# Boundedness ofp-Adic Hardy Operators and Their Commutators on p-Adic Central Morrey and BMO Spaces

Authors: Qing Yan Wu; Ling Mi; Zun Wei Fu
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

We obtain the sharp bounds ofp-adic Hardy operators on p-adic central Morrey spaces and p-adic λ-central BMO spaces, respectively. We also establish the λ-central BMO estimates for commutators of p-adic Hardy operators on p-adic central Morrey spaces.


---

## Body

## 1. Introduction

In the past decades, the field ofp-adic numbers has been intensively used in theoretical and mathematical physics (see [1–9] and references therein). As a consequence, new mathematical problems have emerged, among which we refer to [10, 11] for Riesz potentials [12–16], for p-adic pseudodifferential equations, and so forth. In the past few years, there is an increasing interest in the study of harmonic analysis onp-adic field and their various generalizations and the related theory of operators and spaces; see, for example [17–27].For a prime numberp, let ℚp be the field of p-adic numbers. It is defined as the completion of the field of rational numbers ℚ with respect to the non-Archimedean p-adic norm |·|p. This norm is defined as follows: |0|p=0; if any nonzero rational number x is represented as x=pγ(m/n), where γ is an integer and integers m, n are indivisible by p, then |x|p=p-γ. It is easy to see that the norm satisfies the following properties:
(1)|xy|p=|x|p|y|p,|x+y|p≤max{|x|p,|y|p}.
Moreover, if |x|p≠|y|p, then |x+y|p=max{|x|p,|y|p}. It is well known that ℚp is a typical model of non-Archimedean local fields. From the standard p-adic analysis [7], we see that any nonzero p-adic number x∈ℚp can be uniquely represented in the canonical series as follows:
(2)x=pγ∑j=0∞‍ajpj,γ=γ(x)∈ℤ,
where aj are integers, 0≤aj≤p-1, and a0≠0. The series (2) converges in the p-adic norm since |ajpj|p=p-j.The spaceℚpn consists of points x=(x1,x2,…,xn), where xj∈ℚp, j=1,2,…,n. The p-adic norm on ℚpn is
(3)|x|p∶=max1≤j≤n|xj|p,x∈ℚpn.
Denote by Bγ(a)={x∈ℚpn:|x-a|p≤pγ} the ball with center at a∈ℚpn and radius pγ and by Sγ(a):={x∈ℚpn:|x-a|p=pγ} the sphere with center at a∈ℚpn and radius pγ, γ∈ℤ. It is clear that Sγ(a)=Bγ(a)∖Bγ-1(a), and
(4)Bγ(a)=⋃k≤γ‍Sk(a),⋃γ=-∞+∞‍Bγ(a)=⋃γ=-∞+∞‍Sγ(a)=ℚpn.
We set Bγ(0)=Bγ and Sγ(0)=Sγ.Sinceℚpn is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure dx on ℚpn, which is unique up to positive constant multiple and is translation invariant. We normalize the measure dx by the equality
(5)∫B0(0)‍dx=|B0(0)|H=1,
where |E|H denotes the Haar measure of a measurable subset E of ℚpn. By simple calculation, we can obtain that
(6)|Bγ(a)|H=pγn,|Sγ(a)|H=pγn(1-p-n)
for any a∈ℚpn. For a more complete introduction to the p-adic field, see [27] or [7].The well-known Hardy’s integral inequality [28] tells us that, for 1<q<∞,
(7)∥Hf∥Lq(ℝ+)≤qq-1∥f∥Lq(ℝ+),
where the classical Hardy operator is defined by
(8)Hf(x)∶=1x∫0x‍f(t)dt
for nonnegative integral function f on ℝ+, and the constant q/(q-1) is the best possible. Thus the norm of Hardy operator on Lq(ℝ+) is
(9)∥H∥Lq(ℝ+)→Lq(ℝ+)=qq-1.Faris [29] introduced the following n-dimensional Hardy operator, for nonnegative function f on ℝn,
(10)ℋf(x)∶=1Ωn|x|n∫|y|<|x|‍f(y)dy,
where Ωn is the volume of the unit ball in ℝn. Christ and Grafakos [30] obtained that the norm of ℋ on Lq(ℝn) is
(11)∥ℋ∥Lq(ℝn)→Lq(ℝn)=qq-1,
which is the same as that of the 1-dimensional Hardy operator.In [31], Fu et al. obtained the precise norm of m-linear Hardy operators on weighted Lebesgue spaces and central Morrey spaces. Fu et al. [32] introduced p-adic Hardy operators and got the sharp estimates of p-adic Hardy operators on p-adic weighted Lebesgue spaces. Moreover, they proved that the commutators generated by the p-adic Hardy operators and the central BMO functions are bounded on p-adic weighted Lebesgue spaces and p-adic Herz spaces see; [33] for more information about Herz spaces. Ren and Tao [34] Yu and Lu [35] studied the boundedness of commutators of Hardy type on some spaces.Inspired by these results, in this paper we will establish the sharp estimates ofp-adic Hardy operators on p-adic central Morrey and λ-central BMO spaces. Furthermore, we will discuss the boundedness for commutators of p-adic Hardy operators and λ-central BMO functions on p-adic central Morrey spaces.Definition 1.
For a functionf on ℚpn, we define the p-adic Hardy operator as follows:
(12)ℋpf(x)=1|x|pn∫B(0,|x|p)‍f(t)dt,x∈ℚpn∖{0},
where B(0,|x|p) is a ball in ℚpn with center at 0∈ℚpn and radius |x|p.Morrey [36] introduced the Lq,λ(ℝn) spaces to study the local behavior of solutions to second-order elliptic partial differential equations. The p-adic Morrey space is defined as follows.Definition 2.
Let1≤q<∞ and let λ≥-1/q. The p-adic Morrey space Lq,λ(ℚpn) is defined by
(13)Lq,λ(ℚpn)={f∈Llocq(ℚpn):∥f∥Lq,λ(ℚpn)<∞},
where
(14)∥f∥Lq,λ(ℚpn)=supa∈ℚpn,γ∈ℤ(1|Bγ(a)|H1+λq∫Bγ(a)‍|f(x)|qdx)1/q.Remark 3.
It is clear thatLq,-1/q(ℚpn)=Lq(ℚpn), Lq,0(ℚpn)=L∞(ℚpn).For some recent developments of Morrey spaces and their related function spaces onℝn, we refer the reader to [37]. In 2000, Alvarez et al. [38] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced λ-central bounded mean oscillation spaces and central Morrey spaces, respectively. Next, we introduce their p-adic versions.Definition 4.
Letλ∈ℝ and let 1<q<∞. The p-adic central Morrey space B˙q,λ(ℚpn) is defined by
(15)∥f∥B˙q,λ(ℚpn)∶=supγ∈ℤ(1|Bγ|H1+λq∫Bγ‍|f(x)|qdx)1/q<∞,
where Bγ=Bγ(0).Remark 5.
It is clear that(16)B˙q,-1/q(ℚpn)=Lq(ℚpn).
When λ<-1/q, the space B˙q,λ(ℚpn) reduces to {0}; therefore, we can only consider the case λ≥-1/q. If 1≤q1<q2<∞, by Hölder’s inequality,
(17)B˙q2,λ(ℚpn)⊂B˙q1,λ(ℚpn)
for λ∈ℝ.Definition 6.
Letλ<1/n and let 1<q<∞. The space CBMOq,λ(ℚpn) is defined by the condition
(18)∥f∥CBMOq,λ(ℚpn)∶=supγ∈ℤ(1|Bγ|H1+λq∫Bγ‍|f(x)-fBγ|qdx)1/q<∞,
where fBγ=(1/|Bγ|H)∫Bγ‍f(x)dx.Remark 7.
Whenλ=0, the space CBMOq,λ(ℚpn) is just CBMOq(ℚpn), which is defined in [32]. If 1≤q1<q2<∞, by Hölder’s inequality,
(19)CBMOq2,λ(ℚpn)⊂CBMOq1,λ(ℚpn)
for λ∈ℝ. By the standard proof as that in ℝn, we can see that
(20)∥f∥CBMOq,λ(ℚpn)~supγ∈ℤinfc∈ℂ(1|Bγ|H1+λq∫Bγ‍|f(x)-c|qdx)1/q.Remark 8.
The formulas (18) and (15) yield that B˙q,λ(ℚpn) is a Banach space continuously included in CBMOq,λ(ℚpn).In Section2, we obtain the sharp estimates of p-adic Hardy operators on p-adic central Morrey spaces and p-adic λ-central BMO spaces. Analogous result is also established for p-adic Morrey spaces. In Section 3, we discuss the boundedness of commutators generated by p-adic Hardy operators and p-adic λ-central BMO functions on p-adic central Morrey spaces.We should note that in Euclidean space, when estimating the Hardy operator, one usually discusses its restriction on radical functions. However, onp-adic field, we will consider its restriction on the functions f with f(x)=f(|x|p-1) instead.Throughout this paper the letterC will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.


## 2. Sharp Estimates ofp-Adic Hardy Operator

We get the following precise norms ofp-adic Hardy operators on p-adic central Morrey spaces and p-adic λ-central BMO spaces.Theorem 9.
Let1<q<∞ and let λ≥-1/q. Then
(21)∥ℋp∥B˙q,λ(ℚpn)→B˙q,λ(ℚpn)=1-p-n1-p-n(1+λ).Theorem 10.
Let1<q<∞ and let -1/q<λ<1/n. Then
(22)∥ℋp∥CBMOq,λ(ℚpn)→CBMOq,λ(ℚpn)=1-p-n1-p-n(1+λ).Corollary 11.
Let1<q<∞. Then
(23)∥ℋp∥CBMOq(ℚpn)→CBMOq(ℚpn)=1.Letℒq,λ(ℚpn) denote the subspace of Lq,λ(ℚpn) consisting of all functions f∈Lq,λ(ℚpn) with f(x)=f(|x|p-1). We obtain the sharp estimate of p-adic Hardy operator from ℒq,λ(ℚpn) to Lq,λ(ℚpn).Theorem 12.
Suppose that1<q<∞, -1/q<λ<0. Then ℋp maps ℒq,λ(ℚpn) to Lq,λ(ℚpn) with norm
(24)∥ℋp∥ℒq,λ(ℚpn)→Lq,λ(ℚpn)=1-p-n1-p-n(1+λ).Proof of Theorem9.
Whenλ=-1/q, B˙q,-1/q(ℚpn)=Lq(ℚpn), by Corollary 2.2 in [32],
(25)∥ℋp∥B˙q,λ(ℚpn)→B˙q,λ(ℚpn)=1-p-n1-p-n(1-1/q).
Whenλ>-1/q, we first claim that the operator ℋp and its restriction to the subset of B˙q,λ(ℚpn), which consist of functions g satisfying g(x)=g(|x|p-1), have the same operator norm on B˙q,λ(ℚpn).
In fact, forf∈B˙q,λ(ℚpn), set
(26)g(x)=11-p-n∫|ξ|p=1‍f(|x|p-1ξ)dξ,x∈ℚpn.
It is easy to see that g satisfies that g(x)=g(|x|p-1) and ℋpg=ℋpf. By Hölder’s inequality, for γ∈ℤ, we have
(27)1|Bγ|H1+λq∫Bγ‍|g|qdx=1|Bγ|H1+λq∫Bγ‍|11-p-n∫|ξ|p=1‍f(|x|p-1ξ)dξ|qdx≤1|Bγ|H1+λq×∫Bγ‍1(1-p-n)q(∫|ξ|p=1‍|f(|x|p-1ξ)|qdξ)×(∫|ξ|p=1‍dξ)q/q′dx=1|Bγ|H1+λq∫Bγ‍11-p-n(∫|ξ|p=1‍|f(|x|p-1ξ)|qdξ)dx=1|Bγ|H1+λq(1-p-n)∫Bγ‍|x|p-ndx∫|y|p=|x|p‍|f(y)|qdy=1|Bγ|H1+λq(1-p-n)∫Bγ‍|f(y)|qdy∫|x|p=|y|p‍|x|p-ndx=1|Bγ|H1+λq∫Bγ‍|f(y)|qdy.
Therefore,
(28)∥g∥B˙q,λ(ℚpn)≤∥f∥B˙q,λ(ℚpn).
Consequently,
(29)∥ℋpf∥B˙q,λ(ℚpn)∥f∥B˙q,λ(ℚpn)≤∥ℋpg∥B˙q,λ(ℚpn)∥g∥B˙q,λ(ℚpn),
which implies the claim. In the following, without loss of generality, we may assume that f∈B˙q,λ(ℚpn) satisfies f(x)=f(|x|p-1). Then by Minkowski’s inequality, we have
(30)(1|Bγ|H1+λq∫Bγ‍|ℋpf(x)|qdx)1/q=(1|Bγ|H1+λq∫Bγ‍|1|x|pn∫B(0,|x|p)‍f(t)dt|qdx)1/q=(1|Bγ|H1+λq∫Bγ‍|∫B0‍f(|x|p-1y)dy|qdx)1/q≤[1|Bγ|H1+λq∫Bγ‍(∫B0‍|f(|y|p-1x)|dy)qdx]1/q≤∫B0‍[1|Bγ|H1+λq∫Bγ‍|f(|y|p-1x)|qdx]1/qdy≤∫B0‍[1|Bγ|H1+λq∫B(0,pγ|y|p)‍|f(z)|q|y|p-ndz]1/qdy≤∫B0‍[|y|pnλq|B(0,pγ|y|p)|H1+λq∫B(0,pγ|y|p)‍|f(z)|qdz]1/qdy≤∥f∥B˙q,λ(ℚpn)∫B0‍|y|pnλdy=∥f∥B˙q,λ(ℚpn)(∑k=0∞‍∫|y|p=p-k‍p-knλdy)=1-p-n1-p-n(1+λ)∥f∥B˙q,λ(ℚpn).
Thus,
(31)∥ℋpf∥B˙q,λ(ℚpn)≤1-p-n1-p-n(1+λ)∥f∥B˙q,λ(ℚpn).
On the other hand, takef0(x)=|x|pnλ. Then
(32)1|Bγ|H1+λq∫Bγ‍|x|pnλqdx=p-nγ(1+λq)∑k=-∞γ‍∫Sk‍pnkλqdx=(1-p-n)p-nγ(1+λq)∑k=-∞γ‍pnk(1+λq)=1-p-n1-p-n(1+λq),
where the series converges due to λ>-1/q. Thus, f0∈B˙q,λ(ℚpn) since
(33)ℋpf0(x)=1|x|pn∫B(0,|x|p)‍|t|pnλdt=|x|pnλ∫B0‍|y|pnλdy=1-p-n1-p-n(1+λ)f0(x).
Therefore,
(34)∥ℋp∥B˙q,λ(ℚpn)→B˙q,λ(ℚpn)≥1-p-n1-p-n(1+λ).
Then (31) and (34) imply that
(35)∥ℋp∥B˙q,λ(ℚpn)→B˙q,λ(ℚpn)=1-p-n1-p-n(1+λ).Proof of Theorem10.
As in the proof of Theorem9, we first show that the operator ℋp and its restriction to the subset of CBMOq,λ(ℚpn) consisting of functions g with g(x)=g(|x|p-1) have the same operator norm on CBMOq,λ(ℚpn).
In fact, set(36)g(x)=11-p-n∫|ξ|p=1‍f(|x|p-1ξ)dξ,x∈ℚpn.
Then g(x)=g(|x|p-1) and ℋpg=ℋpf. By change of variable, we get
(37)gBγ=1|Bγ|H∫Bγ‍g(x)dx=1|Bγ|H∫Bγ‍11-p-n∫|ξ|p=1‍f(|x|p-1ξ)dξdx=1|Bγ|H(1-p-n)∫Bγ‍dx∫|y|p=|x|p‍f(y)|x|p-ndy=1|Bγ|H(1-p-n)∫Bγ‍f(y)dy∫|x|p=|y|p‍|x|p-ndx=1|Bγ|H∫Bγ‍f(y)dy=fBγ.
Using Minkowski’s inequality and (37), we have
(38)(1|Bγ|H1+λq∫Bγ‍|g-gBγ|qdx)1/q=(1|Bγ|H1+λq∫Bγ‍|g-fBγ|qdx)1/q=(1|Bγ|H1+λq×∫Bγ‍|11-p-n∫|ξ|p=1‍f(|x|p-1ξ)dξ-fBγ|qdx1|Bγ|H1+λq)1/q≤11-p-n[1|Bγ|H1+λq×∫Bγ‍(∫|ξ|p=1‍|f(|x|p-1ξ)-fBγ|dξ)qdx1|Bγ|H1+λq]1/q≤11-p-n×∫|ξ|p=1‍(1|Bγ|H1+λq∫Bγ‍|f(|x|p-1ξ)-fBγ|qdx)1/qdξ=11-p-n∫|ξ|p=1‍(1|Bγ|H1+λq∫Bγ‍|f(x)-fBγ|qdx)1/qdξ≤∥f∥CBMOq,λ(ℚpn).
Therefore,
(39)∥g∥CBMOq,λ(ℚpn)≤∥f∥CBMOq,λ(ℚpn).
We conclude that
(40)∥ℋpf∥CBMOq,λ(ℚpn)∥f∥CBMOq,λ(ℚpn)≤∥ℋpg∥CBMOq,λ(ℚpn)∥g∥CBMOq,λ(ℚpn).
In the following, without loss of generality, we may assume thatf∈CBMOq,λ(ℚpn) with f(x)=f(|x|p-1). By Fubini theorem, we have
(41)(ℋpf)Bγ=1|Bγ|H∫Bγ‍ℋpf(x)dx=1|Bγ|H∫Bγ‍∫B0‍f(|x|p-1y)dydx=∫B0(1|Bγ|H∫Bγf(|x|p-1y)dx)dy=∫B0‍(1|Bγ|H∫Bγ‍f(|y|p-1x)dx)dy=∫B0‍(1|B(0,|y|ppγ)|H∫B(0,|y|ppγ)‍f(z)dz)dy=∫B0‍fB(0,|y|ppγ)dy.
Then by Minkowski's inequality, we get
(42)(1|Bγ|H1+λq∫Bγ‍|ℋpf(x)-(ℋpf)Bγ|qdx)1/q=(1|Bγ|H1+λq∫Bγ‍|∫B0‍f(|x|p-1y)-fB(0,|y|ppγ)dy|qdx)1/q≤∫B0‍(1|Bγ|H1+λq∫Bγ‍|f(|x|p-1y)-fB(0,|y|ppγ)|qdx)1/qdy=∫B0‍(1|B(0,|y|ppγ)|H1+λq×∫B(0,|y|ppγ)‍|f(z)-fB(0,|y|ppγ)|qdz1|B(0,|y|ppγ)|H1+λq)1/q|y|pnλdy≤∥f∥CBMOq,λ(ℚpn)∫B0‍|y|pnλdy=1-p-n1-p-n(1+λ)∥f∥CBMOq,λ(ℚpn).
Namely,
(43)∥ℋpf∥CBMOq,λ(ℚpn)≤1-p-n1-p-n(1+λ)∥f∥CBMOq,λ(ℚpn).
On the other hand, takef0(x)=|x|pnλ. By Remark 8 and (32), for λ>-1/q, we have f0∈B˙q,λ(ℚpn)⊂CBMOq,λ(ℚpn). Then by (33), we get
(44)(ℋpf0)Bγ=1|Bγ|H∫Bγ‍ℋpf0(x)dx=1|Bγ|H∫Bγ‍1-p-n1-p-n(1+λ)f0(x)dx=1-p-n1-p-n(1+λ)(f0)Bγ.
Therefore,
(45)(1|Bγ|H1+λq∫Bγ‍|ℋpf0(x)-(ℋpf0)Bγ|qdx)1/q=(1|Bγ|H1+λq∫Bγ‍(1-p-n1-p-n(1+λ))q|f0-(f0)Bγ|qdx)1/q=1-p-n1-p-n(1+λ)(1|Bγ|H1+λq∫Bγ‍|f0-(f0)Bγ|qdx)1/q.
We arrive at
(46)∥ℋpf0∥CBMOq,λ(ℚpn)=1-p-n1-p-n(1+λ)∥f0∥CBMOq,λ(ℚpn).
As a result,(47)∥ℋp∥CBMOq,λ(ℚpn)→CBMOq,λ(ℚpn)≥1-p-n1-p-n(1+λ).
Then Theorem 10 follows from (43) and (47).Proof of Theorem12.
Letf∈ℒq,λ(ℚpn). Then f(x)=f(|x|p-1). Using Minkowski’s inequality, we have
(48)(1|Bγ(a)|H1+λq∫Bγ(a)‍|ℋpf(x)|qdx)1/q=(1|Bγ(a)|H1+λq∫Bγ(a)‍|∫B0‍f(|x|p-1y)dy|qdx)1/q≤[1|Bγ(a)|H1+λq∫Bγ(a)‍(∫B0‍|f(|y|p-1x)|dy)qdx]1/q≤∫B0‍[1|Bγ(a)|H1+λq∫Bγ(a)‍|f(|y|p-1x)|qdx]1/qdy≤∫B0‍[1|Bγ(a)|H1+λq∫B(a|y|p-1,pγ|y|p)‍|f(z)|q|y|p-ndz]1/qdy≤∫B0[|y|pnλq|B(a|y|p-1,pγ|y|p)|H1+λq‍×∫B(a|y|p-1,pγ|y|p)‍|f(z)|qdz|y|pnλq|B(a|y|p-1,pγ|y|p)|H1+λq]1/qdy≤∥f∥Lq,λ(ℚpn)∫B0‍|y|pnλdy=1-p-n1-p-n(1+λ)∥f∥Lq,λ(ℚpn).
On the other hand, as in the proof of Theorem9, we take f0(x)=|x|pnλ, and we only need to show that f0∈Lq,λ(ℚpn). Consider the following.
(I) If|a|p>pγ and x∈Bγ(a), then |x|p=max{|x-a|p,|a|p}>pγ. Since -1/q≤λ<0, we have
(49)1|Bγ(a)|H1+λq∫Bγ(a)‍|x|pnλqdx<1|Bγ(a)|H1+λq∫Bγ(a)‍pγnλqdx=1.
(II) If|a|p≤pγ and x∈Bγ(a), then |x|p≤max{|x-a|p,|a|p}≤pγ; therefore, x∈Bγ. Recall that two balls in ℚpn are either disjoint or one is contained in the other (cf. page 21 in [39]). So we have Bγ(a)=Bγ; thus,
(50)1|Bγ(a)|H1+λq∫Bγ(a)‍|x|pnλqdx=1|Bγ|H1+λq∫Bγ‍|x|pnλqdx=p-γn(1+λq)∑k=-∞γ‍∫Sk‍pknλqdx=1-p-n1-p-n(1+λq).
From the previous discussion, we can see that f0∈Lq,λ(ℚpn). Then by (33),
(51)∥ℋpf0∥Lq,λ(ℚpn)=1-p-n1-p-n(1+λ)∥f0∥Lq,λ(ℚpn).
This completes the proof.


## 3. Boundedness for Commutators ofp-Adic Hardy Operators on p-Adic Central Morrey Spaces

The boundedness of commutators is an active topic in harmonic analysis due to its important applications. For example, it can be applied to characterizing some function spaces [40]. In this section, we consider the boundedness for commutators generated by ℋp and λ-central BMO functions on p-adic central Morrey spaces.Definition 13.
Letb∈CBMOq,λ(ℚpn). The commutator of ℋp is defined by
(52)ℋbpf∶=bℋpf-ℋp(bf)
for some suitable functions f.Theorem 14.
Let1<q1<∞,q′1<q2<∞, n(1/q2-1/q1′)<n/q1, 1/q=1/q1+1/q2, λ1>-1/q, 0≤λ2<1/n, and λ=λ1+λ2. If b∈CBMOq2,λ2(ℚpn); then ℋbp is bounded from B˙q1,λ1(ℚpn) to B˙q,λ(ℚpn) and satisfies
(53)∥ℋbpf∥B˙q,λ(ℚpn)≤C∥b∥CBMOq2,λ2(ℚpn)∥f∥B˙q1,λ1(ℚpn).Before the proof of this theorem, we need the following calculations.Lemma 15.
Suppose thatb∈CBMOq,λ(ℚpn) and j,k∈ℤ, λ≥0. Then,
(54)|bBj-bBk|≤pn|j-k|∥b∥CBMOq,λ(ℚpn)max{|Bj|Hλ,|Bk|Hλ}.Proof.
Without loss of generality, we may assume thatk>j. Recall that bBi=(1/|Bi|H)∫Bi‍b(x)dx. By Hölder's inequality, we have
(55)|bBi+1-bBi|≤1|Bi|H∫Bi‍|b(x)-bBi+1|dx≤1|Bi|H∫Bi+1‍|b(x)-bBi+1|dx≤1|Bi|H(∫Bi+1‍|b(x)-bBi+1|qdx)1/q|Bi+1|H1-1/q≤|Bi+1|H1+λ|Bi|H∥b∥CBMOq,λ(ℚpn)=pn|Bi+1|Hλ∥b∥CBMOq,λ(ℚpn).
Therefore,(56)|bBj-bBk|≤∑i=jk-1‍|bBj+1-bBj|≤pn∥b∥CBMOq,λ(ℚpn)∑i=jk-1‍|Bj+1|Hλ≤(k-j)pn∥b∥CBMOq,λ(ℚpn)|Bk|Hλ.Proof of Theorem14.
Assume thatf∈B˙q1,λ1(ℚpn). Fix γ∈ℤ, and by Minkowski’s inequality, we have
(57)(1|Bγ|H1+λq∫Bγ‍|ℋbpf(x)|qdx)1/q=(1|Bγ|H1+λq×∫Bγ‍|1|x|pn∫B(0,|x|p)‍f(y)(b(x)-b(y))dy|qdx1|Bγ|H1+λq)1/q≤(1|Bγ|H1+λq×∫Bγ‍|1|x|pn∫B(0,|x|p)‍f(y)(b(x)-bBγ)dy|qdx1|Bγ|H1+λq)1/q+(1|Bγ|H1+λq×∫Bγ‍|1|x|pn∫B(0,|x|p)‍f(y)(b(y)-bBγ)dy|qdx1|Bγ|H1+λq)1/q∶=I+J.
In the following, we will estimateI and J, respectively. For I, since ℋp is bounded from Lr(ℚpn) to Lr(ℚpn), 1<r<∞ [32], then, by Hölder's inequality (q/q1+q/q2=1), we get
(58)I≤|Bγ|H-1/q-λ(∫Bγ‍|b(x)-bBγ|q2dx)1/q2×(∫Bγ‍|ℋpf(x)|q1dx)1/q1≤|Bγ|H-1/q1-λ1∥b∥CBMOq2,λ2(ℚpn)×(∫ℚpn‍|ℋp(fχBγ)(x)|q1dx)1/q1≤C|Bγ|H-1/q1-λ1∥b∥CBMOq2,λ2(ℚpn)(∫Bγ‍|f(x)|q1dx)1/q1≤C∥b∥CBMOq2,λ2(ℚpn)∥f∥B˙q1,λ1(ℚpn).
Next, let us estimateJ as follows:
(59)Jq≤1|Bγ|H1+λq∫Bγ‍|1|x|pn∫B(0,|x|p)‍f(y)(b(y)-bBγ)dy|qdx≤1|Bγ|H1+λq×∑k=-∞γ‍∫Sk‍p-knq(∫Bk‍|f(y)(b(y)-bBγ)|dy)qdx=1-p-n|Bγ|H1+λq×∑k=-∞γ‍pkn(1-q)(∑j=-∞k‍∫Sj‍|f(y)(b(y)-bBγ)|dy)q=1-p-n|Bγ|H1+λq×∑k=-∞γ‍pkn(1-q)(∑j=-∞k‍∫Sj‍|f(y)(b(y)-bBj)|dy)q+1-p-n|Bγ|H1+λq×∑k=-∞γ‍pkn(1-q)(∑j=-∞k‍∫Sj‍|f(y)(bBj-bBγ)|dy)q∶=J1+J2.
ForJ1, by Hölder’s inequality (1/q1+1/q2+1-1/q=1) and the fact that λ1>-1/q1, λ2>-1/q2, we have
(60)J1≤1-p-n|Bγ|H1+λq∑k=-∞γ‍pkn(1-q)×[∑j=-∞k‍|Bj|H1-1/q(∫Bj‍|f(y)|q1dy)1/q1×(∫Bj‍|b(y)-bBj|q2dy)1/q2∑j=-∞k‍]q≤1-p-n|Bγ|H1+λq∑k=-∞γ‍pkn(1-q)×[∑j=-∞k‍|Bj|H1+λ∥b∥CBMOq2,λ2(ℚpn)∥f∥B˙q1,λ1(ℚpn)]q≤1-p-n|Bγ|H1+λq∥b∥CBMOq2,λ2q∥f∥B˙q1,λ1(ℚpn)q×∑k=-∞γ‍pkn(1-q)(∑j=-∞k‍pjn(1+λ))q=C|Bγ|H1+λq∥b∥CBMOq2,λ2(ℚpn)q×∥f∥B˙q1,λ1(ℚpn)q∑k=-∞γ‍pkn(1+λq)=C∥b∥CBMOq2,λ2(ℚpn)q∥f∥B˙q1,λ1q.
For J2, by Lemma 15 and Hölder’s inequality, we obtain
(61)J2=1-p-n|Bγ|H1+λq∑k=-∞γ‍pkn(1-q)(∑j=-∞k‍∫Sj‍|f(y)(bBj-bBγ)|dy)q≤C|Bγ|H1+λq∑k=-∞γ‍pkn(1-q)×[∑j=-∞k‍(γ-j)∥b∥CBMOq2,λ2|Bγ|Hλ2∫Bj‍|f(y)|dy]q≤C|Bγ|H1+λ1q∥b∥CBMOq2,λ2(ℚpn)q∑k=-∞γ‍pkn(1-q)×[∑j=-∞k‍(γ-j)|Bj|H1-1/q1(∫Bj‍|f(y)|q1dy)1/q1]q≤C|Bγ|H1+λ1q∥b∥CBMOq2,λ2(ℚpn)q∥f∥B˙q1,λ1(ℚpn)q×∑k=-∞γ‍pkn(1-q)[∑j=-∞k‍(γ-j)|Bj|H1+λ1]q=C|Bγ|H1+λ1q∥b∥CBMOq2,λ2(ℚpn)q∥f∥B˙q1,λ1(ℚpn)q×∑k=-∞γ‍pkn(1-q)(γ-k)q|Bk|H(1+λ1)q=C∥b∥CBMOq2,λ2(ℚpn)q∥f∥B˙q1,λ1(ℚpn)q.
Notice that since λ1>-1/q1, then 1+λ1q1>0 and 1+λ1>0.
The above estimates imply that(62)(1|Bγ|H1+λq∫Bγ‍|ℋbpf(x)|qdx)1/q≤C∥b∥CBMOq2,λ2(ℚpn)∥f∥B˙q1,λ1(ℚpn).
Consequently,
(63)∥ℋbpf∥B˙q,λ(ℚpn)≤C∥b∥CBMOq2,λ2(ℚpn)∥f∥B˙q1,λ1(ℚpn).
Theorem 14 is proved.



---

*Source: 359193-2013-09-19.xml*