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

TL;DR

This paper proves a version of Carleson's Theorem for vector-valued functions in the Walsh model, establishing pointwise convergence of Walsh-Fourier series for functions in certain UMD spaces.

Contribution

It extends Carleson's Theorem to vector-valued functions in the Walsh setting, specifically for UMD spaces that are complex interpolations between a UMD space and a Hilbert space.

Findings

Walsh-Fourier series of vector-valued functions converge pointwise under specified conditions.

The result applies to all known examples of UMD spaces satisfying the interpolation condition.

The theorem bridges scalar and vector-valued Fourier analysis in the Walsh model.

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.