On the growth of vector-valued Fourier series Javier ParcetFernando SoriaQuanhua Xu 42A2046G10Growth of Fourier seriesUMD Banach spaces Let $f:\mathbb{T}\to\mathrm{X}$ satisfy \[ \int_{\mathbb{T}}\|f(x)\|_{\mathrm{X}}(\log^{+}\|f(x)\|_{\mathrm{X}})^{1+\delta\ }\dx<\infty, \] where $\mathrm{X}$ is a UMD Banach space and $\delta>0$. Then, we prove that \[ \Big\|\sum_{|k|\le n}\hat{f}(k)e^{2\pi ikx}\Big\|_{\mathrm{X}}=o(\log\log n)\quad\mbox{for almost every }x\in\mathbb{T}. \] In other words, the "little Carleson theorem"${}$ holds for UMD-valued functions. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5135 10.1512/iumj.2013.62.5135 en Indiana Univ. Math. J. 62 (2013) 1765 - 1784 state-of-the-art mathematics http://iumj.org/access/