Pointwise convergence of Walsh--Fourier series of vector-valued functions

Abstract

We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_\theta$ between another UMD space $X$ and a Hilbert space $H$, for some $\theta\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.