Fourier Multiplier Theorems Involving Type and Cotype

In this paper we develop the theory of Fourier multiplier operators (Formula presented.), for Banach spaces X and Y, (Formula presented.) and (Formula presented.) an operator-valued symbol. The case (Formula presented.) has been studied extensively since the 1980s, but far less is known for (Formula presented.). In the scalar setting one can deduce results for (Formula presented.) from the case (Formula presented.). However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for (Formula presented.) other geometric conditions on X and Y, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for (Formula presented.) without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as (Formula presented.) remains constant.