# Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

Authors: Yanqi Yang; Shuangping Tao
Journal: Journal of Function Spaces
 (2017)

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## Abstract

LetT be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ(0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.


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## Body

## 1. Introduction

DenoteSn-1 to be the unit sphere in Rn(n≥2) with normalized Lebesgue measure dσ. The singular integral operator with variable kernel is defined by (1)Tfx=p.v.∫RnΩx,x-yx-ynfydy,where Ω(x,z) satisfies the following conditions: (2)Ωx,λz=Ωx,z,for  anyx,z∈Rn,λ>0,(3)∫Sn-1Ωx,z′dσz′=0,for  anyx∈Rn.As we all know, the singular integrals with variable kernel played an important role in the theory of nondivergent elliptic equations with discontinuous coefficients (see [1, 2]). Some properties for various of the singular integrals with variable kernel have been obtained by authors; for example, see [3–6] and their references. In the Mihlin conditions, Calderón and Zygmund proved the boundedness of T on the L2(Rn) (see [7]).Let0≤γ≤1. For tempered distributions f∈S′(Rn)(n=1,2,…), the fractional differentiation operators Dγ defined by Dγf^(ξ)=ξγf^(ξ); that is, Dγf(x)=(ξγf^(ξ))∨(x).LetIγ be the Riesz potential operator of order γ defined on the space of tempered distributions modulo polynomials by setting Iγf^(ξ)=ξ-γf^(ξ). It is easy to see that a locally integrable function b∈Iγ(BMO)(Rn) if and only if Dγb∈BMO(Rn). Strichartz (see [8]) showed that Iγ(BMO)(Rn) is a space of functions modulo constants which is properly contained in Lipγ(Rn), where γ∈(0,1).DenoteHm to be the space of spherical harmonical homogeneous polynomials of degree m. Let dim⁡Hm=dm and {Ym,j}j=1dm be an orthonormal system of Hm. It is well known that {Ym,j}j=1dm, m=0,1,…, is a complete orthonormal system in L2(Sn-1) (see [9]). Let us expand the function Ω(x,z′) in spherical harmonics (4)Ωx,z′=∑m≥0∑j=1dmam,jxYm,jz′,where (5)am,jx=∫Sn-1Ωx,z′Ym,jz′¯dσz′.If ∫Sn-1Ω(x,z′)dσ(z′)=0, then a0,j=0 for any x∈Rn. Let (6)Tm,jfx=Ym,j·n∗fx.Then T, defined in (1), can be written as (7)Tfx=∑m≥1∑j=1dmam,jxTm,jfx.Let T∗ and T♯ be the adjoint of T and the pseudoadjoint of T, respectively, defined by (8)T∗fx=∑m=1∞∑j=1dm-1mTm,ja¯m,jfx,T♯fx=∑m=1∞∑j=1dm-1ma¯m,jxTm,jfx.Let us give some necessary notations. In the following, unless otherwise stated, for aμ-measurable set E, χE denotes its characteristic function. We use the symbol A≲B to denote that there exists a positive constant C such that A≤CB. For any index p∈(1,∞), we denote by p′ its conjugate index; that is, 1/p+1/p′=1.LetT1 and T2 be the operators defined in (1) which are differentiated by their kernels Ω1(x,y) and Ω2(x,y). Let T1T2, T1∘T2 denote the product and pseudoproduct of T1 and T2, respectively. In [7], Calderón and Zygmund found that these operators are closely related to the second order linear elliptic equations with variable coefficients and established the following boundedness of the operators T1∗, T1♯, T1T2, T1∘T2, and D on Lp(Rn)(1<p<∞).Theorem A (see [7]).Let1<p<∞,Ω1(x,y),Ω2(x,y)∈Cβ(C∞),β>1satisfy (2) and (3). Then(1)
T1D-DT1fLp≲fLp;(2)
T1∗-T1♯DfLp≲fLp;(3)
(T1∘T2-T1T2)DfLp≲fLp.In 2015, Chen and Zhu proved that Theorem A was also true on Weighted Lebesgue space and Morrey space (see [10]). In 2016, Tao and Yang obtained the boundedness of those operators on the weighted Morrey-Herz spaces (see [6]). A natural question is whether these operators also have boundedness on the weak Morrey-Herz spaces. The answer is affirmative. The main purpose of this paper is to generalize the above results to the cases of weak Morrey-Herz spaces
W
M
K
˙
p
,
1
α
,
λ
(
R
n
)
(see Definition7 in the next section).Our main results are stated as follows.Theorem 1.
Let0<γ<1,λ-1<α<λ, and 0<p<∞. Assume that T is defined by (1) and Ω(x,y), which satisfies (2) and (3), meets the following condition: (9)maxj≤2n⁡Dxγ∂j∂yjΩx,yL∞Rn×Sn-1<∞.Then one has(1)
TDγ-DγTfWMK˙p,1α,λRn≲fMK˙p,1α,λRn;(2)
(T∗-T♯)DγfWMK˙p,1α,λ(Rn)≲fMK˙p,1α,λ(Rn).Theorem 2.
Let0<γ<1, λ-1<α<λ, and 0<p<∞. Suppose that Ω1(x,y) and Ω2(x,y) satisfy (2) and (3). If Ω2(x,y) satisfies (9) and Ω1(x,y) satisfies (10)maxj≤2n⁡∂j∂yjΩ1x,yL∞Rn×Sn-1<∞,then one has(11)T1∘T2-T1T2DγfWMK˙p,1α,λRn≲fMK˙p,1α,λRn.Furthermore, we also consider the casesγ=0 and γ=1. As we all know, D is the square root of Laplacian operator and D0 is the identity operator I. In this case, we obtain the following results.Theorem 3.
Letλ-1<α<λ and 0<p<∞. Suppose that Ωi(x,y)(i=1,2) satisfies (2), (3), and (10). Then one has(1)
T1I-IT1fWMK˙p,1α,λRn≲fMK˙p,1α,λRn;(2)
T1∗-T2♯IfWMK˙p,1α,λRn≲fMK˙p,1α,λRn;(3)
(T1∘T2-T1T2)IfWMK˙p,1α,λ(Rn)≲fMK˙p,1α,λ(Rn).Theorem 4.
Letλ-1<α<λ and 0<p<∞. Suppose that Ω(x,y) satisfies (2), (3), and (12)maxj≤2n⁡∇x∂j∂yjΩx,yL∞Rn×Sn-1<∞.Then one has(1)
TD-DTfWMK˙p,1α,λRn≲fMK˙p,1α,λRn;(2)
(T∗-T♯)DfWMK˙p,1α,λ(Rn)≲fMK˙p,1α,λ(Rn).Theorem 5.
Letλ-1<α<λ and 0<p<∞. Suppose that Ω1(x,y) and Ω2(x,y) satisfy (2) and (3). If Ω1(x,y) satisfies (10) and Ω2(x,y) satisfies (12), then one has (13)T1∘T2-T1T2DfWMK˙p,1α,λRn≲fMK˙p,1α,λRn.


## 2. Preliminaries and Main Lemmas

In this section, we shall recall the definitions of the homogeneous Morrey-Herz spaces and weak Morrey-Herz spaces. Furthermore, the weak estimates ofTm,j defined by (6) and a class of Calderón-Zygmund operators will be established on Morrey-Herz spaces.The well-known Morrey spaces, introduced originally by Morrey [11] in relation to the study of partial differential equations, were widely investigated during last decades, including the study of classical operators of harmonic analysis in various generalizations of these spaces. Morrey-type spaces appeared to be quite useful in the study of the local behavior of the solutions of partial differential equations, a priori estimates, and other topics. They are also widely used in applications to regularity properties of solutions to PDE including the study of Navier-Stokes equations (see [12] and references therein). The ideas of Morrey [11] were further developed by Campanato [13]. A more systematic study of these (and even more general) spaces, we refer the readers to see [12, 14–21]. In 1964, Beurling [22] first introduced some fundamental forms of Herz spaces to study convolution algebras. Later Herz [23] gave versions of the spaces defined below in a slightly different setting. Since then, the theory of Herz spaces has been significantly developed, and these spaces have turned out to be quite useful in harmonic analysis. For instance, they were used by Baernstein and Sawyer [24] to characterize the multipliers on the classical Hardy spaces and used by Lu and Yang [25] in the study of partial differential equations. More results and further details can be found in [26–28]. On the basis of above available results, the theory of the homogeneous Morrey-Herz spaces goes back to Lu-Xu [29] who considered the boundedness of a class of sublinear operators; also see [6, 30, 31] for more further results.Next we give the following notation. For eachk∈Z, we denote Bk={x∈Rn:x≤2k} and Ak=Bk-Bk-1, χk=χAk(x).Definition 6 (see [29]).
Letα∈Rn, 0<p≤∞, 0<q<∞, and λ≥0. The homogeneous Morrey-Herz spaces MK˙p,qα,λ(Rn) are defined by (14)MK˙p,qα,λRn=f∈LlocqRn∖0,fMK˙p,qα,λRn<∞,where (15)fMK˙p,qα,λRn=supk0∈Z⁡2-k0λ∑k=-∞k02kαpfχkLqRnp1/p,and the usual modifications should be made when p=∞.In what follows, for anyk∈Z and γ>0, let mk(γ,f)=x∈Ak:fx>γ.Definition 7 (see [29]).
Letα∈Rn, 0<p≤∞, 0<q<∞, and λ≥0. A measurable function f is said to belong to the homogeneous weak Morrey-Herz spaces WMK˙p,qα,λ(Rn), if (16)WMK˙p,qα,λRn=f∈LlocqRn∖0,fWMK˙p,qα,λRn<∞,where (17)fWMK˙p,qα,λRn=supγ>0⁡γsupk0∈Z⁡2-k0λ∑k=-∞k02kαpmkγ,fp/q1/p<∞,where the usual modifications are made when p=∞.Lemma 8 (see [32]).
(
X
,
d
,
μ
) is said to be a homogeneous space. Let 0≤λ≤∞, 0≤l<1, 1<q1<1/l, 1/q2=1/q1-l, λ-1/q2<α<λ+1-1/q1, and 0<p1≤p2<∞, if a sublinear operator Tl meets the following requirements: (18)Tlfx≤C∫XfyμBx0,dx,y1-ldμy,x∉supp⁡f.And Tl is bounded from Lq1(X) to Lq2,∞(X), and then Tl is a bounded operator from MK˙p1,q1α,λ(X) to WMK˙p2,q2α,λ(X). When q1=1, as can be seen from the proof process, the above conclusion is still valid.Lemma 9.
LetTm,j be defined by (6). Then Tm,j is weak (1,1), and (19)x∈Rn:Tm,jf>λ≲mn/21λf1.Proof.
LetTm,jf=Ym,j/·n∗f and Ym,j≲m(n-2)/2 ([7]). Fix λ>0 and f∈L1(Rn). Form the Calderón-Zygmund decomposition of f at height λ. Then for any λ>0, there exists a decomposition of Rn:Rn=Qλ+Pλ such that(1)
fx≤λa.e.x∈Pλ,(2)
Qλ=⋃l=1∞Ql, where Ql are finite overlapping, and (20)λ<1Ql∫Qlfxdx≤2nλ.
We now decomposef as the sum of two functions, g and b, defined by (21)gx=fx,x∉Qλ;1Ql∫Qlfydy,x∈Ql,bx=∑lblx,where (22)blx=fx-1Ql∫QlfydyχQlx.Then g(x)≤2nλ almost everywhere, and bl is supported on Ql and has zero integral. Since Tm,jf=Tm,jg+Tm,jb, (23)x∈Rn:Tm,jfx>λ≤x∈Rn:Tm,jgx>λ2+x∈Rn:Tm,jbx>λ2.
We estimate the first term, in view of the fact thatTm,jfLp(Rn)≲mn/2fLp(Rn) ([10]). Without loss of generality, for p=2, we have (24)x∈Rn:Tm,jgx>λ2≤2λ2∫RnTm,jgx2dx≲mn/24λ2∫Rngx2dx≲mn/22n+2λ∫Rngxdx=mn/22n+2λ∫Rnfxdx.Let 2Ql be the cube with the same center as Ql and whose sides are twice as long, and let (Qλ)∗=⋃l2Ql. Then we have Qλ∗≤2nQλ, and together with characteristic of left in (2), we can obtain (25)Qλ=⋃l=1∞Ql≤∑l=1∞Ql≤1λ∑l=1∞∫Qlfxdx≤1λ∫Qλfxdx≤1λf1.Thus we have (26)x∈Rn:Tm,jbx>λ2≤Qλ∗+x∉Qλ∗:Tm,jbx>λ2≤2nλf1+2λ∫Rn∖Qλ∗Tm,jbxdx.Notice that Tm,jbx≤∑lTm,jblx almost everywhere. Hence, to complete the proof of the weak (1, 1) inequality it will suffice to show that (27)∑l∫Rn∖2QlTm,jblxdx≲f1.
For anybl, x∉2Ql, the formula (28)Tm,jblx=∫QlblyYm,jx-yx-yndyis still valid. Denote the center of Ql by zQl, and then we have (29)∫Rn∖2QlTm,jblxdx=∫Rn∖2Ql∫QlYm,jx-yblyx-yndydx≤∫Rn∖2Ql∫QlblyYm,jx-y1x-yn-1x-zQlndydx≲mn/2∫Qlbly∫Rn∖2Qly-zQlnx-ynx-zQlndxdy≲mn/2∫Qlbly∫Rn∖2QlQlx-zQl2ndxdy.Noticing that (30)∫Rn∖2QlQlx-zQl2ndx=∑k=1∞∫2k+1Ql∖2kQlQlx-zQl2ndx≤∑k=1∞2k+1QlQll2kQl2n=∑k=1∞2-kn+n<∞,it follows that(31)∑l∫Rn∖2QlTm,jblxdx≲mn/2∑l∫Qlblydy≲mn/2f1.Then for any Tm,j, it is clear that (32)x∈Rn:Tm,jbx>λ2≲mn/21λf1.Summing up the estimates above for Tm,jg(x) and Tm,jb(x), we finish the proof of Lemma 9.Lemma 10.
Let0≤λ≤∞, λ-1<α<λ, and 0<p<∞. Let T be a generalized Calderón-Zygmund operator, and then T is bounded from MK˙p,1α,λ(Rn) to WMK˙p,1α,λ(Rn); namely, (33)TfxWMK˙p,1α,λRn≲fxMK˙p,1α,λRn.Proof.
It is well known thatT is weak (1) (e.g., see [33]). Noticing that T satisfying (18) with l=0, then we can obtain Lemma 10 by using Lemma 8.Lemma 11.
Let0≤λ≤∞, λ-1<α<λ, and 0<p<∞; Tm,j defined by (6) is bounded from MK˙p,1α,λ(Rn) to WMK˙p,1α,λ(Rn), and (34)Tm,jfxWMK˙p,1α,λRn≲mn/2fxMK˙p,1α,λRn.Proof.
By applying the fact thatTm,jf(x)=Ym,j/·n∗f(x) and Ym,j≲m(n-2)/2 ([7]), we can easily obtain (35)Tm,jfx≲mn-2/2∫Rnfyx-yndy≲mn/2∫Rnfyx-yndy.Noticing that the interior integral above is meeting the condition in (18) for l=0 and Tm,j is weak (1) on the basis of Lemma 9, it is not difficult to deduce that Tm,j is bounded from MK˙p,1α,λ(Rn) to WMK˙p,1α,λ(Rn). Hence (36)Tm,jfxWMK˙p,1α,λRn≲mn/2fxMK˙p,1α,λRn.This completes the proof of Lemma 11.Lemma 12.
Lett(x) be a homogeneous of degree -n-1 and locally integrable in x>0. Let b∈Lip(Rn) and (37)Kfx=limε↦0⁡∫x-y>εtxbx-byfydy.If t(x)∈C1(Sn-1) and ∫Sn-1t(x)xjdσ(x)=0, j=1,…,n, then, for 0≤λ≤∞, λ-1<α<λ, and 0<p<∞, one has (38)KfWMK˙p,1α,λRn≲∇tL∞Sn-1+tL∞Sn-1∇bL∞fMK˙p,1α,λRn.Proof.
Letk(x,y)=t(x-y)(b(x)-b(y)). For all x,x0,y∈Rn with x-x0≤1/2y-x, then k satisfies the following inequalities: (39)kx,y-kx0,y≲∇tL∞Sn-1∇bL∞x-x0y-x-n-1,kx,y≲tL∞Sn-1∇bL∞y-x-n.This, together with the boundedness of K on L2(Rn) (see [34]), tells us that K is a generalized Calderón-Zygmund operator ([33]). Thus, by applying Lemma 10, we see that K is bounded from MK˙p,1α,λ(Rn) to WMK˙p,1α,λ(Rn) with bound (∇tL∞(Sn-1)+tL∞(Sn-1))∇bL∞ for 0≤λ≤∞, λ-1<α<λ, and 0<p<∞.
Therefore, the proof of Lemma12 is finished.Lemma 13.
Letb∈Lip(Rn) and T be a singular operator which is defined by (40)Tfx=limε↦0⁡∫x-y>εKx-yfydy,where K(x)∈C3(Sn-1), ∫Sn-1K(x)dσ(x)=0, and K(λx)=λ-nK(x), for x∈Rn∖{0}, λ>0, and then one has that, for 0≤λ≤∞, λ-1<α<λ, 0<p<∞, and f∈C0∞, the operator (41)b,T∂f∂xjWMK˙p,1α,λRn≲maxβ≤2∂βKL∞Sn-1∇bL∞fMK˙p,1α,λRn.Proof.
With an argument similar to that used in the proof of Lemma  5.2 in [10], it is not difficult to obtain Lemma 13. Thus, we omit the details here.


## 3. Proofs of Theorems

Proof of Theorem1.
Let(42)Ωx,y=∑m≥1∑j=1dmam,jxYm,jy.From [3], for any x, we can write the coefficients am,j as(43)am,jx=-1nm-nm+n-2-n∫Sn-1Ly′nΩx,y′Ym,jy′dσy′,m≥1,where L(F)=x2ΔF(x).
We will firstly prove conclusion(1). Write (44)TDγ-DγTf=∑m=1∞∑j=1dmam,jTm,jDγ-Dγam,jTm,jf=∑m=1∞∑j=1dmam,jDγTm,j-Dγam,jTm,jf≕∑m=1∞∑j=1dmam,j,DγTm,jf.By (43), it follows that (45)Dγam,jx=-1nm-nm+n-2-n∫Sn-1DxγLy′nΩx,y′Ym,jy′dσy′,m≥1.Then, we have by (9)(46)Dγam,jL∞≲m-2n.Moreover, by the fact that [b,Dγ] is a generalized Calderón-Zygmund operator (see [35]), which is defined by (47)b,Dγfx=Cγ∫Rnbx-byx-yn+γfydy.Thus we can get that [b,Dγ]f(x) is bounded from MK˙p,1α,λ(Rn) to WMK˙p,1α,λ(Rn) by applying Lemma 10; namely, (48)b,DγfWMK˙p,1α,λRn≲DγbBMOfMK˙p,1α,λRn.Then by dm≃mn-2 (see [36]), (46), (48), and Lemma 11, we have (49)TDγ-DγTfWMK˙p,1α,λRn≤∑m=1∞∑j=1dmam,j,DγTm,jfWMK˙p,1α,λRn≲∑m=1∞∑j=1dmDγam,jBMOTm,jfWMK˙p,1α,λRn≲∑m=1∞∑j=1dmmn/2Dγam,jL∞fMK˙p,1α,λRn≲∑m=1∞mn-2mn/2m-2nfMK˙p,1α,λRn≲fMK˙p,1α,λRn.
Now let us turn to estimate(2). By applying the definition of T♯ and T∗ we can deduce that (50)T♯-T∗Dγf=∑m=1∞∑j=1dm-1ma-m,j,Tm,jDγf.In order to estimate WMK˙p,1α,λ(Rn) norm of (T∗-T♯)Dγ, we first consider [b,Tm,j]Dγ for any fixed b∈Iγ(BMO). Noting that b(x)-b(y)=(b(x)-b(z))-(b(y)-b(z)), for any x,y,z∈Rn, then we have (51)b,Tm,jDγf=b,DγTm,jf-Tm,jb,Dγf.Thus, by (48) and Lemma 11, we get (52)Tm,jb,DγfWMK˙p,1α,λRn≲mn/2DγbBMOfMK˙p,1α,λRn.Further, we estimate the WMK˙p,1α,λ(Rn) norm of [b,DγTm,j]. From the fact that [b,DγTm,j]f is a generalized Calderón-Zygmund operator with kernel (see [10]) (53)km,jx,y≲mn/2-1+γDγbBMO1x-yn,then we get by Lemma 10(54)b,DγTm,jfWMK˙p,1α,λRn≲mn/2+γDγbBMOfMK˙p,1α,λRn.Then, combining (52) with (54), we have (55)b,Tm,jDγfWMK˙p,1α,λRn≲b,DγTm,jfWMK˙p,1α,λRn+Tm,jb,DγfWMK˙p,1α,λRn≲mn/2+γDγbBMOfMK˙p,1α,λRn+mn/2DγbBMOfMK˙p,1α,λRn≲mn/2+γDγbBMOfMK˙p,1α,λRn.By estimates (46), (50), and (55), we get (56)T♯-T∗DγfWMK˙p,1α,λRn≤∑m=1∞∑j=1dma¯m,j,Tm,jDγfWMK˙p,1α,λRn≲∑m=1∞∑j=1dmmn/2+γDγa¯m,jBMOfMK˙p,1α,λRn≲∑m=1∞∑j=1dmmn/2+γDγa¯m,jL∞fMK˙p,1α,λRn≲∑m=1∞mn-2mn/2+γm-2nfMK˙p,1α,λRn≲fMK˙p,1α,λRn.Thus we finish the proof of Theorem 1.Proof of Theorem2.
Let(57)T1fx=∫RnΩ1x,x-yx-ynfydy,T2fx=∫RnΩ2x,x-yx-ynfydy.Write (58)Ω1x,y=∑m≥1∑j=1dmam,jxYm,jy,Ω2x,y=∑λ≥1∑μ=1dλbλ,μxYλ,μy,where (59)am,jx=∫Sn-1Ω1x,z′Ym,jz′¯dσz′,bλ,μx=∫Sn-1Ω2x,z′Yλ,μz′¯dσz′.
For anyx∈Rn, with a similar argument used in the proof of Theorem 1 in terms of (9) and (10), we can obtain that (60)am,jL∞≲m-2n,Dγbλ,μL∞≲m-2n.
LetTm,jf(x)=Ym,j/·n∗f(x) and Tλ,μf(x)=Yλ,μ/·n∗f(x). Since Ω1(x,y) and Ω2(x,y) satisfy (3), then we get (61)T1fx=∑m≥1∑j=1dmam,jxTm,jfx,T2fx=∑λ≥1∑μ=1dλbλ,μxTλ,μfx.Write ([10])(62)T1∘T2fx=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jxbλ,μxTm,jTλ,μfx,T1T2fx=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jTm,jbλ,μTλ,μfx.Then (63)T1∘T2-T1T2Dγf=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jbλ,μxTm,j-Tm,jbλ,μxTλ,μDγf=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jbλ,μxTm,j-Tm,jbλ,μxDγTλ,μf=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jbλ,μ,Tm,jDγTλ,μf.Therefore, together with (55), (60), and Lemma 11, we obtain (64)T1∘T2-T1T2DγfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞bλ,μ,Tm,jDγTλ,μfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞Dγbλ,μBMOmn/2+γTλ,μfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞Dγbλ,μL∞mn/2+γλn/2fMK˙p,1α,λRn≲∑m=1∞mn-2m-2nmn/2+γ∑λ=1∞λn-2λ-2nλn/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn.This finishes the proof of Theorem 2.Proof of Theorem3.
We can estimate term (1) exactly as we did for the corresponding boundedness in Theorem1 in the above arguments. Thus, we have only to prove (2) and (3) of Theorem 3. In order to do this, we use the same notations as in the proof of Theorem 2. By using the fact that Ω1(x,y) and Ω2(x,y) satisfy (10), therefore, we have (65)am,jL∞≲m-2n,bλ,μL∞≲λ-2n.Firstly, let us prove (2). As in the proof of Theorem 1, we can get (66)T1♯-T1∗If=∑m=1∞∑j=1dm-1ma¯m,j,Tm,jIf.As [b,Tm,j] is a special Calderón-Zygmund operator, it is bounded from Morrey-Herz spaces MK˙p,1α,λ(Rn) to weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by applying Lemma 10. Thus we have (67)b,Tm,jfWMK˙p,1α,λRn≲mn/2bL∞fMK˙p,1α,λRn.Then by (65), we get (68)T1♯-T1∗IfWMK˙p,1α,λRn≲∑m=1∞mn-2m-3n/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn.Thus conclusion (2) is proved.
We now estimate (3). Write(69)T1∘T2-T1T2If=∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλbλ,μ,Tm,jTλ,μIf.Therefore, by (65), (67), and Lemma 11, we get (70)T1∘T2-T1T2IfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞bλ,μL∞mn/2Tλ,μIfWMK˙p,1α,λRn≲∑m=1∞mn-2m-2nmn/2∑λ=1∞λn-2λ-2nλn/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn.Thus conclusion (3) is also proved. Hence the proof of Theorem 3 is finished.Proof of Theorem4.
In the first place, we will prove conclusion (1). WriteD=∑k=1nRk(∂/∂xk), where Rk denotes the Riesz transform. As in the proof of Theorem 1, we have (71)TD-DTfx=∑m=1∞∑j=1dmam,j,DTm,jfx=∑m=1∞∑j=1dm∑k=1nam,j,Rk∂∂xkTm,jfx=∑m=1∞∑j=1dm∑k=1nRkam,j,∂∂xkTm,jfx+∑m=1∞∑j=1dm∑k=1nam,j,Rk∂∂xkTm,jfx≕I1+I2.We have by Leibniz’s rules that (72)I1=∑m=1∞∑j=1dm∑k=1nRk∂∂xkam,jTm,jf. Thus we deduce from (43) that (73)am,j∂xkx=-1nm-nm+n-2-n∫Sn-1∂xkLy′nΩx,y′Ym,jy′dσy′,m≥1. From this and (12), we get for k=1,…,n, (74)∂am,j∂xkL∞≲m-2n.By using the fact that RkgWMK˙p,1α,λ(Rn)≲gMK˙p,1α,λ(Rn), dm≃mn-2, and Lemma 11, then we have (75)I1WMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑k=1nRk∂∂xkam,jTm,jfWMK˙p,1α,λRn≲∑m=1∞∑j=1dmm-2nmn/2fMK˙p,1α,λRn≲∑m=1∞mn-2m-2nmn/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn.By Lemma 13 and (74), a trivial computation shows that, for I2, (76)I2WMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑k=1n∇am,jL∞Tm,jfWMK˙p,1α,λRn≲∑m=1∞∑j=1dmm-2nmn/2fMK˙p,1α,λRn≲∑m=1∞mn-2m-2nmn/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn. Combining the estimates above, we arrive at the desired boundedness (77)TD-DTfWMK˙p,1α,λRn≲fMK˙p,1α,λRn.We posterior prove conclusion (2). Write D=∑k=1nRk(∂/∂xk); we have (78)T♯-T∗Dfx=∑m=1∞∑j=1dm-1ma¯m,j,Tm,jDfx=∑k=1n∑m=1∞∑j=1dm-1ma¯m,j,Tm,j∂∂xkRkfx.We now turn to estimate the WMK˙p,1α,λ(Rn) norm of [a¯m,j,Tm,j](∂/∂xk)(Rkf). Applying (74), Lemma 13, and the fact that for any multi-index β and x∈Rn∖{0}, m=1,2,…. (see [7]), (79)∂βxmYm,j≤Cnxm-βmβ+n-2/2.Hence, we get (80)a¯m,j,Tm,j∂∂xkRkfWMK˙p,1α,λRn≲∇a¯m,jL∞maxβ≤2⁡∂βYm,jL∞Sn-1RkfWMK˙p,1α,λRn≲m-2nmn/2+1fMK˙p,1α,λRn≲m-3n/2+1fMK˙p,1α,λRn.Combining the estimates of (78) with (80), we have (81)T♯-T∗DfWMK˙p,1α,λRn≲∑m=1∞mn-2m-3n/2+1fMK˙p,1α,λRn≲fMK˙p,1α,λRn.Consequently, the proof of Theorem 4 is completed.Proof of Theorem5.
Similar to the proof of Theorem2, we easily see that (82)T1∘T2-T1T2Df=∑m=1∞∑d=1dm∑λ=1∞∑μ=1dλam,jbλ,μ,Tm,jDTλ,μf,where am,j and bλ,μ are same to occur in the proof of Theorem 2. By (10) and (12), we have (83)am,jL∞≲m-2n,∇bλ,μL∞≲λ-2n.Write D=∑k=1n(∂/∂xk)Rk, and it then follows that (84)T1∘T2-T1T2DfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞bλ,μ,Tm,j∑k=1n∂∂xkRkTλ,μfWMK˙p,1α,λRn≲∑k=1n∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞bλ,μ,Tm,j∂∂xkRkTλ,μfMK˙p,1α,λRn.The above estimates, via Lemma 13, lead to that (85)T1∘T2-T1T2DfWMK˙p,1α,λRn≲∑m=1∞∑j=1dm∑λ=1∞∑μ=1dλam,jL∞∇bλ,μL∞maxβ≤2∂βYm,jL∞Sn-1Tλ,μRkfWMK˙p,1α,λRn.We thus obtain from (79), (83), and Lemma 11 that (86)T1∘T2-T1T2DfWMK˙p,1α,λRn≲∑m=1∞mn/2-1m-2nmn/2+1∑λ=1∞λn/2-1λ-2nλn/2fMK˙p,1α,λRn≲fMK˙p,1α,λRn.Consequently, the proof of Theorem 5 is finished.



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*Source: 4340805-2017-12-31.xml*