研究了 Hardy 算子在 $L^{p}({H}^{n},|x|_{h}^{\alpha}{ d}x)$ 函数空间的有界性问题, 其中 Heisenberg 群记为 ${H}^{n}$. 证明了 Hardy 算子是 $(p,p)$ 型 $(1< p\leqslant \infty)$ 和弱 $(1,1)$ 型, 并得到了 $(p,p)$ 型的最佳常数和弱 $(1,1)$ 型的最佳常数的上下界.

In this paper, the $n$-dimensional Hardy operator with power weight on the Heisenberg group $H^n$ is studied. It is proved that the Hardy operator is a strong type of ($p, p$) ($1<p\leqslant \infty$) and a weak type of (1,1) on $L^p$($H^n$, $|x|_h^a$d$x$) and $L^1$ ($H^n$, $|x|_h^a$d$x$), respectively. Moreover, the results show that such ($p, p$) estimate is sharp, and obtain the upper and the lower bounds of the best constant of weak (1,1) type.