Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson’s theorem: let be a complex interpolation space between an unconditionality of martingale differences (UMD) space and a Hilbert space . For and , the partial sums of the Fourier series of converge to pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form . In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.