, Volume 319, Issue 4, pp 735-758

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

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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 \(T_t\) .

Received July 19, 1999 / Revised July 13, 2000 / Published online February 5, 2001