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

Abstract

In this article, we define a kind of truncated maximal function on the Heisenberg space by Mγcfx=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.