Transference and restriction of maximal multiplier operators on Hardy spaces

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