# Boundedness of Commutators of High-Dimensional Hausdorff Operators

Authors: Guilian Gao; Houyu Jia
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

We prove the boundedness of commutators of high-dimensional Hausdorff operatorsHΦ,b on Lebesgue space with central BMO function or Lipschitz function. Furthermore, the boundedness on Herz spaces and Morrey-Herz spaces is also obtained.


---

## Body

## 1.  Introduction

The one-dimensional Hausdorff operator is defined as(1.1)hΦf(x)=∫0∞Φ(t)tf(xt)dt,
where Φ is a locally integrable function on (0,∞). The operator hΦ has a deep root in the study of the one-dimensional Fourier analysis. Particularly, it is closely related to the summability of the classical Fourier series. The reader can see ([1–3]) to find details of background and recent development of the Hausdorff operator.Recently, Hausdorff operator in the high dimensional spaceℝn was studied. Three extensions of the one-dimensional Hausdorff operator in ℝn were recently introduced and studied in [4]. One of them is the operator(1.2)HΦf(x)=∫RnΦ(|y|-1x)|y|nf(y)dy,
where Φ is a radial function defined on ℝ+. Replacing Φ with(1.3)Φ1(t)=t-nχ(1,∞)(t),Φ2(t)=χ(0,1)(t),HΦf becomes the high dimensional Hardy operator(1.4)Hf(x)=1|x|n∫|y|<|x|f(y)dy,x∈Rn∖{0},
and its adjoint operator(1.5)H*f(x)=∫|y|≥|x|f(y)|y|ndy,
respectively. It is well known that Hardy operators are important operators in Harmonic analysis and a quite number of papers have appeared in [5–9]. And in [10], the authors have obtained the boundedness of commutators of one-dimensional Hausdorff operator with one-sided dyadic CMO functions on the Lebesgue space.These observations motivate us to study the boundedness of commutators ofn-dimensional Hausdorff operators on some function spaces. Let b(x) be a real measurable, locally integrable function; we define the commutators of Hausdorff operators as follows:(1.6)HΦ,bf(x)=b(x)HΦf(x)-HΦ(bf)(x)=∫RnΦ(|y|-1x)|y|n(b(x)-b(y))f(y)dy=∫|y|<|x|Φ(|y|-1x)|y|n(b(x)-b(y))f(y)dy+∫|y|≥|x|Φ(|y|-1x)|y|n(b(x)-b(y))f(y)dy:=HΦ,b1f(x)+HΦ,b2f(x).Definition 1.1 (see [11]).
Let1<q<∞. A function f∈Llocq(Rn) is said to belong to the space CṀOq(ℝn) if
(1.7)‖f‖CṀOq(Rn)=sup⁡r>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.  Obviously, we have BMO(ℝn)⊂CṀOq(ℝn) for all 1≤q<∞, and CṀOq(ℝn)⊂CṀOp(ℝn), 1≤p<q<∞.Definition 1.2 (see [12]).
Let0<γ<1. The Besov-Lipschitz space Λ̇γ(ℝn) is defined by
(1.8)‖b‖Λ̇γ(Rn)=sup⁡x,h∈Rn|b(x+h)-b(x)||h|γ<∞.Herz-type spaces are important function spaces in harmonic analysis. It should be pointed out that Lu and Yang make tremendous contributions on this spaces. Their book (joint with Hu) [13] is the unique research book in this topic. Below, we briefly recall the definition of the Herz-type spaces. We denote by B(x,r) the ball centered at x with radius r. C is a constant which may vary from line to line. For k∈ℤ, let Bk={x∈ℝn:|x|≤2k}, Δk=Bk∖Bk-1, and χk(k∈ℤ) denote the characteristic function of the set Δk.Definition 1.3 (see [13]).
Letα∈ℝ,0<p<∞, and 0<q<∞. The homogeneous Herz space K̇qα,p(ℝn) is defined by
(1.9)K̇qα,p(Rn)={f∈Lloc⁡q(Rn∖{0}):∥f∥K̇qα,p(Rn)<∞},
where
(1.10)‖f‖K̇qα,p(Rn)=(∑k=-∞∞2kαp‖fχk‖Lq(Rn)p)1/p.Obviously,K̇q0,q(ℝn)=Lq(ℝn)  and  K̇qα/q,q(ℝn)=Lq(ℝn,|x|α), so the Herz space is the natural generalization of the Lebesgue spaces with power weight |x|α.Definition 1.4 (see [13]).
Letα∈ℝ,0<p≤∞,0<q<∞  and  λ≥0. The homogeneous Morrey-Herz space MK̇p,qα,λ(ℝn) is defined by
(1.11)MK̇p,qα,λ(Rn)={f∈Lloc⁡q(Rn∖{0}):‖f‖MK̇p,qα,λ(Rn)<∞},
where
(1.12)‖f‖MK̇p,qα,λ(Rn)=sup⁡k0∈Z2-k0λ(∑k=-∞k02kαp‖fχk‖Lq(Rn)p)1/p
with the usual modification made when p=∞.In [14], the Morrey space Mqλ(ℝn) is defined by(1.13)Mqλ(Rn)={f∈Lloc⁡q(Rn):sup⁡λ>0,x∈Rn1rλ∫|x-y|<r|f(y)|qdy<∞}.
Obviously, MK̇p,qα,0(ℝn)=K̇qα,p(ℝn) and Mqλ(ℝn)⊂MK̇q,q0,λ(ℝn).


## 2.  Main Results

Now, we state our main results.Theorem 2.1.
Let1<p<∞, 1<r<min(p,p′), and  b∈CṀOmax(p,pr/(p-r))(ℝn):(a)
if one sets(2.1)AΦ,r=(∫0∞|Φ(t)|r′t-1+n(r′/r)dt)1/r′<∞,
then HΦ,b1 is bounded on Lp(ℝn);(b)
if one sets(2.2)BΦ,r=(∫0∞|Φ(t)|r′t-1+ndt)1/r′<∞,
then HΦ,b2 is bounded on Lp(ℝn);(c)
ifΦ satisfies (2.1) and (2.2), then HΦ,b is bounded on Lp(ℝn).Remark 2.2.
WhenΦ(t)=t-nχ(1,∞)(t), HΦ,bf(x)=HΦ,b1f(x)=Hbf(x). We can check that Φ(t) satisfies (2.1); therefore, the boundedness of commutator of Hardy operator is obtained. When Φ(t)=χ(0,1)(t), HΦ,bf(x)=HΦ,b2f(x)=Hb*f(x). We also know that Φ(t) satisfies (2.2); therefore, we get the boundedness of commutator of the adjoint Hardy operator see [9].Remark 2.3.
Using the same method, we can get the generalized result of Theorem2.1(b). Let 1<p<∞,1<r<p, 0≤λ<min(r′/r,r′/p), and b∈CMOmax(p,pr/(p-r))(ℝn). If
(2.3)BΦ,r,λ=(∫0∞|Φ(t)|r′t-1+nλdt)1/r′<∞,
then HΦ,b2 is bounded on Lp(ℝn).Theorem 2.4.
Let1<p,q<∞,1<r<min(p,q′),1/p-1/q=γ/n, and b∈Λ̇γ(ℝn)(0<γ<1). Then(a)
ifΦ satisfies (2.1), then HΦ,b1 are bounded from Lp(ℝn) to Lq(ℝn);(b)
ifΦ satisfies (2.2), then HΦ,b2 are bounded from Lp(ℝn) to Lq(ℝn);(c)
ifΦ satisfies (2.1) and (2.2), then HΦ,b is bounded from Lp(ℝn) to Lq(ℝn).Remark 2.5.
Just like in Remark2.2, we get Lipschitz estimates for commutator of Hardy operator or the adjoint Hardy operator. See the details in [15].Theorem 2.6.
Let0<p<∞,1<q<∞,1<r<q, and b∈CṀOmax(q,qr/(q-r))(ℝn). Then(a)
ifΦ satisfies (2.1) and α<n(1/r-1/q), then HΦ,b1 is bounded on K̇qα,p(ℝn);(b)
ifΦ satisfies (2.2) and α>-n(1/q-1/r′), then HΦ,b2 is bounded on K̇qα,p(ℝn);(c)
ifΦ satisfies (2.1), (2.2), and -n(1/q-1/r′)<α<n(1/r-1/q), then HΦ,b is bounded on K̇qα,p(ℝn).Remark 2.7.
Letα=0 in Theorem 2.6, then Theorem 2.1 can be obtained.Theorem 2.8.
Let0<p1≤p2<∞, 1<q1, q2<∞, 1<r<q1, 1/q1-1/q2=γ/n, and b∈Λ̇γ(ℝn)(0<γ<1). Then(a)
ifΦ satisfies (2.1) and α<n(1/r-1/q1), then HΦ,b1 is bounded from K̇q1α,p1(ℝn) to K̇q2α,p2(ℝn);(b)
ifΦ satisfies (2.2) and α>-n(1/q2-1/r′), then HΦ,b2 are bounded from K̇q1α,p1(ℝn) to K̇q2α,p2(ℝn);(c)
ifΦ satisfies (2.1), (2.2), and -n(1/q2-1/r′)<α<n(1/r-1/q1), then HΦ,b is bounded from K̇q1α,p1(ℝn) to K̇q2α,p2(ℝn).Remark 2.9.
Letα=0 in Theorem 2.8, then Theorem 2.4 can be obtained.Theorem 2.10.
Let0<p<∞, 1<q<∞, 1<r<q,λ≥0, and b∈CṀOmax(q,qr/(q-r))(ℝn). Then(a)
ifΦ satisfies (2.1) and α<n(1/r-1/q)+λ, then HΦ,b1 is bounded on MK̇p,qα,λ(ℝn);(b)
ifΦ satisfies (2.2) and α>-n(1/q-1/r′)+λ, then HΦ,b2 is bounded on MK̇p,qα,λ(ℝn);(c)
ifΦ satisfies (2.1), (2.2), and -n(1/q-1/r′)+λ<α<n(1/r-1/q)+λ, then HΦ,b is bounded on MK̇p,qα,λ(ℝn).Theorem 2.11.
Letλ≥0,0<p1≤p2<∞,1<q1,q2<∞,1<r<q1,1/q1-1/q2=γ/n, and b∈Λ̇γ(ℝn)(0<γ<1).(a)
ifΦ satisfies (2.1) and α<n(1/r-1/q1)+λ, then HΦ,b1 is bounded from MK̇p1,q1α,λ(ℝn) to MK̇p2,q2α,λ(ℝn);(b)
ifΦ satisfies (2.2) and α>-n(1/q2-1/r′)+λ, then HΦ,b2 is bounded from MK̇p1,q1α,λ(ℝn) to MK̇p2,q2α,λ(ℝn);(c)
ifΦ satisfies (2.1), (2.2), and -n(1/q2-1/r′)+λ<α<n(1/r-1/q1)+λ, then HΦ,b is bounded from MK̇p1,q1α,λ(ℝn) to MK̇p2,q2α,λ(ℝn).


## 3.  Proof of the Main Results

We first give several lemmas.Lemma 3.1.
Let1<r≤p<∞ and λ≥0, then(i)
∫Δi(|Φ(x/|y|)|/|y|n)|f(y)|dy≤CAΦ,r2in(1/r-1/p)|x|-n/r∥fχi∥Lp(ℝn),(ii)
∫Δi(|Φ(x/|y|)|/|y|n)|f(y)|dy≤CBΦ,r,λ2in(λ/r′-1/p)|x|-nλ/r′∥fχi∥Lp(ℝn).Proof.
(i)
By Hölder’s inequality and polar coordinate, we obtain(3.1)∫Δi|Φ(x/|y|)||y|n|f(y)|dy≤(∫Δi|Φ(x/|y|)|r′|y|nr′dy)1/r′(∫Δi|f(y)|rdy)1/r≤(∫|x|/2i|x|/2i-1|Sn-1||Φ(t)|r′t-1+nr′/r|x|n(1-r′)dt)1/r′×(∫Δi|f(y)|pdy)1/p2in(1/r-1/p)≤CAΦ,r2in(1/r-1/p)|x|-n/r‖fχi‖Lp(Rn).(ii)
By Hölder’s inequality and polar coordinate, we have(3.2)∫Δi|Φ(x/|y|)||y|n|f(y)|dy≤(∫Δi|Φ(x/|y|)|r′|y|(n(λ+1)/r′)r′dy)1/r′(∫Δi|f(y)|r|y|(n-n(λ+1)/r′)rdy)1/r≤C(∫2i-12i∫Sn-1|Φ(|x|R-1)|r′R-1-nλdσ(y′)dR)1/r′×2-in(1-(λ+1)/r′)‖fχi‖Lr≤C(∫2i-12i∫Sn-1|Φ(|x|R-1)|r′R-1-nλdσ(y′)dR)1/r′×2in(λ/r′-1/p)‖fχi‖Lp≤CBΦ,r,λ2in(λ/r′-1/p)|x|-nλ/r′‖fχi‖Lp.Lemma 3.2 (see [9]).
Letb be a CṀO1(ℝn) function and i, k∈ℤ, then
(3.3)|b(t)-bBi|≤|b(t)-bBk|+2|i-k|‖b‖CṀO1(Rn).Lemma 3.3 (see [12]).
Let0<γ<1 and f∈Λ̇γ(ℝn). Then
(3.4)|f(x)-f(y)|≤|x-y|γ‖f‖Λ̇γ(Rn)≤(|x|γ+|y|γ)‖f‖Λ̇γ(Rn).Proof of Theorem2.6.
(a) WhenΦ satisfies (2.1) and α<n(1/r-1/q), we get
(3.5)‖(HΦ,b1f)χk‖Lqq≤Cq∫Δk(∑i=-∞k∫Δi|Φ(x/|y|)||y|n|(b(x)-bBk)f(y)|dy)qdx+Cq∫Δk(∑i=-∞k∫Δi|Φ(x/|y|)||y|n|(b(y)-bBk)f(y)|dy)qdx:=I+II.
For I, using Lemma 3.1(i), we have
(3.6)I≤CqAΦ,rq∫Bk|b(x)-bBk|q(∑i=-∞k2in(1/r-1/q)|x|-n/r‖fχi‖Lq)qdx≤CqAΦ,rq‖b‖CṀOq(Rn)q|Bk|(∑i=-∞k2in(1/r-1/q)2-kn/r‖fχi‖Lq)q=CqAΦ,rq‖b‖CṀOq(Rn)q(∑i=-∞k2(i-k)n(1/r-1/q)‖fχi‖Lq)q.
For II, by Lemma 3.2, we have
(3.7)II≤Cq∫Δk(∑i=-∞k∫Δi|Φ(x/|y|)||y|n|(b(y)-bBi)f(y)|dy)qdx+Cq∫Δk(∑i=-∞k∫Δi|Φ(x/|y|)||y|n(k-i)‖b‖CṀO1(Rn)|f(y)|dy)qdx:=II1+II2.
For II1, by Lemma 3.1(i) and Hölder’s inequality, we have
(3.8)II1≤CqAΦ,rq∫Δk(∑i=-∞k|x|-n/r(∫Δi|(b(y)-bBi)f(y)|rdy)1/r)qdx≤CqAΦ,rq∫Δk(∑i=-∞k|x|-n/r(∫Δi|(b(y)-bBi)|rq/(q-r)dy)(q-r)/qr(∫Δi|f(y)|qdy)1/q)q≤CqAΦ,rq(∑i=-∞k2kn(1/q-1/r)‖b‖CṀOqr/(q-r)(Rn)|Bi|(q-r)/qr‖fχi‖Lq)q=CqAΦ,rq‖b‖CṀOqr/(q-r)(Rn)q(∑i=-∞k2(i-k)n(1/)-1/q)‖fχi‖Lq)q.
For II2, by Lemma 3.1(i), we have
(3.9)II2≤CqAΦ,rq‖b‖CṀO1(Rn)q∫Δk(∑i=-∞k(k-i)2in(1/r-1/q)|x|-n/r‖fχi‖Lq)qdx≤CqAΦ,rq‖b‖CṀO1(Rn)q2kn(∑i=-∞k(k-i)2in(1/r-1/q)2-kn/r‖fχi‖Lq)q=CqAΦ,rq‖b‖CṀO1(Rn)q(∑i=-∞k(k-i)2(i-k)n(1/r-1/q)‖fχi‖Lq)q.
Following the estimates ofI,II1, and II2, we can obtain that
(3.10)‖HΦ,b1f‖K̇qα,p(Rn)≤CAΦ,r‖b‖CṀOq(Rn)(∑k=-∞∞2kαp(∑i=-∞k2(i-k)n(1/r-1/q)‖fχi‖Lq)p)1/p+CAΦ,r‖b‖CṀOqr/(q-r)(Rn)(∑k=-∞∞2kαp(∑i=-∞k2(i-k)n(1/r-1/q)‖fχi‖Lq)p)1/p+CAΦ,r‖b‖CṀO1(Rn)(∑k=-∞∞2kαp(∑i=-∞k(k-i)2(i-k)n(1/r-1/q)‖fχi‖Lq)p)1/p:=S.
Obviously,
(3.11)S≤CAΦ,r‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)(∑k=-∞∞2kαp(∑i=-∞k(k-i)2(i-k)n(1/r-1/q)‖fχi‖Lq)p)1/p.
When 0<p≤1, we get
(3.12)Sp≤CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p∑k=-∞∞∑i=-∞k2iαp‖fχi‖Lqp(k-i)p2(i-k)(n/r-n/q-α)p=CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p∑i=-∞∞2iαp‖fχi‖Lqp∑k=i∞(k-i)p2(i-k)(n/r-n/q-α)p=CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p‖f‖K̇qα,pp.
For the case p>1, it follows from Hölder’s inequality that
(3.13)Sp≤CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p∑k=-∞∞∑i=-∞k2iαp‖fχi‖Lqp2(i-k)(n/r-n/q-α)p/2×(∑i=-∞k(k-i)p′2(i-k)(n/r-n/q-α)p′/2)p/p′≤CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p∑i=-∞∞2iαp‖fχi‖Lqp∑k=i∞2(i-k)(n/r-n/q-α)p/2=CpAΦ,rp‖b‖CṀOmax⁡(q,qr/(q-r))(Rn)p‖f‖K̇qα,pp.
When Φ satisfies (2.2) and α>-n(1/q-1/r′), we get
(3.14)‖(HΦ,b2f)χk‖Lq(Rn)q≤Cq∫Δk(∑i=k∞∫Δi|Φ(x/|y|)||y|n|(b(x)-bBk)f(y)|dy)qdx+Cq∫Ck(∑i=k∞∫Δi|Φ(x/|y|)||y|n|(b(y)-bBk)f(y)|dy)qdx:=J+JJ.
For J, using Lemma 3.1(ii), we have
(3.15)J≤CqBΦ,rq∫Δk|b(x)-bBk|q(∑i=k∞2in(1/r′-1/q)|x|-n/r′‖fχi‖Lq)qdx≤CqBΦ,rq‖b‖CṀOq(Rn)q|Bk|(∑i=k∞2in(1/r′-1/q)2-kn/r‖fχi‖Lq)q=CqBΦ,rq‖b‖CṀOq(Rn)q(∑i=k∞2(k-i)n(1/q-1/r′)‖fχi‖Lq)q.
For JJ, by Lemma 3.2, we have
(3.16)JJ≤Cq∫Δk(∑i=k∞∫Δi|Φ(x/|y|)||y|n|(b(y)-bBi)f(y)|dy)qdx+Cq∫Δk(∑i=k∞∫Δi|Φ(x/|y|)||y|n(i-k)‖b‖CṀO1(Rn)|f(y)|dy)qdx:=JJ1+JJ2.
For JJ1, by Lemma 3.1(ii) and Hölder’s inequality, similarly to II1, we have
(3.17)JJ1≤CqBΦ,rq‖b‖CṀOqr/(q-r)(Rn)q(∑i=k∞2(k-i)n(1/q-1/r′)‖fχi‖Lq)q.
For JJ2, also due to Lemma 3.1(ii), we have
(3.18)JJ2≤CqBΦ,rq‖b‖CṀO1(Rn)q∫Δk(∑i=k∞(i-k)2in(1/r′-1/q)|x|-n/r′‖fχi‖Lq)qdx≤CqBΦ,rq‖b‖CṀO1(Rn)q2kn(∑i=k∞(i-k)2in(1/r′-1/q)2-kn/r′‖fχi‖Lq)q=CqBΦ,rq‖b‖CṀO1(Rn)q(∑i=k∞(i-k)2(k-i)n(1/q-1/r′)‖fχi‖Lq)q.
The remaining proof is similar to the proof of (a), so that of (b) can be obtained easily.
(c) WhenΦ satisfies (2.1), (2.2), and -n(1/q-1/r′)<α<n(1/r-1/q), due to
(3.19)HΦ,bf(x)=HΦ,b1f(x)+HΦ,b2f(x),
therefore, by (a) and (b), we can easily get (c).Proof of Theorem2.8.
(a) WhenΦ satisfies (2.1) and α<n(1/r-1/q1), by Lemmas 3.3, 3.1(i), and 1/q1-1/q2=γ/n, we get
(3.20)‖(HΦ,b1f)χk‖Lq2q2≤‖b‖Λ̇γq2∫Δk(∑i=-∞k∫Δi|Φ(x/|y|)||y|n(|x|γ+|y|γ)|f(y)|dy)q2dx≤C‖b‖Λ̇γq2∫Δk(|x|γ∑i=-∞k∫Δi|Φ(x/|y|)||y|n|f(y)|dy)q2dx≤Cq2AΦ,rq2‖b‖Λ̇γq22kn(2kγ-kn/r∑i=-∞k2in(1/r-1/q1)‖fχi‖Lq1)q2=Cq2AΦ,rq2‖b‖Λ̇γq2(∑i=-∞k2(i-k)n(1/r-1/q1)‖fχi‖Lq1)q2.
The remaining proof Is similar to the proof of (a) of Theorem 2.6, so that of (a) can be obtained easily.
(b) WhenΦ satisfies (2.2) and α>-n(1/q2-1/r′), by Lemmas 3.3, 3.1(ii), and (1/q1)-(1/q2)=γ/n, we get
(3.21)‖(HΦ,b2f)χk‖Lq2(Rn)q2≤‖b‖Λ̇γq2∫Δk(∑i=k∞∫Δi|Φ(x/|y|)||y|n(|x|γ+|y|γ)|f(y)|dy)q2dx≤Cq2‖b‖Λ̇γq2∫Δk(∑i=k∞∫Δi|y|γ|Φ(x/|y|)||y|n|f(y)|dy)q2dx≤Cq2BΦ,rq2‖b‖Λ̇γq2∫Δk(∑i=k∞2iγ2in(1/r′-1/q1)|x|-n/r′‖fχi‖Lq1)q2dx≤Cq2BΦ,rq2‖b‖Λ̇γq2(∑i=k∞2(k-i)n(1/q2-1/r′)‖fχi‖Lq1)q2.
The remaining proof is similar to the proof of (a) of Theorem 2.6, so that of (b) can be obtained easily.
(c) The proof is the same as the proof of (c) of Theorem2.6.Proof of Theorem2.10.
We only give the proof of (a) whenλ>0. The proof of (b) is similar to that for (a), and the proof of (c) is similar to that for (c) of Theorem 2.6. By the definition of Morrey-Herz spaces and the estimates for I, II1, and II2 above, we have
(3.22)‖HΦ,b1f‖MK̇p,qα,λ≤CAΦ,r‖b‖CṀOq(Rn)sup⁡k0∈Z2-k0λ(∑k=-∞k02kαp(∑i=-∞k2(i-k)(n/r-n/q)‖fχi‖Lq)p)1/p+CAΦ,r‖b‖CṀOqr/(q-r)(Rn)sup⁡k0∈Z2-k0λ(∑k=-∞k02kαp(∑i=-∞k2(i-k)(n/r-n/q)‖fχi‖Lq)p)1/p+CAΦ,r‖b‖CṀO1(Rn)sup⁡k0∈Z2-k0λ(∑k=-∞k02kαp(∑i=-∞k(k-i)2(i-k)(n/r-n/q)‖fχi‖Lq)p)1/p:=E1+E2+E3.
For E1, noting that α<n(1/r-1/q)+λ, we have
(3.23)E1≤CAΦ,r‖b‖CṀOqsup⁡k0∈Z2-k0λ(∑k=-∞k0(∑i=-∞k2(i-k)(n/r-n/q-α)(∑l=-∞i2lαp‖fχi‖Lqp)1/p)p)1/p≤CAΦ,r‖b‖CṀOqsup⁡k0∈Z2-k0λ(∑k=-∞k02kpλ(∑i=-∞k2(i-k)(n/r-n/q-α+λ)‖f‖MK̇p,qα,λ)p)1/p≤CAΦ,r‖b‖CṀOqsup⁡k0∈Z2-k0λ(∑k=-∞k02kpλ)1/p‖f‖MK̇p,qα,λ=CAΦ,r‖b‖CṀOq(Rn)‖f‖MK̇p,qα,λ(Rn).
Similarly to the proof ofE1, we have
(3.24)E2≤CAΦ,r‖b‖CṀOqr/(q-r)(Rn)‖f‖MK̇p,qα,λ(Rn),E3≤CAΦ,r‖b‖CṀO1(Rn)‖f‖MK̇p,qα,λ(Rn).
This finishes the proof of Theorem 2.10.Proof of Theorem2.11.
The proof is similar to the proof of Theorem2.10, so we omit the details.



---

*Source: 541205-2012-04-18.xml*