Anisotropic Fourier multipliers and singular integrals for vector-valued functions

We show that an anisotropic Mihlin-type condition on the symbol guarantees the boundedness of the associated Fourier multiplier operator on L ^p (R ⁿ ,X) for 1 < p < ∞ and an arbitrary UMD space X. In many cases, this result can be used as a substitute for the Marcinkiewicz–Lizorkin multiplier theorem, which is invalid in general UMD spaces. An application to anisotropic singular integrals is given. Mathematics Subject Classification (2000) 42B15, 42B20, 46E40