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

<jats:p>In this article, we define a kind of truncated maximal function on the Heisenberg space by <jats:inline-formula>
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                                    <mi>B</mi>
                                    <mfenced open="(" close=")">
                                       <mrow>
                                          <mi>x</mi>
                                          <mo>,</mo>
                                          <mi>r</mi>
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                                 </mrow>
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                        <msub>
                           <mrow>
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                                    <mo>,</mo>
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                        <mfenced open="|" close="|">
                           <mrow>
                              <mi>f</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mi>y</mi>
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                        <mi>d</mi>
                        <mi>y</mi>
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                  </jats:inline-formula>. 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 <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mn>1</mn>
                        <mo><</mo>
                        <mi>p</mi>
                        <mo><</mo>
                        <mo>∞</mo>
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                  </jats:inline-formula>, the <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <msup>
                           <mrow>
                              <mi>L</mi>
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                           <mrow>
                              <mi>p</mi>
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                  </jats:inline-formula> norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>p</mi>
                        <mo>=</mo>
                        <mn>1</mn>
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                  </jats:inline-formula>, we get the equivalence of weak norm <jats:inline-formula>
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                        <msup>
                           <mrow>
                              <mi>L</mi>
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                        <mo>⟶</mo>
                        <msup>
                           <mrow>
                              <mi>L</mi>
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                           <mrow>
                              <mn>1</mn>
                              <mrow>
                                 <mo>,</mo>
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                              <mrow>
                                 <mo>∞</mo>
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                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <msup>
                           <mrow>
                              <mover accent="true">
                                 <mi>M</mi>
                                 <mo>̇</mo>
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                           <mrow>
                              <mn>1</mn>
                              <mo>,</mo>
                              <mi>λ</mi>
                           </mrow>
                        </msup>
                        <mo>⟶</mo>
                        <mover accent="true">
                           <mi>W</mi>
                           <mo>̇</mo>
                        </mover>
                        <msup>
                           <mrow>
                              <mi>M</mi>
                           </mrow>
                           <mrow>
                              <mn>1</mn>
                              <mo>,</mo>
                              <mi>λ</mi>
                           </mrow>
                        </msup>
                     </math>
                  </jats:inline-formula>. Those results are generalization of previous work on Euclid spaces.</jats:p>


Introduction
Let f be a locally integrable function on ℝ n . We define the centered Hardy-Littlewood maximal function as and define the uncentered Hardy-Littlewood maximal function as 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. The L 1 ⟶ L 1,∞ and L p ⟶ L p boundness of Hardy-Littlewood maximal functions are well-known [1][2][3][4][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. Let f ∈ L loc ðℝ n Þ be a locally integrable function on ℝ n . We define truncated centered maximal function: and truncated uncentered maximal function: where radiðBÞ is the radium of ball B. 0 ≤ α < β ≤ ∞. β = ∞ means that there is no limit of upper bound of r or radiðBÞ. When α = 0, we write truncated centered maximal function briefly as Then, we have the following theorem [7]: There holds similar results of truncated uncentered maximal function. We also can prove that the operator norm of the truncated maximal function with ðα, βÞ 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]: where Bð0, RÞ is a ball centered in origin with radium R. The central Morrey space _ M q,λ ðℝ n Þ is defined as Set 1 ≤ q < ∞ and 0 ≤ λ ≤ n. For any measurable function f , define weak the central Morrey norm k f k _ WM q,λ ðℝ n Þ : The weak centered Morrey space _ WM q,λ ðℝ n Þ is defined as Zhang et al. proved the following theorem [8]: Those are the results on ℝ n . 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]: 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 group ℍ n is underlying manifold ℝ 2n × ℝ 1 with group law The identity element on ℍ n is 0 ∈ ℝ 2n+1 , and the inverse element x −1 is −x. The Haar measure on ℍ n coincides with the Lebesgue measure on ℝ 2n × ℝ 1 .

2
Journal of Function Spaces ℍ n is a homogeneous group with dilations Set jEj as the measure of any measurable set E in ℍ n . Then where Q = 2n + 2 is called the homogeneous dimension of ℍ n . The metric on ℍ n derived from the norm where For r > 0 and x ∈ ℍ n , define the ball and sphere with center x and radius r on ℍ n as We also have where Ω Q is the volume of the unit ball Bð0, 1Þ on ℍ n , and the area of the sphere Sð0, 1Þ is ω Q = QΩ Q . For further information, readers could refer to [9]. Now, we can present the operators on the Heisenberg group.
For f ∈ L loc ðℍ n Þ, we define the centered Hardy-Littlewood maximal function on the Heisenberg group: and a kind of truncated maximal function on the Heisenberg group: 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.

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 on ℝ n ): Definition 6. Let f be a measurable function on ℍ n . Define distribution function d f : ℍ n ⟶ ½0,+∞Þ as where j * j is the Lebesgue measure on ℝ 2n+1 .
We have such a relation between distribution function and L p norms: This lemma is a basic equivalent expression of L p norm, and readers can find the proof in Proposition 1.1.4 of [3].

Lemma 8.
Let μ be a positive measure on σ-algebra M. If sets fA n g satisfy that A 1 ⊂ A 2 ⊂ A 3 ⋯ ⊂A n ⋯ , and A = ∪ ∞ n=1 A n , then we have Readers can find proofs of Lemma 8 in Theorem 1.19 (d) of [10].
Using Lemma 8, we can obtain the following lemma.

Lemma 9.
Let M c and M c γ be the operators defined in (23) and (24). Then, for any f ∈ L p ðℍ n Þ and λ > 0, we have Proof. Fix x ∈ ℍ n . For any ε, there exists r ε > 0 such that For some large γ, we have

Journal of Function Spaces
Since M c γ f ðxÞ increases as γ becomes larger, then On the other hand, by definition we have Then Set k = 1, 2, 3, ⋯: By (33) and Lemma 8, we obtain that ☐ Lemma 9 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: Proof. By the definition of operator norm, there exists a function f ∈ L p ðℍ n Þ such that Since C ∞ c ðℍ n Þ is dense in L p ðℍ n Þ, then we have g ∈ C ∞ c ðℍ n Þ such that Let A be a upper bound of norm of M c . Then So Pick suitable δ in order that Then, we finish the proof. ☐ Notice that g has compact support. This property is important when we establish equivalence of operator norms.
Proof. By direct computation, we arrive at This lemma is based on dilation. Remind that the dilation on the Heisenberg group is different from that on Euclid spaces. (1) For γ > 0, we have (3) For 0 < λ < n, γ > 0, we have Proof. Based on Lemma 11, we have Then, there stands Taking the supremum over all f with k f k L 1 ðℍ n Þ > 0, we have (2) Using the same method of dilation, it is obvious that Taking the supremum over all f with k f k L p ðℍ n Þ > 0, we have (3) By homogeneous dimension of ℍ n , we obtain that There holds that Then Taking the spermum over all f with k f k _