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

Xiang Li,Xingsong Zhang,Andrea Scapellato

doi:10.1155/2021/7612482

Abstract

In this article, we define a kind of truncated maximal function on the Heisenberg space by M γ c f x = sup 0 < r < γ 1 / m B x , r ∫ B x , r f y d y . 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 L p 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 L 1 ⟶ L 1 , ∞ and M ̇ 1 , λ ⟶ W ̇ M 1 , λ . Those results are generalization of previous work on Euclid spaces.

Highlights

Let f be a locally integrable function on Rn
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

Summary

Introduction

Let f be a locally integrable function on Rn. We define the centered Hardy-Littlewood maximal function as. In 2003, Melas [6] obtained the sharp bound of the one-dimensional centered Hardy-Littlewood maximal function of weak type (1, 1). 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. 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. 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 present the details of our main theorems and the proofs

Preliminaries

Main Theorem