# Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel

Authors: Xiao Yu; Shanzhen Lu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The authors consider the generalized commutator of fractional Hardy operator with a rough kernel as follows:ℋΩ,A,βmf(x)=1/(|x|n-β)∫|y|<|x|‍(Ω(x-y)/|x-y|m-1)Rm(A;x,y)f(y)dy, where Ω∈Lr(𝕊n-1), 0≤β<n, and Rm(A;x,y)=A(x)-∑|γ|<m(1/γ!)DγA(y)(x-y)γ with m∈Z+. The authors prove that ℋΩ,A,βm is bounded on Herz type space and λ-Central Morrey space with m≥1, which is an open problem for m>2.


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## Body

## 1. Introduction

It is well known that the C-Z singular integrals and their commutators have been studied a lot by many mathematicians; please see [1] or [2] for more details. For the generalizations of the commutators of singular integrals, Cohen [3] studied the following generalized commutator TA2 which is defined by
(1)TA2f(x)=∫ℝn‍Ω(x-y)|x-y|n+1(A(x)-A(y)-∇A(y)(x-y))f(y)dy,
where Ω∈L1(𝕊n-1) is homogeneous of degree zero and satisfies the moment condition
(2)∫𝕊n-1‍Ω(x)xγdσ(x)=0
with |γ|=1. Cohen [3] proved that if Ω∈Lip1(𝕊n-1) and ∇A∈BMO, then TA2 is bounded on Lp(ℝn) with 1<p<∞. Later, Cohen and Gosselin [4] considered another type of generalized commutator as follows:
(3)TAmf(x)=∫ℝn‍Ω(x-y)|x-y|n+m-1Rm(A;x,y)f(y)dy,
where Rm(A;x,y)(m∈Z+) is defined by Rm(A;x,y)=A(x)-∑|γ|<m(1/γ!)DγA(y)(x-y)γ, the mth remainder of Taylor series of the function A at y about x, and Ω satisfies the following moment condition:
(4)∫𝕊n-1‍Ω(x)xγdσ(x)=0,
with |γ|=m-1. Obviously, if we choose m=1, TAm becomes [A,T], the commutator of T generalized by A and T. Furthermore, TAm becomes TA2 if we choose m=2.Cohen and Gosselin proved that ifm≥2,Ω∈Lip1(𝕊n-1), and the function A has derivatives of order m-1 in BMO(ℝn), then the operator TAm is bounded on Lp(ℝn) for 1<p<∞. Later, TAm was studied by many mathematicians; please see [5, 6] or [7] for more details. Recently, Wang and Zhang [8] gave a new proof of Wu’s theorem in [9] by using the W1,p estimate for the elliptic equation of divergence form with partially BMO coefficients and the Lp boundedness of the Cohen-Gosselin type generalized commutators proved by Yan in [6]. Furthermore, the method used in [8] is much simpler than that in [9]. Recently, Yu and Tao [7] proved that TAm is bounded on λ-Central Morrey space.Letf be a nonnegative integral on ℝ+: then the Hardy operator is defined by
(5)Hf(x)=1x∫0x‍f(t)dt,x≠0.
In 1920, Hardy [10] proved the following inequality:
(6)∥Hf∥Lp(ℝ+)≤pp-1∥f∥Lp(ℝ+),
where 1<p<∞ and the constant p/(p-1) is the best possible.In 2007, Fu et al. [11] introduced the n-dimensional fractional type Hardy operator ℋβ as follows:
(7)ℋβf(x)=1|x|n-β∫|t|<|x|‍f(t)dt,x∈ℝn∖{0},
where -n<β<n and f is a locally integrable function on ℝn.Obviously, whenβ=0, ℋ0 is just the n-dimensional Hardy operator ℋ which was proposed by Christ and Grafakos in [12].In [11], the authors gave the characterization of the CB˙MOq(ℝn) by the boundedness of the commutator of the fractional type Hardy operator [ℋβ,b] on Herz type spaces. Here the CB˙MOq(ℝn) space is defined by the following.Definition 1 (see [13]).
Let1≤q<∞. A function f∈Llocq(ℝn) is said to belong to the homogeneous Central BMO space CB˙MOq(ℝn) if
(8)∥b∥CB˙MOq(ℝn):=supr>0(1|B(0,r)|∫B(0,r)‍|f(x)-fB|qdx)1/q<∞,
where fB=(1/|B(0,r)|)∫B(0,r)‍f(x)dx.
From [14], we know that BMO(ℝn)⊂CB˙MOq(ℝn) for 1≤q<∞.TheCB˙MOq(ℝn) space can be regarded as the space of bounded mean oscillation, a local version of BMO(ℝn) at the origin. But the famous John-Nirenberg inequality no longer holds in CB˙MOq(ℝn).Now we are interested in the following generalized commutator of Hardy operator:(9)ℋAmf(x)=1|x|n∫|y|<|x|‍1|x-y|m-1f(y)Rm(A;x,y)dy,
where Rm(A;x,y)=A(x)-∑|γ|<m(1/γ!)DγA(y)(x-y)γ and m∈Z+.In 2010, Lu and Zhao [15] proved that when m=2, ℋA2 is bounded on Herz type space and Morrey-Herz type space. Later, Gao and Yu [16] proved that ℋA2 is bounded on λ-Central Morrey spaces. However, we would like to point out that the method used in [15, 16] cannot apply to the case when m>2. An interesting question is whether the boundedness of ℋAm on Herz type space or λ-Central Morrey space still holds with m>2. In this paper, we will use a different method to answer this question. Furthermore, we will consider the generalized commutator of fractional Hardy operator with a rough kernel as follows:
(10)ℋΩ,A,βmf(x)=1|x|n-β∫|y|<|x|‍Ω(x-y)|x-y|m-1f(y)Rm(A;x,y)dy,
where m∈Z+, 0≤β<n, and Ω∈Lr(𝕊n-1).In [17], we prove that ℋΩ,A,βm is bounded from Lp to Lq with 1/p-1/q=β/n. Furthermore, we study the endpoint estimates of ℋΩ,A,βm on H1 spaces with Ω∈Lip1(𝕊n-1) in [17]. In this paper, we will prove that ℋΩ,A,βm is bounded on Herz type space and λ-Central Morrey space when Ω∈Lr(𝕊n-1).


## 2. Boundedness ofℋΩ,A,βm on Herz Type Spaces

In this section, we will give the boundedness ofℋΩ,A,βm on Herz type spaces. First we introduce some notations that will be used throughout this paper.LetBk={x∈ℝn:|x|≤2k}, Ck=Bk∖Bk-1, and χk=χCk for k∈Z: here χCk is the characteristic function of the set Ck.Definition 2 (see [18]).
Letα∈R,0<p,q≤∞. Then the homogeneous Herz type space K˙qα,p(ℝn) is defined by
(11)K˙qα,p(ℝn)={f∈Llocq(ℝn∖{0}):∥f∥K˙qα,p(ℝn)<∞},
where ∥f∥K˙qα,p(ℝn) is defined as
(12)∥f∥K˙qα,p(ℝn)={∑k=-∞∞‍2kαp∥fχk∥Lq(ℝn)p}1/p
with the usual modifications made when p=∞ or q=∞.
Now we show our main results in this section.Theorem 3.
Supposem≥2, Ω∈Lr(𝕊n-1) with 1<r<∞, and A has derivatives of order m-1 in CB˙MOp2 with n<p2<∞. Let 0<s≤p<∞,1<q,p1<∞,1/q=1/p1+1/p2-β/n with 0≤β<n. If  1/r′-1/q-β/n>0,r>p1′, α2=α1+n/p2 and α2 satisfies the following condition:
(13)α2+n/q-n/r′-1/r-n/p2+β<0,
then there exists a constant C, such that
(14)∥ℋΩ,A,βmf∥K˙qα1,s≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2∥f∥K˙p1α2,p.Form=1, ℋΩ,A,β1 is just the commutator of Hardy operator; that is, ℋΩ,A,β1=ℋΩ,βAf(x)=A(x)ℋΩ,βf(x)-ℋΩ,β(Af)(x). We have the following theorem of ℋΩ,βA on Herz type space.Theorem 4.
SupposeΩ∈Lr(𝕊n-1) with 1<r<∞ and A∈CB˙MOp2. Let 0<s≤p<∞,1<q,p1,p2<∞,1/q=1/p1+1/p2-β/n with 0≤β<n. If 1/r′-1/q-β/n>0,r>p1′,  α2=α1+n/p2, and α2 satisfies (13), then there exists a constant C, such that
(15)∥ℋΩ,βAf∥K˙qα1,s≤C∥A∥CB˙MOp2∥f∥K˙p1α2,p.Remark 5.
Comparing Theorems3 and 4, we find that the restrictions on α1 and α2 are more rigid in Theorem 4 than in Theorem 3, which indicates that ℋΩ,A,βm with m≥2 has better properties than the commutators.In order to prove Theorems3 and 4, we need the following lemmas.Lemma 6 (see [19]).
Let1<p1,p2<∞,0≤β<n, and β/n=1/p1-1/p2. If Ω∈Lr(𝕊n-1) with r>p1′, then there exists a constant C independent of f, such that
(16)∥ℋΩ,βf∥Lp2≤C∥f∥Lp1,
where ℋΩ,β is defined by
(17)ℋΩ,βf(x)=1|x|n-β∫|y|<|x|‍Ω(x-y)f(y)dy.
By checking [19] carefully, one can draw the conclusion that if one replaces ℋΩ,βf(x) by ℋ|Ω|,β|f|(x), then (16) still holds.Lemma 7.
Letm≥1,1<p1<p2<∞ and 0≤β<n. If A has derivatives of order m-1 in Lr(ℝn) with 1/p2=1/p1+1/r-β/n and r>p1′, then one has
(18)∥ℋΩ,A,βmf∥Lp2≤C∑|γ|=m-1‍∥DγA∥Lr∥f∥Lp1,
where the constant C is independent of f and A.Proof.
From [20, p. 241], we have the following estimates:
(19)Rm(A;x,y)|x-y|m-1≤Rm-1(A;x,y)|x-y|m-1+C∑|γ|=m-1‍|DγA(y)|≤C∑|γ|=m-1‍((DγA)*(x)+(DγA)*(y)),
where m≥1 and (f)* is the Hardy-Littlewood maximal function of f.
Thus we obtain(20)|ℋΩ,A,βmf(x)|=|1|x|n-β∫|y|<|x|‍Ω(x-y)f(y)|x-y|m-1Rm(A;x,y)dy|≤C1|x|n-β∫|y|<|x|‍|f(y)||Ω(x-y)|×∑|γ|=m-1‍((DγA)*(x)+(DγA)*(y))dy≤C∑|γ|=m-1‍[(DγA)*(x)ℋ|Ω|,β|f|(x)+ℋ|Ω|,β((DγA)*|f|)(x)].
By the above estimates, we can get
(21)(∫ℝn‍|ℋΩ,A,βmf(x)|qdx)1/q≤C∑|γ|=m-1‍((∫ℝn‍|(DγA)*(x)ℋ|Ω|,β|f|(x)|qdx)1/q+(∫ℝn‍|ℋ|Ω|,β((DγA)*|f|)(x)|q)1/q)≤C∑|γ|=m-1‍(I+II).
For the termI, let 1/q=1/r+1/l=1/r+1/p-β/n; then by the Hölder inequality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on Lp spaces, we obtain
(22)I≤(∫ℝn‍(DγA)*(x)rdx)1/r(∫ℝn‍|ℋ|Ω|,β|f|(x)|ldx)1/l≤C∥(DγA)*∥Lr∥f∥Lp≤C∥DγA∥Lr∥f∥Lp.
For the termII, let 1/q=1/t-β/n=1/r+1/p-β/n; then by the Hölder inequality and Lemma 6, we have
(23)II≤C(∫ℝn‍|(DγA)*(x)f(x)|tdx)1/t≤C∥(DγA)*∥Lr∥f∥Lp≤C∥DγA∥Lr∥f∥Lp.
Combining the estimates of I and II, we finish the proof of Lemma 7.Lemma 8 (see [4]).
Letb be a function on ℝn with mth order derivatives in Llocq(ℝn) for some q>n. Then
(24)|Rm(b;x,y)|≤Cm,n|x-y|m×∑|γ|=m‍(1|Q~(x,y)|∫Q~(x,y)‍|Dγb(z)|qdz)1/q,
where Q~(x,y) is the cube centered at x having diameter 5n|x-y|.Lemma 9 (see [5]).
Suppose thatf∈CB˙MOq(ℝn), 1≤q<∞, and r1,r2>0; then
(25)(1|B(0,r1)|∫B(0,r1)‍|f(x)-fB(0,r2)|qdx)1/q≤C(1+|log(r1r2)|)∥f∥CB˙MOq.Proof of Theorem3.
To prove Theorem3, first we split each f as
(26)f(x)=∑i=-∞+∞‍f(x)χi(x)=∑i=-∞+∞‍fi(x);
then we have
(27)∥ℋΩ,A,βmfχk∥Lqq=∫Ck‍(∫B(0,|x|)‍|f(y)||Ω(x-y)||x-y|m-1Rm(A;x,y)dy)q×|x|(β-n)qdx≤∫Ck‍(∑i=-∞k‍∫Ci‍|Ω(x-y)f(y)||x-y|m-1Rm(A;x,y)dy)q×|x|(β-n)qdx≤C∫Ck‍(∑i=-∞k-3‍∫Ci‍|Ω(x-y)f(y)||x-y|m-1Rm(A;x,y)dy)q×|x|(β-n)qdx+C∫Ck‍(∑i=k-2k‍∫Ci‍|Ω(x-y)f(y)||x-y|m-1Rm(A;x,y)dy)q×|x|(β-n)qdx=C(I1+I2).
For the termI1, we denote Ak(x)=A(x)-∑|γ|=m-1(1/γ!)mBk(DγA)xγ; then it is easy to check Rm(A;x,y)=Rm(Ak;x,y). By the fact that x∈Ck, y∈Ci with i≤k-3, we have |x-y|~|x|~2k. As p2>n, then by Lemmas 8 and 9, we obtain
(28)|Rm(Ak;x,y)|≤|Rm-1(Ak;x,y)|+∑|γ|=m-1‍1γ!|DγAk(x)||x-y|m-1≤C|x-y|m-1∑|γ|=m-1‍{(1|Q~(x,y)|×∫Q~(x,y)‍|DγAk(z)|p2dz(1|Q~(x,y)|)1/p2+|DγAk(y)|(1|Q~(x,y)|}≤C|x-y|m-1∑|γ|=m-1‍(∥DγA∥CB˙MOp2+|DγAk(y)|).
As |x-y|~|x|~2k and 1-1/p1-1/p2-1/r=1/r′-1/q-β/n>0, then by the Hölder inequality, we have
(29)I1≤C∫Ck‍(∑|γ|=m-1‍∑i=-∞k-3‍∫Ci‍|Ω(x-y)f(y)|×∑|γ|=m-1‍(∥DγA∥CB˙MOp2+|DγAk(y)|)dy)q×|x|(β-n)qdx≤C2-k(n-β)q∫Ck‍(∑i=-∞k-3‍∫Ci‍|f(y)||Ω(x-y)|×∑|γ|=m-1‍(∥DγA∥CB˙MOp2+|DγAk(y)|∥DγA∥CB˙MOp2)dy∑i=-∞k-3‍)qdx≤C2-k(n-β)q∫Ck‍∑i=-∞k-3‍(∫Ci‍|f(y)|p1dy)q/p1×(∫Ci‍|Ω(x-y)|rdy)q/r×(∑|γ|=m-1‍∫Ci‍(∥DγA∥CB˙MOp2+|DγAk(y)|)p2dy)q/p2×|Ci|q(1-1/p1-1/p2-1/r)dx.
As
(30)(∫Ci(∥DγA∥CB˙MOp2+|DγAk(y)|)p2dy)1/p2≤(∫Ci‍(|DγAk(y)-mBk(DγA)(y)|∥DγA∥CB˙MOp2+|DγAk(y)-mBk(DγA)(y)|)p2dy∫Ci‍)1/p2≤C∥DγA∥CB˙MOp2|Ci||k-i|,∫Ci‍|Ω(x-y)|rdy≤∫|x|-2i|x|+2i‍∫𝕊n-1‍|Ω(y′)|rdσ(y′)rn-1dr≤C2kn-k+i,
we obtain the following estimates:
(31)I1≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2q|k-i|q2-knq+kβq×2in(1-1/p1-1/p2-1/r)q×2((k(n-1)+i)/r)q+(in/p2)q×∫Ck‍∑i=-∞k-3‍∥fi∥Lp1qdx≤C∑i=-∞k-3(∑|γ|=m-1‍|k-i|2in(1-1/p1-1/p2-1/r)‍×2-kn+kβ+(k(n-1)+i)/r+in/p2+kn/q×∥fi∥Lp1∑|γ|=m-1‍∥DγA∥CB˙MOp2)q.
For the termI2, we choose ϕ∈C0∞(ℝn) satisfying suppϕ⊂B(0,4) as well as ϕ≡1 in B(0,2) and we set L=max{∥Dγϕ∥L∞,|γ|≤m-1}. Let y0∈Bk+4 and Akϕ(x)=Rm-1(Ak;x,y0)ϕ(2-kx). Then it is easy to see that Rm(A;x,y)=Rm(Akϕ;x,y)=Rm(Ak;x,y) for x∈Bk and y∈Bi with k-2≤i≤k. Thus we get
(32)ℋΩ,A,βmfi(x)=ℋΩ,Akϕ,βmfi(x)=ℋΩ,Ak,βmfi(x).
Thus by Lemma 7, we have
(33)I2≤C∑i=k-2k‍∫Ck‍(1|x|n-β∫|y|<|x|‍|Ω(x-y)fi(y)||x-y|m-1×Rm(Akϕ;x,y)dy|Ω(x-y)fi(y)||x-y|m-1)qdx≤C∑i=k-2k‍∥ℋΩ,Akϕ,βmfi∥Lqq≤C∑i=k-2k‍∥fi∥Lp1q∑|γ|=m-1‍∥DγAkϕ∥Lp2q.
From [5, p.80], we have the following estimates:
(34)∥DγAkϕ∥Lp2≤C2kn/p2∑|γ|=m-1‍∥DγA∥CB˙MOp2,
where C is dependent on L. Thus we get
(35)I2≤C∑i=k-2k‍∥fi∥Lp1q2knq/p2∑|γ|=m-1‍∥DγA∥CB˙MOp2q.
As 0<s≤p<∞, we obtain the following estimates:
(36){∑k=-∞+∞‍2kα1s∥ℋΩ,A,βmfχk∥Lqs}1/s≤{∑k=-∞+∞‍2kα1p∥ℋΩ,A,βmfχk∥Lqp}1/p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2×{∑k=-∞+∞‍2kα1p×∑i=-∞k-3‍(|k-i|2in(1-1/p1-1/p2-1/r)∥fi∥Lp1×2-kn+kβ+(k(n-1)+i)/r+in/p2)+kn/q)p∑k=-∞+∞‍}1/p+C∑|γ|=m-1‍∥DγA∥CB˙MOp2×{∑k=-∞+∞‍2kα1p∑i=k-2k‍∥fi∥Lp1p2(kn/p2)p}1/p=C(A+B).
For the term A, we have
(37)A=∑|γ|=m-1‍∥DγA∥CB˙MOp2×{∑k=-∞+∞‍2kα1p×∑i=-∞k-3‍(|k-i|2in(1-1/p1-1/p2-1/r)2-kn+kβ×2(k(n-1)+i)/r+in/p2+kn/q∥fi∥Lp1|k-i|2in(1-1/p1-1/p2-1/r)2-kn+kβ)p∑k=-∞+∞‍}1/p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2×{∑k=-∞+∞∑i=-∞k-3‍‍(|k-i|2(k-i)(n/q-n/r′+β-1/r-n/p2)×2(k-i)(α1+n/p2)×2i(α1+n/p2)∥fi∥Lp1)p∑i=-∞k-3‍}1/p.
When 0<p≤1, by condition (13), we get
(38)Ap≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍(∑i=-∞k-3‍2(k-i)(n/q-n/r′-1/r-n/p2+β)×2(k-i)(α1+n/p2)×|k-i|2i(α1+n/p2)∥fi∥Lp1∑i=-∞k-3‍)p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍2iα2p(∑i=-∞k-3‍|k-i|×2(k-i)(n/q-n/r′-1/r-n/p2+β+α2)×∥fi∥Lp1∑i=-∞k-3‍)p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p∥f∥K˙p1α2,pp.
When p≥1, by the Hölder inequality and condition (13), we have
(39)Ap≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍2kα1p(∑i=-∞k-3‍|k-i|×2(k-i)(n/q-n/r′-1/r-n/p2+β)×2kn/p2∥fi∥Lp1∑i=-∞k-3‍)p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍(∑i=-∞k-3‍|k-i|2kα12(k-i)(n/q-n/r′-1/r-n/p2+β)×2kn/p2∥fi∥Lp1∑i=-∞k-3‍)p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍(∑i=-∞k-3‍|k-i|2iα22(k-i)(n/q-n/r′-1/r-n/p2+β+α2)×∥fi∥Lp1∑i=-∞k-3‍)p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑k=-∞+∞‍∑i=-∞k-3‍2iα2p∥fi∥Lp1p2p(k-i)(n/q-n/r′-1/r-n/p2+β+α2)/p×(∑i=-∞k-3‍|k-i|p′2((k-i)/p′)(n/q-n/r′-1/r-n/p2+β+α2)p′)p/p′≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p×∑i=-∞+∞∑k=i+3∞‍2ip(α1+n/p2)2(k-i)(n/q-n/r′-1/r-n/p2+β+α2)∥fi∥Lp1×(∑i=-∞k-3‍|k-i|p′2(k-i)(n/q-n/r′-1/r-n/p2+β+α2))p/p′≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p∑i=-∞+∞‍2iα2p∥fi∥Lp1p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2p∥f∥K˙p1α2,pp.
For the term B, we have the following estimates:
(40)B≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2{∑k=-∞+∞‍2kα1p∑i=k-2k‍2knp/p2∥fi∥Lp1p}1/p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2(∑i=-∞+∞‍2iα2p∥fi∥Lp1p)1/p≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2∥f∥K˙p1α2,p.
Combining the estimates of A and B, we finish the proof of Theorem 3.Proof of Theorem4.
The proof of Theorem4 is quite similar and much easier than Theorem 3 and we omit the details here.


## 3. Boundedness ofℋΩ,A,βm on λ-Central Morrey Spaces

In [21], Wiener gave a way to describe the behavior of a function at the infinity. Later, Beurling [22] extended Weiner’s idea and introduced a pair of dual Banach spaces, Aq and Bq′, with 1/q+1/q′=1. In [23], Feichtinger proved that Bq can be described as
(41)∥f∥Bq=supk≥0(2-kn/q∥fχk∥Lq)<∞,
where χ0 is the characterization of the unit ball {x∈ℝn:|x|≤1} and χk is defined as in Section 2.Now by duality, the spaceAq, which is called the Beurling algebra, can be described by
(42)∥f∥Aq=∑k=0∞‍2kn/q′∥fχk∥Lq<∞.Later, Chen and Lau [24] as well as García-Cuerva [25] introduced atomic spaces HAq(ℝn) associated with the Buerling algebra Aq and the dual space of HAq(ℝn) can be described by
(43)∥f∥CBMOq:=supR≥1(1|B(0,R)|∫B(0,R)‍|f(x)-fB(0,R)|qdx)1/q<∞;
here the CBMOq can be regarded as the inhomogeneous central BMO spaces.In 2000, Alvarez et al. [26] introduced the λ-Central bounded mean oscillation space and λ-Central Morrey space, respectively.Definition 10 (see [26]).
Givenλ<1/n,1<q<∞, then a function f∈Llocq(ℝn) is said to belong to the λ-Central bounded mean oscillation space CB˙MOq,λ(ℝn) if
(44)∥f∥CB˙MOq,λ:=supR>0(1|B(0,R)|1+λq∫B(0,R)‍|f(x)-fB(0,R)|qdx)1/q<∞.Definition 11 (see [26]).
Letλ∈ℝ and 1<q<∞. Then the λ-Central Morrey space E˙q,λ(ℝn) is defined by
(45)∥f∥E˙q,λ=supR>0(1|B(0,R)|1+λq∫B(0,R)‍|f(x)|qdx)1/q<∞.From [27], we know that if 1<q1<q2<∞, we obtain E˙q2,λ⊂E˙q1,λ for λ∈R and CB˙MOq2,λ⊂CB˙MOq1,λ for λ<1/n. Furthermore, when λ<-1/q, E˙q,λ reduces to {0} and CB˙MOq,λ reduces to the space of constant functions. When λ=-1/q, CB˙MOq,λ coincides with Lq(ℝn) modulo constant and E˙q,λ=Lq.In 2011, Fu et al. [19] proved the boundedness of the commutator of fractional Hardy operator with a rough kernel on λ-Central Morrey space. Later, Fu et al. [28] proved the boundness of the weighted Hardy operator and its commutator on λ-Central Morrey space. In this paper, we will give the boundedness of ℋΩ,A,βm on λ-Central Morrey space with m≥1.Our results can be stated as follows.Theorem 12.
Supposem≥2, n<p2<∞,1<p1<∞, 1/q=1/p1+1/p2-β/n with 0≤β<n and λ=λ1+λ2+β/n. Let Ω∈Lr(𝕊n-1) with 1/r′>β/n+1/q, and A has derivatives of order m-1 in CBM˙Op2,λ2. If r>p1′, λ1>-1/p1, and qλ+1>0, then one has
(46)∥ℋΩ,A,βmf∥E˙q,λ≤C∑|γ|=m-1‍∥DγA∥CBM˙Op2,λ2∥f∥E˙p1,λ1,
where the constant C is independent of f and A.For the casem=1, we have the following theorem.Theorem 13.
Suppose1<p1,p2<∞, 1/q=1/p1+1/p2-β/n with 0≤β<n and λ=λ1+λ2+β/n. Let Ω∈Lr(𝕊n-1) with 1/r′>β/n+1/q and A∈CBM˙Op2,λ2. If λ1>-1/p1,r>p1′, and qλ+1>0, then one has
(47)∥ℋΩ,βAf∥E˙q,λ≤C∥A∥CBM˙Op2,λ2∥f∥E˙p1,λ1,
where the constant C is independent of f and A.In order to prove Theorems12 and 13, by a standard argument, we have the following lemma.Lemma 14 (see [16]).
Supposef∈CBM˙Op,λ, 1≤p<∞, λ<1/n, and r1,r2∈R+; then
(48)(1|B(0,r1)|1+pλ∫B(0,r1)‍|f(x)-fB(0,r2)|pdx)1/p≤C(1+|logr1r2|)∥f∥CBM˙Op,λ.Proof of Theorem12.
For anyR>0, we denote B(0,R) by B and B(0,kR) by kB for any k∈Z+. Thus we have the following estimates:
(49)1|B|∫B‍|ℋΩ,A,βmf(x)|qdx≤1|B|∫B‍|1|x|n-β∫B(0,|x|)‍|Ω(x-y)f(y)||x-y|m-1×|Rm(A;x,y)|dy|Ω(x-y)f(y)||x-y|m-1|qdx=1|B|∑k=-∞0‍∫2kB∖2k-1B‍|1|x|n-β∑i=-∞k‍∫2iB∖2i-1B‍|Ω(x-y)f(y)||x-y|m-1×|Rm(A;x,y)|dy∑i=-∞k‍|qdx≤1|B|∑k=-∞0‍∫2kB∖2k-1B‍|1|x|n-β∑i=-∞k-3‍∫2iB∖2i-1B‍|Ω(x-y)f(y)||x-y|m-1×|Rm(A;x,y)|dy∑i=-∞k-3‍|qdx+1|B|×∑k=-∞0‍∫2kB∖2k-1B‍|1|x|n-β∑i=k-2k‍∫2iB∖2i-1B‍|Ω(x-y)f(y)||x-y|m-1×|Rm(A;x,y)|dy∑i=-∞k-3‍|qdx=I+II.
For the termI, let A~(x)=A(x)-∑|γ|=m-1(1/γ!)m2kB(DγA)xγ. Then it is easy to see Rm(A;x,y)=Rm(A~;x,y). As p2>n, then by Lemmas 8 and 9, we have
(50)|Rm(A~;x,y)|≤|Rm-1(A~;x,y)|+∑|γ|=m-1‍1γ!|DγA~(y)||x-y|m-1≤C|x-y|m-1∑|γ|=m-1‍[|Q~|λ2∥DγA∥CB˙MOp2,λ2+|Dγ(A~)(y)||Q~|λ2],
where Q~(x,y) is the cube centered at x and having diameter 5n|x-y|.
Asx∈2kB∖2k-1B and y∈2iB∖2i-1B with i≤k-3, we have |x-y|~|x|~|2kB|1/n and |x-y|≥C1|2iB|1/n.
Thus by the Hölder inequality and the condition1-1/p1-1/p2-1/r=1/r′-β/n-1/q>0, we have
(51)I≤C|B|-1×∑k=-∞0‍∫2kB∖2k-1B‍|1|x|n-β×∑i=-∞k-3‍∫2iB∖2i-1B‍|Ω(x-y)f(y)||x-y|m-1×|x-y|m-1×∑|γ|=m-1‍(|Q~|λ2∥DγA∥CB˙MOp2,λ2+|DγA~(y)||Q~|λ2)dy1|x|n-β|qdx≤C|B|-1×∑k=-∞0‍∫2kB∖2k-1B‍|1|x|n-β×∑i=-∞k-3‍∫2iB∖2i-1B‍|Ω(x-y)f(y)|×∑|γ|=m-1‍(|Q~|λ2×∥DγA∥CB˙MOp2,λ2+|DγA~(y)||Q~|λ2)dy1|x|n-β|qdx≤C|B|-1×∑k=-∞0‍∫2kB∖2k-1B‍1|x|(n-β)q×|∑i=-∞k-3‍(∫2iB∖2i-1B‍|Ω(x-y)|rdy)1/r×(∫2iB∖2i-1B‍∑|γ|=m-1‍(|x-y|λ2n×∥DγA∥CB˙MOp2,λ2+|DγA~(y)||x-y|λ2n)p2dy∫2iB∖2i-1B‍∑|γ|=m-1‍)1/p2×(∫2iB∖2i-1B‍|f(y)|p1dy)1/p1×|2iB|1-1/p1-1/p2-1/r∑i=-∞k-3‍|qdx.
Note the following fact:
(52)(∫2iB∖2i-1B‍∑|γ|=m-1‍(|x-y|λ2n∥DγA∥CB˙MOp2,λ2+|DγA~(y)||x-y|λ2n)p2dy∫2iB∖2i-1B‍∑|γ|=m-1‍)1/p2≤C∑|γ|=m-1‍[(∫2iB‍|DγA(y)-mQ~(DγA)|p2dy)1/p2∥DγA∥CB˙MOp2,λ2|2kB|λ2|2iB|1/p2+(∫2iB‍|DγA(y)-mQ~(DγA)|p2dy)1/p2]≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2|2iB|1/p2(|2kB|λ2+|2iB|λ2),
where the last inequality follows from Lemma 14 and the fact |x-y|≥C1|2iB|1/n. Thus by the condition 1+qλ>0 and λ1+1>λ1+1/p1>0, we have
(53)I≤C|B|-1×∑k=-∞0‍|2kB|(β/n-1)q×∫2kB∖2k-1B‍(∑i=-∞k-3‍∥f∥E˙p1,λ1|2iB|1/p1+λ1|2iB|1/r×∑|γ|=m-1∥DγA∥CB˙MOp2,λ2|2iB|1/p2‍×(|2kB|λ2+|2iB|λ2)×|2iB|1-1/p1-1/p2-1/r∑i=-∞k-3‍)qdx≤C∥f∥E˙p1,λ1q∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q|B|-1×∑k=-∞0‍|2kB|(β/n-1)q|2kB||2kB|λ2q×(∑i=-∞k-3‍|2iB|1/p1+λ1×|2iB|1/p2|2iB|1/r|2iB|1-1/p1-1/p2-1/r∑i=-∞k-3‍)q≤C∥f∥E˙p1,λ1q∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q×|B|-1+βq/n-q+λ2q+1+λ1q+q×∑k=-∞0‍2nkβq/n-knq+kn+kλ2nq+kλ1nq+knq≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q∥f∥E˙p1,λ1q|B|qλ.
To estimate the termII, we adopt some basic ideas from the estimates of the term I2 in Theorem 3. First, we denote Rm(A;x,y)=Rm(Akϕ;x,y)=Rm(Ak;x,y) for x∈2kB and y∈2iB with k-2≤i≤k, where Akϕ(x)=Rm-1(Ak;,x,y0)ϕ(|x-y0|-1x) with y0∈2iB∖2i-1B. Here Ak(x)=A(x)-∑|γ|=m-1(1/γ!)m2kB(DγA)xγ and ϕ is defined as in Section 2.
AsΩ∈Lr(𝕊n-1) with r>p1′, by Lemma 7, we get
(54)II≤1|B|∑k=-∞0‍∑i=k-2k‍∥ℋΩ,Akϕ,βmfi∥qq≤C|B|∑k=-∞0‍∑i=k-2k‍∥fi∥Lp1q∑|γ|=m-1‍∥DγAkϕ∥Lp2q,
where fi is defined by fi(x)=f(x)χ2iB∖2i-1B(x).
ForDγAkϕ, as |x-y0|≤C|y0|, then by Lemmas 8 and 14, we have the following estimates:
(55)|DγAkϕ(x)|≤∑|μ|+|ν|=m-1‍Cμν|Rm-1-|μ|(DμAk;x,y0)|×|Dνϕ(|x-y0|-1x)||x-y0|-|ν|≤C∑|μ|+|ν|=m-1‍|x-y0|m-1-|μ|-|ν|×∑|γ′|=m-1-|μ|‍(1|Q~(x,y0)|∫Q~(x,y0)‍|Dγ′(DμAk)(z)|p2dz)1/p2×χ|x|≤|x-y0|(x)≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2|x-y0|λ2nχ|x|≤C|y0|(x).
Thus we have
(56)∑|γ|=m-1‍∥DγAkϕ∥Lp2q≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q|2iB|qλ2|2iB|q/p2.
By the above estimates, we obtain
(57)II≤C|B|∑k=-∞0‍∑i=k-2k‍(1|2iB|1+p1λ1∫2iB‍|f(x)|p1dx)q/p1×|2iB|q/p1+qλ1×∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q|2iB|qλ2+q/p2≤C∑k=-∞0‍∑i=k-2k‍2in(1+λq)×∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q∥f∥E˙p1,λ1q|B|qλ≤C∑|γ|=m-1‍∥DγA∥CB˙MOp2,λ2q∥f∥E˙p1,λ1q|B|qλ.
Combining the estimates of I and II and by the definition of E˙q,λ, we finish the proof of Theorem 12.Proof of Theorem13.
The proof of Theorem13 is quite similar but much simpler than Theorem 12 and we omit the details here.



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*Source: 431692-2013-11-13.xml*