Abstract:

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.

Key words: Hardy operator, Heisenberg group, power weight, sharp estimate