article

Title: On the growth of vector-valued Fourier series

Authors: Javier Parcet, Fernando Soria and Quanhua Xu

Issue: Volume 62 (2013), Issue 6, 1765-1784

Abstract:

$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.$