# Endpoint Estimates for Fractional Hardy Operators and  Their Commutators on Hardy Spaces

Authors: Jiang Zhou; Dinghuai Wang
Journal: Journal of Function Spaces
 (2014)

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## Abstract

(
H
p
ℝ
n
,
L
q
ℝ
n
) bounds of fractional Hardy operators are obtained. Moreover, the estimates for commutators of fractional Hardy operators on Hardy spaces are worked out. It is also proved that the commutators of fractional Hardy operators are mapped from the Herz-type Hardy spaces into the Herz spaces. The estimates for multilinear commutators of fractional Hardy operators are also discussed.


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## Body

## 1. Introduction

The most fundamental averaging operator is the Hardy operators defined by(1)H(f)(x)=1x∫0x‍f(t)dt,
where the function f is nonnegative integrable on ℝ+=(0,∞) and x>0. A classical inequality, due to Hardy et al. [1], states that
(2)∥H(f)∥Lp(ℝ+)≤pp-1∥f∥Lp(ℝ+),
holds for 1<p<∞ and the constant p/(p-1) is the best possible.The Hardy integral inequality has received considerable attention. A number of papers involved its alternative proofs, generalizations, variants, and applications. Among numerous papers dealing with such inequalities, we choose to refer to the papers [2–4].Letf be a locally integrable function on ℝn,  0≤β<n. In [5], Fu et al. defined n-dimensional fractional Hardy operators ℋβ:
(3)ℋβf(x)=1|x|n-β∫|y|<|x|‍f(y)dy.
When β=0, ℋβ is just n-dimensional Hardy operators ℋ; see [3] for more details.It is well known that averaging operators play an important role in harmonic analysis. For example, the Hardy-Littlewood maximal operators control many kinds of operators in analysis. Therefore, the study of Hardy operators is meaningful and has been fully discussed (see [4, 6–11]). In 2012, Zhao et al. [11] have shown the following results: when b∈BMO(ℝn),  (1)ℋ maps H1(ℝn) into L1(ℝn);  (2)   [b,ℋ] maps H1(ℝn) into L1(ℝn) was false; (3)   [b,ℋ] maps H1(ℝn)L1,∞(ℝn). Furthermore, (3) is also true for b∈CM˙Oq(ℝn), 1≤q<∞. Recently, fractional Hardy operators were studied by many authors (see [5, 12, 13]). In 2007, Fu et al. [5] have obtained that the commutator [b,ℋβ] is bounded from K˙q1γ,p1(ℝn) to K˙q2γ,p2(ℝn), where γ<n(1-1/q1). In 2013, Lu et al. [13] have proved that fractional Hardy operators Hβ map Lp(ℝn) into Lq(ℝn), where 1<p≤n/β, 1/q=(1/p)-(β/n), and ∥ℋβ∥L1→L(n/(n-β)),∞=1. Inspired by the above, we consider the endpoint estimates for fractional Hardy operators and their commutators on Hardy-type spaces.This paper is organized as follows. In the second section, we give the (Hp(ℝn), Lq(ℝn)) bounds of fractional Hardy operators. In the third section, we obtain the estimates for commutators of fractional Hardy operators on Hardy spaces. In Section 4, we consider the case on Herz-type Hardy spaces. In Section 5, we obtain the estimate for multilinear commutators of fractional Hardy operators.Throughout this paper,C1~C2 denotes that C1 is equivalent to C2, which means there exist two positive constants c1 and c2 such that c1C1≤C2≤c2C1. For 1<p, and p′<∞, p and p′ are conjugate indices; that is, (1/p)+(1/p′)=1.Let us introduce some definitions below.Definition 1.
Let1≤q<∞ and b∈Lloc⁡(ℝn). One says that b∈CM˙Oq(ℝn) if and only if
(4)sup⁡r>0(1|B(0,r)|∫B(0,r)‍|b(x)-bB(0,r)|qdx)1/q<∞,
where bB(0,r)=(1/|B(0,r)|)∫B(0,r)‍b(x)dx. The CM˙Oq norm of b is defined by
(5)∥b∥CM˙Oq(ℝn)=sup⁡r>0(1|B(0,r)|∫B(0,r)‍|b(x)-bB(0,r)|qdx)1/q<∞.Remark 2.
When1≤q<∞, BMO(ℝn)⊆CM˙Oq(ℝn); when 1≤p<q<∞, CM˙Oq(ℝn)⊆CM˙Op(ℝn). We choose to refer to papers [5, 9].Definition 3.
The Lipschitz spaceLipα(ℝn) is the space of functions f satisfying
(6)∥f∥Lipα(ℝn):=sup⁡x,h∈ℝn,h≠0|f(x+h)-f(x)||h|α<∞,
where 0<α≤1.Remark 4.
When0<α<1, Lipα(ℝn)=∧˙α(ℝn), where ∧˙α(ℝn) is the homogeneous Besov-Lipschitz space.Definition 5 (see [14]).
Let0<p≤1; a function is called a(p,∞,s)-atom, where s≥[n(1/p-1)], if it satisfies the following conditions:
(7)(1)supp⁡(a)⊂B(x0,r);(2)∥a∥L∞≤|B(x0,r)|-1/p;(3)∫ℝn‍a(x)xγdx=0,where0≤|γ|≤s.As a proper subspace ofLp(ℝn), the atomic Hardy space Hp(ℝn) is defined by
(8)Hp(ℝn)={f∈S′(ℝn):f(x)=S′∑k‍λkak(x),∑k‍|λk|p<∞},
where each ak is a (p,∞,s)-atom and f is a tempered distribution. Set Hp(ℝn) norm of f by
(9)∥f∥Hp(ℝn):=inf⁡{(∑k∞‍|λk|p)1/p},
where the infimum has taken over all the decompositions of f=∑k‍λkak as above.Given a positive integerm and 1≤i≤m, we denote by Cim the family of all finite subsets σ={σ(1),σ(2),…,σ(i)} of {1,2,…,m} of j different elements. For σ∈Cim, set σc={1,…,m}∖σ. For b→=(b1,b2,…,bm) and σ={σ1,σ2,…,σi}∈Cjm, set b→σ=(bσ1,…,bσj), bσ=bσ1⋯bσj, and ∥b→σ∥Lipα(ℝn)=∥bσ1∥Lipα(ℝn)⋯∥bσi∥Lipα(ℝn).Definition 6.
Letbi (i=1,2,…,m) be a locally integrable functions, and 0<p≤1. A bounded measurable function a on ℝn is called a(p,b→)-atom, if(i)
supp⁡a⊂B=B(x0,r),(ii)
∥a∥L∞≤|B(x0,r)|-1/p,(iii)
∫B‍a(y)dy=∫B‍a(y)∏j∈σ‍bjdy=0 for any σ∈Cim, 1≤i≤m.A temperate distribution (see [15, 16]) f is said to belong to Hb→p(ℝn), if, in the Schwartz distribution sense, it can be written as
(10)f(x)=∑i‍λiai(x),
where ai is (p,b→)-atom, λi∈C, and ∑i‍|λi|p<∞. Moreover, ∥f∥Hb→p=inf⁡(∑i‍|λi|p)1/p, where the infimum has taken over all the decompositions of f as above.Remark 7.
Whenm=1, Hb→1(ℝn)=Hb1(ℝn).
LetBk={x∈ℝ:|x|<2k} and Ek=Bk∖Bk-1 for k∈ℤ. Denote χk=χEk.Definition 8.
Letγ∈ℝ, 0<p  and  q≤∞.(i)
The homogeneous Herz spaceK˙qγ,p(ℝn) is defined by
(11)K˙qγ,p(ℝn)={f:f∈Lloc⁡q(ℝn∖{0}),∥f∥K˙qγ,p<∞},
where
(12)∥f∥K˙qγ,p(ℝn)={∑k∈ℤ‍2kγp∥fχk∥Lq(ℝn)p}1/p.(ii)
The nonhomogeneous Herz spaceKqγ,p(ℝn) is defined by
(13)Kqγ,p(ℝn)={f:f∈Lloc⁡q(ℝn∖{0}),∥f∥K˙qγ,p<∞},
where
(14)∥f∥Kqγ,p(ℝn)={∑k=0∞‍2kγp∥fχk∥Lq(ℝn)p}1/p
(the usual modifications are made when p=∞).Remark 9.
When0<q≤∞,  K˙q0,q(ℝn)=Kq0,q(ℝn)=Lq(ℝn), and K˙qγ/q(ℝn)=L|x|γq(ℝn).Definition 10 (see [15]).
Letγ∈ℝ, and 0<p<q<∞.(i)
The homogeneous Herz-type Hardy spaceHK˙qγ,p(ℝn) is defined by
(15)HK˙qγ,p(ℝn)={f∈S′(ℝn):G(f)∈K˙qγ,p(ℝn)},
and we define ∥f∥HK˙qγ,p(ℝn)=∥G(f)∥K˙qγ,p(ℝn).(ii)
The nonhomogeneous Herz-type Hardy spaceHKqγ,p(ℝn) is defined by
(16)HKqγ,p(ℝn)={f∈S′(ℝn):G(f)∈Kqγ,p(ℝn)},
and we define ∥f∥HKqγ,p(ℝn)=∥G(f)∥Kqγ,p(ℝn).Remark 11.
When0<p<∞,  HK˙p0,p(ℝn)=Kp0,p(ℝn)=Hp(ℝn) and HK˙pγ/p(ℝn)=L|x|γp(ℝn). And when 1<q<∞, we know that HK˙qγ,p(ℝn)=K˙qγ,p(ℝn)andHKqγ,p(ℝn)=Kqγ,p(ℝn), where -n/q<γ<n(1-1/q). However, when γ≥n(1-1/q),  HK˙qγ,p(ℝn)≠K˙qγ,p(ℝn) and HKqγ,p(ℝn)≠Kqγ,p(ℝn) (see [17, 18]).


## 2. (Hp(ℝn), Lq(ℝn)) Bounds of Fractional Hardy Operators

Theorem 12.
Let0<p≤1  and  1/q=1/p-β/n.  ℋβ maps Hp(ℝn) into Lq(ℝn).Proof.
Assume thata is an atom of Hp(ℝn) and satisfies the following conditions: (i)  supp⁡(a)⊂B(x0,r), (ii)  ∥a∥L∞≤|B(x0,r)|-1/p, and (iii)  ∫‍a(x)xγdx=0, where 0≤|γ|≤s, s≥[n(1/p-1)]. We now take a~(x)=a(x+x0); then a~ satisfies: (i)   supp⁡(a~)⊂B(0,r), (ii)  ∥a~∥L∞≤|B(0,r)|-1/p, and (iii)  ∫‍a~(x)dx=0.
Suppose thatsupp⁡a~⊂B(0,r) for r>0. Consider
(17)∫ℝn‍|ℋβ(a~)(x)|qdx=∫ℝn‍|1|x|n-β∫|y|<|x|‍a~(y)dy|qdx=∫B(0,r)‍|1|x|n-β∫|y|<|x|‍a~(y)dy|qdx+∫ℝn∖B(0,r)‍|1|x|n-β∫|y|<|x|‍a~(y)dy|qdx=∫B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)dy|qdx+∫ℝn∖B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)dy|qdx:=I1+I2,
where we used the condition supp⁡(a)⊂B(0,r). For I1, we have the following estimate
(18)I1≤∫B(0,r)‍1|x|(n-β)q(∫{|y|<|x|}∩B(0,r)‍|a~(y)|dy)qdx≤∫B(0,r)‍1|x|(n-β)q1|B(0,r)|q/p(∫|y|<|x|‍dy)qdx≤C1|B(0,r)|q/p∫B(0,r)‍1|x|(n-β)q|x|nqdx≤C1|B(0,r)|q/p∫B(0,r)‍|x|βqdx≤C1rnq/p·rn+βq≤C<∞,
where we used the condition ∥a~∥L∞≤|B(0,r)|-1/p and 1/q=(1/p)-(β/n). For I2, since x∈ℝn∖B(0,r), we have {y∈ℝn:|y|<|x|}⊃{y∈ℝn:y∈B(0,r)}. Then
(19)I2=∫ℝn∖B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)dy|qdx=∫ℝn∖B(0,r)‍|1|x|n-β∫B(0,r)‍a~(y)dy|qdx=0.
The proof is completed.


## 3. Estimates for Commutators of Fractional Hardy Operators on Hardy Spaces

Definition 13 (see [5, 19]).
Letb be a locally integrable function on ℝn. The commutator of n-dimensional Hardy operators is defined by
(20)[b,ℋ]f=bℋf-ℋ(bf).
Meanwhile, the commutators ofn-dimensional fractional Hardy operators are defined by
(21)[b,ℋβ]f=bℋβf-ℋβ(bf).In general, the properties of commutator are worse than those of the operators themselves (e.g., the Hardy operators [11] and the singular integral operators [20]). Therefore, when b is in Lipα(ℝn), we prove that [b,ℋβ] is not bounded from H1(ℝn) to Ln/(n-β-α)(ℝn). Furthermore, we conclude that the commutator maps from H1(ℝn) to Ln/(n-β-α),∞(ℝn) and the commutator maps Hbp(ℝn) into Lq(ℝn), where 0<p≤1 and 1/q=1/p-(β+α)/n.Proposition 14.
Ifb∈Lipα(ℝn),  q=n/(n-β-α) and 0<q<∞, then [b,ℋβ] is not bounded from H1(ℝn) to Lq(ℝn).Proof.
We give the proof only for the casen=1, then q=1/(1-β-α). Taking b(x)=(|x|α-2α)χ(2,∞)(x) and f0(x)=χ(0,2)(x)-χ(-2,0)(x), it is easy to see that b∈Lipα(ℝn). Then for x>3, we have the following estimate:
(22)|[b,ℋβ]f(x)|=|1|x|1-β∫0x‍(b(x)-b(y))f0(y)dy|=1x1-β∫02‍(xα-2α-0)×1dy≥Cαx1-β-α.
Here Cα is the constant dependent on α. So we get
(23)∫ℝ1‍|[b,ℋβ]f(x)|qdx≥∫3∞‍(2x1-β-α)qdx=∞.Theorem 15.
Letb∈Lipα(ℝn),  q=n/(n-β-α), and 0<q<∞ then [b,ℋβ] maps H1(ℝn) into Lq,∞(ℝn).Proof.
It is enough to prove that(24)λ|{x∈ℝn:|[b,ℋβ](a~)(x)|>λ}|1/q≤C∥b∥Lipα(ℝn)∥a~∥H1(ℝn)
holds for any a~ defined in the proof of Theorem 12 and the constant C is independent of a~. Suppose that supp⁡a~⊂B(0,r) for r>0. Then
(25)|{x∈ℝn:|[b,ℋβ](a~)(x)|>λ}|1/q≤max⁡⁡{1,n(1/q)-1}×{|{x∈B(0,r):|[b,ℋβ](a~)(x)|>λ2}|1/q+|{x∈ℝn∖B(0,r):|[b,ℋβ](a~)(x)|>λ2}|1/q}.
For the first term, |y|<|x| implies |x-y|≤|x|+|y|≤2|x|; then we have
(26)λq|{x∈B(0,r):|[b,ℋβ](a~)(x)|>λ2}|≤C∫B(0,r)‍|[b,ℋβ](a~)(x)|qdx≤C∫B(0,r)‍|1|x|n-β∫|y|<|x|‍a~(y)(b(x)-b(y))dy|qdx≤C∥b∥Lipα(ℝn)q×∫B(0,r)‍(1|x|n-β∫|y|<|x|‍|a~(y)||x-y|αdy)qdx≤C∥b∥Lipα(ℝn)q∫B(0,r)‍1|B(0,r)|q×(1|x|n-β∫|y|<|x|‍|x-y|αdy)qdx≤C∥b∥Lipα(ℝn)q1|B(0,r)|q∫B(0,r)‍|x|βq+αqdx≤C∥b∥Lipα(ℝn)qr-nq·rn+αq+βq=C∥b∥Lipα(ℝn)q,
where q=n/(n-β-α). For the last term, by x∈ℝn∖B(0,r), we have
(27)|1|x|n-β∫|y|<r‍(b(x)-b(y))a~(y)dy|≤1|x|n-β∫|y|<r‍|b(x)-b(y)||a~(y)|dy≤C∥b∥Lipβ(ℝn)1|x|n-β∫|y|<r‍|x-y|α1|B(0,r)|dy≤C∥b∥Lipβ(ℝn)1|x|n-β∫|y|<r‍|x|α1|B(0,r)|dy≤C∥b∥Lipβ(ℝn)1|x|n-β-α.
So we obtain that
(28)|{x∈ℝn∖B(0,r):|[b,ℋβ](a~)(x)>λ2|}|≤|{x∈ℝn∖B(0,r):C∥b∥Lipβ(ℝn)1|x|n-β-α>λ}|=C∫r(C∥b∥Lipβ(ℝn)/λ)1/(n-α-β)‍tn-1dt≤C(∥b∥Lipβ(ℝn)λ)n/(n-α-β)=C(∥b∥Lipβ(ℝn)λ)q.
Combining all the above estimates, we complete the proof of Theorem 15.Theorem 16.
Letb∈Lipα(ℝn),  0<p≤1 and 1/q=1/p-(β+α)/n and 0<q<∞; then [b,ℋβ] maps Hbp(ℝn) into Lq(ℝn).Proof.
Similar to the proof above, suppose thatsupp⁡a~⊂B(0,r) for r>0. Consider
(29)∫ℝn‍|[b,ℋβ](a~)(x)|qdx=∫ℝn‍|1|x|n-β∫|y|<|x|‍a~(y)(b(x)-b(y))dy|qdx=∫B(0,r)‍|∫{|y|<|x|}∩B(0,r)‍1|x|n-β×∫{|y|<|x|}∩B(0,r)‍a~(y)(b(x)-b(y))dy|qdx+∫ℝn∖B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)×(b(x)-b(y))dy1|x|n-β∫{|y|<|x|}∩B(0,r)‍|qdx≤∫B(0,r)‍|∫{|y|<|x|}∩B(0,r)‍1|x|n-β×∫{|y|<|x|}∩B(0,r)‍a~(y)(b(x)-b(y))dy|qdx+C∫ℝn∖B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)b(x)dy|qdx+C∫ℝn∖B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)b(y)dy|qdx:=J1+J2+J3.
For J1, by the fact that |y|<|x| implies |x-y|≤|x|+|y|≤2|x|, we have that
(30)J1=∫B(0,r)‍|1|x|n-β∫{|y|<|x|}∩B(0,r)‍a~(y)(b(x)-b(y))dy|qdx≤C∥b∥Lipα(ℝn)q×∫B(0,r)‍|1|x|n-β1|B(0,r)|1/p∫|y|<|x|‍|x|αdy|qdx≤C∥b∥Lipα(ℝn)q∫B(0,r)‍||x|β+α1|B(0,r)|1/p|qdx≤C∥b∥Lipα(ℝn)q.
By ∫‍a~(y)dy=∫‍a~(y)b(y)dy=0, we obtain that J2=J3=0. Then [b,ℋβ] maps Hbp(ℝn) into Lq(ℝn).Theorem 17.
Letb∈Lipα(ℝn), 1/q=1/p-(β+α)/n, and 0<p≤n/(n+α); 0<q<∞, then the following two conditions are equivalent:(i)
[b,ℋβ] is bounded from Hp(ℝn) to Lq(ℝn);(ii)
for alla(p,∞,0)-atom, ∫ℝn‍a(y)b(y)dy=0.Proof.
By Theorem16, it is clear that (ii) ⇒ (i) is obvious. We only need to prove (i) ⇒ (ii). We assume that supp⁡a⊂B=B(x0,r) for r>0. Let
(31)v1(x)=χB[b,ℋβ]a(x),v2(x)=χℝn∖B(x)b(x)1|x|n-β∫B‍a(y)dy,v3(x)=χℝn∖B(x)1|x|n-β∫B‍a(y)b(y)dy.
Then
(32)[b,ℋβ]a(x)=v1(x)+v2(x)-v3(x).
Obviously, ∥v1∥q≤C, and ∥v2∥q=0. For N>2,
(33)C≥∥v3∥q≥|∫B‍a(y)b(y)dy|{∫r<|x-x0|≤Nr‍1|x|(n-β)qdx}1/q.
By 0<p≤n/(n+α), we have (n-β)q≤n.
WhenN→∞, ∫r<|x-x0|≤Nr‍(1/|x|(n-β)q)dx→∞; then
(34)∫B‍a(y)b(y)dy=0.Whenb is in CM˙O(ℝn), the commutator [b,ℋβ] has the similar properties.Proposition 18.
Ifb∈BMO(ℝn)  and  q=n/(n-β), then [b,ℋβ] is not bounded from H1(ℝn) to Lq(ℝn).Theorem 19.
Letb∈CM˙Omax⁡{sq,s}(ℝn)  and  q=n/(n-β),  1<s<∞, then [b,ℋβ] maps H1(ℝn) into Lq,∞(ℝn).Remark 20.
In 2013, Yu and Lu [21] have given the analogous results of Proposition 18 and Theorem 19.Theorem 21.
Letb∈CM˙Omax⁡{sq,s}(ℝn),  0<p≤1 and 1<s<∞, 1/q=1/p-β/n; then [b,ℋβ] is bounded from Hbp(ℝn) to Lq(ℝn).Theorem 22.
Letb∈CM˙Omax⁡{sq,s}(ℝn) and 1/q=1/p-β/n, 1<s<∞, and 0<p≤1; then the following two conditions are equivalent:(i)
[b,ℋβ] maps Hp(ℝn) into Lq(ℝn);(ii)
for alla(p,∞,0)-atom, ∫ℝn‍a(y)b(y)dy=0.


## 4. Estimates for Commutators of Fractional Hardy Operators on Herz-Type Hardy Spaces

The boundedness of[b,ℋβ] on the Herz spaces K˙qγ,p(ℝn) has been obtained as the following, where γ<n(1-1/q1).Proposition 23 (see [22]).
Letβ≥0, 0<p1≤p2<∞, 1/q2=1/q1-(β+α)/n, 1<q1, q2<∞, 1/q1+1/q1′=1 and b∈Lipα(ℝn); then γ<n/q1′ implies that [b,ℋβ] is bounded from K˙q1γ,p1(ℝn) to K˙q2γ,p2(ℝn).Proposition 24 (see [5]).
Letβ≥0, 0<p1≤p2<∞, 1/q2=1/q1-β/n, 1<q1<∞, 1/q1+1/q1′=1 and b∈CM˙Omax⁡(q2,q1′)(ℝn); then γ<n/q1′ implies that [b,ℋβ] is bounded from K˙q1γ,p1(ℝn) to K˙q2γ,p2(ℝn).In this section, we discuss the caseγ≥n(1-1/q1).Definition 25.
Letγ∈ℝ and 1<q<∞. A function is called central (γ,q)-atom, if it satisfies the following conditions:(i)
supp⁡a⊂B(0,r);(ii)
∥a∥q≤|B(0,r)|-γ/n;(iii)
∫ℝn‍a(x)xsdx=0, where |s|≤[γ-n(1-1/q)].Lemma 26 (see [18]).
Let0<p<∞, 1<q<∞ and γ≥n(1-1/q).  Consider that f∈HK˙qγ,p(ℝn) if and only if
(35)f=∑k∈ℤ‍λkak,inthesenseofS′(ℝn),
where each ak is a central (γ,q)-atom with the support Bk and ∑k∈ℤ‍|λk|p<∞. Moreover,
(36)∥f∥HK˙qγ,p(ℝn)~inf⁡{(∑k∈ℤ‍|λk|p)1/p},
where the infimum has taken over all above decompositions of f.Theorem 27.
Letb∈Lipα(ℝn), γ≥n(1-1/q1)+α,  0<p<∞, 1<q1, q2<∞, 1/q2=1/q1-(β+α)/n, then the following two conditions are equivalent:(i)
[b,ℋβ] maps HK˙q1γ,p(ℝn) into K˙q2γ,p(ℝn);(ii)
for all centrala(γ,q1)-atom, ∫ℝn‍b(y)a(y)dy=0.Proof.
(i)⇒ (ii). Let a be a central (γ,q1)-atom, with the support Bk. In fact, if supp⁡a(x)⊂B(0,r), let k∈ℤ such that 2k-1<r≤2k, and for C(n,q1,α)=2-γ, set a′=C(n,q1,α)a; then a′ is a central (γ,q1)-atom with the support Bk.
Similar to the proof of Theorem16, we have
(37)v1(x)=χBk[b,ℋβ]a(x),v2(x)=χℝn∖Bk(x)b(x)1|x|n-β∫B‍a(y)dy,v3(x)=χℝn∖Bk(x)1|x|n-β∫B‍a(y)b(y)dy.
Then
(38)[b,ℋβ]a(x)=v1(x)+v2(x)-v3(x).
Using the fact that [b,ℋβ] maps Lq1(ℝn) into Lq2(ℝn)(39)∥v1∥K˙qγ,p(ℝn)≤(∑j=-∞k+1‍2jγp∥χj([b,ℋβ]a)∥q2p)1/p≤C(∑j=-∞k+1‍2jγp∥a∥q1p)1/p≤C.
When |x|≥2k+1, we have ∥v2∥K˙q2γ,p(ℝn)=0. On the other hand, ∥a∥HK˙q1γ,p(ℝn)≤1, so ∥[b,ℋβ]a∥K˙q2γ,p≤C. Combining all the above estimates, we have ∥v3∥K˙q2γ,p(ℝn)≤C. For N>3, by γ≥n(1-1/q1+α), we have that
(40)Cp≥∑j=k+2∞‍2jγp(∫Ej‍|1|x|n-β∫|y|<|x|‍b(y)a(y)dy|q2dx)p/q2≥∑j=k+3k+N‍2jγp(|∫ℝn‍b(y)a(y)dy|∫Ej‍|x|(β-n)q2dx)p/q2≥C∑j=k+3k+N‍2j(γ-n(1-1/q1)+α)p|∫ℝn‍b(y)a(y)dy|p.
Set N→∞; then
(41)∫ℝn‍b(y)a(y)dy=0.
(ii)⇒ (i). Similar to the previous proof, by ∫ℝn‍b(y)a(y)dy=0, we obtain ∥v3∥K˙q2γ,p(ℝn)=0. Combining ∥v1∥K˙q2γ,p(ℝn)≤C and ∥v2∥K˙q2γ,p(ℝn)=0,  [b,ℋβ] maps HK˙q1γ,p(ℝn) into K˙q2n(1-1/q1)+β+α,p(ℝn).Whenb is in CM˙O(ℝn), the commutator [b,ℋβ] also has the similar properties.Theorem 28.
Letb∈CM˙Omax⁡{sq,s}(ℝn),  γ≥n(1-1/q1),  1<s<∞, 0<p<∞, 1<q1, q2<∞ and 1/q2=1/q1-β/n; then the following two conditions are equivalent:(i)
[b,ℋβ] maps HK˙q1γ,p(ℝn) into K˙q2γ,p(ℝn);(ii)
for all centrala(γ,q1)-atom, ∫ℝn‍b(y)a(y)dy=0.


## 5. Estimates for Multilinear Commutators of Fractional Hardy Operators

Definition 29.
The multilinear commutator of fractional Hardy operators is defined by(42)[b→,ℋβ]=1|x|n-β∫|y|<|x|‍∏i=1m‍(bi(x)-bi(y))f(y)dy,
where b→=(b1,b2,…,bm) is a m-dimensional vector. When m=0, [b→,ℋβ]=ℋβ. When m=1, [b→,ℋβ]=[b,ℋβ].The study of multilinear operators is motivated by a mere quest to generalize the theory of linear operators and by their natural appearance in analysis (see [23–25]). In this section, we consider the multilinear commutators of fractional Hardy operators on Hardy spaces.Theorem 30.
Letb→=(b1,b2,…,bm),  bi∈Lipα(ℝn),  1/q=1/p-(β+mα)/n, 0<p≤1 and 0<q<∞; then [b→,ℋβ] maps Hb→p(ℝn) to Lq(ℝn).Proof.
Similar to the proof above, whenx∈ℝn∖B(0,r), we have
(43)∫ℝn∖B(0,r)‍|[b→,ℋβ](a~)(x)|qdx=0,
so it is enough to prove that
(44)(∫B(0,r)‍|[b→,ℋβ](a~)(x)|qdx)1/q≤C∥b∥Lipα(ℝn)∥a~∥Hb→1.
By bi∈Lipα(ℝn), we obtain |bi(x)-bi(y)|≤∥bi∥Lipα(ℝn)|x-y|α, i=1,2,…,m,   and
(45)(∫B(0,r)‍|[b→,ℋβ](a~)(x)|qdx)1/q≤(∫B(0,r)‍|×∏i=1m‍(bi(x)-bi(y))dy|q1|x|n-β∫|y|<|x|‍a~(y)×∏i=1m‍(bi(x)-bi(y))dy|qdx)1/q≤∏i=1m‍∥bi∥Lipα(ℝn)×(∫B(0,r)‍|1|x|n-β×∫|y|<|x|‍|a~(y)||x-y)|mαdy1|x|n-β∫|y|<|x|‍|qdx∫B(0,r)‍|1|x|n-β∫|y|<|x|‍|)1/q≤C∏i=1m∥bi∥Lipα(ℝn)‍×(∫B(0,r)‍|∫|y|<|x|‍r-n/p1|x|n-β×∫|y|<|x|‍r-n/p·2|x|mαdy1|x|n-β|qdx)1/q≤C∏i=1m∥bi∥Lipα(ℝn)‍×(∫B(0,r)‍|1|x|n-β∫|y|<|x|‍r-n/p+mαdy|qdx)1/q≤C∏i=1m‍∥bi∥Lipα(ℝn)(∫B(0,r)‍|rβ·r-n/p+mα|qdx)1/q≤C∏i=1m∥bi∥Lipα(ℝn),‍
where |x|<r and 1/q=(1/p)-((β+mα)/n); we complete the proof of Theorem 30.Theorem 31.
Letb→=(b1,b2,…,bm),  bi∈Lipα(ℝn),  q=n/(n-β-mα), and 0<q<∞; then [b→,ℋβ] maps Hp(ℝn) into Lq,∞(ℝn).Proof.
It is similar to the proof of Theorem15.Whenb is in CM˙O(ℝn), we suppose the multilinear commutator of fractional Hardy operators maps H1(ℝn) into Ln/(n-β),∞, but we cannot prove it. However, we get the following result.Theorem 32.
Letb→=(b1,b2,…,bm) and bi∈CM˙Omax⁡{siq,si}(ℝn), 1/q=1/p-β/n, 0<p≤1, 1<si<∞, then [b→,ℋβ] maps Hb→p(ℝn) into Lq(ℝn).Proof.
Similar to the proof of Theorem12, it is enough to prove that
(46)∥[b→,ℋβ](a~)(x)∥Lq≤C∏i=1m‍∥bi∥CM˙O(ℝn)∥a~∥Hb→1,
where a~ is a center (p,b→)-atom supported on a ball B=B(0,r). We write
(47)[b→,ℋβ](a~)(x)=1|x|n-β∫|y|<|x|‍a~(y)∏i=1m‍(bi(x)-bi(y))dy=1|x|n-β∫|y|<|x|‍a~(y)∑i=0m‍∑σ∈Cim‍∏i∈σ‍(bi(x)-λi)×∏i∈σc‍(λi-bi(y))dy=1|x|n-β∏i=1m‍(bi(x)-λi)∫|y|<|x|‍a~(y)dy+∑i=1m-1‍∑σ∈Cim‍1|x|n-β∏i∈σ‍(bi(x)-λi)×∫|y|<|x|‍a~(y)∏i∈σc‍(λi-bi(y))dy+1|x|n-β∫|y|<|x|‍a~(y)∏i=1m‍(λi-bi(y))dy:=K1+K2+K3,
where λi=(bi)B. By Minkowski inequality, we have
(48)∥[b→,ℋβ](a~)(x)∥Lq=∫ℝn‍|[b→,ℋβ](a~)(x)|qdx≤(∫ℝn‍|K1+K2+K3|qdx)1/q≤(∫ℝn‍|K1|qdx)1/q+(∫ℝn‍|K2|qdx)1/q+(∫ℝn‍|K3|qdx)1/q≤(∫B(0,r)‍|K1|qdx)1/q+(∫ℝn∖B(0,r)‍|K1|qdx)1/q+(∫B(0,r)‍|K2|qdx)1/q+(∫ℝn∖B(0,r)‍|K2|qdx)1/q+(∫B(0,r)‍|K3|qdx)1/q+(∫ℝn∖B(0,r)‍|K3|qdx)1/q:=K11+K12+K21+K22+K31+K32.
For x∈B(0,r), we have |x|<r; then
(49)K11=(∫B(0,r)‍|1|x|n-β∏i=1m‍(bj(x)-λi)×∫|y|<|x|‍a~(y)dy|qdx)1/q≤C(∫B(0,r)‍|1|x|n-β·∏i=1m‍|bi(x)-λi|·|x|n·|B(0,r)|-1/p∏i=1m‍|qdx)1/q≤Cr(β-n/p)(∫B(0,r)‍∏i=1m‍|bi(x)-λi|qdx)1/q≤C(1B(0,r)∫B(0,r)‍∏i=1m‍|bi(x)-λi|qdx)1/q≤C∏i=1m‍∥bi∥CM˙Osiq(ℝn),K31=(∫B(0,r)‍|∫|y|<|x|‍a~(y)∏i=1m‍1|x|n-β×∫|y|<|x|‍a~(y)∏i=1m‍(λi-bi(y))dy|qdx)1/q≤C(∫B(0,r)‍|∏i=1m‍1|x|n-β·|B(0,r)|-1×∫|y|<|x|‍∏i=1m‍(λi-bi(y))dy|qdx)1/q≤C(∏i=1m‍∫B(0,r)|∏i=1m‍|B(0,r)|-1/p·|x|β·1B(0,|x|)‍×∫|y|<|x|‍∏i=1m‍(λi-bi(y))dy|qdx)1/q≤C(∫B(0,r)‍|·∏i=1m‍∥b∥CM˙Osi(ℝn)|q|B(0,r)|-1/p·|x|β·∏i=1m‍∥b∥CM˙Osi(ℝn)|qdx)1/q≤C∏i=1m‍∥b∥CM˙Osi(ℝn)(r(β-n/p)q∫B(0,r)‍dx)1/q≤C∏i=1m‍∥b∥CM˙Osi(ℝn).
For K21,
(50)(∫B(0,r)‍|1|x|n-β∏i∈σ‍(bi(x)-λi)×∫|y|<|x|‍a~(y)∏i∈σc‍(λi-bi(y))dy|qdx)1/q≤C(∫B(0,r)‍|∏i∈σc‍1rn/p-β×∏i∈σ‍(bi(x)-λi)∏i∈σc‍∥b∥CM˙Osi(ℝn)|qdx)1/q≤C∏i∈σc‍∥b∥CM˙Osi(ℝn)(r-n∫B(0,r)‍|∏i∈σ‍(bi(x)-λi)|qdx)1/q≤C∏i∈σc‍∥b∥CM˙Osi(ℝn)·∏i∈σ‍∥b∥CM˙Osiq(ℝn)≤C∏i=1m‍∥b∥CM˙Omax⁡⁡{siq,si}(ℝn).
For x∈ℝn∖B(0,r), we have |x|≥r; then {y:|y|<|x|}⋂‍{y:|y|<r}={y:|y|<r}. By the condition of ∫B‍a(y)dy=∫B‍a(y)∏j∈σ‍bjdy=0 for any σ∈Cim, 1≤j≤m, we have that
(51)∫|y|<r‍a~(y)∏i∈σc‍(λi-bi(y))dy=∫|y|<r‍a~(y)∏i=1m‍(λi-bi(y))dy=0,
so,
(52)K12=K22=K32=0.
Combining all the above estimates, we complete the proof of Theorem 32.



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*Source: 272864-2014-02-18.xml*