Multiplier transformations on ${\displaystyle H^p}$ spaces

The authors obtain some multiplier theorems on ${\displaystyle H^p}$ spaces analogous to the classical ${\displaystyle L^p}$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\displaystyle {\left(Tf\right)}^x=\lambda \left(x\right)f̂\left(x\right)}$ ${\displaystyle (\lambda \in C\left({ℝ}^n\right))}$ is bounded on ${\displaystyle H^p\left({ℝ}^n\right)}$ if and only if the restriction ${\displaystyle {\{\lambda (\epsilon m)\}}_{m\in \Lambda }}$ is an ${\displaystyle H^p\left(T^n\right)}$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in ${\displaystyle {ℝ}^n}$.