Operator-valued Fourier multiplier theorems on L _p -spaces on \( \mathbb{T}^d \)

Abstract.

We establish operator-valued Fourier multiplier theorems on L _p -spaces on \( \mathbb{T}^d \). The conditions on the multipliers depend on the geometry of the underlying Banach spaces (UMD property and property \( (\alpha)) \) and the growth rate (estimated by means of R-boundedness) at infinity of the partial derivatives of the multipliers. We also give an application of the obtained Fourier multiplier theorems to L _p -maximal regularity for a second order problem.