Mikhlin's Theorem for Operator–Valued Fourier Multipliers in n Variables

An operator–valued Mikhlin theorem is proved for multipliers of the form M : ℝⁿ → ℒ︁(X, Y) where X and Y are UMD spaces. The usual norm bounds of the classical Mikhlin condition are replaced by R–bounds. Furthermore, the concept of R–bounded variation is introduced to generalize the Marcinkiewicz Fourier multiplier Theorem to the operator–valued setting.