  In the recent work [23], one studied Fourier multiplies on graded Lie groups
defined via group Fourier transform, proving that Fourier multiplier operators
satisfying H\"{o}rmander type conditions on graded Lie groups are $L^p$ bounded
for $1<p<\infty$ and are of weak-type $(1,1)$. In this paper we investigate the
$H^p \rightarrow L^p$ and $H^p \rightarrow H^p$ boundedness of such Fourier
multiplier operators for $0<p\leq 1$, where $H^p$ is the Hardy space on graded
Lie groups.
