Fourier Series in Banach spaces and Maximal Regularity

We consider Fourier series of functions in L-p(0, 2 pi; X) where X is a Banach space. In particular, we show that the Fourier series of each function in L-p(0, 2 pi; X) converges unconditionally if and only if p = 2 and X is a Hilbert space. For operator-valued multipliers we present the Marcinkiewicz theorem and give applications to differential equations. In particular, we characterize maximal regularity (in a slightly different version than the usual one) by R-sectoriality. Applications to non-autonomous problems are indicated.