# Weighted Estimates for Commutators ofn-Dimensional Rough Hardy Operators

Authors: Zhuanxi Ren; Shuangping Tao
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We establish the weighted estimates for the commutatorsHΩ,b and HΩ,b∗ which are generated by the n-dimensional rough Hardy operators and central BMO functions on the weighted
Lebesgue spaces, the weighted Herz spaces and the weighted Morrey-Herz spaces. Furthermore, the weighted Lipschitz estimates are also obtained.


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## Body

## 1. Introduction

The classical Hardy operator and its adjoint operator are defined, respectively, by(1)hf(x)∶=1x∫0xf(y)dy,h*f(x)∶=∫x∞f(y)ydyx>0.
Hardy proved the following most celebrated inequality in [1]:
(2)∥hf∥Lp(ℝ+)≤pp-1∥f∥Lp(ℝ+),∥h*f∥Lq(ℝ+)≤pp-1∥f∥Lq(ℝ+),1<p<∞.
Moreover,
(3)∥h∥Lp(ℝ+)→Lp(ℝ+)=∥h*∥Lq(ℝ+)→Lq(ℝ+)=pp-1,
where 1/p+1/q=1. Hardy’s inequality has received considerable attention. In 1995, Christ and Grafakos obtained it on ℝn. Firstly, we recall that the definitions of the n-dimensional Hardy operator and its adjoint operator given by Christ and Grafakos are as follows:
(4)Hf(x)=1|x|n∫|y|<|x|‍f(y)dy,x∈ℝn∖{0},H*f(x)=∫|y|≥|x|‍f(y)|y|ndy.
Let 1<p, q<∞. Then Christ and Grafakos’s results in [2] are
(5)∥Hf∥Lp(ℝn)≤pνnp-1∥f∥Lp(ℝn),∥H*f∥Lq(ℝn)≤pνnp-1∥f∥Lq(ℝn),
where νn=πn/2/Γ(1+n/2) and the constant pνn/(p-1) is the best.Letf be a nonnegative integrable function on ℝn. The n-dimensional rough Hardy operator and its adjoint operator are defined, respectively, by
(6)HΩf(x)∶=1|x|n∫|y|<|x|Ω(x-y)f(y)dy,x∈ℝn∖{0},(7)HΩ*f(x):=∫|y|≥|x|Ω(x-y)f(y)|y|ndy,
where Ω∈Ls(Sn-1), 1<s≤∞, is homogeneous of degree zero. Then the commutators generated by HΩ or HΩ* and a locally integrable function b are defined, respectively, as follows:
(8)HΩbf(x)∶=1|x|n∫|y|<|x|(b(x)-b(y))Ω(x-y)f(y)dy,x∈ℝn∖{0},(9)HΩ*,bf(x):=∫|y|≥|x|(b(x)-b(y))Ω(x-y)f(y)|y|ndy.In [3], it was proved that the commutators HΩb were bounded on the Lebesgue spaces and Herz spaces if b∈CM˙Omax{q,u}(ℝn). Recently, Gao obtained in [4] that HΩb is also bounded from the Morrey-Herz spaces MK˙p1,q1α,λ(ℝn) to MK˙p2,q2α,λ(ℝn) if b∈CM˙Omax{q2,u}(ℝn). On the other hand, Gao and Wang in [5] had established the weighted estimates on weighted Lebesgue and Herz-type spaces for the commutators Hb and Hb* which are generated by n-dimensional Hardy operator H and H* and b∈Lipβ,ω. It is easy to see that HΩb=Hb and HΩ*,b=Hb* when Ω=1. A natural question is whether commutators of n-dimensional standard rough Hardy operators HΩb and HΩ*,b also have boundedness on these weighted spaces. The answer is affirmative. The main purpose of this paper is to generalize the above results on the weighted Lebesgue spaces, the weighted Herz spaces, and the weighted Morrey-Herz spaces.First let us recall some standard definitions and notations before introducing our main results. The classicalAp weighted theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of the Hardy-Littlewood maximal functions in [6]. A weight ω is a locally integrable function on ℝn which takes values in (0,∞) at almost everywhere. Let B=B(y,r) denote the ball with the center y and radius r. C is a constant which may vary from line to line. For k∈ℤ, let Bk={x∈ℝn:|x|≤2k} and Δk=Bk∖Bk-1. We use χk to denote the characteristic functions of the set Δk. We also denote the weighted measure of E by ω(E); that is, ω(E)=∫E‍ω(x)dx. And let q′ be the conjugate index of q whenever q≥1, 1/q+1/q′=1.Definition 1.
We say thatω∈Ap, 1<p<∞, if
(10)(1|B|∫Bω(x)dx)(1|B|∫Bω(x)-1/(p-1)dx)p-1≤CforeveryballB⊂ℝn,
where C is a positive constant which is independent of the choice of B.For the casep=1, ω∈A1, if
(11)1|B|∫Bω(x)dx≤C·essinfx∈Bω(x)foreveryballB⊂ℝn.
A weight function ω∈A∞ if it satisfies the Ap condition for some 1<p<∞.Definition 2 (see [7]).
Letα∈ℝ, 0<p, q≤∞, and ω1 and ω2 a weighted function. Then the homogeneous weighted Herz space K˙qα,p(ω1,ω2) is defined by
(12)K˙qα,p(ω1,ω2)={f∈Llocq(ℝn∖{0},ω2):∥f∥K˙qα,p(ω1,ω2)<∞},
where
(13)∥f∥K˙qα,p(ω1,ω2)=(∑k=-∞∞ω1(Bk)αp/n∥fχk∥Lq(ω2)p)1/p,
with the usual modification made when p=∞ or q=∞.Obviously, whenω1=ω2=1, K˙qα,p(ω1,ω2)=K˙qα,p(ℝn); when α=0, p=q, K˙qα,p(ω1,ω2)=Lp(ω2).Definition 3 (see [8]).
Letα∈ℝ, λ≥0, 0<p, q≤∞, and ω1 and ω2 a weighted function. Then the homogeneous weighted Morrey-Herz space MK˙p,qα,λ(ω1,ω2) is defined by
(14)MK˙p,qα,λ(ω1,ω2)={f∈Llocq(ℝn∖{0},ω2):∥f∥MK˙p,qα,λ(ω1,ω2)<∞},
where
(15)∥f∥MK˙p,qα,λ(ω1,ω2)=supk0∈Zω1(Bk0)-λ/n×(∑k=-∞k0ω1(Bk)αp/n∥fχk∥Lq(ω2)p)1/p,
with the usual modification made when p=∞ or q=∞.Obviously, whenλ=0, MK˙p,qα,λ(ω1,ω2)=K˙qα,p(ω1,ω2); when ω1=ω2=1, MK˙p,qα,λ(ω1,ω2)=MK˙p,qα,λ(ℝn).Definition 4 (see [9]).
Let1≤p≤∞, 0<β<1, and ω a weighted function. We say that a locally integrable function f belongs to the weighted Lipschitz space Lipβ,ωp if
(16)supB⊂ℝn1ω(B)β/n(1ω(B)∫B|f(x)-fB|pω(x)1-pdx)1/p≤C<∞,
where fB=(1/|B|)∫B‍f(x)dx.The smallest boundC satisfying conditions above is then taken to be the norm of f in this space and is denoted by ∥f∥Lipβ,ωp. We also put Lipβ,ω=Lipβ,ω1. Obviously, for the case ω=1, the space Lipβ,ω is the classical Lipschitz space Lipβ. If ω∈A1(ℝn), then García-Cuerva in [9] proved that the space Lipβ,ωp coincide for any 1<p≤∞ and the norms of ∥f∥Lipβ,ωp are equivalent with respect to different values of p. That is ∥f∥Lipβ,ωp~∥f∥Lipβ,ω.Definition 5 (see [5]).
Let1≤p<∞, and ω a weighted function. A function f∈Llocp(ℝn) is said to belong to the weighted central BMO space CM˙Op(ω) if
(17)∥f∥CM˙Op(ω)=supr>0(1ω(B(0,r))∫B(0,r)|f(x)-fB|pω(x)1-pdx)1/p≤C<∞,
where fB=(1/|B(0,r)|)∫B(0,r)‍f(x)dx.Obviously,CM˙Opq(ω)⊆CM˙Op(ω)(1≤p<q<∞). When ω=1, CM˙Op(ω)=CM˙Op(ℝn).The organization of this paper is as follows. In Section2, we shall present our main results. Finally, in Section 3, we shall give the proofs of theorems.


## 2. Main Results

Now, we present our main results as follows.Theorem 6.
Let1<p, q<∞,  ω∈A1, and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that HΩb and HΩ*,b are defined by (9) and b∈CM˙Opmax{q,q′}(ω); then HΩb and HΩ*,b are bounded from Lq(ω) to Lq(ω1-q).Theorem 7.
Let1<p, q<∞, 0<p1≤p2≤∞, ω∈A1, and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that b∈CM˙Opmax{q,q′}(ω); then(a)
HΩb is bounded from K˙qα,p1(ω,ω) to K˙qα,p2(ω,ω1-q) if α<nδ/q′;(b)
HΩ*,b is bounded from K˙qα,p1(ω,ω) to K˙qα,p2(ω,ω1-q) if α>-nδ/q.Remark 8.
δ appearing in Theorem 7 and the following theorems is defined by the Lemma 16 in the next section.Remark 9.
Letα=0 and p1=p2=q in Theorem 7; then Theorem 6 can be obtained.Theorem 10.
Let1<p, q<∞, 0<p1≤p2≤∞, ω∈A1 and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that b∈CM˙Opmax{q,q′}(ω) and λ≥0, then(a)
HΩb is bounded from MK˙p1,qα,λ(ω,ω) to MK˙p2,qα,λ(ω,ω1-q) if α<nδ/q′+λ;(b)
HΩ*,b is bounded from MK˙p1,qα,λ(ω,ω) to MK˙p2,qα,λ(ω,ω1-q) if α>-nδ/q+λ.Theorem 11.
Let1<p, q<∞, 1/q=(1/p)-(β/n), ω∈A1, and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that b∈Lipβ,ω for 0<β<1; then HΩb and HΩ*,b are bounded from Lp(ω) to Lq(ω1-q).Theorem 12.
Let0<p1≤p2<∞, 1<q1, q2<∞, 1/q2=(1/q1)-(β/n), ω∈A1, and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that b∈Lipβ,ω for 0<β<1; then(a)
HΩb is bounded from K˙q1α,p1(ω,ω) to K˙q2α,p2(ω,ω1-q2) if α<nδ/q1′;(b)
HΩ*,b is bounded from K˙q1α,p1(ω,ω) to K˙q2α,p2(ω,ω1-q2) if α>-nδ/q2.Remark 13.
Letα=0, p1=q1=p, and p2=q2=q in Theorem 12; then Theorem 11 can be obtained.Theorem 14.
Let0<p1≤p2<∞, 1<q1, q2<∞, 1/q2=(1/q1)-(β/n), ω∈A1, and Ω∈Ls(Sn-1) with homogeneous of degree zero for some 1<s≤∞. Suppose that b∈Lipβ,ω for 0<β<1 and λ≥0; then(a)
HΩb is bounded from MK˙p1,q1α,λ(ω,ω) to MK˙p2,q2α,λ(ω,ω1-q2) if α<nδ/q1′+λ;(b)
HΩ*,b is bounded from MK˙p1,q1α,λ(ω,ω) to MK˙p2,q2α,λ(ω,ω1-q2) if α>-nδ/q2+λ.Remark 15.
AsΩ=1, our results are consistent with the main results in [5].


## 3. Proofs of the Main Results

In this section, we shall give the proofs of Theorems7, 10, 12, and 14. In order to do this, we shall need the following lemmas.Lemma 16 (see [10]).
Letω∈A1. Then there exist constants C1, C2, and 0<δ<1 depending only on A1-constant of ω, such that for any measurable subset  E of a ball B,
(18)C1|E||B|≤ω(E)ω(B)≤C2(|E||B|)δ.Remark 17.
Ifω∈A1, it is easy to see from Lemma 16 that there exists a constant C and δ(0<δ<1) such that ω(Bk)/ω(Bj)≤C2(k-j)n as k>j and ω(Bk)/ω(Bj)≤C2(k-j)nδ as k≤j.Lemma 18 (see [5]).
Letω∈A1 and b∈CM˙Op(ω). Then there exists a constant C such that, for any j>k,
(19)|bBj-bBk|≤C(j-k)∥b∥CM˙Op(ω)ω(Bk)|Bk|.Lemma 19 (see [11]).
Letω∈A1 and b∈Lipβ,ω. Then there exists a constant C such that, for any j>k,
(20)|bBj-bBk|≤C(j-k)∥b∥Lipβ,ωω(Bj)β/nω(Bk)|Bk|.Lemma 20 (see [11]).
Letω∈A1. Then, for any 1≤p<∞,
(21)∫Bω(x)1-p′dx≤C|B|p′ω(B)1-p′,
where 1/p+1/p′=1.We are now in a position to give the proof of Theorem7.Proof of Theorem7.
(a) From the definition in (9), we readily see that
(22)∥(HΩbf)χk∥Lq(ω1-q)q≤C2-kqn∫Δk(∑i=-∞k∫Δi|f(y)Ω(x-y)×(b(x)-bBk)|dy∑i=-∞k∫Δi|f(y)Ω(x-y))q×ω(x)1-qdx+C2-kqn∫Δk(∑i=-∞k∫Δi|f(y)Ω(x-y)×(b(y)-bBk)|dy∑i=-∞k∫Δi|f(y)Ω(x-y))q×ω(x)1-qdx=I+II.
ForI, noticing that ω∈A1⊂Aq and then by Lemma 20 and Hölder’s inequality it follows that
(23)∫Δi|f(y)|dy≤(∫Δi|f(y)|qω(y)dy)1/q(∫Δiω(y)1-q′dy)1/q′≤∥fχi∥Lq(ω)|Bi|ω(Bi)-1/q.
When Ω∈L∞(Sn-1), by inequality (23) and Remark 17, we have
(24)I≤C2-kqn∫Δk|b(x)-bBk|qω(x)1-q×(∑i=-∞k∫Δi|f(y)Ω(x-y)|dy)qdx≤C2-kqn∥b∥CM˙Oq(ω)qω(Bk)×(∑i=-∞k∥Ω∥L∞(Sn-1)∫Δi|f(y)|dy)q≤C∥Ω∥L∞(Sn-1)q2-kqn∥b∥CM˙Oq(ω)qω(Bk)×(∑i=-∞k∥fχi∥Lq(ω)|Bi|ω(Bi)-1/q)q=C∥Ω∥L∞(Sn-1)q∥b∥CM˙Oq(ω)q×(∑i=-∞k2(i-k)n(ω(Bk)ω(Bi))1/q∥fχi∥Lq(ω))q≤C∥Ω∥L∞(Sn-1)q∥b∥CM˙Oq(ω)q×(∑i=-∞k2(i-k)n/q′∥fχi∥Lq(ω))q.
When Ω∈Ls(Sn-1), 1<s<∞, also by inequality (23) and Remark 17 we can obtain that
(25)I≤C2-kqn∫Δk‍|b(x)-bBk|qω(x)1-q×(∑i=-∞k∫Δi|f(y)Ω(x-y)|dy)qdx≤C2-kqn∥b∥CM˙Oq(ω)qω(Bk)×(∑i=-∞k(∫Δi|f(y)|s′dy)1/s′×(∫Δi|Ω(x-y)|sdy)1/s)q≤C2-kqn∥b∥CM˙Oq(ω)qω(Bk)×(∑i=-∞k∥Ω∥Ls(Sn-1)∫Δi|f(y)|dy)q≤C∥Ω∥Ls(Sn-1)q∥b∥CM˙Oq(ω)q×(∑i=-∞k2-knω(Bk)1/q∥fχi∥Lq(ω)|Bi|ω(Bi)-1/q)q=C∥Ω∥Ls(Sn-1)q∥b∥CM˙Oq(ω)q×(∑i=-∞k2(i-k)n(ω(Bk)ω(Bi))1/q∥fχi∥Lq(ω))q≤C∥Ω∥Ls(Sn-1)q∥b∥CM˙Oq(ω)q×(∑i=-∞k2(i-k)n/q′∥fχi∥Lq(ω))q.
Hence, for 1<s≤∞, applying inequalities (24) and (25), we have
(26)I≤C∥b∥CM˙Oq(ω)q(∑i=-∞k2(i-k)n/q′∥fχi∥Lq(ω))q.
Now we turn our attention to the estimate of II. According to the conditions, we can further decompose II as
(27)II≤C2-kqn×∫Δk(∑i=-∞k∫Δi‍|f(y)Ω(x-y)(b(y)-bBi)|dy)q×ω(x)1-qdx+C2-kqn∫Δk(∑i=-∞k∫Δi|f(y)Ω(x-y)×(bBk-bBi)|dy∑i=-∞k∫Δi|f(y)Ω(x-y))q×ω(x)1-qdx=II1+II2.
Below we shall give the estimates of II1. By Hölder’s inequality, we can deduce
(28)∫Δi‍|f(y)(b(y)-bBi)|dy≤(∫Δi|f(y)|qω(y)dy)1/q×(∫Δi‍|b(y)-bBi|q′ω(y)-q′/qdy)1/q′≤∥fχi∥Lq(ω)∥b∥CM˙Oq′(ω)ω(Bi)1/q′.
When Ω∈L∞(Sn-1), using Lemma 20, we can obtain the following result by inequality (28) and Remark 17 that
(29)II1≤C2-kqn∥Ω∥L∞(Sn-1)q×∫Δk(∑i=-∞k∫Δi|f(y)(b(y)-bBi)|dy)qω(x)1-qdx≤C∥Ω∥L∞(Sn-1)q2-kqn|Bk|qω(Bk)1-q×(∑i=-∞k∥fχi∥Lq(ω)∥b∥CM˙Oq′(ω)ω(Bi)1/q′)q=C∥Ω∥L∞(Sn-1)q∥b∥CM˙Oq′(ω)q×(∑i=-∞k(ω(Bi)ω(Bk))1-1/q∥fχi∥Lq(ω))q≤C∥Ω∥L∞(Sn-1)q∥b∥CM˙Oq′(ω)q×(∑i=-∞k2(i-k)nδ/q′∥fχi∥Lq(ω))q.
When Ω∈Ls(Sn-1), 1<s<∞, noticing inequality (28) and then by using Hölder’s inequality, Lemma 20, and Remark 17, we can obtain that
(30)II1≤C2-kqn×∫Δk(∑i=-∞k(∫Δi|f(y)(b(y)-bBi)|s′dy)1/s′×(∫Δi|Ω(x-y)|sdy)1/s∑i=-∞k∫Δi|f(y)Ω(x-y))qω(x)1-qdx≤C2-kqn∫Δkω(x)1-qdx×(∑i=-∞k∥Ω∥Ls(Sn-1)×∫Δi|f(y)(b(y)-bBi)|dy∑i=-∞k∥Ω∥Ls(Sn-1))q≤C2-kqn|Bk|qω(Bk)1-q×(∑i=-∞k∥Ω∥Ls(Sn-1)∫Δi|f(y)(b(y)-bBi)|dy)q≤C∥Ω∥Ls(Sn-1)q2-kqn|Bk|qω(Bk)1-q×(∑i=-∞k∥fχi∥Lq(ω)∥b∥CM˙Oq′(ω)ω(Bi)1/q′)q≤C∥Ω∥Ls(Sn-1)q∥b∥CM˙Oq′(ω)q×(∑i=-∞k(ω(Bi)ω(Bk))1-1/q∥fχi∥Lq(ω))q≤C∥Ω∥Ls(Sn-1)q∥b∥CM˙Oq′(ω)q×(∑i=-∞k2(i-k)nδ/q′∥fχi∥Lq(ω))q.
Hence, for 1<s≤∞, by estimates (29) and (30), we get that
(31)II1≤C∥b∥CM˙Oq′(ω)q(∑i=-∞k2(i-k)nδ/q′∥fχi∥Lq(ω))q.
Now, let us deal with the last term II2. When Ω∈L∞(Sn-1), noting inequality (23), by using Lemma 18, Lemma 20, and Remark 17, we have that
(32)II2≤C2-kqn×∫Δk(∑i=-∞k∫Δi|ω(Bi)|Bi|f(y)Ω(x-y)(k-i)×∥b∥CM˙Op(ω)ω(Bi)|Bi||dy∑i=-∞k∫Δi|ω(Bi)|Bi|f(y)Ω(x-y)(k-i))q×ω(x)1-qdx≤C2-kqn∥b∥CM˙Op(ω)q|Bk|qω(Bk)1-q×(∑i=-∞k(k-i)∥Ω∥L∞(Sn-1)ω(Bi)|Bi|∫Δi|f(y)|dy)q≤C∥Ω∥L∞(Sn-1)q∥b∥CM˙Op(ω)qω(Bk)1-q×(∑i=-∞k(k-i)ω(Bi)|Bi|∥fχi∥Lq(ω)|Bi|ω(Bi)-1/q)q=C∥Ω∥L∞(Sn-1)q∥b∥CM˙Op(ω)q×(∑i=-∞k(k-i)(ω(Bi)ω(Bk))1-1/q∥fχi∥Lq(ω))q≤C∥Ω∥L∞(Sn-1)q∥b∥CM˙Op(ω)q×(∑i=-∞k(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))q.
When Ω∈Ls(Sn-1), 1<s<∞, by using Lemma 18, inequalities (21) and (23), Remark 17, and an application of Hölder’s inequality give us that
(33)II2≤C2-kqn×∫Δk‍(∑i=-∞k∫Δi|ω(Bi)|Bi|f(y)Ω(x-y)×(k-i)∥b∥CM˙Op(ω)ω(Bi)|Bi||dy)q×ω(x)1-qdx≤C2-kqn∥b∥CM˙Op(ω)q×∫Δk(∑i=-∞k(k-i)ω(Bi)|Bi|(∫Δi‍|f(y)|s′dy)1/s′×(∫Δi|Ω(x-y)|sdy)1/s)q×ω(x)1-qdx≤C2-kqn∥b∥CM˙Op(ω)q|Bk|qω(Bk)1-q×(∑i=-∞k(k-i)∥Ω∥Ls(Sn-1)ω(Bi)|Bi|∫Δi|f(y)|dy)q≤C∥Ω∥Lq(Sn-1)q∥b∥CM˙Op(ω)qω(Bk)1-q×(∑i=-∞k(k-i)ω(Bi)|Bi|∥fχi∥Lq(ω)|Bi|ω(Bi)-1/q)q=C∥Ω∥Ls(Sn-1)q∥b∥CM˙Op(ω)q×(∑i=-∞k(k-i)(ω(Bi)ω(Bk))1-1/q∥fχi∥Lq(ω))q≤C∥Ω∥Ls(Sn-1)q∥b∥CM˙Op(ω)q×(∑i=-∞k(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))q.
As before, for 1<s≤∞, using inequalities (32) and (33), we can get
(34)II2≤C∥b∥CM˙Op(ω)q(∑i=-∞k(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))q.
Summing up the above estimates for I, II1, and II2, it is true that
(35)∥HΩbf∥K˙qα,p2(ω,ω1-q)≤(∑k=-∞∞‍ω(Bk)αp1/n∥(HΩbf)χk∥Lq(ω1-q)p1)1/p1≤C∥b∥CM˙Oq(ω)×(∑k=-∞∞ω(Bk)αp1/n×(∑i=-∞k2(i-k)n/q′∥fχi∥Lq(ω))p1)1/p1+C∥b∥CM˙Oq′(ω)×(∑k=-∞∞ω(Bk)αp1/n×(∑i=-∞k2(i-k)nδ/q′∥fχi∥Lq(ω))p1)1/p1+C∥b∥CM˙Op(ω)×(∑k=-∞∞ω(Bk)αp1/n×(∑i=-∞k‍(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))p1)1/p1∶=S.
Consequently,(36)Sp1≤C∥b∥CM˙Opmax{q,q′}(ω)p1∑k=-∞∞ω(Bk)αp1/n×(∑i=-∞k(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))p1≤C∥b∥CM˙Opmax{q,q′}(ω)p1×∑k=-∞∞(∑k(k-i)ω(Bi)α/n2(i-k)(nδ/q′-α)×∥fχi∥Lq(ω)∑)p1.
We shall consider two cases. As 0<p1≤1, noticing that α<nδ/q′, we can deduce
(37)Sp1≤C∥b∥CM˙Opmax{q,q′}(ω)p1×∑k=-∞∞∑i=-∞k(k-i)p1ω(Bi)αp1/n2(i-k)(nδ/q′-α)p1×∥fχi∥Lq(ω)p1=C∥b∥CM˙Opmax{q,q′}(ω)p1×∑i=-∞∞ω(Bi)αp1/n∥fχi∥Lq(ω)p1×∑k=i∞(k-i)p12(i-k)(nδ/q′-α)p1=C∥b∥CM˙Opmax{q,q′}(ω)p1∥f∥K˙qα,p1(ω,ω)p1.
For the case of 1<p1<∞, it follows from Hölder’s inequality and α<nδ/q′ that
(38)Sp1≤C∥b∥CM˙Opmax{q,q′}(ω)p1×∑k=-∞∞∑i=-∞kω(Bi)αp1/n2(i-k)(nδ/q′-α)p1/2×∥fχi∥Lq(ω)p1×(∑i=-∞k‍(k-i)p1′2(i-k)(nδ/q′-α)p1′/2)p1/p1′=C∥b∥CM˙Opmax{q,q′}(ω)p1×∑i=-∞∞ω(Bi)αp1/n∥fχi∥Lq(ω)p1∑k=i∞2(i-k)(nδ/q′-α)p1/2=C∥b∥CM˙Opmax{q,q′}(ω)p1∥f∥K˙qα,p1(ω,ω)p1.
Therefore, we conclude the proof of Theorem 7(a).
(b) Following the same procedure as that of Theorem7(a), we can also show the conclusion of (b); the details are omitted here.Proof of Theorem10.
We only give the proof of (a) asλ>0. The proof of (b) is similar to that for (a). By using (15) and the above estimates for I, II1, and II2 as those of the proof of Theorem 7, we can obtain that
(39)∥HΩbf∥MK˙p2,qα,λ(ω,ω1-q)≤C∥b∥CM˙Oq(ω)supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)αp2/n×(∑i=-∞k‍2(i-k)n/q′∥fχi∥Lq(ω))p2)1/p2+C∥b∥CM˙Oq′(ω)supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0‍ω(Bk)αp2/n×(∑i=-∞k2(i-k)nδ/q′∥fχi∥Lq(ω))p2)1/p2+C∥b∥CM˙Op(ω))supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)αp2/n×(∑i=-∞k(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))p2)1/p2≤C∥b∥CM˙Opmax{q,q′}(ω)supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0‍ω(Bk)αp2/n×(∑i=-∞k‍(k-i)2(i-k)nδ/q′∥fχi∥Lq(ω))p2)1/p2≤C∥b∥CM˙Opmax{q,q′}(ω)supk0∈Zω(Bk0)-λ/n×(∑1p2∑k=-∞k0ω(Bk)λp2/n×(∑∑i=-∞k(k-i)2(i-k)nδ/q′×(ω(Bk)ω(Bi))(α-λ)/nω(Bi)-λ/n×(∑l=-∞iω(Bl)αp1/n∥fχl∥Lq(ω)p1)1/p1)p2)1/p2≤C∥b∥CM˙Opmax{q,q′}(ω)supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)λp2/n×(∑i=-∞k(k-i)2(i-k)(nδ/q′-α+λ)×∥f∥MK˙p1,qα,λ(ω,ω)∑i=-∞k(k-i))p2)1/p2=C∥b∥CM˙Opmax{q,q′}(ω)supk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)λp2/n)1/p2∥f∥MK˙p1,qα,λ(ω,ω)≤C∥b∥CM˙Opmax{q,q′}(ω)∥f∥MK˙p1,qα,λ(ω,ω).
This completes the proof of Theorem 10.Proof of Theorem12.
(a) Letα<nδ/q1′. We follow the strategy of the proof of Theorem 7; we can also write
(40)∥(HΩbf)χk∥Lq2(ω1-q2)q2≤C2-knq2∫Δk(∑i=-∞k∫Δi(b(x)-bBk)×Ω(x-y)f(y)∣dy∑i=-∞k∫Δi(b(x)-bBk))q2×ω(x)1-q2dx+C2-knq2∫Δk(∑i=-∞k‍∫Δi‍(b(y)-bBk)×Ω(x-y)f(y)∣dy∑i=-∞k‍∫Δi‍(b(y)-bBk))q2×ω(x)1-q2dx∶=J+JJ.
For J, when Ω∈L∞(Sn-1), noticing that 1/q1=1/q2+β/n and ω∈A1⊂Aq1, by inequality (23) and Remark 17, we obtain that
(41)J≤C2-knq2∫Δk|b(x)-bBk|q2ω(x)1-q2×(∑i=-∞k∥Ω∥L∞(Sn-1)∫Δi|f(y)|dy)q2dx≤C∥Ω∥L∞(Sn-1)q22-knq2∥b∥Lipβ,ωq2×ω(Bk)1+βq2/n(∑i=-∞k∫Δi|f(y)|dy)q2≤C∥Ω∥L∞(Sn-1)q22-knq2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=-∞k∥fχi∥Lq1(ω)|Bi|ω(Bi)1/q1′-1)q2=C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)n(ω(Bk)ω(Bi))1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)n/q1′∥fχi∥Lq1(ω))q2.
As Ω∈Ls(Sn-1), 1<s<∞, observing that 1/q1=1/q2+β/n and ω∈A1⊂Aq1, by the inequality (23) and Remark 17, again using Hölder’s inequality, we can obtain that
(42)J≤C2-knq2∫Δk|b(x)-bBk|q2ω(x)1-q2×(∑i=-∞k(∫Δi|Ω(x-y)|sdy)1/s×(∫Δi|f(y)|s′dy)1/s′)q2dx≤C2-knq2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=-∞k∥Ω∥Ls(Sn-1)∫Δi|f(y)|dy)q2≤C∥Ω∥Ls(Sn-1)q22-knq2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=-∞k∥fχi∥Lq1(ω)|Bi|ω(Bi)-1/q1)q2=C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)n(ω(Bk)ω(Bi))1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)n/q1′∥fχi∥Lq1(ω))q2.
Therefore, for 1<s≤∞, by estimates (41) and (42), we obtain
(43)J≤C∥b∥Lipβ,ωq2(∑i=-∞k2(i-k)n/q1′∥fχi∥Lq1(ω))q2.
Now we come to estimate the other term JJ. Again, we shall further decompose JJ as
(44)JJ≤C2-knq2×∫Δk‍(∑i=-∞k‍∫Δi|(b(y)-bBi)Ω(x-y)f(y)|dy)q2×ω(x)1-q2dx+C2-knq2×∫Δk(∑i=-∞k‍∫Δi|(bBk-bBi)Ω(x-y)f(y)|dy)q2×ω(x)1-q2dx∶=JJ1+JJ2.
We first turn to deal with the term JJ1. By Hölder’s inequality, we can obtain that
(45)∫Δi|f(y)(b(y)-bBi)|dy≤(∫Δi‍|f(y)|q1ω(y)dy)1/q1×(∫Δi‍|b(y)-bBi|q1′ω(y)-q1′/q1dy)1/q1′≤∥b∥Lipβ,ωω(Bi)1/q1′+β/n∥fχi∥Lq1(ω).
As Ω∈L∞(Sn-1), noticing that 1/q1=1/q2+β/n and ω∈A1⊂Aq2′, we can deduce from inequality (45), Lemma 20, and Remark 17 that
(46)JJ1≤C2-knq2×∫Δk(∑i=-∞k∥Ω∥L∞(Sn-1)∫Δi|(b(y)-bBi)f(y)|dy)q2×ω(x)1-q2dx≤C2-knq2|Bk|q2ω(Bk)1-q2×(∑i=-∞k∥Ω∥L∞(Sn-1)∫Δi|(b(y)-bBi)f(y)|dy)q2≤C∥Ω∥L∞(Sn-1)q22-knq2|Bk|q2ω(Bk)1-q2×(∑i=-∞k∥b∥Lipβ,ωω(Bi)1/q1′+β/n∥fχi∥Lq1(ω))q2=C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(ω(Bi)ω(Bk))1-1/q2∥fχi∥Lq1(ω))q2≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)nδ/q2′∥fχi∥Lq1(ω))q2.
As Ω∈Ls(Sn-1), observing that 1/q1=1/q2+β/n and ω∈A1⊂Aq2′ and then by using Hölder’s inequality, inequality (45), Lemma 20, and Remark 17, we can obtain that
(47)JJ1≤C2-knq2×∫Δk(∑i=-∞k(∫Δi|Ω(x-y)|sdy)1/s×(∫Δi‍|(b(y)-bBi)f(y)|s′dy)1/s′)q2×ω(x)1-q2dx≤C2-knq2∫Δkω(x)1-q2dx×(∑i=-∞k∥Ω∥Ls(Sn-1)∫Δi|(b(y)-bBi)f(y)|dy)q2≤C2-knq2∫Δkω(x)1-q2dx×(∑i=-∞k∥Ω∥Ls(Sn-1)∥b∥Lipβ,ωω(Bi)1/q1′+β/n∥fχi∥Lq1(ω))q2≤C∥Ω∥Ls(Sn-1)q22-knq2|Bk|q2ω(Bk)1-q2×(∑i=-∞k∥b∥Lipβ,ωω(Bi)1/q1′+β/n∥fχi∥Lq1(ω))q2=C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(ω(Bi)ω(Bk))1-1/q2∥fχi∥Lq1(ω))q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k2(i-k)nδ/q2′∥fχi∥Lq1(ω))q2.
Hence, for 1<s≤∞, by combining inequalities (46) and (47), we have
(48)JJ1≤C∥b∥Lipβ,ωq2(∑i=-∞k2(i-k)nδ/q2′∥fχi∥Lq1(ω))q2.
Finally, let us deal with the term JJ2. When Ω∈L∞(Sn-1), noticing that ω∈A1⊂Aq2′, by using Lemma 19, inequality (23), and Remark 17, we can get that
(49)JJ2≤C2-knq2×∫Δk(∑i=-∞k∫Δi(k-i)∥b∥Lipβ,ωω(Bk)β/nω(Bi)|Bi|×|Ω(x-y)f(y)|dy∑)q2×ω(x)1-q2dx≤C∥b∥Lipβ,ωq22-knq2×∫Δk(∑i=-∞k(k-i)ω(Bk)β/nω(Bi)|Bi|∥Ω∥L∞(Sn-1)×∫Δi‍|f(y)|dy∫)q2ω(x)1-q2dx≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2ω(Bk)1-q2×(∑i=-∞k(k-i)ω(Bk)β/n×ω(Bi)|Bi||Bi|ω(Bi)-1/q1∥fχi∥Lq1(ω))q2=C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(k-i)(ω(Bi)ω(Bk))1-1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(k-i)2(i-k)nδ/q1′∥fχi∥Lq1(ω))q2.
When Ω∈Ls(Sn-1), observing that ω∈A1⊂Aq2′, it follows from Hölder’s inequality, Lemma 19, inequality (23), and Remark 17 that
(50)JJ2≤C2-knq2×∫Δk(∑i=-∞k∫Δi(k-i)∥b∥Lipβ,ωω(Bk)β/n×ω(Bi)|Bi||Ω(x-y)f(y)|dy)q2×ω(x)1-q2dx≤C∥b∥Lipβ,ωq22-knq2×∫Δk(∑i=-∞k(k-i)ω(Bk)β/nω(Bi)|Bi|×(∫Δi|f(y)|s′dy)1/s′×(∫Δi|Ω(x-y)|sdy)1/s)q2×ω(x)1-q2dx≤C∥b∥Lipβ,ωq2ω(Bk)1-q2×(∑i=-∞k(k-i)ω(Bk)β/nω(Bi)|Bi|×∥Ω∥Ls(Sn-1)∫Δi‍|f(y)|dy)q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2(Bk)1-q2×(∑i=-∞k(k-i)ω(Bk)β/n×ω(Bi)|Bi||Bi|ω(Bi)-1/q1∥fχi∥Lq1(ω)∑)q2=C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(k-i)(ω(Bi)ω(Bk))1-1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=-∞k(k-i)2(i-k)nδ/q1′∥fχi∥Lq1(ω))q2.
As before, for1<s≤∞, from inequalities (49), and (50), we obtain that
(51)JJ2≤C∥b∥Lipβ,ωq2×(∑i=-∞k‍(k-i)2(i-k)nδ/q1′∥fχi∥Lq1(ω))q2.
The remaining proof is similar to the proof of Theorem 7 in case (a); we omit the details here. Thus we complete the proof of Theorem 12 in case (a).
(b) Whenα>-nδ/q2, by a direct calculation, we can see
(52)∥(HΩ*,bf)χk∥Lq2(ω1-q2)q2≤C∫Δk(∑i=k∞‍2-in×∫Δi‍(b(x)-bBk)Ω(x-y)f(y)∣dy)q2×ω(x)1-q2dx+C∫Δk(∑i=k∞2-in×∫Δi(b(y)-bBk)Ω(x-y)f(y)∣dy)q2×ω(x)1-q2dx∶=J′+JJ′.
When Ω∈L∞(Sn-1), note that 1/q1=1/q2+β/n and ω∈A1⊂Aq1. Applying Hölder’s inequality, inequality (23), and Remark 17, we can deduce
(53)J′≤C∫Δk|b(x)-bBk|q2ω(x)1-q2×(∑i=k∞‍2-in∥Ω∥L∞(Sn-1)∫Δi‍|f(y)|dy)q2dx≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=k∞2-in∫Δi|f(y)|dy)q2≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=k∞2-in∥fχi∥Lq1(ω)|Bi|ω(Bi)1/q1′-1)q2=C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=k∞(ω(Bk)ω(Bi))1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥L∞(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=k∞2(k-i)nδ/q1∥fχi∥Lq1(ω))q2.
When Ω∈Ls(Sn-1), 1<s<∞, observe that 1/q1=1/q2+β/n and ω∈A1⊂Aq1; then by using Hölder’s inequality, inequality (23), and Remark 17, we can obtain
(54)J′≤C∫Δk|b(x)-bBk|q2ω(x)1-q2×(∑i=k∞2-in(∫Δi‍|Ω(x-y)|sdy)1/s×(∫Δi‍|f(y)|s′dy)1/s′)q2dx≤C∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=k∞2-in∥Ω∥Ls(Sn-1)∫Δi|f(y)|dy)q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2ω(Bk)1+βq2/n×(∑i=k∞2-in∥fχi∥Lq1(ω)|Bi|ω(Bi)-1/q1)q2=C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=k∞‍(ω(Bk)ω(Bi))1/q1∥fχi∥Lq1(ω))q2≤C∥Ω∥Ls(Sn-1)q2∥b∥Lipβ,ωq2×(∑i=k∞2(k-i)nδ/q1∥fχi∥Lq1(ω))q2.
Hence, for 1<s≤∞, by estimates (53) and (54), we get
(55)J′≤C∥b∥Lipβ,ωq2(∑i=k∞2(k-i)nδ/q1∥fχi∥Lq1(ω))q2.
For JJ′, using the same arguments as those of JJ, we can see
(56)JJ′≤C∫Δk(∑i=k∞2-in×∫Δi|(b(y)-bBi)Ω(x-y)f(y)|dy)q2×ω(x)1-q2dx+C∫Δk(∑i=k∞2-in×∫Δi|(bBk-bBi)Ω(x-y)f(y)|dy)q2×ω(x)1-q2dx∶=JJ1′+JJ2′.
Similar to the estimate of JJ1, JJ2 of (a), we have
(57)JJ1′≤C∥b∥Lipβ,ωq2(∑i=k∞2(k-i)n/q2∥fχi∥Lq1(ω))q2JJ2′≤C∥b∥Lipβ,ωq2(∑i=k∞2(k-i)nδ/q2∥fχi∥Lq1(ω))q2.
The remaining proof is similar to the proof of (a) of Theorem 7, so that the proof of (b) can be obtained easily. So we conclude the proof of Theorem 12.Proof of Theorem14.
(a) Letα<nδ/q1′+λ. By the definition of Morry-Herz spaces and combining inequalities (43), (48), and (51) in the above, it is not difficult to see that
(58)∥HΩbf∥MK˙p2,q2α,λ(ω,ω1-q2)≤C∥b∥Lipβ,ωsupk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)αp2/n×(∑i=-∞k2(i-k)n/q1′∥fχi∥Lq1(ω))p2)1/p2+C∥b∥Lipβ,ωsupk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)αp2/n×(∑i=-∞k2(i-k)nδ/q2′∥fχi∥Lq1(ω))p2)1/p2+C∥b∥Lipβ,ωsupk0∈Zω(Bk0)-λ/n×(∑k=-∞k0ω(Bk)αp2/n×(∑i=-∞k‍(k-i)2(i-k)nδ/q1′∥fχi∥Lq1(ω))p2)1/p2∶=E1+E2+E3.
Therefore, by applying the similar argument as that in the proof of Theorem 10, we can obtain that
(59)E1≤C∥b∥Lipβ,ω∥f∥MK˙p1,q1α,λ(ω,ω),asα<nq1′+λ.E2≤C∥b∥Lipβ,ω∥f∥MK˙p1,q1α,λ(ω,ω),asα<nδq2′+λ.E3≤C∥b∥Lipβ,ω∥f∥MK˙p1,q1α,λ(ω,ω),asα<nδq1′+λ.
Thus, when α<nδ/q1′+λ, we have
(60)∥HΩbf∥MK˙p2,q2α,λ(ω,ω1-q2)≤C∥b∥Lipβ,ω∥f∥MK˙p1,q1α,λ(ω,ω).
The proof of Theorem 10 (a) is completed.
(b) The proof of case (b) can be obtained similarly, so we omit the details here.



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*Source: 568202-2013-04-11.xml*