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

We prove a version of Carleson’s Theorem in the Walsh model for vector-valued functions: For $1 \lt p \lt \infty$, 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.

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T.H. was supported by the European Union through the ERC Starting Grant “Analytic-probabilistic methods for borderline singular integrals”, and by the Academy of Finland via project Nos. 130166 and 133264 and via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333). M.L. was supported in part by the NSF grant 0968499, and a grant from the Simons Foundation (#229596 to Michael Lacey).

Received 1 February 2012

Published 5 July 2018