In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^{p}(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(G)$ and $L^{p}(G) \rightarrow L^{p}(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved.

Let $G$ be a complex reductive linear algebraic group and $G_0 \subseteq G$ a real form. Suppose $P$ is a parabolic subgroup of $G$ and assume that $P$ has a Levi factor $L$ such that $G_0 \cap L = L_0$ is a real form of $L$. Using the minimal globalization $V_{\min}$ of a finite length admissible representation for $L_0$, one can define a homogeneous analytic vector bundle on the $G_0$ orbit $S$ of $P$ in the generalized flag manifold $Y = G/P$. Let $A(P, V_{\min})$ denote the corresponding sheaf of polarized sections. In this article we analyze the $G_0$ representations obtained on the compactly supported sheaf cohomology groups $H^p_c(S,A(P, V_{\min}))$.