Let H~P(R~n), 0p≤1, be the real Hardy spaces, and H~P(T~n) be the periodic counter-parts. We prove in this paper that if m(x) is an H~P(R~n) multiplier, then ? = {m_}k∈x~n isan H~P(T~n) multiplier. On the other hand, if m(x) is continuous on R~n{0} and ?={m(sk)}k∈Z~n forms a class of multipliers on H~P(T~n) with their multiplier norms uniformlybounded in s0, then m is an H~P(R~n) multiplier. And as an immediate application of theseresults, the "restriction theorem" for H~P(R~n) multipliers to lower-dimensional spaces isestablished.
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