Variational Carleson operators in UMD spaces

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