MULTIPLIERS ON REAL HARDY-SPACES

Let H(p)(R(n)), 0 < p less-than-or-equal-to 1, be the real Hardy spaces, and H(p)(T(n)) be the periodic counterparts. We prove in this paper that if m(x) is an H(p)(R(n)) multiplier, then m approximately = {m(k)}k is-an-element-of z(n) is an H(p)(T(n)) multiplier. On the other hand, if m(x) is continuous on R(n)/{0} and m(s) approximately = {m(sk)}k is-an-element-of Z(n) forms a class of multipliers on H(p)(T(n)) with their multiplier norms uniformly bounded in s > 0, then m is an H(p)(R(n)) multiplier. And as an immediate application of these results, the "restriction theorem" for H(p)(R(n)) multipliers to lower-dimensional spaces is established.