# The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group

Authors: Xiang Li; Xingsong Zhang
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this article, we define a kind of truncated maximal function on the Heisenberg space byMγcfx=sup0<r<γ1/mBx,r∫Bx,rfydy. The equivalence of operator norm between the Hardy-Littlewood maximal function and the truncated maximal function on the Heisenberg group is obtained. More specifically, when 1<p<∞, the Lp norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When p=1, we get the equivalence of weak norm L1⟶L1,∞ and Ṁ1,λ⟶ẆM1,λ. Those results are generalization of previous work on Euclid spaces.


---

## Body

## 1. Introduction

Letf be a locally integrable function on ℝn. We define the centered Hardy-Littlewood maximal function as
(1)Mcfx=supr>01mBx,r∫Bx,rfydmy,and define the uncentered Hardy-Littlewood maximal function as
(2)Mfx=supB∋x1mB∫Bfydmy.The Hardy-Littlewood maximal functions play an important role in harmonic analysis. Their boundness and sharp bounds are important since a variety of operators are controlled by maximal functions. TheL1⟶L1,∞ and Lp⟶Lp boundness of Hardy-Littlewood maximal functions are well-known [1–5]. However, sharp bounds are very hard to obtain. For a long time, we only know how the sharp bounds of Hardy-Littlewood maximal functions behave when the dimension n changes. In 2003, Melas [6] obtained the sharp bound of the one-dimensional centered Hardy-Littlewood maximal function of weak type (1, 1). But it is hard to apply his method to higher dimensional cases. No result has been stated for general cases.Now, we introduce another point of view. In [7], Wei et al. defined a truncated maximal function:Definition 1.
Letf∈Llocℝn be a locally integrable function on ℝn. We define truncated centered maximal function:
(3)Mα,βcfx=supα<r<β1mBx,r∫Bx,rfydy,and truncated uncentered maximal function:
(4)Mα,βfx=supB∋x,α<radiB<β1mB∫Bfydy,where radiB is the radium of ball B. 0≤α<β≤∞. β=∞ means that there is no limit of upper bound of r or radiB.
Whenα=0, we write truncated centered maximal function briefly as
(5)Mγcfx=sup0<r<γ1mBx,r∫Bx,rfydy.Then, we have the following theorem [7]:Theorem 2.
Forγ>0, there is
(6)McL1ℝn⟶L1,∞ℝn=MγcL1ℝn⟶L1,∞ℝn.
For1<p≤∞, γ>0, there is
(7)McLpℝn⟶Lpℝn=MγcLpℝn⟶Lpℝn.
There holds similar results of truncated uncentered maximal function. We also can prove that the operator norm of the truncated maximal function withα,β is only related to C=α/β. When C tends to zero, the norm tends to that of the Hardy-Littlewood maximal function.We choose to study the truncated maximal operator and maximal operator because there are some differences among them while their operator norms are equivalent. For example, for locally integrable functions with compact support, their truncated maximal functions have compact support while their Hardy-Littlewood maximal functions may not have. Based on the above reasons, it will be easier to study the truncated maximal functions sometimes. We hope that equivalence and difference among truncated maximal operator and maximal operator may bring new thoughts and methods.In addition, Zhang et al. obtained the equivalence of operator norm between the truncated maximal function and the Hardy-Littlewood function on Morrey spaces in [8]:Definition 3.
Set1≤q<∞ and 0≤λ≤n. For f∈Llocqℝn, define the central Morrey norm fṀq,λℝn:
(8)fṀq,λRn=supR>01Rλ∫B0,Rfxqdx1/q,where B0,R is a ball centered in origin with radium R.
The central Morrey spaceṀq,λℝn is defined as
(9)Ṁq,λℝn=f∈Llocqℝn:fṀq,λℝn<∞.
Set1≤q<∞ and 0≤λ≤n. For any measurable function f, define weak the central Morrey norm fẆMq,λℝn:
(10)fẆMq,λℝn=supR>01Rλ/qsups>0sy∈B0,R:fy>s1/q.
The weak centered Morrey spaceẆMq,λℝn is defined as
(11)ẆMq,λℝn=measurablefunctionf:fẆMq,λℝn<∞.Zhang et al. proved the following theorem [8]:Theorem 4.
For0<λ<n, γ>0, there is
(12)McṀ1,λℝn⟶ẆM1,λℝn=MγcṀ1,λℝn⟶ẆM1,λℝn.
For1<p<∞, 0<λ<n, γ>0, there is
(13)McṀp,λℝn⟶Ṁp,λℝn=MγcṀp,λℝn⟶Ṁp,λℝn.Those are the results onℝn. Nowdays, the researchers concern about the classic operators on more abstract background such as p-adic fields and the Heisenberg group. Some researchers have already obtained the boundness of the Hardy-Littlewood maximal function on the Heisenberg group, one example is the following theorem in [9]:Theorem 5.
There exists constantA such that for lnn≫1, we have
(14)McfL1,∞ℍn≤AnfL1ℍn.
We can see that there is still no specific sharp bound. In order to apply the methods of truncated maximal functions, we are going to establish the equivalence of operator norm between the Hardy-Littlewood maximal functions and truncated maximal functions on the Heisenberg group.We outline some basic information of the Heisenberg group. The Heisenberg groupℍn is underlying manifold ℝ2n×ℝ1 with group law
(15)x1,x2,⋯,x2n,x2n+1x1′,x2′,⋯,x2n′,x2n+1′=x1+x1′,x2+x2′,⋯,x2n+x2n′,x2n+1+x2n+1′+2∑j=1nxj′xn+j−xjxn+j′.The identity element onℍn is 0∈ℝ2n+1, and the inverse element x−1 is −x. The Haar measure on ℍn coincides with the Lebesgue measure on ℝ2n×ℝ1.ℍn is a homogeneous group with dilations
(16)δrx1,x2,⋯,x2n,x2n+1=rx1,rx2,⋯,rx2n,r2x2n+1,r>0.SetE as the measure of any measurable set E in ℍn. Then
(17)δrE=rQE,dδrx=rQdx,where Q=2n+2 is called the homogeneous dimension of ℍn.The metric onℍn derived from the norm
(18)xh=∑i=12nxi2+x2n+121/4,where x=x1,x2,⋯,x2n,x2n+1, is given by
(19)dp,q=dq−1p,0=q−1ph.Forr>0 and x∈ℍn, define the ball and sphere with center x and radius r on ℍn as
(20)Bx,r=y∈ℍn:dx,y<r,Sx,r=y∈ℍn:dx,y=r.We also have(21)Bx,r=B0,r=ΩQrQ,where
(22)ΩQ=2πn+1/2Γn/2n+1ΓnΓn+1/2.ΩQ is the volume of the unit ball B0,1 on ℍn, and the area of the sphere S0,1 is ωQ=QΩQ.For further information, readers could refer to [9].Now, we can present the operators on the Heisenberg group.Forf∈Llocℍn, we define the centered Hardy-Littlewood maximal function on the Heisenberg group:
(23)Mcfx=supr>01Bx,r∫Bx,rfydy,and a kind of truncated maximal function on the Heisenberg group:
(24)Mγcfx=sup0<r<γ1mBx,r∫Bx,rfydy.We will establish equivalence of their operator norms on the Heisenberg group. In the next section, we introduce some preliminaries and basic lemmas. In the third section, we present the details of our main theorems and the proofs.


## 2. Preliminaries

In this section, we present some preliminaries and basic lemmas.First, we give the definition of distribution function on the Heisenberg group (similar to that onℝn):Definition 6.
Letf be a measurable function on ℍn. Define distribution function df:ℍn⟶0,+∞ as
(25)dfα=x∈ℍn:fx>α,where ∗ is the Lebesgue measure on ℝ2n+1.We have such a relation between distribution function andLp norms:Lemma 7.
Set0<p<∞ and f∈Lpℍn. Then, we have
(26)fLpℍnp=p∫0∞αp−1dfαdα.
This lemma is a basic equivalent expression ofLp norm, and readers can find the proof in Proposition 1.1.4 of [3].Lemma 8.
Letμ be a positive measure on σ-algebra M. If sets An satisfy that A1⊂A2⊂A3⋯⊂An⋯, and A=∪n=1∞An, then we have
(27)limn⟶∞μAn=μA.
Readers can find proofs of Lemma8 in Theorem 1.19 (d) of [10].
Using Lemma8, we can obtain the following lemma.Lemma 9.
LetMc and Mγc be the operators defined in (23) and (24). Then, for any f∈Lpℍn and λ>0, we have
(28)dMcfλ=limγ⟶∞dMγcfλ.Proof.
Fixx∈ℍn. For any ε, there exists rε>0 such that
(29)1mBx,rε∫Bx,rεfydy>Mcfx−ε.
For some largeγ, we have
(30)Mγcfx≥1mBx,rε∫Bx,rεfydy>Mcfx−ε.
SinceMγcfx increases as γ becomes larger, then
(31)limγ⟶∞Mγcfx≥Mcfx.
On the other hand, by definition we have(32)Mγcfx≤Mcfx.
Then(33)limγ⟶∞Mγcfx=Mcfx.
Setk=1,2,3,⋯:
(34)Ak=x∈:Mkcfx>λ,A=x∈:Mcfx>λ.
By (33) and Lemma 8, we obtain that
(35)dMcfλ=A=limk⟶∞Ak=limk⟶∞dMkcfλ=limγ⟶∞dMγcfλ.☐Lemma9 implies that when γ tends to infinite, the distribution tends to equal. It will be important in the proof of weak-type operator norms. For strong-type operator norms, we have the following lemma:Lemma 10.
1<p<∞. Fixing ε>0, there is a function g∈Cc∞ℍn such that
(36)McgLpℍngLpℍn≥McLpℍn⟶Lpℍn−ε.Proof.
By the definition of operator norm, there exists a functionf∈Lpℍn such that
(37)McfLpℍnfLpℍn≥McLpℍn⟶Lpℍn−ε/2.
SinceCc∞ℍn is dense in Lpℍn, then we have g∈Cc∞ℍn such that
(38)f−gLpℍn<δ.
LetA be a upper bound of norm of Mc. Then
(39)Mcf−gLpℍn<Af−gLpℍn<Aδ.
So(40)McgLpℍngLpℍn≥McfLpℍn−Mcf−gLpℍnfLpℍn+f−gLpℍn≥McfLpℍn−AδfLpℍn+δ.
Pick suitableδ in order that
(41)McfLpℍn−AδfLpℍn+δ≥McfLpℍnfLpℍn−ε/2≥McLpℍn⟶Lpℍn−ε.
Then, we finish the proof.☐Notice thatg has compact support. This property is important when we establish equivalence of operator norms.We need the following key lemma:Lemma 11.
Letγ>0. Then
(42)Mγcfδγx=M1cτγfx,where τγfx=fδγx.Proof.
By direct computation, we arrive at(43)Mγcfγx=sup0<r<γ1ΩQrQ∫y<rfδγx−ydy=sup0<r<γγQΩQrQ∫y<r/γfδγx−δγydy=sup0<r/γ<11ΩQr/γQ∫y<r/γτγfx−ydy=M1cτγfx.
This lemma is based on dilation. Remind that the dilation on the Heisenberg group is different from that on Euclid spaces.☐Notice that Lemma11 implies the relation between Mγc and M1c. Connecting it with the definition of operator norm, we obtain the following Lemma 12:Lemma 12.
(1)
Forγ>0, we have(44)M1cL1ℍn⟶L1,∞ℍn=MγcL1ℍn⟶L1,∞ℍn.(2)
For1<p≤∞, γ>0, we have(45)M1cLpℍn⟶Lpℍn=MγcLpℍn⟶Lpℍn.(3)
For0<λ<n, γ>0, we have(46)M1cṀ1,λℍn⟶ẆM1,λℍn=MγcṀ1,λℍn⟶ẆM1,λℍn.(4)
For1<p<∞, 0<λ<n, γ>0, we have(47)M1cṀp,λℍn⟶Ṁp,λℍn=MγcṀp,λℍn⟶Ṁp,λℍn.Proof.
Based on Lemma11, we have
(1)
Forλ>0(48)x:M1cτγfx>λ=x:Mγcfδγx>λ=δ1γx:Mγcfx>λ=1γQx:Mγcfx>λ.
Then, there stands(49)1γQsupλ>0x:Mγcfx>λfL1ℍn=supλ>0x:M1cτγfx>λfL1ℍn=1γQsupλ>0x:M1cτγfx>λτγfL1ℍn.
Taking the supremum over allf with fL1ℍn>0, we have
(50)M1cL1ℍn⟶L1,∞ℍn=MγcL1ℍn⟶L1,∞ℍn.(2)
Using the same method of dilation, it is obvious that(51)MγcfLpℍnfLpℍn=Mγcfδγ·Lpℍnfδγ·Lpℍn=M1cτγfLpℍnτγfLpℍn.
Taking the supremum over allf with fLpℍn>0, we have
(52)M1cLpℍn⟶Lpℍn=MγcLpℍn⟶Lpℍn.(3)
By homogeneous dimension ofℍn, we obtain that(53)fẆM1,λℍn=γQ−λτγfẆM1,λℍn.
There holds that(54)MγcfẆM1,λℍn=supR>01Rλsups>0sy∈B0,R:Mγcfy>s=supR>01Rλsups>0sγQδ1γy:y∈B0,R,Mγcfy>s=supR>01Rλsups>0sγQzh<Rγ:Mγcfδγz>s=supR>01Rλsups>0sγQzh<Rγ:M1cτγfz>s=γQ−λsupR>01R/γλsups>0sγQzh<Rγ:M1cτγfz>s=γQ−λsupR>01Rλsups>0sγQzh<R:M1cτγfz>s=γQ−λM1cτγfẆM1,λℍn.
Then(55)MγcfẆM1,λℍnfṀ1,λℍn=M1cτγfẆM1,λℍnτγfṀ1,λℍn.
Taking the spermum over allf with fṀ1,λℍn>0, we have
(56)M1cṀ1,λℍn⟶ẆM1,λℍn=MγcṀ1,λℍn⟶ẆM1,λℍn.(4)
By homogeneous dimension ofℍn, we obtain that(57)fṀp,λℍn=γQ−λ/pτγfṀp,λℍn.
So we get that(58)MγcfṀp,λℍnp=supR>01Rλ∫B0,RMγcfxpdx=supR>01Rλ∫B0,R/γMγcfδγypγQdy=supR>01Rλ∫B0,R/γM1cτγfypγQdy=γQ−λsupR>01R/λλ∫B0,R/γM1cτγfypdy=γQ−λM1cτγfṀp,λℍnp.
Then(59)MγcfṀp,λℍnfṀp,λℍn=M1cτγfṀp,λℍnτγfṀp,λℍn.
Taking the spermum over allf with fṀp,λℍn>0, we have
(60)M1cṀp,λℍn⟶Ṁp,λℍn=MγcṀp,λℍn⟶Ṁp,λℍn.
Now, we finish the proof.☐Lemma12 suggests that the operator norm is unrelated to the limit of radius γ.


## 3. Main Theorem

In this section, we present our main results. We divide them into four theorems:Theorem 13.
Forγ>0, we have
(61)McL1ℍn⟶L1,∞ℍn=MγcL1ℍn⟶L1,∞ℍn.Theorem 14.
For1<p≤∞, γ>0, we have
(62)∥Mc∥Lpℍn⟶Lpℍn=∥Mγc∥Lpℍn⟶Lpℍn.Theorem 15.
For0<λ<n, γ>0, we have
(63)McṀ1,λℍn⟶ẆM1,λℍn=MγcṀ1,λℍn⟶ẆM1,λℍn.Theorem 16.
For1<p<∞, 0<λ<n, γ>0, we have
(64)McṀp,λℍn⟶Ṁp,λℍn=MγcṀp,λℍn⟶Ṁp,λℍn.Remark 17.
The four theorems only concern about equivalence of operator norms. We do not need that the norms are less than infinite, neither can we ensure that the norms are finite by our theorems. Readers can get results of upper bounds from such articles like [9], but the results are neither necessary nor sufficient with our theorems like Theorem 14.In the process of the proof, the general idea is that the norm of truncated maximal functions is less than that of the Hardy-Littlewood maximal function from definition. For someγ, the norm of the Hardy-Littlewood maximal function will be less than the norm of some truncated maximal functions from the definition of supremum. But according to Lemma 12, the norm of truncated maximal function is unrelated to γ. Thus, we can establish the equivalence of operator norms from that idea.Proof of Theorem 18.
By definition of truncated maximal functions, we always haveMγcf≤Mcf for any γ>0; combining with Proposition 1.1.3(1) of [3], we can obtain that
(65)supλ>0λlimγ⟶∞dMγcfλ≤supλ>0λdMcfλ.
On the other hand, by Lemma9, we have
(66)supλ>0λdMcfλ=supλ>0λlimγ⟶∞dMγcfλ.
For anyε>0, there exists λ0>0, such that
(67)supλ>0λdMcfλ−ε<λ0dMcfλ0<supλ>0λdMcfλ.
Obviously(68)supλ>0λlimγ⟶∞dMγcfλ≥limγ⟶∞λ0dMγcfλ0≥supλ>0λdMcfλ−ε.
Then(69)supλ>0λlimγ⟶∞dMγcfλ≥supλ>0λdMcfλ.
So we obtain(70)limγ⟶∞supλ>0λdMγcfλ=supλ>0λdMcfλ.
Combining with Lemma12, we have
(71)McL1ℍn⟶L1,∞ℍn=limγ⟶∞MγcL1ℍn⟶L1,∞ℍn=limγ⟶∞M1cL1ℍn⟶L1,∞ℍn=M1cL1ℍn⟶L1,∞ℍn=MγcL1ℍn⟶L1,∞ℍn.
So, we finish the proof.Proof of Theorem 19.
By definition of truncated maximal functions, we obtain that(72)∥Mc∥Lpℍn⟶Lpℍn≥∥Mγc∥Lpℍn⟶Lpℍn.
Now, we only need to prove that(73)McLpℍn⟶Lpℍn≤MγcLpℍn⟶Lpℍn.
By Lemma10, fixing ε>0, there exists g∈Cc∞ℍn, so that
(74)McgLpℍngLpℍn≥McLpℍn⟶Lpℍn−ε.
Suppose that the support ofg is in the ball x∈ℍn:xh<R.
Sinceg has compact support, we can find S>0 such that
(75)Mcgχ·≥SLpℍn≤εgLpℍn.
Fixγ0>R+S, when xh<S,
(76)Mcgx=Mγ0cgx.
So(77)Mγ0cgLpℍn≥Mγ0cgχ·≤SLpℍn=Mcgχ·≤SLpℍn≥McgLpℍn−Mcgχ·≥SLpℍn≥McLpℍn⟶LpℍngLpℍn−2εgLpℍn.
Then(78)M1cLpℍn⟶Lpℍn=Mγ0cLpℍn⟶Lpℍn≥McLpℍn⟶Lpℍn−2ε.
For anyγ, by Lemma 12(79)MγcLpℍn⟶Lpℍn=M1cLpℍn⟶Lpℍn≥McLpℍn⟶Lpℍn−2ε.
So we can obtain (73).
Therefore, we have(80)McLpℍn⟶Lpℍn=MγcLpℍn⟶Lpℍn.Proof of Theorem 20.
Using Lemmas9 and 12, we directly compute that
(81)McṀ1,λℍn⟶ẆM1,λℍn=supfṀ1,λℍn>0McfẆM1,λℍnfṀ1,λℍn=supfṀ1,λℍn>0supR>01/Rλsups>0sy∈B0,R:Mcfy>sfṀ1,λℍn=supfṀ1,λℍn>0supR>01/Rλlimγ⟶∞sups>0s∣y∈B0,R:Mγcfy>s∣fṀ1,λℍn=limγ⟶∞supfṀ1,λℍn>0supR>01/Rλsups>0sy∈B0,R:Mγcfy>sfṀ1,λℍn=limγ⟶∞MγcṀ1,λℍn⟶ẆM1,λℍn=M1cṀ1,λℍn⟶ẆM1,λℍn=MγcṀ1,λℍn⟶ẆM1,λℍn.
We can immediately obtain Theorem15.Proof of Theorem 21.
By definition of truncated maximal functions, we obtain that(82)McṀp,λℍn⟶Ṁp,λℍn≥MγcṀp,λℍn⟶Ṁp,λℍn.
It is remained to be proved that(83)McṀp,λℍn⟶Ṁp,λℍn≤MγcṀp,λℍn⟶Ṁp,λℍn.
Using distribution functions and Lemma9, we have
(84)1Rλ∫B0,RMcfxpdx=1RλMcfχB0,RLpℍnp=pRλ∫0∞αp−1x∈B0,R:Mcfx>αdα=pRλ∫0∞αp−1limγ⟶∞B0,R∩x∈ℍn:Mγcfx>αdα=limγ⟶∞pRλ∫0∞αp−1B0,R∩x∈ℍn:Mγcfx>αdα=limγ⟶∞1Rλ∫B0,RMγcfxpdx.
Taking the supremum overR, combining with Lemma 12, we have
(85)McṀp,λℍn≤limγ⟶∞MγcṀp,λℍn=limγ⟶∞MγcṀp,λℍn⟶Ṁp,λℍnfṀp,λℍn=M1cṀp,λℍn⟶Ṁp,λℍnfṀp,λℍn=MγcṀp,λℍn⟶Ṁp,λℍnfṀp,λℍn.
So we obtain (83).
Then(86)McṀp,λℍn⟶Ṁp,λℍn=MγcṀp,λℍn⟶Ṁp,λℍn.
So the proof is accomplished.We only present the details of centered maximal function and its truncation. In fact, for uncentered case, there still holds similar results:Theorem 22.
(1)
Forγ>0, we have(87)ML1ℍn⟶L1,∞ℍn=MγL1ℍn⟶L1,∞ℍn.(2)
For1<p≤∞, γ>0, we have(88)MLpℍn⟶Lpℍn=MγLpℍn⟶Lpℍn.(3)
For0<λ<n, γ>0, we have(89)MṀ1,λℍn⟶ẆM1,λℍn=MγṀ1,λℍn⟶ẆM1,λℍn.(4)
For1<p<∞, 0<λ<n, γ>0, we have(90)MṀp,λℍn⟶Ṁp,λℍn=MγṀp,λℍn⟶Ṁp,λℍn.
We save the details since the methods are the same.Remark 23.
For general cases of truncated maximal function, we also can define them on the Heisenberg group:(91)Mα,βfx=supα<r<β1mBx,r∫Bx,rfydmy.Using the same methods, we can prove such theorem:Theorem 24.
1<p≤∞. γ1,γ2,γ3,γ4 is positive number, and
(92)γ1γ2=γ3γ4=C.
Then, the norms ofMγ1,γ2 and Mγ3,γ4 have such relations:
(1)(93)Mγ1,γ2L1ℍn⟶L1,∞ℍn=Mγ3,γ4L1ℍn⟶L1,∞ℍn.(2)
For1<p≤∞(94)Mγ1,γ2Lpℍn⟶Lpℍn=Mγ3,γ4Lpℍn⟶Lpℍn.(3)
For0<λ<n, we have(95)Mγ1,γ2Ṁ1,λℍn⟶ẆM1,λℍn=Mγ3,γ4Ṁ1,λℍn⟶ẆM1,λℍn.(4)
For1<p<∞, 0<λ<n, we have(96)Mγ1,γ2Ṁp,λℍn⟶Ṁp,λℍn=Mγ3,γ4Ṁp,λℍn⟶Ṁp,λℍn.
The main theorems can be seen as the special caseC=0.



---

*Source: 7612482-2021-08-24.xml*