Variational Carleson operators in UMD spaces

We prove $L^p$-boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate of convergence of partial Fourier integrals of vector-valued functions. Our proof relies on bounds on wave packet embeddings into outer Lebesgue spaces on the time-frequency-space $\mathbb{R}^3_+$, which are the focus of this paper.