Centered Hardy-Littlewood maximal functions on Heisenberg type groups

Abstract:

In this paper, by establishing uniform lower bounds for the Poisson kernel and $(-\Delta )^{-\frac 12}$ on the Heisenberg type group $\mathbb {H}(2n,m)$ with $m \geq 2$, which follow from the various properties of Bessel functions and Legendre functions, we prove that there exists a constant $A>0$ such that, for all $f\in L^1(\mathbb {H}(2n,m))$ and all $n,m \in \mathbb {N}^*$ satisfying $4 \leq m^2\ll \log n$, we have $\|M_{K}f\|_{L^{1,\infty }}\le An\|f\|_1$, where $M_{K}$ denotes the centered Hardy-Littlewood maximal function defined by the Korányi norm. For the centered Hardy-Littlewood maximal function $M_{CC}$ defined by the Carnot-Carathédory distance, we prove $\|M_{CC}f\|_{L^{1,\infty }}\le A(m)n\|f\|_1$ holds for some constant $A(m)$ independent of $n$.