The equivalence of operator norm between the Hardy-Littlewood maximal function and truncated maximal function on the Heisenberg group (Q2230014)

Summary: In this article, we define a kind of truncated maximal function on the Heisenberg space by \(\mathbb{M}_\gamma^cf( x)= \sup_{0<r<\gamma}(1/m(B(x,r)))\int_{B(x,r)}|f(y)|dy\). 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 \(1<p<\infty\), 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 \(L^1\longrightarrow L^{1,\infty}\) and \(\dot{M}^{1,\lambda}\longrightarrow\dot{W}M^{1,\lambda}\). Those results are generalization of previous work on Euclid spaces.