The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group. (24th August 2021) - Li, Xiang; Zhang, Xingsong
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.