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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^p\left({ℝ}^n\right)$ and $H^p{(}^n)$, 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty }\left({ℝ}^n\right)$ function m is a maximal multiplier on $H^p\left({ℝ}^n\right)$ if and only if it is a maximal multiplier on $H^p{(}^n)$. As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered.