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<∞, and a UMD space Y, the Walsh–Fourier series of f∈Lp(0,1;Y) converges pointwise, provided that Y is a complex interpolation space Y=[X,H]θ between another UMD space X and a Hilbert space H, for some θ∈(0,1). Apparently, all known examples of UMD spaces satisfy this condition.