Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

Abstract.

We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal $L_p$–regularity for the generator of an analytic semigroup $T_t$ in terms of the R–boundedness of the resolvent of A or the semigroup $T_t$.