)I221≲supε>0εθ∑k∈ℕ0∑φ∈ℤ−2k−φnq10−n+kα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Mk2kα2∞2σ−knfσq2·p21+ε1/p21+ε≕B1B2.
The termB1 is equal to A1 and for B2 we use similar arguments as for I212, replacing α20 with α2∞.
For the termI222, by Hölder’s inequality and Lemma 2, we have
(53)νk0,k−2¯,Mk≲∑φ=0k−2∑σ∈Mk2k−φn/q1∞−nfφq1·2σ−knfσq2·.
Taking into consideration (53) and applying Hölder’s inequality, we get
(54)I222≲supε>0εθ∑k∈ℕ0∑φ=0k−22k−φnq1∞−n+kα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Mk2kα2∞2σ−knfσq2·p21+ε1/p21+ε≕B1′B2′.
Noting thatB2′=B2 and estimating B1′ in a similar fashion as I212, we obtain I22≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn, from which we get
(55)I2≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Estimation forI3: we have
(56)I3≲supε>0εθ∑k∈ℤ−2kα0p1+ενkLk,Nkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkLk,Nkp1+ε1/p1+ε≕I31+I32.
Forφ∈Lk, σ∈Nk, x∈Rk, y1∈Rφ, and y2∈Rσ, we get
(57)x−y1≲x−y1≳2k,x−y2≥y2−x≳2σ,from which, taking into account the elementary inequality 2k+2σ2n≳2kn+σn, it follows
(58)Tfφ,fσx≲2−kn∫ℝnfφy1dy12−σn∫ℝnfσy2dy2,x∈Rk.
By Hölder’s inequality, Lemma2, q20≲q2∞, and k∈ℤ−, we obtain
(59)νkLk,Nk≲∑φ∈Lk2k−φn/q10−nfφq1·∑σ∈Nk2k−σn/q2∞fσq2·.
To estimate the termI31, using (59) and the Hölder inequality, we obtain
(60)I31≲supε>0εθ∑k∈ℤ−∑φ∈Lk2k−φn/q10−n+α102φα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ−∑σ∈Nk2k−σn/q2∞+α202σα20fσq2·p21+ε1/p21+ε≕I311I312.
We haveI311≲f1K̇q1·α1·,p1,θℝn, since I311=I111.
As forI312, we split as follow
(61)I312≲supε>0εθ∑k∈ℤ−∑σ=k+2−12k−σn/q2∞+α202σα20fσq2·p21+ε1/p21+ε+supε>0εθ∑k∈ℤ−∑σ∈ℕ02k−σn/q2∞+α202σα20fσq2·p21+ε1/p21+ε≕D1+D2.
By Hölder’s inequality, Fubini’s theorem for series, the inequalityn/q2∞+α20>0 and 2−p21+ε<2−p2, we obtain
(62)D1≲supε>0εθ∑k∈ℤ−∑σ=k+2−12σα20p21+εfσq2·p21+ε2k−σn/q2∞+α20p21+ε2×∑σ=k+2−12k−σn/q2∞+α20p21+ε′2p21+εp21+ε′1/p21+ε≲supε>0εθ∑k∈ℤ−∑σ=k+2−12σα202k−σn/q2∞+α20p21+ε2fσq2·p21+ε1/p21+ε≲supε>0εθ∑σ=−∞−12σα20p21+εfσq2·p21+ε∑k=−∞σ−22k−σn/q2∞+α2021/p21+ε≲f2K̇q2·α2·,p2,θℝn.
ForD2, using the same argument as above and the inequality α20⩽α2∞, we obtain
(63)D2≲supε>0εθ∑k∈ℤ−∑σ∈ℕ02σα20p21+εfσq2·p21+ε2k−σn/q2∞+α20p21+ε21/p21+ε≲supε>0εθ∑k∈ℤ−∑σ∈ℕ02k−σn/q2∞+α20p21+ε2∑j=σ∞2jα2∞p21+εf2χRjq2·p21+ε1/p21+ε≲supε>0∑k∈ℤ−∑σ∈ℕ02k−σn/q2∞+α20p21+ε2f2K̇q2·α2·,p2,θℝnp21+ε1/p21+ε≲f2K̇q2·α2·,p2,θℝn.
SoI31≲f1K̇q1·α1,p1,θℝnf2K̇q2·α2,p2,θℝn.
The termI32 can be estimated as
(64)I32≲supε>0εθ∑k∈ℕ02kα∞p1+ενkℤ−,Nkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενk0,k−2¯,Nkp1+ε1/p1+ε≕I321+I322.
Fork∈ℕ0, φ∈ℤ−, σ∈Nk, q10≲q1∞, and Lemma 2, we obtain
(65)νkℤ−,Nk≲∑φ∈ℤ−2k−φn/q10−nfφq1·∑σ∈Nk2k−σn/q2∞fσq2·.
By (65) and Hölder’s inequality, we get
(66)I321≲supε>0εθ∑k∈ℕ0∑φ∈ℤ−2k−φnq10−n+α1∞2φα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Nk2k−σnq2∞+α2∞2σα2∞fσq2·p21+ε1/p21+ε≕D1′D2′.
Note thatD1′=A1. The estimate for D2′ can be obtained in a similar way as for I312, by replacing α20 with α2∞.
For estimateI322, by Hölder’s inequality and Lemma 2, we have
(67)νk0,k−2¯,Nk≲∑φ=0k−2∑σ∈Nk2k−φn/q1∞−nfφq1·2k−σn/q2∞fσq2·.
Using (67) and Hölder’s inequality, we obtain
(68)I322≲supε>0εθ∑k∈ℕ0∑φ=0k−22k−φn/q1∞−n+α1∞2φα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Nk2k−σn/q2∞+α2∞2σα2∞fσq2·p21+ε1/p21+ε≕D1′′D2′′.
We haveD2′′=D2′ and the estimate for the term D1′′ is similar to B1′. Taking all the estimates into account yields
(69)I3≲∥f1∥K̇q1·α1·,p1,θℝn∥f2∥K̇q2·α2·,p2,θℝn.
Estimation forI5: We have
(70)I5≲supε>0εθ∑k∈ℤ−2kα0p1+ενkMk,Mkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkMk,Mkp1+ε1/p1+ε≕I51+I52.
By theLq·-boundedness of T, see Lemma 7, we obtain that
(71)νkMk,Mk≲∑φ∈Mk∑σ∈Mkfφq1·fσq2·.
By Hölder’s inequality, we have(72)I51≲supε>0εθ∑k∈ℤ−∑φ∈Mk2kα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ−∑σ∈Mk2kα20fσq2·p21+ε1/p21+ε≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Similarly, we can obtain similar estimate forI52, replacing αi0 by αi∞ with k∈ℕ0. Therefore
(73)I5≲∥f1∥K̇q1·α1·,p1,θℝn∥f2∥K̇q2·α2·,p2,θℝn.
Estimation forI6: we have
(74)I6≲supε>0εθ∑k∈ℤ−2kα0p1+ενkMk,Nkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkMk,Nkp1+ε1/p1+ε≕I61+I62.
Fork∈ℤ−, φ∈Mk, σ∈Nk, and x∈Rk, we have
(75)x−y2≤y2−x2σ,yielding
(76)Tfφ,fσx≤2−kn∫ℝnfφy1dy12−σn∫ℝnfσy2dy2.
By Hölder’s inequality, Lemma2, k∈ℤ−, and inequality q20⩽q2∞, we obtain
(77)νkMk,Nk≲∑φ∈Mk∑σ∈Nk2φ−knfφq1·2k−σn/q2∞fσq2·.
By (77) and invoking Hölder’s inequality, we get
(78)I61≲supε>0εθ∑k∈ℤ−∑φ∈Mk2φ−kn2kα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ−∑σ∈Nk2k−σn/q2∞+α202σα20fσq2·p21+ε1/p21+ε≕I611I612.
Note that the estimateI611 is equal to that of I51 and I612 is obtained by similar argument used in I312.
Fork∈ℕ0, we have
(79)νkMk,Nk≲∑φ∈Mk∑σ∈Nk2φ−knfφq1·2k−σn/q2∞fσq2·.
Using (79) and Hölder’s inequality, we get
(80)I62≲supε>0εθ∑k∈ℕ0∑φ∈Mk2φ−kn2kα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Nk2k−σnq2∞+α2∞2σα2∞fσq2·p21+ε1/p21+ε≕I621I622.
The estimateI621 is similar to I52 and I622=D2′′. Hence
(81)I6≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Estimation forI9: we estimate I9 as usual
(82)I9≲supε>0εθ∑k∈ℤ−2kα0p1+ενkNk,Nkp1+ε1/p1+ε+supε>0εθ∑k∈ℕ02kα∞p1+ενkNk,Nkp1+ε1/p1+ε≕I91+I92.
Forφ∈Nk, y1∈Rφ, and x∈Rk, we have ∣x−y1∣2φ, whereas for σ∈Nk, y2∈Rσ, and x∈Rk, we have ∣x−y2∣2σ. Under the assumptions for the obtained inequalities, we have
(83)Tfφ,fσx≲2−φn∫ℝnfφy1dy12−σn∫ℝnfσy2dy2.
Whenk∈ℤ−, by Hölder’s inequality, Lemma 2, and qj0⩽qj∞, we obtain
(84)νkNk,Nk≲∑φ∈Nk∑σ∈Nk2k−φn/q1∞2k−σn/q2∞fφq1·fσq2·.
Thus(85)I91≲supε>0εθ∑k∈ℤ−∑φ∈Nk2k−φn/q1∞+α102φα10fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℤ−∑σ∈Nk2k−σn/q1∞+α102σα20fσq2·p21+ε1/p21+ε≕I911I912.
The estimates for the termsI911 and I912 are obtained in the same way as for I312. Therefore
(86)I91≤Cf1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Finally, for the termI92, we have
(87)I92≲supε>0εθ∑k∈ℕ0∑φ∈Nk2k−φn/q1∞+α1∞2φα1∞fφq1·p11+ε1/p11+ε×supε>0εθ∑k∈ℕ0∑σ∈Nk2k−σn/q2∞+α2∞2σα2∞fσq2·p21+ε1/p21+ε≕I921I922.
The termsI921 and I922 can be estimated in a similar manner as for that I322. Thus
(88)I9≲f1K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn.
Taking into account the estimates (44), (55), (69), (73), (81), and (88), we get
(89)Tf1,f2K̇q·α·,p,θℝn≲∥f1∥K̇q1·α1·,p1,θℝnf2K̇q2·α2·,p2,θℝn,which completes the proof.



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*Source: 4845507-2022-03-28.xml*