Operator-Valued Fourier Multipliers on Multi-dimensional Hardy Spaces

The author establishes operator-valued Fourier multiplier theorems on multi-dimensional Hardy spaces Hp(Td;X),where 1 ≤ p < ∞,d ∈ N,and X is an AUMD Banach space having the property (α).The suffcient condition on the multiplier is a Marcinkiewicz type condition of order 2 using Rademacher boundedness of sets of bounded linear operators.It is also shown that the assumption that X has the property (α) is necessary when d ≥ 2 even for scalar-valued multipliers.When the underlying Banach space does not have the property (α),a suffcient condition on the multiplier of Marcinkiewicz type of order 2 using a notion of d-Rademacher boundedness is also given.